var bibbase_data = {"data":"<img src=\"//bibbase.org/img/ajax-loader.gif\" id=\"spinner\"\n  style=\"display: none;\" alt=\"Loading..\" />\n\n<div id=\"bibbase\">\n\n  <script type=\"text/javascript\">\n    var bibbase = {\n      params: {\"bib\":\"https://www2.math.upenn.edu/~strain/RMS3web.bib\",\"theme\":\"simple\",\"authorFirst\":\"1\",\"fullnames\":\"1\",\"jsonp\":\"1\",\"host\":\"bibbase.org\"},\n      data: [{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"We prove time decay of solutions to the Muskat equation in 2D and in 3D. We consider the norm $\\\\|f\\\\|_{s}(t)$. In this paper, for the 3D Muskat problem, given initial data $f_{0}∈ H^{l}(ℝ^{2})$ for some $l≥ 3$ such that $\\\\|f_{0}\\\\|_{1} < k_{0}$ for a constant $k_{0} ≈ 1/5$, we prove uniform in time bounds of $\\\\|f\\\\|_{s}(t)$ for $-d < s < l-1$ and assuming $\\\\|f_{0}\\\\|_{ν} < ∞$ we prove time decay estimates of the form $\\\\|f\\\\|_{s}(t) łesssim (1+t)^{-s+ν}$ for $0 ≤ s ≤ l-1$ and $-d ≤ ν < s$. These large time decay rates are the same as the optimal rate for the linear Muskat equation. We also prove analogous results in 2D.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Patel\"],\"firstnames\":[\"Neel\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1080/03605302.2017.1321661\",\"eprint\":\"1610.05271\",\"fjournal\":\"Communications in Partial Differential Equations\",\"issn\":\"0360-5302\",\"journal\":\"Comm. Partial Differential Equations\",\"keywords\":\"Fluid mechanics, Free boundary problems, Muskat problem\",\"mrclass\":\"76D03 (35B40 35Q35 76S05)\",\"mrnumber\":\"3683311\",\"mrreviewer\":\"Youcef Amirat\",\"number\":\"6\",\"pages\":\"977–999\",\"title\":\"Large time decay estimates for the Muskat equation\",\"url_arxiv\":\"https://arxiv.org/abs/1610.05271\",\"volume\":\"42\",\"year\":\"2017\",\"zblnumber\":\"1378.35245\",\"bdsk-url-1\":\"https://doi.org/10.1080/03605302.2017.1321661\",\"bibtex\":\"@article{161005271,\\n\\tabstract = {We prove time decay of solutions to the Muskat equation in 2D and in 3D. We consider the norm $\\\\|f\\\\|_{s}(t)$.  In this paper, for the 3D Muskat problem, given initial data $f_{0}\\\\in H^{l}(\\\\mathbb{R}^{2})$ for some $l\\\\geq 3$ such that $\\\\|f_{0}\\\\|_{1} < k_{0}$ for a constant $k_{0} \\\\approx 1/5$, we prove uniform in time bounds of $\\\\|f\\\\|_{s}(t)$ for $-d < s < l-1$ and assuming $\\\\|f_{0}\\\\|_{\\\\nu} < \\\\infty$ we prove time decay estimates of the form $\\\\|f\\\\|_{s}(t) \\\\lesssim (1+t)^{-s+\\\\nu}$ for $0 \\\\leq s \\\\leq l-1$ and $-d \\\\leq \\\\nu < s$.  These large time decay rates are the same as the optimal rate for the linear Muskat equation.  We also prove analogous results in 2D.},\\n\\tauthor = {Patel, Neel and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1080/03605302.2017.1321661},\\n\\teprint = {1610.05271},\\n\\tfjournal = {Communications in Partial Differential Equations},\\n\\tissn = {0360-5302},\\n\\tjournal = {Comm. Partial Differential Equations},\\n\\tkeywords = {Fluid mechanics, Free boundary problems, Muskat problem},\\n\\tmrclass = {76D03 (35B40 35Q35 76S05)},\\n\\tmrnumber = {3683311},\\n\\tmrreviewer = {Youcef Amirat},\\n\\tnumber = {6},\\n\\tpages = {977--999},\\n\\ttitle = {Large time decay estimates for the {M}uskat equation},\\n\\turl_arxiv = {https://arxiv.org/abs/1610.05271},\\n\\tvolume = {42},\\n\\tyear = {2017},\\n\\tzblnumber = {1378.35245},\\n\\tbdsk-url-1 = {https://doi.org/10.1080/03605302.2017.1321661}}\\n\\n\",\"author_short\":[\"Patel, N.\",\"Strain, R. M.\"],\"key\":\"161005271\",\"id\":\"161005271\",\"bibbaseid\":\"patel-strain-largetimedecayestimatesforthemuskatequation-2017\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1610.05271\"},\"keyword\":[\"Fluid mechanics\",\"Free boundary problems\",\"Muskat problem\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":4,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this paper we consider gravity-capillarity Muskat bubbles in 2D. We obtain a new approach to improve our result in i̧teGancedoGarciaJuarezPatelStrain23. Due to a new bubble-adapted formulation, the improvement is two fold. We significantly condense the proof and we now obtain the global well-posedness result for Muskat bubbles in critical regularity.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Gancedo\"],\"firstnames\":[\"Francisco\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"García-Juárez\"],\"firstnames\":[\"Eduardo\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Patel\"],\"firstnames\":[\"Neel\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\"],\"suffixes\":[]}],\"eprint\":\"2312.14323\",\"keywords\":\"Fluid mechanics, Free boundary problems, critical regularity, Muskat problem\",\"month\":\"Dec\",\"note\":\"preprint\",\"pages\":\"26 pages\",\"title\":\"On Nonlinear Stability of Muskat Bubbles\",\"url_arxiv\":\"https://arxiv.org/abs/2312.14323\",\"year\":\"2023\",\"bibtex\":\"@article{GancedoGarciaJuarezPatelStrain24,\\n\\tabstract = { In this paper we consider gravity-capillarity Muskat bubbles in 2D. We obtain\\na new approach to improve our result in \\\\cite{GancedoGarciaJuarezPatelStrain23}. Due to a new bubble-adapted\\nformulation, the improvement is two fold.  We significantly condense the proof\\nand we now obtain the global well-posedness result for Muskat bubbles in\\ncritical regularity.},\\n\\tauthor = {Gancedo, Francisco and Garc\\\\'{\\\\i}a-Ju\\\\'{a}rez, Eduardo and Patel, Neel and Strain, Robert},\\n\\teprint = {2312.14323},\\n\\tkeywords = {Fluid mechanics, Free boundary problems, critical regularity, Muskat problem},\\n\\tmonth = {Dec},\\n\\tnote = {preprint},\\n\\tpages = {26 pages},\\n\\ttitle = {{On Nonlinear Stability of Muskat Bubbles}},\\n\\turl_arxiv = {https://arxiv.org/abs/2312.14323},\\n\\tyear = {2023}}\\n\\n\",\"author_short\":[\"Gancedo, F.\",\"García-Juárez, E.\",\"Patel, N.\",\"Strain, R.\"],\"key\":\"GancedoGarciaJuarezPatelStrain24\",\"id\":\"GancedoGarciaJuarezPatelStrain24\",\"bibbaseid\":\"gancedo-garcajurez-patel-strain-onnonlinearstabilityofmuskatbubbles-2023\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/2312.14323\"},\"keyword\":[\"Fluid mechanics\",\"Free boundary problems\",\"critical regularity\",\"Muskat problem\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":1,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this paper, we study the dynamics of fluids in porous media governed by Darcy's law: the Muskat problem. We consider the setting of two immiscible fluids of different densities and viscosities under the influence of gravity in which one fluid is completely surrounded by the other. This setting is gravity unstable because along a portion of the interface, the denser fluid must be above the other. Surprisingly, even without capillarity, the circle-shaped bubble is a steady state solution moving with vertical constant velocity determined by the density jump between the fluids. Taking advantage of our discovery of this steady state, we are able to prove global in time existence and uniqueness of dynamic bubbles of nearly circular shapes under the influence of surface tension. We prove this global existence result for low regularity initial data. Moreover, we prove that these solutions are instantly analytic and decay exponentially fast in time to the circle.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Gancedo\"],\"firstnames\":[\"Francisco\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"García-Juárez\"],\"firstnames\":[\"Eduardo\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Patel\"],\"firstnames\":[\"Neel\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\"],\"suffixes\":[]}],\"doi\":\"10.1090/memo/1455\",\"eprint\":\"1902.02318\",\"fjournal\":\"Memoirs of the American Mathematical Society\",\"issn\":\"0065-9266,1947-6221\",\"journal\":\"Mem. Amer. Math. Soc.\",\"keywords\":\"Fluid mechanics, Free boundary problems, Muskat problem\",\"mrclass\":\"99-06\",\"mrnumber\":\"4679708\",\"number\":\"1455\",\"pages\":\"87 pages\",\"title\":\"Global Regularity for Gravity Unstable Muskat Bubbles\",\"url\":\"https://doi.org/10.1090/memo/1455\",\"url_arxiv\":\"https://arxiv.org/abs/1902.02318\",\"volume\":\"292\",\"year\":\"2023\",\"bdsk-url-1\":\"https://doi.org/10.1090/memo/1455\",\"bibtex\":\"@article{GancedoGarciaJuarezPatelStrain23,\\n\\tabstract = {In this paper, we study the dynamics of fluids in porous media governed by Darcy's law: the Muskat problem. We consider the setting of two immiscible fluids of different densities and viscosities under the influence of gravity in which one fluid is completely surrounded by the other. This setting is gravity unstable because along a portion of the interface, the denser fluid must be above the other. Surprisingly, even without capillarity, the circle-shaped bubble is a steady state solution moving with vertical constant velocity determined by the density jump between the fluids. Taking advantage of our discovery of this steady state, we are able to prove global in time existence and uniqueness of dynamic bubbles of nearly circular shapes under the influence of surface tension. We prove this global existence result for low regularity initial data. Moreover, we prove that these solutions are instantly analytic and decay exponentially fast in time to the circle.},\\n\\tauthor = {Gancedo, Francisco and Garc\\\\'{\\\\i}a-Ju\\\\'{a}rez, Eduardo and Patel, Neel and Strain, Robert},\\n\\tdoi = {10.1090/memo/1455},\\n\\teprint = {1902.02318},\\n\\tfjournal = {Memoirs of the American Mathematical Society},\\n\\tissn = {0065-9266,1947-6221},\\n\\tjournal = {Mem. Amer. Math. Soc.},\\n\\tkeywords = {Fluid mechanics, Free boundary problems, Muskat problem},\\n\\tmrclass = {99-06},\\n\\tmrnumber = {4679708},\\n\\tnumber = {1455},\\n\\tpages = {87 pages},\\n\\ttitle = {Global {R}egularity for {G}ravity {U}nstable {M}uskat {B}ubbles},\\n\\turl = {https://doi.org/10.1090/memo/1455},\\n\\turl_arxiv = {https://arxiv.org/abs/1902.02318},\\n\\tvolume = {292},\\n\\tyear = {2023},\\n\\tbdsk-url-1 = {https://doi.org/10.1090/memo/1455}}\\n\\n\",\"author_short\":[\"Gancedo, F.\",\"García-Juárez, E.\",\"Patel, N.\",\"Strain, R.\"],\"key\":\"GancedoGarciaJuarezPatelStrain23\",\"id\":\"GancedoGarciaJuarezPatelStrain23\",\"bibbaseid\":\"gancedo-garcajurez-patel-strain-globalregularityforgravityunstablemuskatbubbles-2023\",\"role\":\"author\",\"urls\":{\"Paper\":\"https://doi.org/10.1090/memo/1455\",\" arxiv\":\"https://arxiv.org/abs/1902.02318\"},\"keyword\":[\"Fluid mechanics\",\"Free boundary problems\",\"Muskat problem\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"We study the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time well-posedness for arbitrary initial data in the scaling critical Besov space $Ḃ^{3/2}_{2,1}(\\\\mathbb{T} ; ℝ^2)$. We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancellation structure.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Cameron\"],\"firstnames\":[\"Stephen\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"blurbreport\":\"This paper studies the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time well-posedness for arbitrary initial data in the scaling critical Besov space $Ḃ^{3/2}_{2,1}(\\\\mathbb{T} ; ℝ^2)$. We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancellation structure.\",\"doi\":\"10.1002/cpa.22139\",\"eprint\":\"2112.00692\",\"fjournal\":\"Communications on Pure and Applied Mathematics\",\"issn\":\"0010-3640,1097-0312\",\"journal\":\"Comm. Pure Appl. Math.\",\"keywords\":\"Fluid mechanics, Free boundary problems, Peskin problem, Fluid-Structure interface, critical regularity, immersed boundary problem\",\"mrclass\":\"35Q35 (35C15 35R11 35R35 76D07)\",\"mrnumber\":\"4673875\",\"number\":\"2\",\"pages\":\"901–989\",\"title\":\"Critical local well-posedness for the fully nonlinear Peskin problem\",\"url_arxiv\":\"https://arxiv.org/abs/2112.00692\",\"volume\":\"77\",\"year\":\"2024\",\"bdsk-url-1\":\"https://doi.org/10.1002/cpa.22139\",\"bibtex\":\"@article{2112.00692,\\n\\tabstract = {We study the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid.  This is known as the Stokes immersed boundary problem and also as the Peskin problem.   We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem.  In this case we prove local in time well-posedness for arbitrary initial data in the scaling critical Besov space $\\\\dot{B}^{3/2}_{2,1}(\\\\mathbb{T} ; \\\\mathbb{R}^2)$.  We additionally prove the optimal higher order smoothing effects for the solution.  To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancellation structure.},\\n\\tauthor = {Cameron, Stephen and Strain, Robert M.},\\n\\tblurbreport = {This paper studies the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time well-posedness for arbitrary initial data in the scaling critical Besov space $\\\\dot{B}^{3/2}_{2,1}(\\\\mathbb{T} ; \\\\mathbb{R}^2)$. We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancellation structure.},\\n\\tdoi = {10.1002/cpa.22139},\\n\\teprint = {2112.00692},\\n\\tfjournal = {Communications on Pure and Applied Mathematics},\\n\\tissn = {0010-3640,1097-0312},\\n\\tjournal = {Comm. Pure Appl. Math.},\\n\\tkeywords = {Fluid mechanics, Free boundary problems, Peskin problem, Fluid-Structure interface, critical regularity, immersed boundary problem},\\n\\tmrclass = {35Q35 (35C15 35R11 35R35 76D07)},\\n\\tmrnumber = {4673875},\\n\\tnumber = {2},\\n\\tpages = {901--989},\\n\\ttitle = {Critical local well-posedness for the fully nonlinear {P}eskin problem},\\n\\turl_arxiv = {https://arxiv.org/abs/2112.00692},\\n\\tvolume = {77},\\n\\tyear = {2024},\\n\\tbdsk-url-1 = {https://doi.org/10.1002/cpa.22139}}\\n\\n\",\"author_short\":[\"Cameron, S.\",\"Strain, R. M.\"],\"key\":\"2112.00692\",\"id\":\"2112.00692\",\"bibbaseid\":\"cameron-strain-criticallocalwellposednessforthefullynonlinearpeskinproblem-2024\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/2112.00692\"},\"keyword\":[\"Fluid mechanics\",\"Free boundary problems\",\"Peskin problem\",\"Fluid-Structure interface\",\"critical regularity\",\"immersed boundary problem\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":4,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"This paper introduces the \\\\em3D Peskin problem: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow. We obtain the equations that model this free boundary problem and show that they admit a boundary integral reduction, providing an evolution equation for the elastic interface. We consider general nonlinear elastic laws, i.e., the fully nonlinear Peskin problem, and prove that the problem is well-posed in low-regularity Hölder spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time.\",\"author\":[{\"firstnames\":[\"Eduardo\"],\"propositions\":[],\"lastnames\":[\"García-Juárez\"],\"suffixes\":[]},{\"firstnames\":[\"Po-Chun\"],\"propositions\":[],\"lastnames\":[\"Kuo\"],\"suffixes\":[]},{\"firstnames\":[\"Yoichiro\"],\"propositions\":[],\"lastnames\":[\"Mori\"],\"suffixes\":[]},{\"firstnames\":[\"Robert\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Strain\"],\"suffixes\":[]}],\"eprint\":\"2301.12153\",\"keywords\":\"Fluid mechanics, Free boundary problems, Peskin problem, Fluid-Structure interface, critical regularity, immersed boundary problem\",\"month\":\"Jan\",\"note\":\"preprint\",\"pages\":\"93 pages\",\"title\":\"Well-Posedness of the 3D Peskin Problem\",\"url_arxiv\":\"https://arxiv.org/abs/2301.12153\",\"year\":\"2023\",\"bibtex\":\"@article{Peskin3D,\\n\\tabstract = {This paper introduces the {\\\\em{3D Peskin problem}}: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow. We obtain the equations that model this free boundary problem and show that they admit a boundary integral reduction, providing an evolution equation for the elastic interface. We consider general nonlinear elastic laws, i.e., the fully nonlinear Peskin problem, and prove that the problem is well-posed in low-regularity H\\\\\\\"older spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time.},\\n\\tauthor = {Eduardo Garc{\\\\'{i}}a-Ju{\\\\'{a}}rez and Po-Chun Kuo and Yoichiro Mori and Robert M. Strain},\\n\\teprint = {2301.12153},\\n\\tkeywords = {Fluid mechanics, Free boundary problems, Peskin problem, Fluid-Structure interface, critical regularity, immersed boundary problem},\\n\\tmonth = {Jan},\\n\\tnote = {preprint},\\n\\tpages = {93 pages},\\n\\ttitle = {{Well-Posedness of the 3D Peskin Problem}},\\n\\turl_arxiv = {https://arxiv.org/abs/2301.12153},\\n\\tyear = {2023}}\\n\\n\",\"author_short\":[\"García-Juárez, E.\",\"Kuo, P.\",\"Mori, Y.\",\"Strain, R. M.\"],\"key\":\"Peskin3D\",\"id\":\"Peskin3D\",\"bibbaseid\":\"garciajurez-kuo-mori-strain-wellposednessofthe3dpeskinproblem-2023\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/2301.12153\"},\"keyword\":[\"Fluid mechanics\",\"Free boundary problems\",\"Peskin problem\",\"Fluid-Structure interface\",\"critical regularity\",\"immersed boundary problem\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":14,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"This paper is concerned with the relativistic Boltzmann equation without angular cutoff. We establish the global-in-time existence, uniqueness and asymptotic stability for solutions nearby the relativistic Maxwellian to the special relativistic Boltzmann equation without any angular cutoff. We work in the case of a spatially periodic box. We assume the generic hard-interaction and soft-interaction conditions on the collision kernel that were derived by Dudyński and Ekiel-Je$\\\\̇text{z}}$ewska (Comm. Math. Phys. \\\\textbf115(4):607–629, 1988), and our assumptions include the case of Israel particles (J. Math. Phys. \\\\textbf4:1163–1181, 1963). In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. We further derive the relativistic analogue of the Carleman dual representation of Boltzmann collision operator. This is the first global existence and stability result for relativistic Boltzmann equation without angular cutoff and this resolves the open question of perturbative global existence without the Grad's angular cut-off assumption.\",\"author\":[{\"firstnames\":[\"Jin\",\"Woo\"],\"propositions\":[],\"lastnames\":[\"Jang\"],\"suffixes\":[]},{\"firstnames\":[\"Robert\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Strain\"],\"suffixes\":[]}],\"date-modified\":\"2024-03-20 16:23:02 -0400\",\"doi\":\"10.1007/s40818-022-00137-2\",\"eprint\":\"2103.15885\",\"fjournal\":\"Annals of PDE. Journal Dedicated to the Analysis of Problems from Physical Sciences\",\"journal\":\"Ann. PDE\",\"keywords\":\"relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory\",\"number\":\"2\",\"pages\":\"167 pages\",\"title\":\"Asymptotic Stability of the Relativistic Boltzmann Equation without Angular Cut-off\",\"url_arxiv\":\"https://arxiv.org/abs/2103.15885\",\"volume\":\"8\",\"year\":\"2022\",\"bdsk-url-1\":\"https://doi.org/10.1007/s40818-022-00137-2\",\"bibtex\":\"@article{2103.15885,\\n\\tabstract = {This paper is concerned with the relativistic Boltzmann equation without angular cutoff.  We establish the global-in-time existence, uniqueness and asymptotic stability for solutions nearby the relativistic Maxwellian to the special relativistic Boltzmann equation without any angular cutoff.  We work in the case of a spatially periodic box. We assume the generic hard-interaction and  soft-interaction conditions on the collision kernel that were derived by Dudy\\\\'nski and Ekiel-Je$\\\\dot{\\\\text{z}}$ewska (Comm. Math. Phys. \\\\textbf{115}(4):607--629, 1988), and our assumptions include the case of Israel particles (J. Math. Phys. \\\\textbf{4}:1163--1181, 1963).   In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator.   We further derive the relativistic analogue of the Carleman dual representation of Boltzmann collision operator.  This is the first global existence and stability result for relativistic Boltzmann equation without angular cutoff and this resolves the open question of perturbative global existence without the Grad's angular cut-off assumption.},\\n\\tauthor = {{J}in {W}oo {J}ang and {R}obert {M}. {S}train},\\n\\tdate-modified = {2024-03-20 16:23:02 -0400},\\n\\tdoi = {10.1007/s40818-022-00137-2},\\n\\teprint = {2103.15885},\\n\\tfjournal = {Annals of PDE. Journal Dedicated to the Analysis of Problems from Physical Sciences},\\n\\tjournal = {Ann. {PDE}},\\n\\tkeywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\\n\\tnumber = {2},\\n\\tpages = {167 pages},\\n\\ttitle = {{Asymptotic Stability of the Relativistic Boltzmann Equation without Angular Cut-off}},\\n\\turl_arxiv = {https://arxiv.org/abs/2103.15885},\\n\\tvolume = {8},\\n\\tyear = {2022},\\n\\tbdsk-url-1 = {https://doi.org/10.1007/s40818-022-00137-2}}\\n\\n\",\"author_short\":[\"Jang, J. W.\",\"Strain, R. M.\"],\"key\":\"2103.15885\",\"id\":\"2103.15885\",\"bibbaseid\":\"jang-strain-asymptoticstabilityoftherelativisticboltzmannequationwithoutangularcutoff-2022\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/2103.15885\"},\"keyword\":[\"relativistic Boltzmann equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":5,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"The Peskin problem models the dynamics of a closed elastic filament immersed in an incompressible fluid. In this paper, we consider the case when the inner and outer viscosities are possibly different. This viscosity contrast adds further non-local effects to the system through the implicit non-local relation between the net force and the free interface. We prove the first global well-posedness result for the Peskin problem in this setting. The result applies for medium size initial interfaces in critical spaces and shows instant analytic smoothing. We carefully calculate the medium size constraint on the initial data. These results are new even without viscosity contrast.\",\"author\":[{\"firstnames\":[\"Eduardo\"],\"propositions\":[],\"lastnames\":[\"García-Juárez\"],\"suffixes\":[]},{\"firstnames\":[\"Yoichiro\"],\"propositions\":[],\"lastnames\":[\"Mori\"],\"suffixes\":[]},{\"firstnames\":[\"Robert\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Strain\"],\"suffixes\":[]}],\"doi\":\"10.2140/apde.2023.16.785\",\"eprint\":\"2009.03360\",\"fjournal\":\"Analysis & PDE\",\"journal\":\"Anal. PDE\",\"keywords\":\"Fluid mechanics, Free boundary problems, Peskin problem, Fluid-Structure interface, critical regularity, immersed boundary problem\",\"number\":\"3\",\"pages\":\"785–838\",\"title\":\"The Peskin Problem with Viscosity Contrast\",\"url_arxiv\":\"https://arxiv.org/abs/2009.03360\",\"volume\":\"16\",\"year\":\"2023\",\"bdsk-url-1\":\"https://doi.org/10.2140/apde.2023.16.785\",\"bibtex\":\"@article{2009.03360,\\n\\tabstract = {The Peskin problem models the dynamics of a closed elastic filament immersed\\nin an incompressible fluid. In this paper, we consider the case when the inner\\nand outer viscosities are possibly different. This viscosity contrast adds\\nfurther non-local effects to the system through the implicit non-local relation\\nbetween the net force and the free interface. We prove the first global\\nwell-posedness result for the Peskin problem in this setting. The result\\napplies for medium size initial interfaces in critical spaces and shows instant\\nanalytic smoothing. We carefully calculate the medium size constraint on the\\ninitial data. These results are new even without viscosity contrast.},\\n\\tauthor = {Eduardo Garc{\\\\'{i}}a-Ju{\\\\'{a}}rez and Yoichiro Mori and Robert M. Strain},\\n\\tdoi = {10.2140/apde.2023.16.785},\\n\\teprint = {2009.03360},\\n\\tfjournal = {Analysis \\\\& PDE},\\n\\tjournal = {Anal. PDE},\\n\\tkeywords = {Fluid mechanics, Free boundary problems, Peskin problem, Fluid-Structure interface, critical regularity, immersed boundary problem},\\n\\tnumber = {3},\\n\\tpages = {785--838},\\n\\ttitle = {{The Peskin Problem with Viscosity Contrast}},\\n\\turl_arxiv = {https://arxiv.org/abs/2009.03360},\\n\\tvolume = {16},\\n\\tyear = {2023},\\n\\tbdsk-url-1 = {https://doi.org/10.2140/apde.2023.16.785}}\\n\\n\",\"author_short\":[\"García-Juárez, E.\",\"Mori, Y.\",\"Strain, R. M.\"],\"key\":\"2009.03360\",\"id\":\"2009.03360\",\"bibbaseid\":\"garciajurez-mori-strain-thepeskinproblemwithviscositycontrast-2023\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/2009.03360\"},\"keyword\":[\"Fluid mechanics\",\"Free boundary problems\",\"Peskin problem\",\"Fluid-Structure interface\",\"critical regularity\",\"immersed boundary problem\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":6,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"Although the nuclear fusion process has received a great deal of attention in recent years, the amount of the mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the <i>Vlasov-Maxwell</i> system in a two-dimensional annulus when a huge (<i>but finite-in-time</i>) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. This work is the first step to a more generalized work on the three dimensional Tokamak structure. The highlight of this work is from the physical assumptions on the external magnetic potential well which remains finite <i>within a finite time interval</i> and from that we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical-coordinate-forms of the <i>Vlasov-Maxwell</i> system.\",\"author\":[{\"firstnames\":[\"Jin\",\"Woo\"],\"propositions\":[],\"lastnames\":[\"Jang\"],\"suffixes\":[]},{\"firstnames\":[\"Robert\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Strain\"],\"suffixes\":[]},{\"firstnames\":[\"Tak\",\"Kwong\"],\"propositions\":[],\"lastnames\":[\"Wong\"],\"suffixes\":[]}],\"doi\":\"10.3934/krm.2021039\",\"eprint\":\"2111.04583\",\"fjournal\":\"Kinetic and Related Models\",\"journal\":\"Kinet. Relat. Models\",\"keywords\":\"Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems\",\"note\":\"published online November 25, 2021\",\"pages\":\"36 pages\",\"title\":\"Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus\",\"url_arxiv\":\"https://arxiv.org/abs/2111.04583\",\"year\":\"2021\",\"bdsk-url-1\":\"https://doi.org/10.3934/krm.2021039\",\"bibtex\":\"@article{JangStrainWong2021,\\n\\tabstract = {Although the nuclear fusion process has received a great deal of attention in recent years, the amount of the mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the \\\\textit{Vlasov-Maxwell} system in a two-dimensional annulus when a huge (\\\\textit{but finite-in-time}) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. This work is the first step to a more generalized work on the three dimensional Tokamak structure. The highlight of this work is from the physical assumptions on the external magnetic potential well which remains finite \\\\textit{within a finite time interval} and from that we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical-coordinate-forms of the \\\\textit{Vlasov-Maxwell} system.},\\n\\tauthor = {{J}in {W}oo {J}ang and {R}obert {M}. {S}train and {T}ak {K}wong {W}ong},\\n\\tdoi = {10.3934/krm.2021039},\\n\\teprint = {2111.04583},\\n\\tfjournal = {Kinetic and Related Models},\\n\\tjournal = {Kinet. Relat. Models},\\n\\tkeywords = {Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},\\n\\tnote = {published online November 25, 2021},\\n\\tpages = {36 pages},\\n\\ttitle = {{Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus}},\\n\\turl_arxiv = {https://arxiv.org/abs/2111.04583},\\n\\tyear = {2021},\\n\\tbdsk-url-1 = {https://doi.org/10.3934/krm.2021039}}\\n\\n\",\"author_short\":[\"Jang, J. W.\",\"Strain, R. M.\",\"Wong, T. K.\"],\"key\":\"JangStrainWong2021\",\"id\":\"JangStrainWong2021\",\"bibbaseid\":\"jang-strain-wong-magneticconfinementforthe2daxisymmetricrelativisticvlasovmaxwellsysteminanannulus-2021\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/2111.04583\"},\"keyword\":[\"Kinetic Theory\",\"relativistic Kinetic Theory\",\"Vlasov-Maxwell systems\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":5,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"This article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in the center-of-momentum coordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular we mention the widely used cancellation lemma by Alexandre et al. (Arch. Ration. Mech. Anal. 152(4):327–355, 2000). In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentum $p$ to the post collisional momentum $p'$; specifically we calculate the determinant for $p↦ u = θ p'+(1-θ )p$ for $θ ∈ [0,1]$. Afterwards we give an upper-bound for this determinant that has no singularity in both $p$ and $q$ variables. Next we give an example where we prove that the Jacobian goes to zero in a specific pointwise limit. We further explain the results of our numerical study which shows that the Jacobian determinant has a very large number of distinct points at which it is machine zero. This generalizes the work of Glassey-Strauss (Transport Theory Statist. Phys. 20(1):55–68, 1991) and Guo-Strain (Comm. Math. Phys. 310(3):649–673, 2012). These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Chapman\"],\"firstnames\":[\"James\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Jang\"],\"firstnames\":[\"Jin\",\"Woo\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"doi\":\"10.1007/s00220-021-04101-2\",\"eprint\":\"2006.02540\",\"fjournal\":\"Communications in Mathematical Physics\",\"journal\":\"Comm. Math. Phys.\",\"keywords\":\"relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory\",\"pages\":\"1913–1943\",\"published\":\"published online May 07, 2021\",\"title\":\"On the Determinant Problem for the Relativistic Boltzmann Equation\",\"url_arxiv\":\"https://arxiv.org/abs/2006.02540\",\"volume\":\"384\",\"year\":\"2021\",\"bdsk-url-1\":\"https://doi.org/10.1007/s00220-021-04101-2\",\"bibtex\":\"@article{ChapmanJangStrain2020,\\n\\tabstract = {This article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in the center-of-momentum coordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular we mention the widely used cancellation lemma by Alexandre et al. (Arch. Ration. Mech. Anal. 152(4):327--355, 2000). In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentum $p$ to the post collisional momentum $p'$; specifically we calculate the determinant for $p\\\\mapsto u = \\\\theta p'+(1-\\\\theta )p$ for $\\\\theta \\\\in [0,1]$. Afterwards we give an upper-bound for this determinant that has no singularity in both $p$ and $q$ variables. Next we give an example where we prove that the Jacobian goes to zero in a specific pointwise limit. We further explain the results of our numerical study which shows that the Jacobian determinant has a very large number of distinct points at which it is machine zero. This generalizes the work of Glassey-Strauss (Transport Theory Statist. Phys. 20(1):55--68, 1991) and Guo-Strain (Comm. Math. Phys. 310(3):649--673, 2012). These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates.},\\n\\tauthor = {Chapman, James and Jang, Jin Woo and Strain, Robert M.},\\n\\tdoi = {10.1007/s00220-021-04101-2},\\n\\teprint = {2006.02540},\\n\\tfjournal = {Communications in Mathematical Physics},\\n\\tjournal = {Comm. Math. Phys.},\\n\\tkeywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\\n\\tpages = {1913--1943},\\n\\tpublished = {published online May 07, 2021},\\n\\ttitle = {{On the Determinant Problem for the Relativistic Boltzmann Equation}},\\n\\turl_arxiv = {https://arxiv.org/abs/2006.02540},\\n\\tvolume = {384},\\n\\tyear = {2021},\\n\\tbdsk-url-1 = {https://doi.org/10.1007/s00220-021-04101-2}}\\n\\n\",\"author_short\":[\"Chapman, J.\",\"Jang, J. W.\",\"Strain, R. M.\"],\"key\":\"ChapmanJangStrain2020\",\"id\":\"ChapmanJangStrain2020\",\"bibbaseid\":\"chapman-jang-strain-onthedeterminantproblemfortherelativisticboltzmannequation-2021\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/2006.02540\"},\"keyword\":[\"relativistic Boltzmann equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":11,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"This report succinctly summarizes results proved in the authors' recent work (2019) where the unique existence of solutions to the Boltzmann equation without angular cut-off and the Landau equation with Coulomb potential are studied in a perturbation framework. A major feature is the use of the Wiener space $A(Ω)$, which can be expected to play a similar role to $L^∞$. Compared to the $L^2$-based solution spaces that were employed for prior known results, this function space enables us to establish a new global existence theory. One further feature is that, not only an initial value problem, but also an initial boundary value problem whose boundary conditions can be regarded as physical boundaries in some simple situation, are considered for both equations. In addition to unique existence, large-time behavior of the solutions and propagation of spatial regularity are also proved. In the end of report, key ideas of the proof will be explained in a concise way.\",\"author\":[{\"firstnames\":[\"Renjun\"],\"propositions\":[],\"lastnames\":[\"Duan\"],\"suffixes\":[]},{\"firstnames\":[\"Shuangqian\"],\"propositions\":[],\"lastnames\":[\"Liu\"],\"suffixes\":[]},{\"firstnames\":[\"Shota\"],\"propositions\":[],\"lastnames\":[\"Sakamoto\"],\"suffixes\":[]},{\"firstnames\":[\"Robert\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Strain\"],\"suffixes\":[]}],\"fjournal\":\"RIMS Kokyuroku Bessatsu\",\"journal\":\"RIMS Kokyuroku Bessatsu\",\"keywords\":\"Boltzmann equation, Landau equation, Kinetic Theory, non-cutoff\",\"note\":\"proceedings of symposium on ``Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations''\",\"pages\":\"29–46\",\"title\":\"Global solutions to the Boltzmann equation without angular cutoff and the Landau equation with Colomb potential\",\"urlpdf\":\"http://hdl.handle.net/2433/260676\",\"volume\":\"B82\",\"year\":\"2020\",\"bdsk-url-1\":\"http://www.kurims.kyoto-u.ac.jp/ kenkyubu/bessatsu.html\",\"bibtex\":\"@article{RIMSbessatsu2020,\\n\\tabstract = {This report succinctly summarizes results proved in the authors' recent work (2019) where the unique existence of solutions to the Boltzmann equation without angular cut-off and the Landau equation with Coulomb potential are studied in a perturbation framework. A major feature is the use of the Wiener space $A(\\\\Omega)$, which can be expected to play a similar role to $L^\\\\infty$. Compared to the $L^2$-based solution spaces that were employed for prior known results, this function space enables us to establish a new global existence theory. One further feature is that, not only an initial value problem, but also an initial boundary value problem whose boundary conditions can be regarded as physical boundaries in some simple situation, are considered for both equations. \\nIn addition to unique existence, large-time behavior of the solutions and propagation of spatial regularity are also proved. In the end of report, key ideas of the proof will be explained in a concise way.},\\n\\tauthor = {Renjun Duan and Shuangqian Liu and Shota Sakamoto and Robert M. Strain},\\n\\tfjournal = {RIMS K{{o}}ky{{u}}roku {B}essatsu},\\n\\tjournal = {RIMS K{{o}}ky{{u}}roku {B}essatsu},\\n\\tkeywords = {Boltzmann equation, Landau equation, Kinetic Theory, non-cutoff},\\n\\tnote = {proceedings of symposium on ``Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations''},\\n\\tpages = {29--46},\\n\\ttitle = {{G}lobal solutions to the {B}oltzmann equation without angular cutoff and the {L}andau equation with {C}olomb potential},\\n\\turlpdf = {http://hdl.handle.net/2433/260676},\\n\\tvolume = {B82},\\n\\tyear = {2020},\\n\\tbdsk-url-1 = {http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu.html}}\\n\\n\",\"author_short\":[\"Duan, R.\",\"Liu, S.\",\"Sakamoto, S.\",\"Strain, R. M.\"],\"key\":\"RIMSbessatsu2020\",\"id\":\"RIMSbessatsu2020\",\"bibbaseid\":\"duan-liu-sakamoto-strain-globalsolutionstotheboltzmannequationwithoutangularcutoffandthelandauequationwithcolombpotential-2020\",\"role\":\"author\",\"urls\":{\"Pdf\":\"http://hdl.handle.net/2433/260676\"},\"keyword\":[\"Boltzmann equation\",\"Landau equation\",\"Kinetic Theory\",\"non-cutoff\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":6,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"This paper proves the existence of small-amplitude global-in-time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well-known works (Guo, 2002) and (Gressman-Strain-2011, AMUXY-2012) on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem to obtain global solutions in an $L^∞_{x,v}$ framework, similar to that in (Guo-2010), for the Boltzmann equation with cutoff in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity-diffusion-type collision operator in the non-cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem. In the present work we introduce a new function space with low regularity in the spatial variable to treat the problem in cases when the spatial domain is either a torus, or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the $L^∞_T L^2_v$ norm, in velocity and time, of the distribution function is in the Wiener algebra $A(Ω)$ in the spatial variables. Besides the construction of global solutions in these function spaces, we additionally study the large-time behavior of solutions for both hard and soft potentials, and we further justify the property of propagation of regularity of solutions in the spatial variables.\",\"author\":[{\"firstnames\":[\"Renjun\"],\"propositions\":[],\"lastnames\":[\"Duan\"],\"suffixes\":[]},{\"firstnames\":[\"Shuangqian\"],\"propositions\":[],\"lastnames\":[\"Liu\"],\"suffixes\":[]},{\"firstnames\":[\"Shota\"],\"propositions\":[],\"lastnames\":[\"Sakamoto\"],\"suffixes\":[]},{\"firstnames\":[\"Robert\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Strain\"],\"suffixes\":[]}],\"doi\":\"10.1002/cpa.21920\",\"eprint\":\"1904.12086\",\"fjournal\":\"Communications on Pure and Applied Mathematics\",\"journal\":\"Comm. Pure Appl. Math.\",\"keywords\":\"Boltzmann equation, Landau equation, Kinetic Theory, non-cutoff\",\"number\":\"5\",\"pages\":\"932–1020\",\"title\":\"Global mild solutions of the Landau and non-cutoff Boltzmann equations\",\"url_arxiv\":\"https://arxiv.org/abs/1904.12086\",\"volume\":\"74\",\"year\":\"2021\",\"bdsk-url-1\":\"https://arxiv.org/abs/1904.12086\",\"bibtex\":\"@article{1904.12086,\\n\\tabstract = {This paper proves the existence of small-amplitude global-in-time unique mild\\nsolutions to both the Landau equation including the Coulomb potential and the\\nBoltzmann equation without angular cutoff. Since the well-known works (Guo,\\n2002) and (Gressman-Strain-2011, AMUXY-2012) on the construction of classical\\nsolutions in smooth Sobolev spaces which in particular are regular in the\\nspatial variables, it still remains an open problem to obtain global solutions\\nin an $L^\\\\infty_{x,v}$ framework, similar to that in (Guo-2010), for the\\nBoltzmann equation with cutoff in general bounded domains. One main difficulty\\narises from the interaction between the transport operator and the\\nvelocity-diffusion-type collision operator in the non-cutoff Boltzmann and\\nLandau equations; another major difficulty is the potential formation of\\nsingularities for solutions to the boundary value problem. In the present work\\nwe introduce a new function space with low regularity in the spatial variable\\nto treat the problem in cases when the spatial domain is either a torus, or a\\nfinite channel with boundary. For the latter case, either the inflow boundary\\ncondition or the specular reflection boundary condition is considered. An\\nimportant property of the function space is that the $L^\\\\infty_T L^2_v$ norm,\\nin velocity and time, of the distribution function is in the Wiener algebra\\n$A(\\\\Omega)$ in the spatial variables. Besides the construction of global\\nsolutions in these function spaces, we additionally study the large-time\\nbehavior of solutions for both hard and soft potentials, and we further justify\\nthe property of propagation of regularity of solutions in the spatial\\nvariables.},\\n\\tauthor = {Renjun Duan and Shuangqian Liu and Shota Sakamoto and Robert M. Strain},\\n\\tdoi = {10.1002/cpa.21920},\\n\\teprint = {1904.12086},\\n\\tfjournal = {Communications on Pure and Applied Mathematics},\\n\\tjournal = {Comm. Pure Appl. Math.},\\n\\tkeywords = {Boltzmann equation, Landau equation, Kinetic Theory, non-cutoff},\\n\\tnumber = {5},\\n\\tpages = {932--1020},\\n\\ttitle = {Global mild solutions of the {L}andau and non-cutoff {B}oltzmann equations},\\n\\turl_arxiv = {https://arxiv.org/abs/1904.12086},\\n\\tvolume = {74},\\n\\tyear = {2021},\\n\\tbdsk-url-1 = {https://arxiv.org/abs/1904.12086}}\\n\\n\",\"author_short\":[\"Duan, R.\",\"Liu, S.\",\"Sakamoto, S.\",\"Strain, R. M.\"],\"key\":\"1904.12086\",\"id\":\"1904.12086\",\"bibbaseid\":\"duan-liu-sakamoto-strain-globalmildsolutionsofthelandauandnoncutoffboltzmannequations-2021\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1904.12086\"},\"keyword\":[\"Boltzmann equation\",\"Landau equation\",\"Kinetic Theory\",\"non-cutoff\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":10,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"The Muskat problem models the filtration of two incompressible immiscible fluids of different characteristics in porous media. In this paper, we consider both the 2D and 3D setting of two fluids of different constant densities and different constant viscosities. In this situation, the related contour equations are non-local, not only in the evolution system, but also in the implicit relation between the amplitude of the vorticity and the free interface. Among other extra difficulties, no maximum principles are available for the amplitude and the slopes of the interface in $L^∞$. We prove global in time existence results for medium size initial stable data in critical spaces. We also enhance previous methods by showing smoothing (instant analyticity), improving the medium size constant in 3D, together with sharp decay rates of analytic norms. The found technique is twofold, giving ill-posedness in unstable situations for very low regular solutions.\",\"author\":[{\"firstnames\":[\"Francisco\"],\"propositions\":[],\"lastnames\":[\"Gancedo\"],\"suffixes\":[]},{\"firstnames\":[\"Eduardo\"],\"propositions\":[],\"lastnames\":[\"García-Juárez\"],\"suffixes\":[]},{\"firstnames\":[\"Neel\"],\"propositions\":[],\"lastnames\":[\"Patel\"],\"suffixes\":[]},{\"firstnames\":[\"Robert\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Strain\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1016/j.aim.2019.01.017\",\"eprint\":\"1710.11604\",\"fjournal\":\"Advances in Mathematics\",\"issn\":\"0001-8708\",\"journal\":\"Adv. Math.\",\"keywords\":\"Fluid mechanics, Free boundary problems, Muskat problem\",\"mrclass\":\"76S05\",\"mrnumber\":\"3899970\",\"pages\":\"552–597\",\"title\":\"On the Muskat problem with viscosity jump: Global in time results\",\"url_arxiv\":\"https://arxiv.org/abs/1710.11604\",\"volume\":\"345\",\"year\":\"2019\",\"zblnumber\":\"07021548\",\"bdsk-url-1\":\"https://doi.org/10.1016/j.aim.2019.01.017\",\"bibtex\":\"@article{GGPS19,\\n\\tabstract = {The Muskat problem models the filtration of two incompressible immiscible\\nfluids of different characteristics in porous media. In this paper, we consider\\nboth the 2D and 3D setting of two fluids of different constant densities and\\ndifferent constant viscosities. In this situation, the related contour\\nequations are non-local, not only in the evolution system, but also in the\\nimplicit relation between the amplitude of the vorticity and the free\\ninterface. Among other extra difficulties, no maximum principles are available\\nfor the amplitude and the slopes of the interface in $L^\\\\infty$. We prove\\nglobal in time existence results for medium size initial stable data in\\ncritical spaces. We also enhance previous methods by showing smoothing (instant\\nanalyticity), improving the medium size constant in 3D, together with sharp\\ndecay rates of analytic norms. The found technique is twofold, giving\\nill-posedness in unstable situations for very low regular solutions.},\\n\\tauthor = {Francisco Gancedo and Eduardo Garc{\\\\'{i}}a-Ju{\\\\'{a}}rez and Neel Patel and Robert M. Strain},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1016/j.aim.2019.01.017},\\n\\teprint = {1710.11604},\\n\\tfjournal = {Advances in Mathematics},\\n\\tissn = {0001-8708},\\n\\tjournal = {Adv. Math.},\\n\\tkeywords = {Fluid mechanics, Free boundary problems, Muskat problem},\\n\\tmrclass = {76S05},\\n\\tmrnumber = {3899970},\\n\\tpages = {552--597},\\n\\ttitle = {On the {M}uskat problem with viscosity jump: {G}lobal in time results},\\n\\turl_arxiv = {https://arxiv.org/abs/1710.11604},\\n\\tvolume = {345},\\n\\tyear = {2019},\\n\\tzblnumber = {07021548},\\n\\tbdsk-url-1 = {https://doi.org/10.1016/j.aim.2019.01.017}}\\n\\n\",\"author_short\":[\"Gancedo, F.\",\"García-Juárez, E.\",\"Patel, N.\",\"Strain, R. M.\"],\"key\":\"GGPS19\",\"id\":\"GGPS19\",\"bibbaseid\":\"gancedo-garciajurez-patel-strain-onthemuskatproblemwithviscosityjumpglobalintimeresults-2019\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1710.11604\"},\"keyword\":[\"Fluid mechanics\",\"Free boundary problems\",\"Muskat problem\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":4,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this paper we prove the global existence, uniqueness, optimal large time decay rates, and uniform gain of analyticity for the exponential PDE $h_t=Δ e^{-Δ h}$ in the whole space $ℝ^d_x$. We assume the initial data is of medium size in the critical Wiener algebra $Δ h ∈ A(ℝ^d)$. This exponential PDE was derived in (Krug, Dobbs, and Majaniemi in 1995) and more recently in (Marzuola and Weare 2013).\",\"author\":[{\"firstnames\":[\"Jian-Guo\"],\"propositions\":[],\"lastnames\":[\"Liu\"],\"suffixes\":[]},{\"firstnames\":[\"Robert\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Strain\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-08 14:21:35 -0600\",\"doi\":\"10.4171/IFB/417\",\"eprint\":\"1805.02246\",\"fjournal\":\"Interfaces and Free Boundaries. Mathematical Analysis, Computation and Applications.\",\"issn\":\"1463-9963\",\"journal\":\"Interfaces Free Bound.\",\"keywords\":\"Free boundary problems, Materials science\",\"mrclass\":\"35K25 (35B40 35K55 35K65 74A50)\",\"mrnumber\":\"3951578\",\"number\":\"1\",\"pages\":\"61–86\",\"title\":\"Global stability for solutions to the exponential PDE describing epitaxial growth\",\"url_arxiv\":\"https://arxiv.org/abs/1805.02246\",\"volume\":\"21\",\"year\":\"2019\",\"zblnumber\":\"07084773\",\"bdsk-url-1\":\"https://doi.org/10.4171/IFB/417\",\"bibtex\":\"@article{LS2019,\\n\\tabstract = {In this paper we prove the global existence, uniqueness, optimal large time\\ndecay rates, and uniform gain of analyticity for the exponential PDE\\n$h_t=\\\\Delta e^{-\\\\Delta h}$ in the whole space $\\\\mathbb{R}^d_x$. We assume the\\ninitial data is of medium size in the critical Wiener algebra $\\\\Delta h \\\\in\\nA(\\\\mathbb{R}^d)$. This exponential PDE was derived in (Krug, Dobbs, and\\nMajaniemi in 1995) and more recently in (Marzuola and Weare 2013).},\\n\\tauthor = {Jian-Guo Liu and Robert M. Strain},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-08 14:21:35 -0600},\\n\\tdoi = {10.4171/IFB/417},\\n\\teprint = {1805.02246},\\n\\tfjournal = {Interfaces and Free Boundaries. Mathematical Analysis, Computation and Applications.},\\n\\tissn = {1463-9963},\\n\\tjournal = {Interfaces Free Bound.},\\n\\tkeywords = {Free boundary problems, Materials science},\\n\\tmrclass = {35K25 (35B40 35K55 35K65 74A50)},\\n\\tmrnumber = {3951578},\\n\\tnumber = {1},\\n\\tpages = {61--86},\\n\\ttitle = {Global stability for solutions to the exponential {PDE} describing epitaxial growth},\\n\\turl_arxiv = {https://arxiv.org/abs/1805.02246},\\n\\tvolume = {21},\\n\\tyear = {2019},\\n\\tzblnumber = {07084773},\\n\\tbdsk-url-1 = {https://doi.org/10.4171/IFB/417}}\\n\\n\",\"author_short\":[\"Liu, J.\",\"Strain, R. M.\"],\"key\":\"LS2019\",\"id\":\"LS2019\",\"bibbaseid\":\"liu-strain-globalstabilityforsolutionstotheexponentialpdedescribingepitaxialgrowth-2019\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1805.02246\"},\"keyword\":[\"Free boundary problems\",\"Materials science\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this paper we study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. Despite it's physical importance, this equation has not received a lot of mathematical attention we think due to the extreme complexity of the relativistic structure of the kernel of the collision operator. In this paper we first largely decompose the structure of the relativistic Landau collision operator. After that we prove the global Entropy dissipation estimate. Then we prove the propagation of any polynomial moment for a weak solution. Lastly we prove the existence of a true weak solution for a large class of initial data.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Tasković\"],\"firstnames\":[\"Maja\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1016/j.jfa.2019.04.007\",\"eprint\":\"1806.08720\",\"fjournal\":\"Journal of Functional Analysis\",\"issn\":\"0022-1236\",\"journal\":\"J. Funct. Anal.\",\"keywords\":\"relativistic Landau equation, Landau equation, Kinetic Theory, relativistic Kinetic Theory\",\"mrclass\":\"82D10 (35A01 35B45 35Q75)\",\"mrnumber\":\"3959729\",\"number\":\"4\",\"pages\":\"1139–1201\",\"title\":\"Entropy dissipation estimates for the relativistic Landau equation, and applications\",\"url_arxiv\":\"https://arxiv.org/abs/1806.08720\",\"volume\":\"277\",\"year\":\"2019\",\"zblnumber\":\"07066836\",\"bdsk-url-1\":\"https://doi.org/10.1016/j.jfa.2019.04.007\",\"bibtex\":\"@article{StrainTas2019,\\n\\tabstract = {In this paper we study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. Despite it's physical importance, this equation has not received a lot of mathematical attention we think due to the extreme complexity of the relativistic structure of the kernel of the collision operator. In this paper we first largely decompose the structure of the relativistic Landau collision operator. After that we prove the global Entropy dissipation estimate. Then we prove the propagation of any polynomial moment for a weak solution. Lastly we prove the existence of a true weak solution for a large class of initial data.},\\n\\tauthor = {Strain, Robert M. and Taskovi{\\\\'{c}}, Maja},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1016/j.jfa.2019.04.007},\\n\\teprint = {1806.08720},\\n\\tfjournal = {Journal of Functional Analysis},\\n\\tissn = {0022-1236},\\n\\tjournal = {J. Funct. Anal.},\\n\\tkeywords = {relativistic Landau equation, Landau equation, Kinetic Theory, relativistic Kinetic Theory},\\n\\tmrclass = {82D10 (35A01 35B45 35Q75)},\\n\\tmrnumber = {3959729},\\n\\tnumber = {4},\\n\\tpages = {1139--1201},\\n\\ttitle = {{E}ntropy dissipation estimates for the relativistic {L}andau equation, and applications},\\n\\turl_arxiv = {https://arxiv.org/abs/1806.08720},\\n\\tvolume = {277},\\n\\tyear = {2019},\\n\\tzblnumber = {07066836},\\n\\tbdsk-url-1 = {https://doi.org/10.1016/j.jfa.2019.04.007}}\\n\\n\",\"author_short\":[\"Strain, R. M.\",\"Tasković, M.\"],\"key\":\"StrainTas2019\",\"id\":\"StrainTas2019\",\"bibbaseid\":\"strain-taskovi-entropydissipationestimatesfortherelativisticlandauequationandapplications-2019\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1806.08720\"},\"keyword\":[\"relativistic Landau equation\",\"Landau equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":6,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"We prove the uniqueness of weak solutions to the spatially homogeneous special relativistic Landau equation under the conditional assumption that the solution satisfies $(p^0)^7 F(t,p) ∈ L^1 ([0,T]; L^∞)$. The existence of standard weak solutions to the relativistic Landau equation has been shown recently in (Strain-Tasković 2019, 1806.08720).\",\"author\":[{\"firstnames\":[\"Robert\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Strain\"],\"suffixes\":[]},{\"firstnames\":[\"Zhenfu\"],\"propositions\":[],\"lastnames\":[\"Wang\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1090/qam/1545\",\"eprint\":\"1903.05301\",\"fjournal\":\"Quarterly of Applied Mathematics\",\"journal\":\"Quart. Appl. Math.\",\"keywords\":\"relativistic Landau equation, Landau equation, Kinetic Theory, relativistic Kinetic Theory\",\"mrnumber\":\"4042221\",\"pages\":\"107–145\",\"title\":\"Uniqueness of Bounded Solutions for the Homogeneous Relativistic Landau Equation with Coulomb Interactions\",\"url_arxiv\":\"https://arxiv.org/abs/1903.05301\",\"volume\":\"78\",\"year\":\"2019\",\"zblnumber\":\"1427.82044\",\"bdsk-url-1\":\"https://arxiv.org/abs/1903.05301\",\"bibtex\":\"@article{1903.05301,\\n\\tabstract = {We prove the uniqueness of weak solutions to the spatially homogeneous special relativistic Landau equation under the conditional assumption that the solution satisfies $(p^0)^7 F(t,p) \\\\in L^1 ([0,T]; L^\\\\infty)$.  The existence of standard weak solutions to the relativistic Landau equation has been shown recently in (Strain-Taskovi{\\\\'{c}} 2019, 1806.08720).},\\n\\tauthor = {Robert M. Strain and Zhenfu Wang},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1090/qam/1545},\\n\\teprint = {1903.05301},\\n\\tfjournal = {Quarterly of Applied Mathematics},\\n\\tjournal = {Quart. Appl. Math.},\\n\\tkeywords = {relativistic Landau equation, Landau equation, Kinetic Theory, relativistic Kinetic Theory},\\n\\tmrnumber = {4042221},\\n\\tpages = {107--145},\\n\\ttitle = {Uniqueness of Bounded Solutions for the Homogeneous Relativistic {L}andau Equation with {C}oulomb Interactions},\\n\\turl_arxiv = {https://arxiv.org/abs/1903.05301},\\n\\tvolume = {78},\\n\\tyear = {2019},\\n\\tzblnumber = {1427.82044},\\n\\tbdsk-url-1 = {https://arxiv.org/abs/1903.05301}}\\n\\n\",\"author_short\":[\"Strain, R. M.\",\"Wang, Z.\"],\"key\":\"1903.05301\",\"id\":\"1903.05301\",\"bibbaseid\":\"strain-wang-uniquenessofboundedsolutionsforthehomogeneousrelativisticlandauequationwithcoulombinteractions-2019\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1903.05301\"},\"keyword\":[\"relativistic Landau equation\",\"Landau equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":1,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"This paper considers the three dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an $L^2$ maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants. Furthermore we refine the estimates from our prior paper to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in critical spaces, giving solutions with bounded slope and time integrable bounded curvature.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Constantin\"],\"firstnames\":[\"Peter\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Córdoba\"],\"firstnames\":[\"Diego\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Gancedo\"],\"firstnames\":[\"Francisco\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Rodríguez-Piazza\"],\"firstnames\":[\"Luis\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1353/ajm.2016.0044\",\"eprint\":\"1310.0953\",\"fjournal\":\"American Journal of Mathematics\",\"issn\":\"0002-9327\",\"journal\":\"Amer. J. Math.\",\"keywords\":\"Fluid mechanics, Free boundary problems, Muskat problem\",\"mrclass\":\"35Q35 (76S05)\",\"mrnumber\":\"3595492\",\"mrreviewer\":\"Maria Specovius-Neugebauer\",\"number\":\"6\",\"pages\":\"1455–1494\",\"title\":\"On the Muskat problem: global in time results in 2D and 3D\",\"url_arxiv\":\"https://arxiv.org/abs/1310.0953\",\"volume\":\"138\",\"year\":\"2016\",\"zblnumber\":\"1369.35053\",\"bdsk-url-1\":\"https://doi.org/10.1353/ajm.2016.0044\",\"bibtex\":\"@article{CCGRS16,\\n\\tabstract = {This paper considers the three dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an $L^2$ maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants.  Furthermore we refine the estimates from our prior paper to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in critical spaces, giving solutions with bounded slope and time integrable bounded curvature.},\\n\\tauthor = {Constantin, Peter and C\\\\'{o}rdoba, Diego and Gancedo, Francisco and Rodr\\\\'{i}guez-Piazza, Luis and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1353/ajm.2016.0044},\\n\\teprint = {1310.0953},\\n\\tfjournal = {American Journal of Mathematics},\\n\\tissn = {0002-9327},\\n\\tjournal = {Amer. J. Math.},\\n\\tkeywords = {Fluid mechanics, Free boundary problems, Muskat problem},\\n\\tmrclass = {35Q35 (76S05)},\\n\\tmrnumber = {3595492},\\n\\tmrreviewer = {Maria Specovius-Neugebauer},\\n\\tnumber = {6},\\n\\tpages = {1455--1494},\\n\\ttitle = {On the {M}uskat problem: global in time results in 2{D} and 3{D}},\\n\\turl_arxiv = {https://arxiv.org/abs/1310.0953},\\n\\tvolume = {138},\\n\\tyear = {2016},\\n\\tzblnumber = {1369.35053},\\n\\tbdsk-url-1 = {https://doi.org/10.1353/ajm.2016.0044}}\\n\\n\",\"author_short\":[\"Constantin, P.\",\"Córdoba, D.\",\"Gancedo, F.\",\"Rodríguez-Piazza, L.\",\"Strain, R. M.\"],\"key\":\"CCGRS16\",\"id\":\"CCGRS16\",\"bibbaseid\":\"constantin-crdoba-gancedo-rodriguezpiazza-strain-onthemuskatproblemglobalintimeresultsin2dand3d-2016\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1310.0953\"},\"keyword\":[\"Fluid mechanics\",\"Free boundary problems\",\"Muskat problem\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"We consider the relativistic Vlasov-Maxwell system with data of unrestricted size and without compact support in momentum space. In the two dimensional and the two-and-a-half dimensional cases, Glassey-Schaeffer proved (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:331–354, 1998; Arch Ration Mech Anal. 141:355–374, 1998) that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the 2D and the $2\\\\frac 12$D cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the $L^∞_x$ norms of the electromagnetic fields compared to (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:355–374, 1998). In the three dimensional case, we apply Strichartz estimates and moment bounds to show that a regular solution can be extended as long as $\\\\|(p^0)^{θ} f \\\\|_{L^q_xL^1_{p}}$ remains bounded for $θ>{2/q}$ and $q ∈ (2, ∞]$. This improves previous results of Pallard (Indiana Univ Math J 54(5):1395–1409, 2005; Commun Math Sci 13(2):347–354, 2015).\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Luk\"],\"firstnames\":[\"Jonathan\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-08 13:24:39 -0600\",\"doi\":\"10.1007/s00205-015-0899-1\",\"fjournal\":\"Archive for Rational Mechanics and Analysis\",\"issn\":\"0003-9527\",\"journal\":\"Arch. Ration. Mech. Anal.\",\"keywords\":\"Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems\",\"mrclass\":\"35Q83 (35Q61 35Q75 35Q82 76Y05 82D10)\",\"mrnumber\":\"3437855\",\"mrreviewer\":\"Ł ukasz Pł ociniczak\",\"number\":\"1\",\"pages\":\"445–552\",\"title\":\"Strichartz estimates and moment bounds for the relativistic Vlasov-Maxwell system\",\"url_arxiv1\":\"https://arxiv.org/abs/1406.0168\",\"url_arxiv2\":\"https://arxiv.org/abs/1406.0169\",\"volume\":\"219\",\"year\":\"2016\",\"zblnumber\":\"1337.35150\",\"bdsk-url-1\":\"https://doi.org/10.1007/s00205-015-0899-1\",\"bibtex\":\"@article{MR3437855,\\n\\tabstract = {We consider the relativistic Vlasov-Maxwell system with data of unrestricted size and without compact support in momentum space. In the two dimensional and the two-and-a-half dimensional cases, Glassey-Schaeffer proved (Commun Math Phys 185:257--284, 1997; Arch Ration Mech Anal 141:331--354, 1998; Arch Ration Mech Anal. 141:355--374, 1998) that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the 2D and the $2\\\\frac 12$D cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the $L^\\\\infty_x$ norms of the electromagnetic fields compared to (Commun Math Phys 185:257--284, 1997; Arch Ration Mech Anal 141:355--374, 1998).  In the three dimensional case, we apply Strichartz estimates and moment bounds to show that a regular solution can be extended as long as \\n$\\\\|(p^0)^{\\\\theta} f \\\\|_{L^q_xL^1_{p}}$ remains bounded for $\\\\theta>{2/q}$ and $q \\\\in (2, \\\\infty]$. This improves previous results of Pallard (Indiana Univ Math J 54(5):1395--1409, 2005; Commun Math Sci 13(2):347--354, 2015).},\\n\\tauthor = {Luk, Jonathan and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-08 13:24:39 -0600},\\n\\tdoi = {10.1007/s00205-015-0899-1},\\n\\tfjournal = {Archive for Rational Mechanics and Analysis},\\n\\tissn = {0003-9527},\\n\\tjournal = {Arch. Ration. Mech. Anal.},\\n\\tkeywords = {Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},\\n\\tmrclass = {35Q83 (35Q61 35Q75 35Q82 76Y05 82D10)},\\n\\tmrnumber = {3437855},\\n\\tmrreviewer = {\\\\L ukasz P\\\\l ociniczak},\\n\\tnumber = {1},\\n\\tpages = {445--552},\\n\\ttitle = {Strichartz estimates and moment bounds for the relativistic {V}lasov-{M}axwell system},\\n\\turl_arxiv1 = {https://arxiv.org/abs/1406.0168},\\n\\turl_arxiv2 = {https://arxiv.org/abs/1406.0169},\\n\\tvolume = {219},\\n\\tyear = {2016},\\n\\tzblnumber = {1337.35150},\\n\\tbdsk-url-1 = {https://doi.org/10.1007/s00205-015-0899-1}}\\n\\n\",\"author_short\":[\"Luk, J.\",\"Strain, R. M.\"],\"key\":\"MR3437855\",\"id\":\"MR3437855\",\"bibbaseid\":\"luk-strain-strichartzestimatesandmomentboundsfortherelativisticvlasovmaxwellsystem-2016\",\"role\":\"author\",\"urls\":{\" arxiv1\":\"https://arxiv.org/abs/1406.0168\",\" arxiv2\":\"https://arxiv.org/abs/1406.0169\"},\"keyword\":[\"Kinetic Theory\",\"relativistic Kinetic Theory\",\"Vlasov-Maxwell systems\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":1,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this article we will introduce a new model to describe the leading order behavior of an ideal and axisymmetric fluid moving in a very narrow domain. After providing a formal derivation of the model, we will prove the well-posedness and provide a rigorous mathematical justification for the formal derivation under a new sign condition. Finally, a blowup result regarding this model will be discussed as well.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wong\"],\"firstnames\":[\"Tak\",\"Kwong\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-08 13:17:31 -0600\",\"doi\":\"10.1016/j.jde.2015.11.023\",\"eprint\":\"1505.06281\",\"fjournal\":\"Journal of Differential Equations\",\"issn\":\"0022-0396\",\"journal\":\"J. Differential Equations\",\"keywords\":\"Fluid mechanics\",\"mrclass\":\"35Q31 (35B30 35Q35 76B03)\",\"mrnumber\":\"3437599\",\"mrreviewer\":\"Franck Sueur\",\"number\":\"5\",\"pages\":\"4619–4656\",\"title\":\"Axisymmetric flow of ideal fluid moving in a narrow domain: a study of the axisymmetric hydrostatic Euler equations\",\"url_arxiv\":\"https://arxiv.org/abs/1505.06281\",\"volume\":\"260\",\"year\":\"2016\",\"zblnumber\":\"1333.35170\",\"bdsk-url-1\":\"https://doi.org/10.1016/j.jde.2015.11.023\",\"bibtex\":\"@article{MR3437599,\\n\\tabstract = {In this article we will introduce a new model to describe the leading order behavior of an ideal and axisymmetric fluid moving in a very narrow domain. After providing a formal derivation of the model, we will prove the well-posedness and provide a rigorous mathematical justification for the formal derivation under a new sign condition. Finally, a blowup result regarding this model will be discussed as well.},\\n\\tauthor = {Strain, Robert M. and Wong, Tak Kwong},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-08 13:17:31 -0600},\\n\\tdoi = {10.1016/j.jde.2015.11.023},\\n\\teprint = {1505.06281},\\n\\tfjournal = {Journal of Differential Equations},\\n\\tissn = {0022-0396},\\n\\tjournal = {J. Differential Equations},\\n\\tkeywords = {Fluid mechanics},\\n\\tmrclass = {35Q31 (35B30 35Q35 76B03)},\\n\\tmrnumber = {3437599},\\n\\tmrreviewer = {Franck Sueur},\\n\\tnumber = {5},\\n\\tpages = {4619--4656},\\n\\ttitle = {Axisymmetric flow of ideal fluid moving in a narrow domain: a study of the axisymmetric hydrostatic {E}uler equations},\\n\\turl_arxiv = {https://arxiv.org/abs/1505.06281},\\n\\tvolume = {260},\\n\\tyear = {2016},\\n\\tzblnumber = {1333.35170},\\n\\tbdsk-url-1 = {https://doi.org/10.1016/j.jde.2015.11.023}}\\n\\n\",\"author_short\":[\"Strain, R. M.\",\"Wong, T. K.\"],\"key\":\"MR3437599\",\"id\":\"MR3437599\",\"bibbaseid\":\"strain-wong-axisymmetricflowofidealfluidmovinginanarrowdomainastudyoftheaxisymmetrichydrostaticeulerequations-2016\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1505.06281\"},\"keyword\":[\"Fluid mechanics\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":0,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this paper for either the sharp front Surface Quasi-Geostrophic equation or the Muskat problem we rule out the `'splash singularity'' blow-up scenario; in other words we prove that the contours evolving from either of these systems can not intersect at a single point while the free boundary remains smooth. Splash singularities have been shown to hold for the free boundary incompressible Euler equation in the form of the water waves contour evolution problem (Castro A, et al. (2012) Proc Natl Acad Sci USA 109:733-738). Our result confirms the numerical simulations in (Córdoba D, et al. (2005) Proc Natl Acad Sci USA 102:5949-5952) where it is shown that the curvature blows up due to the contours collapsing at a point. Here we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse. Another conclusion that we provide is a better understanding of the work (Córdoba D, Gancedo F (2010) Comm Math Phys 299:561-575) in which squirt singularities are ruled out; in this case a positive volume of fluid between the contours can not be ejected in finite time.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Gancedo\"],\"firstnames\":[\"Francisco\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1073/pnas.1320554111\",\"eprint\":\"1309.4023\",\"fjournal\":\"Proceedings of the National Academy of Sciences of the United States of America\",\"issn\":\"0027-8424\",\"journal\":\"Proc. Natl. Acad. Sci. USA\",\"keywords\":\"Fluid mechanics, Free boundary problems, Muskat problem\",\"mrclass\":\"76S05 (35B35 76D27 76Txx 86A05)\",\"mrnumber\":\"3181769\",\"mrreviewer\":\"José Miguel Pacheco Castelao\",\"number\":\"2\",\"pages\":\"635–639\",\"publisher\":\"Proceedings of the National Academy of Sciences\",\"title\":\"Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem\",\"url_arxiv\":\"https://arxiv.org/abs/1309.4023\",\"volume\":\"111\",\"year\":\"2014\",\"zblnumber\":\"1355.76065\",\"bdsk-url-1\":\"https://www.pnas.org/content/111/2/635\",\"bdsk-url-2\":\"https://doi.org/10.1073/pnas.1320554111\",\"bibtex\":\"@article{MR3181769,\\n\\tabstract = {In this paper for either the sharp front Surface Quasi-Geostrophic equation or the Muskat problem we rule out the `'splash singularity'' blow-up scenario; in other words we prove that the contours evolving from either of these systems can not intersect at a single point while the free boundary remains smooth. Splash singularities have been shown to hold for the free boundary incompressible Euler equation in the form of the water waves contour evolution problem (Castro A, et al. (2012) Proc Natl Acad Sci USA 109:733-738). Our result confirms the numerical simulations in (C{\\\\'o}rdoba D, et al. (2005) Proc Natl Acad Sci USA 102:5949-5952) where it is shown that the curvature blows up due to the contours collapsing at a point. Here we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse. Another conclusion that we provide is a better understanding of the work (C{\\\\'o}rdoba D, Gancedo F (2010) Comm Math Phys 299:561-575) in which squirt singularities are ruled out; in this case a positive volume of fluid between the contours can not be ejected in finite time.},\\n\\tauthor = {Gancedo, Francisco and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1073/pnas.1320554111},\\n\\teprint = {1309.4023},\\n\\tfjournal = {Proceedings of the National Academy of Sciences of the United States of America},\\n\\tissn = {0027-8424},\\n\\tjournal = {Proc. Natl. Acad. Sci. USA},\\n\\tkeywords = {Fluid mechanics, Free boundary problems, Muskat problem},\\n\\tmrclass = {76S05 (35B35 76D27 76Txx 86A05)},\\n\\tmrnumber = {3181769},\\n\\tmrreviewer = {Jos\\\\'{e} Miguel Pacheco Castelao},\\n\\tnumber = {2},\\n\\tpages = {635--639},\\n\\tpublisher = {Proceedings of the National Academy of Sciences},\\n\\ttitle = {Absence of splash singularities for surface quasi-geostrophic sharp fronts and the {M}uskat problem},\\n\\turl_arxiv = {https://arxiv.org/abs/1309.4023},\\n\\tvolume = {111},\\n\\tyear = {2014},\\n\\tzblnumber = {1355.76065},\\n\\tbdsk-url-1 = {https://www.pnas.org/content/111/2/635},\\n\\tbdsk-url-2 = {https://doi.org/10.1073/pnas.1320554111}}\\n\\n\",\"author_short\":[\"Gancedo, F.\",\"Strain, R. M.\"],\"key\":\"MR3181769\",\"id\":\"MR3181769\",\"bibbaseid\":\"gancedo-strain-absenceofsplashsingularitiesforsurfacequasigeostrophicsharpfrontsandthemuskatproblem-2014\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1309.4023\"},\"keyword\":[\"Fluid mechanics\",\"Free boundary problems\",\"Muskat problem\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":0,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"The global existence of solutions to the relativistic Vlasov-Maxwell system given sufficiently regular finite energy initial data is a longstanding open problem. The main result of Glassey-Strauss 1986 shows that a solution $(f, E, B)$ remains $C^1$ as long as the momentum support of $f$ remains bounded. Alternate proofs were later given by Bouchut-Golse-Pallard 2003 and Klainerman-Staffilani 2002. We show that only the boundedness of the momentum support of $f$ ıt after projecting to any two dimensional plane is needed for $(f, E, B)$ to remain $C^1$. \",\"author\":[{\"propositions\":[],\"lastnames\":[\"Luk\"],\"firstnames\":[\"Jonathan\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1007/s00220-014-2108-8\",\"eprint\":\"1406.0165\",\"fjournal\":\"Communications in Mathematical Physics\",\"issn\":\"0010-3616\",\"journal\":\"Comm. Math. Phys.\",\"keywords\":\"Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems\",\"mrclass\":\"35Q75 (35Q83 81T20)\",\"mrnumber\":\"3248056\",\"mrreviewer\":\"Calvin Tadmon\",\"number\":\"3\",\"pages\":\"1005–1027\",\"title\":\"A new continuation criterion for the relativistic Vlasov-Maxwell system\",\"url_arxiv\":\"https://arxiv.org/abs/1406.0165\",\"volume\":\"331\",\"year\":\"2014\",\"zblnumber\":\"1309.35174\",\"bdsk-url-1\":\"https://doi.org/10.1007/s00220-014-2108-8\",\"bibtex\":\"@article{MR3248056,\\n\\tabstract = {The global existence of solutions to the relativistic Vlasov-Maxwell system given sufficiently regular finite energy initial data is a longstanding open problem.  The main result of Glassey-Strauss 1986 shows that a solution $(f, E, B)$ remains $C^1$ as long as the momentum support of $f$ remains bounded.  Alternate proofs were later given by Bouchut-Golse-Pallard 2003 and Klainerman-Staffilani 2002.  We show that only the boundedness of the momentum support of $f$ {\\\\it after projecting to any two dimensional plane} is needed  for $(f, E, B)$ to remain $C^1$. },\\n\\tauthor = {Luk, Jonathan and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1007/s00220-014-2108-8},\\n\\teprint = {1406.0165},\\n\\tfjournal = {Communications in Mathematical Physics},\\n\\tissn = {0010-3616},\\n\\tjournal = {Comm. Math. Phys.},\\n\\tkeywords = {Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},\\n\\tmrclass = {35Q75 (35Q83 81T20)},\\n\\tmrnumber = {3248056},\\n\\tmrreviewer = {Calvin Tadmon},\\n\\tnumber = {3},\\n\\tpages = {1005--1027},\\n\\ttitle = {A new continuation criterion for the relativistic {V}lasov-{M}axwell system},\\n\\turl_arxiv = {https://arxiv.org/abs/1406.0165},\\n\\tvolume = {331},\\n\\tyear = {2014},\\n\\tzblnumber = {1309.35174},\\n\\tbdsk-url-1 = {https://doi.org/10.1007/s00220-014-2108-8}}\\n\\n\",\"author_short\":[\"Luk, J.\",\"Strain, R. M.\"],\"key\":\"MR3248056\",\"id\":\"MR3248056\",\"bibbaseid\":\"luk-strain-anewcontinuationcriterionfortherelativisticvlasovmaxwellsystem-2014\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1406.0165\"},\"keyword\":[\"Kinetic Theory\",\"relativistic Kinetic Theory\",\"Vlasov-Maxwell systems\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":3,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"We prove that $k$-th order derivatives of perturbative classical solutions to the hard and soft potential Boltzmann equation (without the angular cut-off assumption) in the whole space, ${\\\\mathbb R}^{n}_x$ with ${n \\\\ge 3}$, converge in large time to the global Maxwellian with the optimal decay rate of $t^{-\\\\frac{1}{2}(k+r+\\\\frac{n}{2}-\\\\frac{n}{r})}$ in the $L^r_x(L^2_{v})$-norm for any $2≤ r≤ ∞$. These results hold for any $r ∈ (0, n/2]$ as long as initially $\\\\| f_0\\\\|_{Ḃ^{-r,∞}_2 L^2_{v}} < ∞$. In the hard potential case, we prove faster decay results in the sense that if $\\\\| \\\\mathbf{P} f_0\\\\|_{Ḃ^{-r,∞}_2 L^2_{v}} < ∞$ and $\\\\| \\\\{\\\\mathbf{I} - \\\\mathbf{P}\\\\} f_0\\\\|_{Ḃ^{-r+1,∞}_2 L^2_{v}} < ∞$ for $r ∈ (n/2, (n+2)/2]$ then the solution decays the global Maxwellian in $L^2_v(L^2_x)$ with the optimal large time decay rate of $t^{-\\\\frac{1}{2} r}$.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Sohinger\"],\"firstnames\":[\"Vedran\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-15 15:47:31 -0400\",\"doi\":\"10.1016/j.aim.2014.04.012\",\"eprint\":\"1206.0027\",\"fjournal\":\"Advances in Mathematics\",\"issn\":\"0001-8708\",\"journal\":\"Adv. Math.\",\"keywords\":\"Boltzmann equation, non-cutoff, Kinetic Theory\",\"mrclass\":\"35Q20 (35B40 35F25 76P05 82C40)\",\"mrnumber\":\"3213301\",\"mrreviewer\":\"Cecil Pompiliu Grünfeld\",\"msc2010\":\"35Q20 35R11 76P05 82C40 35B65 26A33\",\"pages\":\"274–332\",\"publisher\":\"Elsevier (Academic Press), San Diego, CA\",\"title\":\"The Boltzmann equation, Besov spaces, and optimal time decay rates in $ℝ_x^n$\",\"url_arxiv\":\"https://arxiv.org/abs/1206.0027\",\"volume\":\"261\",\"year\":\"2014\",\"zblnumber\":\"1293.35195\",\"bdsk-url-1\":\"https://doi.org/10.1016/j.aim.2014.04.012\",\"bibtex\":\"@article{MR3213301,\\n\\tabstract = {We prove that $k$-th order derivatives of perturbative classical solutions to the hard and soft potential Boltzmann equation (without the angular cut-off assumption) in the whole space, ${\\\\mathbb R}^{n}_x$ with ${n \\\\ge 3}$, converge in large time to the global Maxwellian with the optimal decay rate of   $t^{-\\\\frac{1}{2}(k+r+\\\\frac{n}{2}-\\\\frac{n}{r})}$\\nin the $L^r_x(L^2_{v})$-norm for any $2\\\\leq r\\\\leq \\\\infty$.   These results hold for any $r \\\\in (0, n/2]$ as long as initially $\\\\| f_0\\\\|_{\\\\dot{B}^{-r,\\\\infty}_2 L^2_{v}} < \\\\infty$.   In the hard potential case, we prove faster decay results in the sense that if $\\\\| \\\\mathbf{P} f_0\\\\|_{\\\\dot{B}^{-r,\\\\infty}_2 L^2_{v}} < \\\\infty$ and \\n$\\\\| \\\\{\\\\mathbf{I} - \\\\mathbf{P}\\\\} f_0\\\\|_{\\\\dot{B}^{-r+1,\\\\infty}_2 L^2_{v}} < \\\\infty$ for $r \\\\in (n/2, (n+2)/2]$  then the solution decays the global Maxwellian in $L^2_v(L^2_x)$ with the optimal large time decay rate of $t^{-\\\\frac{1}{2} r}$.},\\n\\tauthor = {Sohinger, Vedran and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-15 15:47:31 -0400},\\n\\tdoi = {10.1016/j.aim.2014.04.012},\\n\\teprint = {1206.0027},\\n\\tfjournal = {Advances in Mathematics},\\n\\tissn = {0001-8708},\\n\\tjournal = {Adv. Math.},\\n\\tkeywords = {Boltzmann equation, non-cutoff, Kinetic Theory},\\n\\tmrclass = {35Q20 (35B40 35F25 76P05 82C40)},\\n\\tmrnumber = {3213301},\\n\\tmrreviewer = {Cecil Pompiliu Gr\\\\\\\"unfeld},\\n\\tmsc2010 = {35Q20 35R11 76P05 82C40 35B65 26A33},\\n\\tpages = {274--332},\\n\\tpublisher = {Elsevier (Academic Press), San Diego, CA},\\n\\ttitle = {The {B}oltzmann equation, {B}esov spaces, and optimal time decay rates in {$\\\\mathbb{R}_x^n$}},\\n\\turl_arxiv = {https://arxiv.org/abs/1206.0027},\\n\\tvolume = {261},\\n\\tyear = {2014},\\n\\tzblnumber = {1293.35195},\\n\\tbdsk-url-1 = {https://doi.org/10.1016/j.aim.2014.04.012}}\\n\\n\",\"author_short\":[\"Sohinger, V.\",\"Strain, R. M.\"],\"key\":\"MR3213301\",\"id\":\"MR3213301\",\"bibbaseid\":\"sohinger-strain-theboltzmannequationbesovspacesandoptimaltimedecayratesinxn-2014\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1206.0027\"},\"keyword\":[\"Boltzmann equation\",\"non-cutoff\",\"Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"The spatially homogeneous Boltzmann equation has been studied extensively in the Newtonian case, but not much is known for the special relativistic case. In this paper, we address several issues for the spatially homogeneous Boltzmann equation for relativistic particles. We first derive the relativistic version of the Povzner inequality. Using this, we study the Cauchy problem and investigate how the polynomial and exponential moments in $L^1$ are propagated. Several key differences between the relativistic and the Newtonian cases are confronted and discussed.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Yun\"],\"firstnames\":[\"Seok-Bae\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-08 11:57:02 -0600\",\"doi\":\"10.1137/130923531\",\"fjournal\":\"SIAM Journal on Mathematical Analysis\",\"issn\":\"0036-1410\",\"journal\":\"SIAM J. Math. Anal.\",\"keywords\":\"relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory\",\"mrclass\":\"35Q20 (76P05 76Y05)\",\"mrnumber\":\"3166961\",\"mrreviewer\":\"Piotr Biler\",\"number\":\"1\",\"pages\":\"917–938\",\"title\":\"Spatially homogeneous Boltzmann equation for relativistic particles\",\"urlpdf\":\"https://strain.math.upenn.edu/preprints/92353.pdf\",\"volume\":\"46\",\"year\":\"2014\",\"zblnumber\":\"1321.35127\",\"bdsk-url-1\":\"https://doi.org/10.1137/130923531\",\"bibtex\":\"@article{MR3166961,\\n\\tabstract = {The spatially homogeneous Boltzmann equation has been studied extensively in the Newtonian case, but not much is known for the special relativistic case. In this paper, we address several issues for the spatially homogeneous Boltzmann equation for relativistic particles. We first derive the relativistic version of the Povzner inequality. Using this, we study the Cauchy problem and investigate how the polynomial and exponential moments in $L^1$ are propagated. Several key differences between the relativistic and the Newtonian cases are confronted and discussed.},\\n\\tauthor = {Strain, Robert M. and Yun, Seok-Bae},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-08 11:57:02 -0600},\\n\\tdoi = {10.1137/130923531},\\n\\tfjournal = {SIAM Journal on Mathematical Analysis},\\n\\tissn = {0036-1410},\\n\\tjournal = {SIAM J. Math. Anal.},\\n\\tkeywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\\n\\tmrclass = {35Q20 (76P05 76Y05)},\\n\\tmrnumber = {3166961},\\n\\tmrreviewer = {Piotr Biler},\\n\\tnumber = {1},\\n\\tpages = {917--938},\\n\\ttitle = {Spatially homogeneous {B}oltzmann equation for relativistic particles},\\n\\turlpdf = {https://strain.math.upenn.edu/preprints/92353.pdf},\\n\\tvolume = {46},\\n\\tyear = {2014},\\n\\tzblnumber = {1321.35127},\\n\\tbdsk-url-1 = {https://doi.org/10.1137/130923531}}\\n\\n\",\"author_short\":[\"Strain, R. M.\",\"Yun, S.\"],\"key\":\"MR3166961\",\"id\":\"MR3166961\",\"bibbaseid\":\"strain-yun-spatiallyhomogeneousboltzmannequationforrelativisticparticles-2014\",\"role\":\"author\",\"urls\":{\"Pdf\":\"https://strain.math.upenn.edu/preprints/92353.pdf\"},\"keyword\":[\"relativistic Boltzmann equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":1,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2(ℝ)$ maximum principle, in the form of a new ``log'' conservation law which is satisfied by the equation for the interface. Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy $\\\\|f_0\\\\|_{L^∞}<∞$ and $\\\\|∂_x f_0\\\\|_{L^∞}<1$. We take advantage of the fact that the bound $\\\\|∂_x f_0\\\\|_{L^∞}<1$ is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law. Lastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance $\\\\| f\\\\|_1 łe 1/5$. Previous results of this sort used a small constant $ε ≪1$ which was not explicit.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Constantin\"],\"firstnames\":[\"Peter\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Córdoba\"],\"firstnames\":[\"Diego\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Gancedo\"],\"firstnames\":[\"Francisco\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.4171/JEMS/360\",\"eprint\":\"1007.3744\",\"fjournal\":\"Journal of the European Mathematical Society (JEMS)\",\"issn\":\"1435-9855\",\"journal\":\"J. Eur. Math. Soc. (JEMS)\",\"keywords\":\"Fluid mechanics, Free boundary problems, Muskat problem\",\"mrclass\":\"35Q35 (35B50 35D35 76S05)\",\"mrnumber\":\"2998834\",\"mrreviewer\":\"Weiran Sun\",\"number\":\"1\",\"pages\":\"201–227\",\"title\":\"On the global existence for the Muskat problem\",\"url_arxiv\":\"https://arxiv.org/abs/1007.3744\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/ccgsMuskat.pdf\",\"volume\":\"15\",\"year\":\"2013\",\"zblnumber\":\"1258.35002\",\"bdsk-url-1\":\"https://doi.org/10.4171/JEMS/360\",\"bibtex\":\"@article{CCGS13,\\n\\tabstract = {The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities.\\nIn this work we prove three results.  First we prove an $L^2(\\\\mathbb{R})$ maximum principle, in the form of a new ``log''  conservation law which is satisfied by the equation for the interface.  Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy \\n$\\\\|f_0\\\\|_{L^\\\\infty}<\\\\infty$ and $\\\\|\\\\partial_x f_0\\\\|_{L^\\\\infty}<1$. We take advantage of the fact that the bound $\\\\|\\\\partial_x f_0\\\\|_{L^\\\\infty}<1$ is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.\\nLastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance $\\\\| f\\\\|_1 \\\\le 1/5$. Previous results of this sort used a small constant $\\\\epsilon \\\\ll1$ which was not explicit.},\\n\\tauthor = {Constantin, Peter and C\\\\'{o}rdoba, Diego and Gancedo, Francisco and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.4171/JEMS/360},\\n\\teprint = {1007.3744},\\n\\tfjournal = {Journal of the European Mathematical Society (JEMS)},\\n\\tissn = {1435-9855},\\n\\tjournal = {J. Eur. Math. Soc. (JEMS)},\\n\\tkeywords = {Fluid mechanics, Free boundary problems, Muskat problem},\\n\\tmrclass = {35Q35 (35B50 35D35 76S05)},\\n\\tmrnumber = {2998834},\\n\\tmrreviewer = {Weiran Sun},\\n\\tnumber = {1},\\n\\tpages = {201--227},\\n\\ttitle = {On the global existence for the {M}uskat problem},\\n\\turl_arxiv = {https://arxiv.org/abs/1007.3744},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/ccgsMuskat.pdf},\\n\\tvolume = {15},\\n\\tyear = {2013},\\n\\tzblnumber = {1258.35002},\\n\\tbdsk-url-1 = {https://doi.org/10.4171/JEMS/360}}\\n\\n\",\"author_short\":[\"Constantin, P.\",\"Córdoba, D.\",\"Gancedo, F.\",\"Strain, R. M.\"],\"key\":\"CCGS13\",\"id\":\"CCGS13\",\"bibbaseid\":\"constantin-crdoba-gancedo-strain-ontheglobalexistenceforthemuskatproblem-2013\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1007.3744\",\" pdf\":\"https://strain.math.upenn.edu/preprints/ccgsMuskat.pdf\"},\"keyword\":[\"Fluid mechanics\",\"Free boundary problems\",\"Muskat problem\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":4,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"We present the uniform $L^1$-stability estimate for the relativistic Boltzmann equation near vacuum. For this, we explicitly construct a relativistic counterpart of the nonlinear functional which is a linear combination of $L^1$-distance and a collision potential. This functional measures the $L^1$-distance between two continuous mild solutions. When the initial data is sufficiently small and decays exponentially fast, we show that the functional satisfies the uniform stability estimate leading to the uniform $L^1$-stability estimate with respect to initial data.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Ha\"],\"firstnames\":[\"Seung-Yeal\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Jeong\"],\"firstnames\":[\"Eunhee\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-08 14:17:12 -0600\",\"doi\":\"10.3934/cpaa.2013.12.1141\",\"fjournal\":\"Communications on Pure and Applied Analysis\",\"issn\":\"1534-0392\",\"journal\":\"Commun. Pure Appl. Anal.\",\"keywords\":\"relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory\",\"mrclass\":\"35Q20 (35B35 35Q75 82C40)\",\"mrnumber\":\"2982812\",\"mrreviewer\":\"Calvin Tadmon\",\"number\":\"2\",\"pages\":\"1141–1161\",\"title\":\"Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum\",\"urlpdf\":\"https://strain.math.upenn.edu/preprints/cpaa0494.pdf\",\"volume\":\"12\",\"year\":\"2013\",\"zblnumber\":\"1267.35162\",\"bdsk-url-1\":\"https://doi.org/10.3934/cpaa.2013.12.1141\",\"bibtex\":\"@article{MR2982812,\\n\\tabstract = {We present the uniform $L^1$-stability estimate for the relativistic Boltzmann equation near vacuum. For this, we explicitly construct a relativistic counterpart of the nonlinear functional which is a linear combination of  $L^1$-distance and a collision potential. This functional measures the  $L^1$-distance between two continuous mild solutions. When the initial data is sufficiently small and decays exponentially fast, we show that the functional satisfies the uniform stability estimate leading to the uniform  $L^1$-stability estimate with respect to initial data.},\\n\\tauthor = {Ha, Seung-Yeal and Jeong, Eunhee and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-08 14:17:12 -0600},\\n\\tdoi = {10.3934/cpaa.2013.12.1141},\\n\\tfjournal = {Communications on Pure and Applied Analysis},\\n\\tissn = {1534-0392},\\n\\tjournal = {Commun. Pure Appl. Anal.},\\n\\tkeywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\\n\\tmrclass = {35Q20 (35B35 35Q75 82C40)},\\n\\tmrnumber = {2982812},\\n\\tmrreviewer = {Calvin Tadmon},\\n\\tnumber = {2},\\n\\tpages = {1141--1161},\\n\\ttitle = {Uniform {$L^1$}-stability of the relativistic {B}oltzmann equation near vacuum},\\n\\turlpdf = {https://strain.math.upenn.edu/preprints/cpaa0494.pdf},\\n\\tvolume = {12},\\n\\tyear = {2013},\\n\\tzblnumber = {1267.35162},\\n\\tbdsk-url-1 = {https://doi.org/10.3934/cpaa.2013.12.1141}}\\n\\n\",\"author_short\":[\"Ha, S.\",\"Jeong, E.\",\"Strain, R. M.\"],\"key\":\"MR2982812\",\"id\":\"MR2982812\",\"bibbaseid\":\"ha-jeong-strain-uniforml1stabilityoftherelativisticboltzmannequationnearvacuum-2013\",\"role\":\"author\",\"urls\":{\"Pdf\":\"https://strain.math.upenn.edu/preprints/cpaa0494.pdf\"},\"keyword\":[\"relativistic Boltzmann equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":0,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"For the Landau-Poisson system with Coulomb interaction in $ℝ^3_x$, we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Zhu\"],\"firstnames\":[\"Keya\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1007/s00205-013-0658-0\",\"eprint\":\"1202.2471\",\"fjournal\":\"Archive for Rational Mechanics and Analysis\",\"issn\":\"0003-9527\",\"journal\":\"Arch. Ration. Mech. Anal.\",\"keywords\":\"Landau equation, Vlasov-Maxwell systems, Kinetic Theory\",\"mrclass\":\"35Q83 (35A01 35A02 35B40)\",\"mrnumber\":\"3101794\",\"mrreviewer\":\"Jonathan Ben-Artzi\",\"number\":\"2\",\"pages\":\"615–671\",\"title\":\"The Vlasov-Poisson-Landau system in $ℝ^3_x$\",\"url_arxiv\":\"https://arxiv.org/abs/1202.2471\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/2012szLandauP.pdf\",\"volume\":\"210\",\"year\":\"2013\",\"zblnumber\":\"1294.35168\",\"bdsk-url-1\":\"https://doi.org/10.1007/s00205-013-0658-0\",\"bibtex\":\"@article{MR3101794,\\n\\tabstract = {For the Landau-Poisson system with Coulomb interaction in $\\\\mathbb{R}^3_x$, we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close.},\\n\\tauthor = {Strain, Robert M. and Zhu, Keya},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1007/s00205-013-0658-0},\\n\\teprint = {1202.2471},\\n\\tfjournal = {Archive for Rational Mechanics and Analysis},\\n\\tissn = {0003-9527},\\n\\tjournal = {Arch. Ration. Mech. Anal.},\\n\\tkeywords = {Landau equation, Vlasov-Maxwell systems, Kinetic Theory},\\n\\tmrclass = {35Q83 (35A01 35A02 35B40)},\\n\\tmrnumber = {3101794},\\n\\tmrreviewer = {Jonathan Ben-Artzi},\\n\\tnumber = {2},\\n\\tpages = {615--671},\\n\\ttitle = {The {V}lasov-{P}oisson-{L}andau system in {$\\\\mathbb{R}^3_x$}},\\n\\turl_arxiv = {https://arxiv.org/abs/1202.2471},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/2012szLandauP.pdf},\\n\\tvolume = {210},\\n\\tyear = {2013},\\n\\tzblnumber = {1294.35168},\\n\\tbdsk-url-1 = {https://doi.org/10.1007/s00205-013-0658-0}}\\n\\n\",\"author_short\":[\"Strain, R. M.\",\"Zhu, K.\"],\"key\":\"MR3101794\",\"id\":\"MR3101794\",\"bibbaseid\":\"strain-zhu-thevlasovpoissonlandausystemin3x-2013\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1202.2471\",\" pdf\":\"https://strain.math.upenn.edu/preprints/2012szLandauP.pdf\"},\"keyword\":[\"Landau equation\",\"Vlasov-Maxwell systems\",\"Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"Consider steady-state weak solutions to the incompressible Navier-Stokes equations in six spatial dimensions. We prove that the 2D Hausdorff measure of the set of singular points is equal to zero. This problem was mentioned in 1988 by Struwe, during his study of the five dimensional case.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Dong\"],\"firstnames\":[\"Hongjie\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1512/iumj.2012.61.4765\",\"eprint\":\"1101.5580\",\"fjournal\":\"Indiana University Mathematics Journal\",\"issn\":\"0022-2518\",\"journal\":\"Indiana Univ. Math. J.\",\"keywords\":\"Navier-Stokes equations, Fluid mechanics\",\"mrclass\":\"35Q30 (35B65 76D03 76D05)\",\"mrnumber\":\"3129108\",\"mrreviewer\":\"Kazuo Yamazaki\",\"number\":\"6\",\"pages\":\"2211–2229\",\"title\":\"On partial regularity of steady-state solutions to the 6D Navier-Stokes equations\",\"url_arxiv\":\"https://arxiv.org/abs/1101.5580\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/ds6DssNSE.pdf\",\"volume\":\"61\",\"year\":\"2012\",\"zblnumber\":\"1286.35193\",\"bdsk-url-1\":\"https://doi.org/10.1512/iumj.2012.61.4765\",\"bibtex\":\"@article{MR3129108,\\n\\tabstract = {Consider steady-state weak solutions to the incompressible\\nNavier-Stokes equations in six spatial dimensions. We prove that the 2D\\nHausdorff measure of the set of singular points is equal to zero.  This problem was mentioned in 1988 by Struwe, during his study of the five dimensional case.},\\n\\tauthor = {Dong, Hongjie and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1512/iumj.2012.61.4765},\\n\\teprint = {1101.5580},\\n\\tfjournal = {Indiana University Mathematics Journal},\\n\\tissn = {0022-2518},\\n\\tjournal = {Indiana Univ. Math. J.},\\n\\tkeywords = {Navier-Stokes equations, Fluid mechanics},\\n\\tmrclass = {35Q30 (35B65 76D03 76D05)},\\n\\tmrnumber = {3129108},\\n\\tmrreviewer = {Kazuo Yamazaki},\\n\\tnumber = {6},\\n\\tpages = {2211--2229},\\n\\ttitle = {On partial regularity of steady-state solutions to the 6{D} {N}avier-{S}tokes equations},\\n\\turl_arxiv = {https://arxiv.org/abs/1101.5580},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/ds6DssNSE.pdf},\\n\\tvolume = {61},\\n\\tyear = {2012},\\n\\tzblnumber = {1286.35193},\\n\\tbdsk-url-1 = {https://doi.org/10.1512/iumj.2012.61.4765}}\\n\\n\",\"author_short\":[\"Dong, H.\",\"Strain, R. M.\"],\"key\":\"MR3129108\",\"id\":\"MR3129108\",\"bibbaseid\":\"dong-strain-onpartialregularityofsteadystatesolutionstothe6dnavierstokesequations-2012\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1101.5580\",\" pdf\":\"https://strain.math.upenn.edu/preprints/ds6DssNSE.pdf\"},\"keyword\":[\"Navier-Stokes equations\",\"Fluid mechanics\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"incollection\",\"type\":\"incollection\",\"abstract\":\"In this paper, we present a new approach of studying the full dissipative property of the Vlasov-Poisson-Boltzmann system over the whole space. The key part of this approach is to design the interactive functional to capture the dissipation of the system along the degenerate components. The developed approach is generally applicable to other relevant models arising from plasma physics both at the kinetic and fluid levels.\",\"address\":\"Singapore\",\"annote\":\"Proceedings of the HYP2010 conference in Beijing\",\"author\":[{\"firstnames\":[\"Renjun\"],\"propositions\":[],\"lastnames\":[\"Duan\"],\"suffixes\":[]},{\"firstnames\":[\"Robert\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Strain\"],\"suffixes\":[]}],\"booktitle\":\"Hyperbolic problems—theory, numerics and applications. Volume 2\",\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-08 14:12:11 -0600\",\"doi\":\"10/c74v\",\"editor\":[{\"firstnames\":[\"Tatsien\"],\"propositions\":[],\"lastnames\":[\"Li\"],\"suffixes\":[]},{\"firstnames\":[\"Song\"],\"propositions\":[],\"lastnames\":[\"Jiang\"],\"suffixes\":[]}],\"keywords\":\"Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory\",\"mrnumber\":\"3050180\",\"pages\":\"398–405\",\"publisher\":\"World Scientific Publishing and Higher Education Press\",\"series\":\"Ser. Contemp. Appl. Math. CAM\",\"title\":\"On the Full Dissipative Property of the Vlasov-Poisson-Boltzmann System\",\"urlpdf\":\"https://strain.math.upenn.edu/preprints/2012DuanStrainHype.pdf\",\"volume\":\"17\",\"year\":\"2012\",\"zblnumber\":\"1293.35339\",\"bdsk-url-1\":\"https://doi.org/10.1142/9789814417099_0037\",\"bibtex\":\"@incollection{MR3050180,\\n\\tabstract = {In this paper, we present a new approach of studying the full dissipative property of the Vlasov-Poisson-Boltzmann system over the whole space. The key part of this approach is to design the interactive functional to capture the dissipation of the system along the degenerate components. The developed approach is generally applicable to other relevant models arising from plasma physics both at the kinetic and fluid levels.},\\n\\taddress = {Singapore},\\n\\tannote = {Proceedings of the HYP2010 conference in Beijing},\\n\\tauthor = {Renjun Duan and Robert M. Strain},\\n\\tbooktitle = {Hyperbolic problems---theory, numerics and applications. {V}olume 2},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-08 14:12:11 -0600},\\n\\tdoi = {10/c74v},\\n\\teditor = {Tatsien Li and Song Jiang},\\n\\tkeywords = {Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory},\\n\\tmrnumber = {3050180},\\n\\tpages = {398--405},\\n\\tpublisher = {World Scientific Publishing and Higher Education Press},\\n\\tseries = {Ser. Contemp. Appl. Math. CAM},\\n\\ttitle = {On the {F}ull {D}issipative {P}roperty of the {V}lasov-{P}oisson-{B}oltzmann {S}ystem},\\n\\turlpdf = {https://strain.math.upenn.edu/preprints/2012DuanStrainHype.pdf},\\n\\tvolume = {17},\\n\\tyear = {2012},\\n\\tzblnumber = {1293.35339},\\n\\tbdsk-url-1 = {https://doi.org/10.1142/9789814417099_0037}}\\n\\n\",\"author_short\":[\"Duan, R.\",\"Strain, R. M.\"],\"editor_short\":[\"Li, T.\",\"Jiang, S.\"],\"key\":\"MR3050180\",\"id\":\"MR3050180\",\"bibbaseid\":\"duan-strain-onthefulldissipativepropertyofthevlasovpoissonboltzmannsystem-2012\",\"role\":\"author\",\"urls\":{\"Pdf\":\"https://strain.math.upenn.edu/preprints/2012DuanStrainHype.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Vlasov-Maxwell systems\",\"Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":1,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this article we prove a collection of new non-linear and non-local integral inequalities. We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions. Specifically, we include all $α∈ (0, \\\\frac{74}{75})$.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Gressman\"],\"firstnames\":[\"Philip\",\"T.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Krieger\"],\"firstnames\":[\"Joachim\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1016/j.aim.2012.02.017\",\"eprint\":\"1202.4088\",\"fjournal\":\"Advances in Mathematics\",\"issn\":\"0001-8708\",\"journal\":\"Adv. Math.\",\"keywords\":\"Landau equation, Kinetic Theory\",\"mrclass\":\"35K55 (35A01 35B65 35R11)\",\"mrnumber\":\"2914961\",\"mrreviewer\":\"M. Pilar Velasco\",\"number\":\"2\",\"pages\":\"642–648\",\"title\":\"A non-local inequality and global existence\",\"url_arxiv\":\"https://arxiv.org/abs/1202.4088\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/2012gksINEQ.pdf\",\"volume\":\"230\",\"year\":\"2012\",\"zblnumber\":\"1248.35005\",\"bdsk-url-1\":\"https://doi.org/10.1016/j.aim.2012.02.017\",\"bibtex\":\"@article{MR2914961,\\n\\tabstract = {In this article we prove a collection of new non-linear and non-local integral inequalities.  We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions.  Specifically, we include all $\\\\alpha\\\\in (0, \\\\frac{74}{75})$.},\\n\\tauthor = {Gressman, Philip T. and Krieger, Joachim and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1016/j.aim.2012.02.017},\\n\\teprint = {1202.4088},\\n\\tfjournal = {Advances in Mathematics},\\n\\tissn = {0001-8708},\\n\\tjournal = {Adv. Math.},\\n\\tkeywords = {Landau equation, Kinetic Theory},\\n\\tmrclass = {35K55 (35A01 35B65 35R11)},\\n\\tmrnumber = {2914961},\\n\\tmrreviewer = {M. Pilar Velasco},\\n\\tnumber = {2},\\n\\tpages = {642--648},\\n\\ttitle = {A non-local inequality and global existence},\\n\\turl_arxiv = {https://arxiv.org/abs/1202.4088},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/2012gksINEQ.pdf},\\n\\tvolume = {230},\\n\\tyear = {2012},\\n\\tzblnumber = {1248.35005},\\n\\tbdsk-url-1 = {https://doi.org/10.1016/j.aim.2012.02.017}}\\n\\n\",\"author_short\":[\"Gressman, P. T.\",\"Krieger, J.\",\"Strain, R. M.\"],\"key\":\"MR2914961\",\"id\":\"MR2914961\",\"bibbaseid\":\"gressman-krieger-strain-anonlocalinequalityandglobalexistence-2012\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1202.4088\",\" pdf\":\"https://strain.math.upenn.edu/preprints/2012gksINEQ.pdf\"},\"keyword\":[\"Landau equation\",\"Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":6,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In the study of solutions to the relativistic Boltzmann equation, their regularity with respect to the momentum variables has been an outstanding question, even local in time, due to the initially unexpected growth in the post-collisional momentum variables which was discovered in 1991 by Glassey & Strauss. We establish momentum regularity within energy spaces via a new splitting technique and interplay between the Glassey-Strauss frame and the center of mass frame of the relativistic collision operator. In a periodic box, these new momentum regularity estimates lead to a proof of global existence of classical solutions to the two-species relativistic Vlasov-Maxwell-Boltzmann system for charged particles near Maxwellian with hard ball interaction.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Guo\"],\"firstnames\":[\"Yan\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1007/s00220-012-1417-z\",\"eprint\":\"1012.1158\",\"fjournal\":\"Communications in Mathematical Physics\",\"issn\":\"0010-3616\",\"journal\":\"Comm. Math. Phys.\",\"keywords\":\"Boltzmann equation, relativistic Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory, relativistic Kinetic Theory\",\"mrclass\":\"82D10 (35B35 35B65 35Q83 82C24 82C40)\",\"mrnumber\":\"2891870\",\"mrreviewer\":\"Stephen Wollman\",\"number\":\"3\",\"pages\":\"649–673\",\"title\":\"Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system\",\"url_arxiv\":\"https://arxiv.org/abs/1012.1158\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/gsRVMB.pdf\",\"volume\":\"310\",\"year\":\"2012\",\"zblnumber\":\"1245.35130\",\"bdsk-url-1\":\"https://doi.org/10.1007/s00220-012-1417-z\",\"bibtex\":\"@article{MR2891870,\\n\\tabstract = {In the study of solutions to the relativistic Boltzmann equation, their\\nregularity with respect to the momentum variables has been an outstanding\\nquestion, even local in time, due to the initially unexpected growth in the\\npost-collisional momentum variables which was discovered in 1991 by Glassey\\n\\\\& Strauss. We establish momentum regularity within energy\\nspaces via a new splitting technique and interplay between the\\nGlassey-Strauss frame and the center of mass frame of the relativistic collision\\noperator. In a periodic box, these new momentum regularity estimates lead to\\na proof of global existence of classical solutions to the two-species\\nrelativistic Vlasov-Maxwell-Boltzmann system for charged particles near\\nMaxwellian with hard ball interaction.},\\n\\tauthor = {Guo, Yan and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1007/s00220-012-1417-z},\\n\\teprint = {1012.1158},\\n\\tfjournal = {Communications in Mathematical Physics},\\n\\tissn = {0010-3616},\\n\\tjournal = {Comm. Math. Phys.},\\n\\tkeywords = {Boltzmann equation, relativistic Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory, relativistic Kinetic Theory},\\n\\tmrclass = {82D10 (35B35 35B65 35Q83 82C24 82C40)},\\n\\tmrnumber = {2891870},\\n\\tmrreviewer = {Stephen Wollman},\\n\\tnumber = {3},\\n\\tpages = {649--673},\\n\\ttitle = {Momentum regularity and stability of the relativistic {V}lasov-{M}axwell-{B}oltzmann system},\\n\\turl_arxiv = {https://arxiv.org/abs/1012.1158},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/gsRVMB.pdf},\\n\\tvolume = {310},\\n\\tyear = {2012},\\n\\tzblnumber = {1245.35130},\\n\\tbdsk-url-1 = {https://doi.org/10.1007/s00220-012-1417-z}}\\n\\n\",\"author_short\":[\"Guo, Y.\",\"Strain, R. M.\"],\"key\":\"MR2891870\",\"id\":\"MR2891870\",\"bibbaseid\":\"guo-strain-momentumregularityandstabilityoftherelativisticvlasovmaxwellboltzmannsystem-2012\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1012.1158\",\" pdf\":\"https://strain.math.upenn.edu/preprints/gsRVMB.pdf\"},\"keyword\":[\"Boltzmann equation\",\"relativistic Boltzmann equation\",\"Vlasov-Maxwell systems\",\"Kinetic Theory\",\"relativistic Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":0,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this paper we prove the global in time well-posedness of the following non-local diffusion equation with $α ∈ (0,2/3)$: $∂_t u = ((-Δ)^{-1}u)Δ u + α u^2$. The initial condition $u_0$ is positive, radial, and non-increasing with $u_0∈ L^1∩ L^{2+δ}({\\\\mathbb R}^3)$ for some small $δ >0$. There is no size restriction on $u_0$. This model problem appears of interest due to its structural similarity with Landau's equation from plasma physics, and moreover its radically different behavior from the semi-linear Heat equation: $u_t = Δ u + α u^2$. \",\"author\":[{\"propositions\":[],\"lastnames\":[\"Krieger\"],\"firstnames\":[\"Joachim\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1080/03605302.2011.643437\",\"eprint\":\"1012.2890\",\"fjournal\":\"Communications in Partial Differential Equations\",\"issn\":\"0360-5302\",\"journal\":\"Comm. Partial Differential Equations\",\"keywords\":\"Landau equation, Kinetic Theory\",\"mrclass\":\"35K55 (35B30 35B65 35D35 35Q20)\",\"mrnumber\":\"2901061\",\"mrreviewer\":\"Jana Kopfova\",\"number\":\"4\",\"pages\":\"647–689\",\"title\":\"Global solutions to a non-local diffusion equation with quadratic non-linearity\",\"url_arxiv\":\"https://arxiv.org/abs/1012.2890\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/03605302.2011.pdf\",\"volume\":\"37\",\"year\":\"2012\",\"zblnumber\":\"1247.35087\",\"bdsk-url-1\":\"https://doi.org/10.1080/03605302.2011.643437\",\"bibtex\":\"@article{MR2901061,\\n\\tabstract = {In this paper we prove the global in time well-posedness of the following non-local diffusion equation with $\\\\alpha \\\\in (0,2/3)$:\\n$\\\\partial_t u = ((-\\\\Delta)^{-1}u)\\\\Delta u + \\\\alpha u^2$.  The initial condition $u_0$ is positive, radial, and non-increasing with\\n$u_0\\\\in L^1\\\\cap L^{2+\\\\delta}({\\\\mathbb R}^3)$ for some small $\\\\delta >0$.  There is no size restriction on $u_0$.\\nThis model problem appears of interest due to its structural similarity with Landau's equation from plasma physics, and moreover its radically different behavior from the semi-linear Heat equation: $u_t = \\\\Delta u + \\\\alpha u^2$. },\\n\\tauthor = {Krieger, Joachim and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1080/03605302.2011.643437},\\n\\teprint = {1012.2890},\\n\\tfjournal = {Communications in Partial Differential Equations},\\n\\tissn = {0360-5302},\\n\\tjournal = {Comm. Partial Differential Equations},\\n\\tkeywords = {Landau equation, Kinetic Theory},\\n\\tmrclass = {35K55 (35B30 35B65 35D35 35Q20)},\\n\\tmrnumber = {2901061},\\n\\tmrreviewer = {Jana Kopfova},\\n\\tnumber = {4},\\n\\tpages = {647--689},\\n\\ttitle = {Global solutions to a non-local diffusion equation with quadratic non-linearity},\\n\\turl_arxiv = {https://arxiv.org/abs/1012.2890},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/03605302.2011.pdf},\\n\\tvolume = {37},\\n\\tyear = {2012},\\n\\tzblnumber = {1247.35087},\\n\\tbdsk-url-1 = {https://doi.org/10.1080/03605302.2011.643437}}\\n\\n\",\"author_short\":[\"Krieger, J.\",\"Strain, R. M.\"],\"key\":\"MR2901061\",\"id\":\"MR2901061\",\"bibbaseid\":\"krieger-strain-globalsolutionstoanonlocaldiffusionequationwithquadraticnonlinearity-2012\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1012.2890\",\" pdf\":\"https://strain.math.upenn.edu/preprints/03605302.2011.pdf\"},\"keyword\":[\"Landau equation\",\"Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":9,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"For the relativistic Boltzmann equation in $R^3_x$, this work proves the global existence, uniqueness, positivity, and optimal time convergence rates to the relativistic Maxwellian for solutions which start out sufficiently close under the general physical soft potential assumption proposed in 1988.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Zhu\"],\"firstnames\":[\"Keya\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-15 15:47:40 -0400\",\"doi\":\"10.3934/krm.2012.5.383\",\"eprint\":\"1106.1579\",\"fjournal\":\"Kinetic and Related Models\",\"issn\":\"1937-5093\",\"journal\":\"Kinet. Relat. Models\",\"keywords\":\"Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory\",\"mrclass\":\"82C05 (76P05 76Y05)\",\"mrnumber\":\"2911100\",\"mrreviewer\":\"Mark Thompson\",\"number\":\"2\",\"pages\":\"383–415\",\"title\":\"Large-time decay of the soft potential relativistic Boltzmann equation in $\\\\mathbb R^3_x$\",\"url_arxiv\":\"https://arxiv.org/abs/1106.1579\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/szRBwhole.pdf\",\"volume\":\"5\",\"year\":\"2012\",\"zblnumber\":\"1247.76071\",\"bdsk-url-1\":\"https://doi.org/10.3934/krm.2012.5.383\",\"bibtex\":\"@article{MR2911100,\\n\\tabstract = {For the relativistic Boltzmann equation in $R^3_x$, this work proves the global existence, uniqueness, positivity, and optimal time convergence rates to the relativistic Maxwellian for solutions which start out sufficiently close under the general physical soft potential assumption proposed in 1988.},\\n\\tauthor = {Strain, Robert M. and Zhu, Keya},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-15 15:47:40 -0400},\\n\\tdoi = {10.3934/krm.2012.5.383},\\n\\teprint = {1106.1579},\\n\\tfjournal = {Kinetic and Related Models},\\n\\tissn = {1937-5093},\\n\\tjournal = {Kinet. Relat. Models},\\n\\tkeywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory},\\n\\tmrclass = {82C05 (76P05 76Y05)},\\n\\tmrnumber = {2911100},\\n\\tmrreviewer = {Mark Thompson},\\n\\tnumber = {2},\\n\\tpages = {383--415},\\n\\ttitle = {Large-time decay of the soft potential relativistic {B}oltzmann equation in {$\\\\mathbb R^3_x$}},\\n\\turl_arxiv = {https://arxiv.org/abs/1106.1579},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/szRBwhole.pdf},\\n\\tvolume = {5},\\n\\tyear = {2012},\\n\\tzblnumber = {1247.76071},\\n\\tbdsk-url-1 = {https://doi.org/10.3934/krm.2012.5.383}}\\n\\n\",\"author_short\":[\"Strain, R. M.\",\"Zhu, K.\"],\"key\":\"MR2911100\",\"id\":\"MR2911100\",\"bibbaseid\":\"strain-zhu-largetimedecayofthesoftpotentialrelativisticboltzmannequationinmathbbr3x-2012\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1106.1579\",\" pdf\":\"https://strain.math.upenn.edu/preprints/szRBwhole.pdf\"},\"keyword\":[\"Boltzmann equation\",\"relativistic Boltzmann equation\",\"Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space ${\\\\mathbb R}^{n}_x$ with $n \\\\ge 3$. We use the existence theory of global in time nearby Maxwellian solutions from previous work. It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption. For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of $O(t^{-\\\\frac{n}{2}+\\\\frac{n}{2r}})$ in the $L^2_v(L^r_x)$-norm for any $2≤ r≤ ∞$. \",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.3934/krm.2012.5.583\",\"eprint\":\"1011.5561\",\"fjournal\":\"Kinetic and Related Models\",\"issn\":\"1937-5093\",\"journal\":\"Kinet. Relat. Models\",\"keywords\":\"Boltzmann equation, Kinetic Theory, non-cutoff\",\"mrclass\":\"76P05 (26A33 35F20 82Cxx)\",\"mrnumber\":\"2972454\",\"number\":\"3\",\"pages\":\"583–613\",\"title\":\"Optimal time decay of the non cut-off Boltzmann equation in the whole space\",\"url_arxiv\":\"https://arxiv.org/abs/1011.5561\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/SdecaySOFT.pdf\",\"volume\":\"5\",\"year\":\"2012\",\"zblnumber\":\"1383.76414\",\"bdsk-url-1\":\"https://doi.org/10.3934/krm.2012.5.583\",\"bibtex\":\"@article{MR2972454,\\n\\tabstract = {In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space ${\\\\mathbb R}^{n}_x$ with $n \\\\ge 3$.    We use the existence theory of global in time nearby Maxwellian solutions from previous work.  \\nIt has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption.  For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of  $O(t^{-\\\\frac{n}{2}+\\\\frac{n}{2r}})$ in the $L^2_v(L^r_x)$-norm for any $2\\\\leq r\\\\leq \\\\infty$. },\\n\\tauthor = {Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.3934/krm.2012.5.583},\\n\\teprint = {1011.5561},\\n\\tfjournal = {Kinetic and Related Models},\\n\\tissn = {1937-5093},\\n\\tjournal = {Kinet. Relat. Models},\\n\\tkeywords = {Boltzmann equation, Kinetic Theory, non-cutoff},\\n\\tmrclass = {76P05 (26A33 35F20 82Cxx)},\\n\\tmrnumber = {2972454},\\n\\tnumber = {3},\\n\\tpages = {583--613},\\n\\ttitle = {Optimal time decay of the non cut-off {B}oltzmann equation in the whole space},\\n\\turl_arxiv = {https://arxiv.org/abs/1011.5561},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/SdecaySOFT.pdf},\\n\\tvolume = {5},\\n\\tyear = {2012},\\n\\tzblnumber = {1383.76414},\\n\\tbdsk-url-1 = {https://doi.org/10.3934/krm.2012.5.583}}\\n\\n\",\"author_short\":[\"Strain, R. M.\"],\"key\":\"MR2972454\",\"id\":\"MR2972454\",\"bibbaseid\":\"strain-optimaltimedecayofthenoncutoffboltzmannequationinthewholespace-2012\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1011.5561\",\" pdf\":\"https://strain.math.upenn.edu/preprints/SdecaySOFT.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Kinetic Theory\",\"non-cutoff\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":1,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space $R^3_x$. The existence of global in time nearby Maxwellian solutions is known from the work of Strain in 2006. However the asymptotic behavior of these solutions has been a challenging open problem. Building on our previous work on time decay for the simpler Vlasov-Poisson-Boltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of $O(t^{-\\\\frac{3}{2}+\\\\frac{3}{2r}})$ in $L^2_ξ(L^r_x)$-norm for any $2≤ r≤ ∞$ if initial perturbation is smooth enough and decays in space-velocity fast enough at infinity. Moreover, some explicit rates for the electromagnetic field tending to zero are also provided.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Duan\"],\"firstnames\":[\"Renjun\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1002/cpa.20381\",\"eprint\":\"1006.3605\",\"fjournal\":\"Communications on Pure and Applied Mathematics\",\"issn\":\"0010-3640\",\"journal\":\"Comm. Pure Appl. Math.\",\"keywords\":\"Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory\",\"mrclass\":\"82C40 (35B40 35Q20 35Q83 76W05)\",\"mrnumber\":\"2832167\",\"mrreviewer\":\"Stephen Wollman\",\"number\":\"11\",\"pages\":\"1497–1546\",\"title\":\"Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space\",\"url_arxiv\":\"https://arxiv.org/abs/1006.3605\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/dsVMB.pdf\",\"volume\":\"64\",\"year\":\"2011\",\"zblnumber\":\"1244.35010\",\"bdsk-url-1\":\"https://doi.org/10.1002/cpa.20381\",\"bibtex\":\"@article{MR2832167,\\n\\tabstract = {In this paper we study the large-time behavior of classical\\nsolutions to the two-species Vlasov-Maxwell-Boltzmann system in the\\nwhole space $R^3_x$.    The existence of  global in time nearby\\nMaxwellian solutions is known from the work of Strain in 2006.  However\\nthe asymptotic behavior of these solutions  has been a challenging\\nopen problem.  Building on our previous work on time\\ndecay for the simpler Vlasov-Poisson-Boltzmann system, we prove that\\nthese solutions converge to the global Maxwellian with the optimal\\ndecay rate of  $O(t^{-\\\\frac{3}{2}+\\\\frac{3}{2r}})$ in\\n$L^2_\\\\xi(L^r_x)$-norm for any $2\\\\leq r\\\\leq \\\\infty$ if initial\\nperturbation is smooth enough and decays in space-velocity fast\\nenough at infinity. Moreover, some explicit rates for the\\nelectromagnetic field tending to zero are also provided.},\\n\\tauthor = {Duan, Renjun and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1002/cpa.20381},\\n\\teprint = {1006.3605},\\n\\tfjournal = {Communications on Pure and Applied Mathematics},\\n\\tissn = {0010-3640},\\n\\tjournal = {Comm. Pure Appl. Math.},\\n\\tkeywords = {Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory},\\n\\tmrclass = {82C40 (35B40 35Q20 35Q83 76W05)},\\n\\tmrnumber = {2832167},\\n\\tmrreviewer = {Stephen Wollman},\\n\\tnumber = {11},\\n\\tpages = {1497--1546},\\n\\ttitle = {Optimal large-time behavior of the {V}lasov-{M}axwell-{B}oltzmann system in the whole space},\\n\\turl_arxiv = {https://arxiv.org/abs/1006.3605},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/dsVMB.pdf},\\n\\tvolume = {64},\\n\\tyear = {2011},\\n\\tzblnumber = {1244.35010},\\n\\tbdsk-url-1 = {https://doi.org/10.1002/cpa.20381}}\\n\\n\",\"author_short\":[\"Duan, R.\",\"Strain, R. M.\"],\"key\":\"MR2832167\",\"id\":\"MR2832167\",\"bibbaseid\":\"duan-strain-optimallargetimebehaviorofthevlasovmaxwellboltzmannsysteminthewholespace-2011\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1006.3605\",\" pdf\":\"https://strain.math.upenn.edu/preprints/dsVMB.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Vlasov-Maxwell systems\",\"Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"The Vlasov-Poisson-Boltzmann System governs the time evolution of the distribution function for the dilute charged particles in the presence of a self-consistent electric potential force through the Poisson equation. In this paper, we are concerned with the rate of convergence of solutions to equilibrium for this system over $R^3_x$. It is shown that the electric field which is indeed responsible for the lowest-order part in the energy space reduces the speed of convergence and hence the dispersion of this system over the full space is slower than that of the Boltzmann equation without forces, where the exact difference between both power indices in the algebraic rates of convergence is $1/4$. For the proof, in the linearized case with a given non-homogeneous source, Fourier analysis is employed to obtain time-decay properties of the solution operator. In the nonlinear case, the combination of the linearized results and the nonlinear energy estimates with the help of the proper Lyapunov-type inequalities leads to the optimal time-decay rate of perturbed solutions under some conditions on initial data.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Duan\"],\"firstnames\":[\"Renjun\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1007/s00205-010-0318-6\",\"eprint\":\"0912.1742\",\"fjournal\":\"Archive for Rational Mechanics and Analysis\",\"issn\":\"0003-9527\",\"journal\":\"Arch. Ration. Mech. Anal.\",\"keywords\":\"Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory\",\"mrclass\":\"35Q83 (35B40 35Q20 76P05 76X05 82C40 82D10)\",\"mrnumber\":\"2754344\",\"number\":\"1\",\"pages\":\"291–328\",\"title\":\"Optimal time decay of the Vlasov-Poisson-Boltzmann system in $ℝ^3$\",\"url_arxiv\":\"https://arxiv.org/abs/0912.1742\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/dsVPB.pdf\",\"volume\":\"199\",\"year\":\"2011\",\"zblnumber\":\"1232.35169\",\"bdsk-url-1\":\"https://doi.org/10.1007/s00205-010-0318-6\",\"bibtex\":\"@article{MR2754344,\\n\\tabstract = {The Vlasov-Poisson-Boltzmann System governs the time evolution of\\nthe distribution function for the dilute charged particles in the\\npresence of a self-consistent electric potential force through the\\nPoisson equation. In this paper, we are concerned with the rate of\\nconvergence of solutions to equilibrium for this system over $R^3_x$.\\nIt is shown that the electric field which is indeed responsible for\\nthe lowest-order part in the energy space reduces the speed of\\nconvergence and hence the dispersion of this system over the full\\nspace is slower than that of the Boltzmann equation without forces,\\nwhere the exact difference between both power indices in the\\nalgebraic rates of convergence is $1/4$. For the proof, in the\\nlinearized case with a given non-homogeneous source,  Fourier\\nanalysis is employed to obtain time-decay properties of the solution\\noperator. In the nonlinear case, the combination of the linearized\\nresults and the nonlinear energy estimates with the help of the\\nproper Lyapunov-type inequalities leads to the optimal time-decay\\nrate of perturbed solutions under some conditions on initial data.},\\n\\tauthor = {Duan, Renjun and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1007/s00205-010-0318-6},\\n\\teprint = {0912.1742},\\n\\tfjournal = {Archive for Rational Mechanics and Analysis},\\n\\tissn = {0003-9527},\\n\\tjournal = {Arch. Ration. Mech. Anal.},\\n\\tkeywords = {Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory},\\n\\tmrclass = {35Q83 (35B40 35Q20 76P05 76X05 82C40 82D10)},\\n\\tmrnumber = {2754344},\\n\\tnumber = {1},\\n\\tpages = {291--328},\\n\\ttitle = {Optimal time decay of the {V}lasov-{P}oisson-{B}oltzmann system in {$\\\\mathbb{R}^3$}},\\n\\turl_arxiv = {https://arxiv.org/abs/0912.1742},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/dsVPB.pdf},\\n\\tvolume = {199},\\n\\tyear = {2011},\\n\\tzblnumber = {1232.35169},\\n\\tbdsk-url-1 = {https://doi.org/10.1007/s00205-010-0318-6}}\\n\\n\",\"author_short\":[\"Duan, R.\",\"Strain, R. M.\"],\"key\":\"MR2754344\",\"id\":\"MR2754344\",\"bibbaseid\":\"duan-strain-optimaltimedecayofthevlasovpoissonboltzmannsystemin3-2011\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/0912.1742\",\" pdf\":\"https://strain.math.upenn.edu/preprints/dsVPB.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Vlasov-Maxwell systems\",\"Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":0,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials, $r^{-(p-1)}$ with $p>2$, for initial perturbations of the Maxwellian equilibrium states. We more generally cover collision kernels with parameters $s∈ (0,1)$ and $γ$ satisfying $γ > -n$ in arbitrary dimensions $\\\\mathbb{T}^n × ℝ^n$ with $n\\\\ge 2$. Moreover, we prove rapid convergence as predicted by the celebrated Boltzmann $H$-theorem. When $γ \\\\ge -2s$, we have exponential time decay to the Maxwellian equilibrium states. When $γ <-2s$, our solutions decay polynomially fast in time with any rate. These results are completely constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $γ \\\\ge -2s$, as conjectured in Mouhot-Strain. It will be observed that this fundamental equation, derived by both Boltzmann and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the grazing collisions in the Boltzmann theory. \",\"author\":[{\"propositions\":[],\"lastnames\":[\"Gressman\"],\"firstnames\":[\"Philip\",\"T.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1090/S0894-0347-2011-00697-8\",\"eprint\":\"1011.5441\",\"fjournal\":\"Journal of the American Mathematical Society\",\"issn\":\"0894-0347\",\"journal\":\"J. Amer. Math. Soc.\",\"keywords\":\"Boltzmann equation, Kinetic Theory, non-cutoff\",\"mrclass\":\"82C40 (35H20 35Q20 35R11 76P05)\",\"mrnumber\":\"2784329\",\"mrreviewer\":\"Laurent Desvillettes\",\"number\":\"3\",\"pages\":\"771–847\",\"title\":\"Global classical solutions of the Boltzmann equation without angular cut-off\",\"url_arxiv\":\"https://arxiv.org/abs/1011.5441\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/gsNonCut.pdf\",\"volume\":\"24\",\"year\":\"2011\",\"zblnumber\":\"1248.35140\",\"bdsk-url-1\":\"https://doi.org/10.1090/S0894-0347-2011-00697-8\",\"bibtex\":\"@article{MR2784329,\\n\\tabstract = {This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials, $r^{-(p-1)}$ with $p>2$,\\nfor initial perturbations of the Maxwellian equilibrium states.  \\nWe  more generally cover collision kernels with parameters $s\\\\in (0,1)$ and $\\\\gamma$ satisfying \\n$\\\\gamma  > -n$ in arbitrary dimensions $\\\\mathbb{T}^n \\\\times \\\\mathbb{R}^n$ with $n\\\\ge 2$.  \\nMoreover, we prove rapid convergence as predicted by the celebrated Boltzmann $H$-theorem.\\nWhen $\\\\gamma \\\\ge -2s$, we have  exponential time decay to the Maxwellian equilibrium states.  When $\\\\gamma <-2s$, our solutions decay polynomially fast in time with any rate.  \\nThese results are completely constructive.  Additionally, we prove  sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $\\\\gamma  \\\\ge -2s$, as conjectured in  Mouhot-Strain.  It will be observed that this fundamental equation, derived by both Boltzmann  and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world.  Our methods provide a new  understanding of the grazing collisions in the Boltzmann theory. },\\n\\tauthor = {Gressman, Philip T. and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1090/S0894-0347-2011-00697-8},\\n\\teprint = {1011.5441},\\n\\tfjournal = {Journal of the American Mathematical Society},\\n\\tissn = {0894-0347},\\n\\tjournal = {J. Amer. Math. Soc.},\\n\\tkeywords = {Boltzmann equation, Kinetic Theory, non-cutoff},\\n\\tmrclass = {82C40 (35H20 35Q20 35R11 76P05)},\\n\\tmrnumber = {2784329},\\n\\tmrreviewer = {Laurent Desvillettes},\\n\\tnumber = {3},\\n\\tpages = {771--847},\\n\\ttitle = {Global classical solutions of the {B}oltzmann equation without angular cut-off},\\n\\turl_arxiv = {https://arxiv.org/abs/1011.5441},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/gsNonCut.pdf},\\n\\tvolume = {24},\\n\\tyear = {2011},\\n\\tzblnumber = {1248.35140},\\n\\tbdsk-url-1 = {https://doi.org/10.1090/S0894-0347-2011-00697-8}}\\n\\n\",\"author_short\":[\"Gressman, P. T.\",\"Strain, R. M.\"],\"key\":\"MR2784329\",\"id\":\"MR2784329\",\"bibbaseid\":\"gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithoutangularcutoff-2011\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1011.5441\",\" pdf\":\"https://strain.math.upenn.edu/preprints/gsNonCut.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Kinetic Theory\",\"non-cutoff\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":9,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"This article provides sharp constructive upper and lower bound estimates for the Boltzmann collision operator with the full range of physical non cut-off collision kernels ($γ > -n$ and $s∈ (0,1)$) in the trilinear $L^2(ℝ^n)$ energy $( \\\\mathcal{Q}(g,f), f )$. These new estimates prove that, for a very general class of $g(v)$, the global diffusive behavior (on $f$) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works. We further prove new global entropy production estimates with the same anisotropic semi-norm. This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non cut-off Boltzmann collision operator in the energy space $L^2(ℝ^n)$.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Gressman\"],\"firstnames\":[\"Philip\",\"T.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1016/j.aim.2011.05.005\",\"eprint\":\"1007.1276\",\"fjournal\":\"Advances in Mathematics\",\"issn\":\"0001-8708\",\"journal\":\"Adv. Math.\",\"keywords\":\"Boltzmann equation, Kinetic Theory, non-cutoff\",\"mrclass\":\"35Q20 (35B65 35R11 82C40)\",\"mrnumber\":\"2807092\",\"mrreviewer\":\"Francesco Salvarani\",\"number\":\"6\",\"pages\":\"2349–2384\",\"title\":\"Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production\",\"url_arxiv\":\"https://arxiv.org/abs/1007.1276\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/gsNonCutEst.pdf\",\"volume\":\"227\",\"year\":\"2011\",\"zblnumber\":\"1234.35173\",\"bdsk-url-1\":\"https://doi.org/10.1016/j.aim.2011.05.005\",\"bibtex\":\"@article{MR2807092,\\n\\tabstract = {This article provides sharp constructive upper and lower bound estimates for the  \\nBoltzmann collision operator \\nwith the full range of physical non cut-off collision kernels ($\\\\gamma > -n$ and $s\\\\in (0,1)$)\\n in the trilinear $L^2(\\\\mathbb{R}^n)$ energy $( \\\\mathcal{Q}(g,f), f )$.  These new estimates prove that, for a very general class of $g(v)$, the  global diffusive behavior (on $f$) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works.  We further prove new global entropy production estimates with the same anisotropic semi-norm.  \\n This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non cut-off Boltzmann collision operator in the energy space \\n$L^2(\\\\mathbb{R}^n)$.},\\n\\tauthor = {Gressman, Philip T. and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1016/j.aim.2011.05.005},\\n\\teprint = {1007.1276},\\n\\tfjournal = {Advances in Mathematics},\\n\\tissn = {0001-8708},\\n\\tjournal = {Adv. Math.},\\n\\tkeywords = {Boltzmann equation, Kinetic Theory, non-cutoff},\\n\\tmrclass = {35Q20 (35B65 35R11 82C40)},\\n\\tmrnumber = {2807092},\\n\\tmrreviewer = {Francesco Salvarani},\\n\\tnumber = {6},\\n\\tpages = {2349--2384},\\n\\ttitle = {Sharp anisotropic estimates for the {B}oltzmann collision operator and its entropy production},\\n\\turl_arxiv = {https://arxiv.org/abs/1007.1276},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/gsNonCutEst.pdf},\\n\\tvolume = {227},\\n\\tyear = {2011},\\n\\tzblnumber = {1234.35173},\\n\\tbdsk-url-1 = {https://doi.org/10.1016/j.aim.2011.05.005}}\\n\\n\",\"author_short\":[\"Gressman, P. T.\",\"Strain, R. M.\"],\"key\":\"MR2807092\",\"id\":\"MR2807092\",\"bibbaseid\":\"gressman-strain-sharpanisotropicestimatesfortheboltzmanncollisionoperatoranditsentropyproduction-2011\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1007.1276\",\" pdf\":\"https://strain.math.upenn.edu/preprints/gsNonCutEst.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Kinetic Theory\",\"non-cutoff\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":0,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. More specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellian constructed from a class of solutions to the relativistic Euler equations that includes a large subclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Euler equations. \",\"author\":[{\"propositions\":[],\"lastnames\":[\"Speck\"],\"firstnames\":[\"Jared\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1007/s00220-011-1207-z\",\"eprint\":\"1009.5033\",\"fjournal\":\"Communications in Mathematical Physics\",\"issn\":\"0010-3616\",\"journal\":\"Comm. Math. Phys.\",\"keywords\":\"Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory, Fluid mechanics\",\"mrclass\":\"82C40 (35Q20 76P05 76Y05)\",\"mrnumber\":\"2793935\",\"mrreviewer\":\"Stephen Wollman\",\"number\":\"1\",\"pages\":\"229–280\",\"title\":\"Hilbert expansion from the Boltzmann equation to relativistic fluids\",\"url_arxiv\":\"https://arxiv.org/abs/1009.5033\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/ssHilbertExp.pdf\",\"volume\":\"304\",\"year\":\"2011\",\"zblnumber\":\"1221.35271\",\"bdsk-url-1\":\"https://doi.org/10.1007/s00220-011-1207-z\",\"bibtex\":\"@article{MR2793935,\\n\\tabstract = {We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. \\nMore specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellian constructed from a class of solutions to the relativistic Euler equations that includes a large \\nsubclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Euler equations. },\\n\\tauthor = {Speck, Jared and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1007/s00220-011-1207-z},\\n\\teprint = {1009.5033},\\n\\tfjournal = {Communications in Mathematical Physics},\\n\\tissn = {0010-3616},\\n\\tjournal = {Comm. Math. Phys.},\\n\\tkeywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory, Fluid mechanics},\\n\\tmrclass = {82C40 (35Q20 76P05 76Y05)},\\n\\tmrnumber = {2793935},\\n\\tmrreviewer = {Stephen Wollman},\\n\\tnumber = {1},\\n\\tpages = {229--280},\\n\\ttitle = {Hilbert expansion from the {B}oltzmann equation to relativistic fluids},\\n\\turl_arxiv = {https://arxiv.org/abs/1009.5033},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/ssHilbertExp.pdf},\\n\\tvolume = {304},\\n\\tyear = {2011},\\n\\tzblnumber = {1221.35271},\\n\\tbdsk-url-1 = {https://doi.org/10.1007/s00220-011-1207-z}}\\n\\n\",\"author_short\":[\"Speck, J.\",\"Strain, R. M.\"],\"key\":\"MR2793935\",\"id\":\"MR2793935\",\"bibbaseid\":\"speck-strain-hilbertexpansionfromtheboltzmannequationtorelativisticfluids-2011\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1009.5033\",\" pdf\":\"https://strain.math.upenn.edu/preprints/ssHilbertExp.pdf\"},\"keyword\":[\"Boltzmann equation\",\"relativistic Boltzmann equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\",\"Fluid mechanics\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":6,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"It is often the case in mathematical analysis that solving an open problem can be facilitated by finding a new set of coordinates which may illuminate the known difficulties. In this article, we illustrate how to derive an assortment coordinates in which to represent the relativistic Boltzmann collision operator. We show the equivalence between some known representations, and others which seem to be new. One of these representations has been used recently to solve several open problems.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.3934/krm.2011.4.345\",\"eprint\":\"1011.5093\",\"fjournal\":\"Kinetic and Related Models\",\"issn\":\"1937-5093\",\"journal\":\"Kinet. Relat. Models\",\"keywords\":\"Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory\",\"mrclass\":\"82C40 (76P05 83A05)\",\"mrnumber\":\"2765751\",\"mrreviewer\":\"Stephen Wollman\",\"number\":\"1\",\"pages\":\"345–359\",\"title\":\"Coordinates in the relativistic Boltzmann theory\",\"url_arxiv\":\"https://arxiv.org/abs/1011.5093\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/sCCkrm.pdf\",\"volume\":\"4\",\"year\":\"2011\",\"zblnumber\":\"05869610\",\"bdsk-url-1\":\"https://doi.org/10.3934/krm.2011.4.345\",\"bibtex\":\"@article{MR2765751,\\n\\tabstract = {It is often the case in mathematical analysis that solving an open problem can be facilitated by finding a new set of coordinates  which may illuminate the known difficulties. In this article, we illustrate how to derive an assortment coordinates in which to represent the relativistic Boltzmann collision operator.  We show the equivalence between some known representations, and others which seem to be new.  One of these representations has been used recently to solve several open problems.},\\n\\tauthor = {Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.3934/krm.2011.4.345},\\n\\teprint = {1011.5093},\\n\\tfjournal = {Kinetic and Related Models},\\n\\tissn = {1937-5093},\\n\\tjournal = {Kinet. Relat. Models},\\n\\tkeywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\\n\\tmrclass = {82C40 (76P05 83A05)},\\n\\tmrnumber = {2765751},\\n\\tmrreviewer = {Stephen Wollman},\\n\\tnumber = {1},\\n\\tpages = {345--359},\\n\\ttitle = {Coordinates in the relativistic {B}oltzmann theory},\\n\\turl_arxiv = {https://arxiv.org/abs/1011.5093},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/sCCkrm.pdf},\\n\\tvolume = {4},\\n\\tyear = {2011},\\n\\tzblnumber = {05869610},\\n\\tbdsk-url-1 = {https://doi.org/10.3934/krm.2011.4.345}}\\n\\n\",\"author_short\":[\"Strain, R. M.\"],\"key\":\"MR2765751\",\"id\":\"MR2765751\",\"bibbaseid\":\"strain-coordinatesintherelativisticboltzmanntheory-2011\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1011.5093\",\" pdf\":\"https://strain.math.upenn.edu/preprints/sCCkrm.pdf\"},\"keyword\":[\"Boltzmann equation\",\"relativistic Boltzmann equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":1,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"This is a brief announcement of our recent proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation without any angular cutoff, that is, for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential $r^{-(p-1)}$ with $p > 2$, and more generally. We present here a mathematical framework for unique global in time solutions for all of these potentials. We consider it remarkable that this equation, derived by Boltzmann (1) in 1872 and Maxwell (2) in 1867, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects due to grazing collisions.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Gressman\"],\"firstnames\":[\"Philip\",\"T.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-08 11:55:21 -0600\",\"doi\":\"10.1073/pnas.1001185107\",\"fjournal\":\"Proceedings of the National Academy of Sciences of the United States of America\",\"issn\":\"1091-6490\",\"journal\":\"Proc. Natl. Acad. Sci. USA\",\"keywords\":\"Boltzmann equation, Kinetic Theory, non-cutoff\",\"mrclass\":\"82C40 (35Q20)\",\"mrnumber\":\"2629879\",\"number\":\"13\",\"pages\":\"5744–5749\",\"title\":\"Global classical solutions of the Boltzmann equation with long-range interactions\",\"urlpdf\":\"https://strain.math.upenn.edu/preprints/gsPNAS2010.pdf\",\"volume\":\"107\",\"year\":\"2010\",\"zblnumber\":\"1205.82120\",\"bdsk-url-1\":\"https://doi.org/10.1073/pnas.1001185107\",\"bibtex\":\"@article{MR2629879,\\n\\tabstract = {This is a brief announcement of our recent proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation without any angular cutoff, that is, for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential $r^{-(p-1)}$ with $p > 2$, and more generally. We present here a mathematical framework for unique global in time solutions for all of these potentials. We consider it remarkable that this equation, derived by Boltzmann (1) in 1872 and Maxwell (2) in 1867, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects due to grazing collisions.},\\n\\tauthor = {Gressman, Philip T. and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-08 11:55:21 -0600},\\n\\tdoi = {10.1073/pnas.1001185107},\\n\\tfjournal = {Proceedings of the National Academy of Sciences of the United States of America},\\n\\tissn = {1091-6490},\\n\\tjournal = {Proc. Natl. Acad. Sci. USA},\\n\\tkeywords = {Boltzmann equation, Kinetic Theory, non-cutoff},\\n\\tmrclass = {82C40 (35Q20)},\\n\\tmrnumber = {2629879},\\n\\tnumber = {13},\\n\\tpages = {5744--5749},\\n\\ttitle = {Global classical solutions of the {B}oltzmann equation with long-range interactions},\\n\\turlpdf = {https://strain.math.upenn.edu/preprints/gsPNAS2010.pdf},\\n\\tvolume = {107},\\n\\tyear = {2010},\\n\\tzblnumber = {1205.82120},\\n\\tbdsk-url-1 = {https://doi.org/10.1073/pnas.1001185107}}\\n\\n\",\"author_short\":[\"Gressman, P. T.\",\"Strain, R. M.\"],\"key\":\"MR2629879\",\"id\":\"MR2629879\",\"bibbaseid\":\"gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithlongrangeinteractions-2010\",\"role\":\"author\",\"urls\":{\"Pdf\":\"https://strain.math.upenn.edu/preprints/gsPNAS2010.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Kinetic Theory\",\"non-cutoff\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"We study the Cauchy Problem for the relativistic Boltzmann equation with near Vacuum initial data. Unique global in time mild solutions are obtained uniformly in the speed of light parameter $c \\\\ge 1$. We furthermore prove that solutions to the relativistic Boltzmann equation converge to solutions of the Newtonian Boltzmann equation in the limit as $c\\\\to∞$ on arbitrary time intervals $[0,T]$, with convergence rate $1/c^{2-ε}$ for any $ε ∈(0,2)$. This may be the first proof of unique global in time validity of the Newtonian limit for a Kinetic equation.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1137/090762695\",\"eprint\":\"1004.5407\",\"fjournal\":\"SIAM Journal on Mathematical Analysis\",\"issn\":\"0036-1410\",\"journal\":\"SIAM J. Math. Anal.\",\"keywords\":\"Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory\",\"mrclass\":\"82C40 (35Q20 35Q75 76P05)\",\"mrnumber\":\"2679588\",\"mrreviewer\":\"Silvia Lorenzani\",\"number\":\"4\",\"pages\":\"1568–1601\",\"title\":\"Global Newtonian limit for the relativistic Boltzmann equation near vacuum\",\"url_arxiv\":\"https://arxiv.org/abs/1004.5407\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/rBnewt.pdf\",\"volume\":\"42\",\"year\":\"2010\",\"zblnumber\":\"05894999\",\"bdsk-url-1\":\"https://doi.org/10.1137/090762695\",\"bibtex\":\"@article{MR2679588,\\n\\tabstract = {We study the Cauchy Problem for the relativistic Boltzmann equation with near Vacuum initial data. Unique global in time mild solutions are obtained uniformly in the speed of light parameter $c \\\\ge 1$.  We furthermore prove that solutions to the relativistic Boltzmann equation converge to solutions of  the Newtonian Boltzmann equation in the limit as $c\\\\to\\\\infty$ on arbitrary time intervals $[0,T]$, with  convergence rate $1/c^{2-\\\\epsilon}$ for any $\\\\epsilon \\\\in(0,2)$.  This may be the first proof of unique global in time validity of the Newtonian limit for a Kinetic equation.},\\n\\tauthor = {Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1137/090762695},\\n\\teprint = {1004.5407},\\n\\tfjournal = {SIAM Journal on Mathematical Analysis},\\n\\tissn = {0036-1410},\\n\\tjournal = {SIAM J. Math. Anal.},\\n\\tkeywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\\n\\tmrclass = {82C40 (35Q20 35Q75 76P05)},\\n\\tmrnumber = {2679588},\\n\\tmrreviewer = {Silvia Lorenzani},\\n\\tnumber = {4},\\n\\tpages = {1568--1601},\\n\\ttitle = {Global {N}ewtonian limit for the relativistic {B}oltzmann equation near vacuum},\\n\\turl_arxiv = {https://arxiv.org/abs/1004.5407},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/rBnewt.pdf},\\n\\tvolume = {42},\\n\\tyear = {2010},\\n\\tzblnumber = {05894999},\\n\\tbdsk-url-1 = {https://doi.org/10.1137/090762695}}\\n\\n\",\"author_short\":[\"Strain, R. M.\"],\"key\":\"MR2679588\",\"id\":\"MR2679588\",\"bibbaseid\":\"strain-globalnewtonianlimitfortherelativisticboltzmannequationnearvacuum-2010\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1004.5407\",\" pdf\":\"https://strain.math.upenn.edu/preprints/rBnewt.pdf\"},\"keyword\":[\"Boltzmann equation\",\"relativistic Boltzmann equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":5,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in $L^∞_\\\\ell$. If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box. Conditions on the collision kernel are generic; this resolves the open question of global existence for the soft potentials.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1007/s00220-010-1129-1\",\"eprint\":\"1003.4893\",\"fjournal\":\"Communications in Mathematical Physics\",\"issn\":\"0010-3616\",\"journal\":\"Comm. Math. Phys.\",\"keywords\":\"Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory\",\"mrclass\":\"82C40 (35Q20)\",\"mrnumber\":\"2728733\",\"mrreviewer\":\"Stephen Wollman\",\"number\":\"2\",\"pages\":\"529–597\",\"title\":\"Asymptotic stability of the relativistic Boltzmann equation for the soft potentials\",\"url_arxiv\":\"https://arxiv.org/abs/1003.4893\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/sRBsoft.pdf\",\"volume\":\"300\",\"year\":\"2010\",\"zblnumber\":\"1214.35072\",\"bdsk-url-1\":\"https://doi.org/10.1007/s00220-010-1129-1\",\"bibtex\":\"@article{MR2728733,\\n\\tabstract = {In this paper it is  shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in $L^\\\\infty_\\\\ell$.  If the initial data  are continuous then so is the corresponding solution.  We work in the case of a spatially periodic box.  Conditions on the collision kernel are generic; this resolves the open question of global existence for the soft potentials.},\\n\\tauthor = {Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1007/s00220-010-1129-1},\\n\\teprint = {1003.4893},\\n\\tfjournal = {Communications in Mathematical Physics},\\n\\tissn = {0010-3616},\\n\\tjournal = {Comm. Math. Phys.},\\n\\tkeywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\\n\\tmrclass = {82C40 (35Q20)},\\n\\tmrnumber = {2728733},\\n\\tmrreviewer = {Stephen Wollman},\\n\\tnumber = {2},\\n\\tpages = {529--597},\\n\\ttitle = {Asymptotic stability of the relativistic {B}oltzmann equation for the soft potentials},\\n\\turl_arxiv = {https://arxiv.org/abs/1003.4893},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/sRBsoft.pdf},\\n\\tvolume = {300},\\n\\tyear = {2010},\\n\\tzblnumber = {1214.35072},\\n\\tbdsk-url-1 = {https://doi.org/10.1007/s00220-010-1129-1}}\\n\\n\",\"author_short\":[\"Strain, R. M.\"],\"key\":\"MR2728733\",\"id\":\"MR2728733\",\"bibbaseid\":\"strain-asymptoticstabilityoftherelativisticboltzmannequationforthesoftpotentials-2010\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1003.4893\",\" pdf\":\"https://strain.math.upenn.edu/preprints/sRBsoft.pdf\"},\"keyword\":[\"Boltzmann equation\",\"relativistic Boltzmann equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":0,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $ℝ^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the $z$-axis. Suppose the solution satisfies,for some $0 łe ɛ łe 1$ that $|v (x,t)| łe C_* r^{-1+ɛ } |t|^{-ɛ /2}$ for $-T_0łe t < 0$ and a positive finite constant $C_*$ which is allowed to be large, we then prove that $v$ is regular at time zero. \",\"author\":[{\"propositions\":[],\"lastnames\":[\"Chen\"],\"firstnames\":[\"Chiun-Chuan\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Tsai\"],\"firstnames\":[\"Tai-Peng\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Yau\"],\"firstnames\":[\"Horng-Tzer\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1080/03605300902793956\",\"eprint\":\"0709.4230\",\"fjournal\":\"Communications in Partial Differential Equations\",\"issn\":\"0360-5302\",\"journal\":\"Comm. Partial Differential Equations\",\"keywords\":\"Navier-Stokes equations, Fluid mechanics\",\"mrclass\":\"35Q30 (35B40 35B44 35B65 76D05)\",\"mrnumber\":\"2512859\",\"mrreviewer\":\"Thierry Goudon\",\"number\":\"1-3\",\"pages\":\"203–232\",\"title\":\"Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations. II\",\"url_arxiv\":\"https://arxiv.org/abs/0709.4230\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/2009ChenStrainTsaiYau.pdf\",\"volume\":\"34\",\"year\":\"2009\",\"zblnumber\":\"1173.35095\",\"bdsk-url-1\":\"https://doi.org/10.1080/03605300902793956\",\"bibtex\":\"@article{MR2512859,\\n\\tabstract = {Consider axisymmetric strong solutions of the incompressible\\nNavier-Stokes equations in $\\\\mathbb{R}^3$ with non-trivial swirl.  Let $z$\\ndenote the axis of symmetry and $r$ measure the distance to the\\n$z$-axis.  Suppose the solution satisfies,for some $0 \\\\le \\\\varepsilon \\\\le 1$ that $|v (x,t)| \\\\le C_* r^{-1+\\\\varepsilon } |t|^{-\\\\varepsilon /2}$ for $-T_0\\\\le t < 0$ and a positive finite constant $C_*$ which is  allowed to be large,  we then prove that $v$ is regular at time zero. },\\n\\tauthor = {Chen, Chiun-Chuan and Strain, Robert M. and Tsai, Tai-Peng and Yau, Horng-Tzer},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1080/03605300902793956},\\n\\teprint = {0709.4230},\\n\\tfjournal = {Communications in Partial Differential Equations},\\n\\tissn = {0360-5302},\\n\\tjournal = {Comm. Partial Differential Equations},\\n\\tkeywords = {Navier-Stokes equations, Fluid mechanics},\\n\\tmrclass = {35Q30 (35B40 35B44 35B65 76D05)},\\n\\tmrnumber = {2512859},\\n\\tmrreviewer = {Thierry Goudon},\\n\\tnumber = {1-3},\\n\\tpages = {203--232},\\n\\ttitle = {Lower bounds on the blow-up rate of the axisymmetric {N}avier-{S}tokes equations. {II}},\\n\\turl_arxiv = {https://arxiv.org/abs/0709.4230},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/2009ChenStrainTsaiYau.pdf},\\n\\tvolume = {34},\\n\\tyear = {2009},\\n\\tzblnumber = {1173.35095},\\n\\tbdsk-url-1 = {https://doi.org/10.1080/03605300902793956}}\\n\\n\",\"author_short\":[\"Chen, C.\",\"Strain, R. M.\",\"Tsai, T.\",\"Yau, H.\"],\"key\":\"MR2512859\",\"id\":\"MR2512859\",\"bibbaseid\":\"chen-strain-tsai-yau-lowerboundsontheblowuprateoftheaxisymmetricnavierstokesequationsii-2009\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/0709.4230\",\" pdf\":\"https://strain.math.upenn.edu/preprints/2009ChenStrainTsaiYau.pdf\"},\"keyword\":[\"Navier-Stokes equations\",\"Fluid mechanics\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $R^3$ with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown by Caffarelli-Kohn-Nirenberg in 1982 that they could only blow up on the axis of symmetry. Let $z$ denote the axis of symmetry and $r$ measure the distance to the $z$-axis. Suppose the solution satisfies the pointwise scale invariant bound $|v (x,t)| łe C_*{(r^2 -t)^{-1/2}} $ for $-T_0łe t < 0$ and a positive finite constant $C_*$ which is allowed to be large, we then prove that $v$ is regular at time zero. \",\"author\":[{\"propositions\":[],\"lastnames\":[\"Chen\"],\"firstnames\":[\"Chiun-Chuan\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Yau\"],\"firstnames\":[\"Horng-Tzer\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Tsai\"],\"firstnames\":[\"Tai-Peng\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1093/imrn/rnn016\",\"eprint\":\"math/0701796\",\"fjournal\":\"International Mathematics Research Notices. IMRN\",\"issn\":\"1073-7928\",\"journal\":\"Int. Math. Res. Not. IMRN\",\"keywords\":\"Navier-Stokes equations, Fluid mechanics\",\"mrclass\":\"35Q30 (35B40 76D03 76D05)\",\"mrnumber\":\"2429247\",\"mrreviewer\":\"Thierry Goudon\",\"number\":\"9\",\"pages\":\"Art. ID rnn016, 31\",\"title\":\"Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations\",\"url_arxiv\":\"https://arxiv.org/abs/math/0701796\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/2008rnn016.pdf\",\"year\":\"2008\",\"zblnumber\":\"1154.35068\",\"bdsk-url-1\":\"https://doi.org/10.1093/imrn/rnn016\",\"bibtex\":\"@article{MR2429247,\\n\\tabstract = {Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations  in $R^3$ with non-trivial swirl.  Such solutions are not known to be globally defined, but it is shown by Caffarelli-Kohn-Nirenberg in 1982 that they could only blow up on the axis of symmetry.  Let $z$ denote the axis of symmetry and $r$ measure the distance \\nto the $z$-axis.   Suppose the solution satisfies the pointwise scale invariant bound $|v (x,t)| \\\\le C_*{(r^2 -t)^{-1/2}} $ for $-T_0\\\\le t < 0$ and a positive finite constant $C_*$ which is  allowed to be large,  we then prove that $v$ is regular at time zero. },\\n\\tauthor = {Chen, Chiun-Chuan and Strain, Robert M. and Yau, Horng-Tzer and Tsai, Tai-Peng},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1093/imrn/rnn016},\\n\\teprint = {math/0701796},\\n\\tfjournal = {International Mathematics Research Notices. IMRN},\\n\\tissn = {1073-7928},\\n\\tjournal = {Int. Math. Res. Not. IMRN},\\n\\tkeywords = {Navier-Stokes equations, Fluid mechanics},\\n\\tmrclass = {35Q30 (35B40 76D03 76D05)},\\n\\tmrnumber = {2429247},\\n\\tmrreviewer = {Thierry Goudon},\\n\\tnumber = {9},\\n\\tpages = {Art. ID rnn016, 31},\\n\\ttitle = {Lower bound on the blow-up rate of the axisymmetric {N}avier-{S}tokes equations},\\n\\turl_arxiv = {https://arxiv.org/abs/math/0701796},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/2008rnn016.pdf},\\n\\tyear = {2008},\\n\\tzblnumber = {1154.35068},\\n\\tbdsk-url-1 = {https://doi.org/10.1093/imrn/rnn016}}\\n\\n\",\"author_short\":[\"Chen, C.\",\"Strain, R. M.\",\"Yau, H.\",\"Tsai, T.\"],\"key\":\"MR2429247\",\"id\":\"MR2429247\",\"bibbaseid\":\"chen-strain-yau-tsai-lowerboundontheblowuprateoftheaxisymmetricnavierstokesequations-2008\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/math/0701796\",\" pdf\":\"https://strain.math.upenn.edu/preprints/2008rnn016.pdf\"},\"keyword\":[\"Navier-Stokes equations\",\"Fluid mechanics\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":0,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"Consider both soft potentials with angular cutoff and Landau collision kernels in the Boltzmann theory inside a periodic box. We prove that any smooth perturbation near a given Maxwellian approaches to zero at the rate of $e^{-λ t^{p}}$ for some $λ >0$ and $p∈ (0,1)$. Our method is based on an unified energy estimate with appropriate exponential velocity weight. Our results extend the classical Caflisch 1980 result to the case of very soft potential and Coulomb interactions.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Guo\"],\"firstnames\":[\"Yan\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-08 11:40:38 -0600\",\"doi\":\"10.1007/s00205-007-0067-3\",\"fjournal\":\"Archive for Rational Mechanics and Analysis\",\"issn\":\"0003-9527\",\"journal\":\"Arch. Ration. Mech. Anal.\",\"keywords\":\"Boltzmann equation, Landau equation, Kinetic Theory\",\"mrclass\":\"82B05 (35B45 35F25)\",\"mrnumber\":\"2366140\",\"mrreviewer\":\"Simone Calogero\",\"number\":\"2\",\"pages\":\"287–339\",\"read\":\"0\",\"title\":\"Exponential decay for soft potentials near Maxwellian\",\"urlpdf\":\"https://strain.math.upenn.edu/preprints/2005SGed.pdf\",\"volume\":\"187\",\"year\":\"2008\",\"zblnumber\":\"1130.76069\",\"bdsk-url-1\":\"https://doi.org/10.1007/s00205-007-0067-3\",\"bibtex\":\"@article{MR2366140,\\n\\tabstract = {Consider both soft potentials with angular cutoff and Landau collision kernels in the Boltzmann theory inside a periodic box. We prove that any smooth perturbation near a given Maxwellian approaches to zero at the rate of $e^{-\\\\lambda t^{p}}$ for some $\\\\lambda >0$ and  $p\\\\in (0,1)$. Our method is based on an unified energy estimate with appropriate exponential velocity\\nweight. Our results extend the classical Caflisch 1980 result to the case of very soft potential and Coulomb interactions.},\\n\\tauthor = {Strain, Robert M. and Guo, Yan},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-08 11:40:38 -0600},\\n\\tdoi = {10.1007/s00205-007-0067-3},\\n\\tfjournal = {Archive for Rational Mechanics and Analysis},\\n\\tissn = {0003-9527},\\n\\tjournal = {Arch. Ration. Mech. Anal.},\\n\\tkeywords = {Boltzmann equation, Landau equation, Kinetic Theory},\\n\\tmrclass = {82B05 (35B45 35F25)},\\n\\tmrnumber = {2366140},\\n\\tmrreviewer = {Simone Calogero},\\n\\tnumber = {2},\\n\\tpages = {287--339},\\n\\tread = {0},\\n\\ttitle = {Exponential decay for soft potentials near {M}axwellian},\\n\\turlpdf = {https://strain.math.upenn.edu/preprints/2005SGed.pdf},\\n\\tvolume = {187},\\n\\tyear = {2008},\\n\\tzblnumber = {1130.76069},\\n\\tbdsk-url-1 = {https://doi.org/10.1007/s00205-007-0067-3}}\\n\\n\",\"author_short\":[\"Strain, R. M.\",\"Guo, Y.\"],\"key\":\"MR2366140\",\"id\":\"MR2366140\",\"bibbaseid\":\"strain-guo-exponentialdecayforsoftpotentialsnearmaxwellian-2008\",\"role\":\"author\",\"urls\":{\"Pdf\":\"https://strain.math.upenn.edu/preprints/2005SGed.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Landau equation\",\"Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":5,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this paper we prove new constructive coercivity estimates for the Boltzmann collision operator without cutoff, that is for \\\\em long-range interactions. In particular we give a generalized sufficient condition for the existence of a spectral gap which involves both the growth behavior of the collision kernel at large relative velocities and its singular behavior at grazing and frontal collisions. It provides in particular existence of a spectral gap and estimates on it for interactions deriving from the hard potentials $ϕ(r)=r^{-(s-1)}$, $s \\\\ge 5$ or the so-called moderately soft potentials $ϕ(r)=r^{-(s-1)}$, $3< s < 5$, (without angular cutoff). In particular this paper recovers (by constructive means), improves and extends previous results of Pao 1974. We also obtain constructive coercivity estimates for the Landau collision operator for the optimal coercivity norm pointed out in Guo 2002 and we formulate a conjecture about a unified necessary and sufficient condition for the existence of a spectral gap for Boltzmann and Landau linearized collision operators. \",\"author\":[{\"propositions\":[],\"lastnames\":[\"Mouhot\"],\"firstnames\":[\"Clément\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1016/j.matpur.2007.03.003\",\"eprint\":\"math/0607495\",\"fjournal\":\"Journal de Mathématiques Pures et Appliquées. Neuvième Série\",\"issn\":\"0021-7824\",\"journal\":\"J. Math. Pures Appl. (9)\",\"keywords\":\"Boltzmann equation, Landau equation, Kinetic Theory, non-cutoff\",\"mrclass\":\"82C40 (47G20 76P05)\",\"mrnumber\":\"2322149\",\"mrreviewer\":\"Cédric Villani\",\"number\":\"5\",\"pages\":\"515–535\",\"title\":\"Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff\",\"url_arxiv\":\"https://arxiv.org/abs/math/0607495\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/2007mouhotStrain.pdf\",\"volume\":\"87\",\"year\":\"2007\",\"zblnumber\":\"1388.76338\",\"bdsk-url-1\":\"https://doi.org/10.1016/j.matpur.2007.03.003\",\"bibtex\":\"@article{MR2322149,\\n\\tabstract = {In this paper we prove new constructive coercivity estimates for the Boltzmann collision operator without cutoff, that is for {\\\\em long-range} interactions. \\nIn particular we give a generalized sufficient condition for the existence of a spectral gap which involves both the growth behavior of the collision kernel at large relative velocities and its singular behavior at grazing and frontal collisions. It provides in particular existence of a spectral gap and estimates on it for interactions deriving from the hard potentials $\\\\phi(r)=r^{-(s-1)}$, $s \\\\ge 5$ or the so-called moderately soft potentials $\\\\phi(r)=r^{-(s-1)}$, $3< s < 5$, (without angular cutoff). \\nIn particular this paper recovers (by constructive means), improves and extends previous results of Pao 1974. We also obtain constructive coercivity estimates for \\nthe Landau collision operator for the optimal coercivity norm pointed out in Guo 2002 and we formulate a conjecture about a unified necessary and sufficient condition \\nfor the existence of a spectral gap for Boltzmann and Landau linearized collision operators. },\\n\\tauthor = {Mouhot, Cl{\\\\'{e}}ment and Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1016/j.matpur.2007.03.003},\\n\\teprint = {math/0607495},\\n\\tfjournal = {Journal de Math{\\\\'{e}}matiques Pures et Appliqu{\\\\'{e}}es. Neuvi{\\\\`e}me S{\\\\'{e}}rie},\\n\\tissn = {0021-7824},\\n\\tjournal = {J. Math. Pures Appl. (9)},\\n\\tkeywords = {Boltzmann equation, Landau equation, Kinetic Theory, non-cutoff},\\n\\tmrclass = {82C40 (47G20 76P05)},\\n\\tmrnumber = {2322149},\\n\\tmrreviewer = {C\\\\'{e}dric Villani},\\n\\tnumber = {5},\\n\\tpages = {515--535},\\n\\ttitle = {Spectral gap and coercivity estimates for linearized {B}oltzmann collision operators without angular cutoff},\\n\\turl_arxiv = {https://arxiv.org/abs/math/0607495},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/2007mouhotStrain.pdf},\\n\\tvolume = {87},\\n\\tyear = {2007},\\n\\tzblnumber = {1388.76338},\\n\\tbdsk-url-1 = {https://doi.org/10.1016/j.matpur.2007.03.003}}\\n\\n\",\"author_short\":[\"Mouhot, C.\",\"Strain, R. M.\"],\"key\":\"MR2322149\",\"id\":\"MR2322149\",\"bibbaseid\":\"mouhot-strain-spectralgapandcoercivityestimatesforlinearizedboltzmanncollisionoperatorswithoutangularcutoff-2007\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/math/0607495\",\" pdf\":\"https://strain.math.upenn.edu/preprints/2007mouhotStrain.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Landau equation\",\"Kinetic Theory\",\"non-cutoff\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"The Balescu-Lenard equation from plasma physics is widely considered to include a highly accurate correction to Landau's fundamental collision operator. Yet so far it has seen very little mathematical study. We perform an extensive linearized analysis of this equation, which includes determining the asymptotic behavior of the new components of the linearized operator and establishing time decay rates for the linearized equation. \",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1080/03605300601088609\",\"eprint\":\"math/0603490\",\"fjournal\":\"Communications in Partial Differential Equations\",\"issn\":\"0360-5302\",\"journal\":\"Comm. Partial Differential Equations\",\"keywords\":\"Kinetic Theory, Balescu-Lenard equation\",\"mrclass\":\"82D10 (76P05 82C40)\",\"mrnumber\":\"2372479\",\"mrreviewer\":\"Stephen Wollman\",\"number\":\"10-12\",\"pages\":\"1551–1586\",\"title\":\"On the linearized Balescu-Lenard equation\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/2007BalescuLenard.pdf\",\"volume\":\"32\",\"year\":\"2007\",\"zblnumber\":\"1128.76068\",\"bdsk-url-1\":\"https://doi.org/10.1080/03605300601088609\",\"bibtex\":\"@article{MR2372479,\\n\\tabstract = {The Balescu-Lenard equation  from  plasma physics is widely considered to include a highly accurate correction to Landau's fundamental collision operator.   \\nYet so far it has seen very little mathematical study. We perform an extensive linearized analysis of this equation, \\nwhich includes determining the asymptotic behavior of the new components of the linearized operator and establishing time decay rates for the linearized equation.  },\\n\\tauthor = {Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1080/03605300601088609},\\n\\teprint = {math/0603490},\\n\\tfjournal = {Communications in Partial Differential Equations},\\n\\tissn = {0360-5302},\\n\\tjournal = {Comm. Partial Differential Equations},\\n\\tkeywords = {Kinetic Theory, Balescu-Lenard equation},\\n\\tmrclass = {82D10 (76P05 82C40)},\\n\\tmrnumber = {2372479},\\n\\tmrreviewer = {Stephen Wollman},\\n\\tnumber = {10-12},\\n\\tpages = {1551--1586},\\n\\ttitle = {On the linearized {B}alescu-{L}enard equation},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/2007BalescuLenard.pdf},\\n\\tvolume = {32},\\n\\tyear = {2007},\\n\\tzblnumber = {1128.76068},\\n\\tbdsk-url-1 = {https://doi.org/10.1080/03605300601088609}}\\n\\n\",\"author_short\":[\"Strain, R. M.\"],\"key\":\"MR2372479\",\"id\":\"MR2372479\",\"bibbaseid\":\"strain-onthelinearizedbalesculenardequation-2007\",\"role\":\"author\",\"urls\":{\" pdf\":\"https://strain.math.upenn.edu/preprints/2007BalescuLenard.pdf\"},\"keyword\":[\"Kinetic Theory\",\"Balescu-Lenard equation\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"By direct interpolation of a family of smooth energy estimates for solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann type equations, we show convergence to Maxwellian with any polynomial rate in time. Our results not only resolve the important open problem for both the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system for charged particles, but also lead to a simpler alternative proof of recent decay results for soft potentials as well as the Coulombic interaction, with precise decay rate depending on the initial conditions. \",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Guo\"],\"firstnames\":[\"Yan\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-08 11:41:10 -0600\",\"doi\":\"10.1080/03605300500361545\",\"fjournal\":\"Communications in Partial Differential Equations\",\"issn\":\"0360-5302\",\"journal\":\"Comm. Partial Differential Equations\",\"keywords\":\"Boltzmann equation, Landau equation, relativistic Landau equation, Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems\",\"mrclass\":\"82B40 (82C40 82D05 82D10)\",\"mrnumber\":\"2209761\",\"mrreviewer\":\"Hai-Liang Li\",\"number\":\"1-3\",\"pages\":\"417–429\",\"title\":\"Almost exponential decay near Maxwellian\",\"urlpdf\":\"https://strain.math.upenn.edu/preprints/2005SGaed.pdf\",\"volume\":\"31\",\"year\":\"2006\",\"zblnumber\":\"1096.82010\",\"bdsk-url-1\":\"https://doi.org/10.1080/03605300500361545\",\"bibtex\":\"@article{MR2209761,\\n\\tabstract = {By direct interpolation of a family of smooth energy estimates for solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann type equations, we show convergence to Maxwellian with any polynomial rate in time. Our results not only resolve the important open problem for both the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system for charged particles, but also lead to a simpler alternative proof of recent decay results for soft potentials as well as the Coulombic interaction, with precise decay rate depending on the initial conditions.\\n},\\n\\tauthor = {Strain, Robert M. and Guo, Yan},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-08 11:41:10 -0600},\\n\\tdoi = {10.1080/03605300500361545},\\n\\tfjournal = {Communications in Partial Differential Equations},\\n\\tissn = {0360-5302},\\n\\tjournal = {Comm. Partial Differential Equations},\\n\\tkeywords = {Boltzmann equation, Landau equation, relativistic Landau equation, Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},\\n\\tmrclass = {82B40 (82C40 82D05 82D10)},\\n\\tmrnumber = {2209761},\\n\\tmrreviewer = {Hai-Liang Li},\\n\\tnumber = {1-3},\\n\\tpages = {417--429},\\n\\ttitle = {Almost exponential decay near {M}axwellian},\\n\\turlpdf = {https://strain.math.upenn.edu/preprints/2005SGaed.pdf},\\n\\tvolume = {31},\\n\\tyear = {2006},\\n\\tzblnumber = {1096.82010},\\n\\tbdsk-url-1 = {https://doi.org/10.1080/03605300500361545}}\\n\\n\",\"author_short\":[\"Strain, R. M.\",\"Guo, Y.\"],\"key\":\"MR2209761\",\"id\":\"MR2209761\",\"bibbaseid\":\"strain-guo-almostexponentialdecaynearmaxwellian-2006\",\"role\":\"author\",\"urls\":{\"Pdf\":\"https://strain.math.upenn.edu/preprints/2005SGaed.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Landau equation\",\"relativistic Landau equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\",\"Vlasov-Maxwell systems\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"The Vlasov-Maxwell-Boltzmann system is a fundamental model to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-07-13 15:27:26 -0400\",\"doi\":\"10.1007/s00220-006-0109-y\",\"eprint\":\"math/0512002\",\"fjournal\":\"Communications in Mathematical Physics\",\"issn\":\"0010-3616\",\"journal\":\"Comm. Math. Phys.\",\"keywords\":\"Boltzmann equation, Kinetic Theory, Vlasov-Maxwell systems\",\"mrclass\":\"82D10 (35A05 35F20 35Q60 76P05 76X05)\",\"mrnumber\":\"2259206\",\"mrreviewer\":\"Simone Calogero\",\"number\":\"2\",\"pages\":\"543–567\",\"title\":\"The Vlasov-Maxwell-Boltzmann system in the whole space\",\"url_pdf\":\"https://strain.math.upenn.edu/preprints/2005Sws.pdf\",\"volume\":\"268\",\"year\":\"2006\",\"zblnumber\":\"1129.35022\",\"bdsk-url-1\":\"https://doi.org/10.1007/s00220-006-0109-y\",\"bibtex\":\"@article{MR2259206,\\n\\tabstract = {The Vlasov-Maxwell-Boltzmann system is a fundamental model to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians.},\\n\\tauthor = {Strain, Robert M.},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-07-13 15:27:26 -0400},\\n\\tdoi = {10.1007/s00220-006-0109-y},\\n\\teprint = {math/0512002},\\n\\tfjournal = {Communications in Mathematical Physics},\\n\\tissn = {0010-3616},\\n\\tjournal = {Comm. Math. Phys.},\\n\\tkeywords = {Boltzmann equation, Kinetic Theory, Vlasov-Maxwell systems},\\n\\tmrclass = {82D10 (35A05 35F20 35Q60 76P05 76X05)},\\n\\tmrnumber = {2259206},\\n\\tmrreviewer = {Simone Calogero},\\n\\tnumber = {2},\\n\\tpages = {543--567},\\n\\ttitle = {The {V}lasov-{M}axwell-{B}oltzmann system in the whole space},\\n\\turl_pdf = {https://strain.math.upenn.edu/preprints/2005Sws.pdf},\\n\\tvolume = {268},\\n\\tyear = {2006},\\n\\tzblnumber = {1129.35022},\\n\\tbdsk-url-1 = {https://doi.org/10.1007/s00220-006-0109-y}}\\n\\n\",\"author_short\":[\"Strain, R. M.\"],\"key\":\"MR2259206\",\"id\":\"MR2259206\",\"bibbaseid\":\"strain-thevlasovmaxwellboltzmannsysteminthewholespace-2006\",\"role\":\"author\",\"urls\":{\" pdf\":\"https://strain.math.upenn.edu/preprints/2005Sws.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Kinetic Theory\",\"Vlasov-Maxwell systems\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":3,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"We discuss recent work proving exponential time decay rates to equilibrium for Boltzmann equations such as the soft potentials, Landau's equation and the lin- earized Balescu-Lenard model. We also mention a proof of existence and unique- ness of solutions near Maxwellian to the Vlasov-Maxwell-Boltzmann system in the whole space. Some of these projects are joint work with Yan Guo.\",\"abstractow\":\"The Oberwolfach meeting described here aimed at presenting the latest mathematical results in the field of kinetic theory (both classical and quantum). There were 50 participants, among which 15 young participants (PhD students, post-docs or young assistant professors). Two of them (M.-P. Gualdani and R. Strain) were invited within the program \\\"US Junior Oberwolfach Fellows\\\": they are promising young researchers working in the US. The program of the meeting was made in such a way that a lot of time remained for people to meet informally and discuss about scientific issues. It also ensured that almost everybody attended all (or most of) the scheduled talks. The program was structured in the following way: three subtopics were defined (relationships between micro/meso/macroscopic models; kinetic theory for complex particles: granular media, coagulation/fragmentation, chemotaxis and sprays; quantum mechanical kinetic theory). For each subtopic, there were a few (2 or 3) longer talks (about 40 minutes) by senior participants. For those talks, a specific effort of clarity was asked to the speakers, and the subject had to be rather broad. Then, shorter talks of about 20 minutes were planned, on more specialized issues. Finally, a special (much longer) talk in two parts was presented by one of the participants (C. Villani), in order to describe in a very didactical way an emerging link between kinetic theory, optimal transport, and Riemannian geometry. Some participants did not give a talk within this program, but organized an informal discussion around a poster, with an audience composed of specially interested people.\",\"author\":[{\"firstnames\":[\"Robert\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Strain\"],\"suffixes\":[]}],\"booktitle\":\"Classical and quantum mechanical models of many-particle systems. Workshop held December 3–9, 2006.\",\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-08 13:59:27 -0600\",\"doi\":\"10.4171/owr/2006/54\",\"editor\":[{\"firstnames\":[\"Anton\"],\"propositions\":[],\"lastnames\":[\"Arnold\"],\"suffixes\":[]},{\"firstnames\":[\"Carlo\"],\"propositions\":[],\"lastnames\":[\"Cercignani\"],\"suffixes\":[]},{\"firstnames\":[\"Laurent\"],\"propositions\":[],\"lastnames\":[\"Desvillettes\"],\"suffixes\":[]}],\"fjournal\":\"Oberwolfach Reports\",\"issn\":\"1660-8933; 1660-8941/e\",\"journal\":\"Oberwolfach Rep.\",\"keywords\":\"Boltzmann equation, Landau equation, relativistic Landau equation, Kinetic Theory, relativistic Kinetic Theory\",\"language\":\"English\",\"msc2010\":\"82B40 81-06 82-06 81S30 00B05\",\"number\":\"4\",\"pages\":\"3189–3258\",\"publisher\":\"European Mathematical Society (EMS) Publishing House, Zurich; Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach\",\"title\":\"Recent results on existence, uniqueness and asymptotic decay rates for collisional kinetic models\",\"urlpdf\":\"https://strain.math.upenn.edu/preprints/rms2006ow.pdf\",\"volume\":\"3\",\"year\":\"2006\",\"zblnumber\":\"1177.82057\",\"bdsk-url-1\":\"https://doi.org/10.4171/owr/2006/54\",\"bibtex\":\"@article{Arnold2006,\\n\\tabstract = {We discuss recent work proving exponential time decay rates to equilibrium for Boltzmann equations such as the soft potentials, Landau's equation and the lin- earized Balescu-Lenard model. We also mention a proof of existence and unique- ness of solutions near Maxwellian to the Vlasov-Maxwell-Boltzmann system in the whole space. Some of these projects are joint work with Yan Guo.},\\n\\tabstractow = {The Oberwolfach meeting described here aimed at presenting the latest mathematical results in the field of kinetic theory (both classical and quantum). There were 50 participants, among which 15 young participants (PhD students, post-docs or young assistant professors). Two of them (M.-P. Gualdani and R. Strain) were invited within the program \\\"US Junior Oberwolfach Fellows\\\": they are promising young researchers working in the US. The program of the meeting was made in such a way that a lot of time remained for people to meet informally and discuss about scientific issues. It also ensured that almost everybody attended all (or most of) the scheduled talks. The program was structured in the following way: three subtopics were defined (relationships between micro/meso/macroscopic models; kinetic theory for complex particles: granular media, coagulation/fragmentation, chemotaxis and sprays; quantum mechanical kinetic theory). For each subtopic, there were a few (2 or 3) longer talks (about 40 minutes) by senior participants. For those talks, a specific effort of clarity was asked to the speakers, and the subject had to be rather broad. Then, shorter talks of about 20 minutes were planned, on more specialized issues. Finally, a special (much longer) talk in two parts was presented by one of the participants (C. Villani), in order to describe in a very didactical way an emerging link between kinetic theory, optimal transport, and Riemannian geometry. Some participants did not give a talk within this program, but organized an informal discussion around a poster, with an audience composed of specially interested people.},\\n\\tauthor = {Robert M. Strain},\\n\\tbooktitle = {{Classical and quantum mechanical models of many-particle systems. Workshop held December 3--9, 2006.}},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-08 13:59:27 -0600},\\n\\tdoi = {10.4171/owr/2006/54},\\n\\teditor = {Anton {Arnold} and Carlo {Cercignani} and Laurent {Desvillettes}},\\n\\tfjournal = {{Oberwolfach Reports}},\\n\\tissn = {1660-8933; 1660-8941/e},\\n\\tjournal = {{Oberwolfach Rep.}},\\n\\tkeywords = {Boltzmann equation, Landau equation, relativistic Landau equation, Kinetic Theory, relativistic Kinetic Theory},\\n\\tlanguage = {English},\\n\\tmsc2010 = {82B40 81-06 82-06 81S30 00B05},\\n\\tnumber = {4},\\n\\tpages = {3189--3258},\\n\\tpublisher = {European Mathematical Society (EMS) Publishing House, Zurich; Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach},\\n\\ttitle = {{R}ecent results on existence, uniqueness and asymptotic decay rates for collisional kinetic models},\\n\\turlpdf = {https://strain.math.upenn.edu/preprints/rms2006ow.pdf},\\n\\tvolume = {3},\\n\\tyear = {2006},\\n\\tzblnumber = {1177.82057},\\n\\tbdsk-url-1 = {https://doi.org/10.4171/owr/2006/54}}\\n\\n\",\"author_short\":[\"Strain, R. M.\"],\"editor_short\":[\"Arnold, A.\",\"Cercignani, C.\",\"Desvillettes, L.\"],\"key\":\"Arnold2006\",\"id\":\"Arnold2006\",\"bibbaseid\":\"strain-recentresultsonexistenceuniquenessandasymptoticdecayratesforcollisionalkineticmodels-2006\",\"role\":\"author\",\"urls\":{\"Pdf\":\"https://strain.math.upenn.edu/preprints/rms2006ow.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Landau equation\",\"relativistic Landau equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":3,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this report, we will describe briefly several recent developments for the Boltzmann equation without the Grad angular cut-off assumption.\",\"abstractow\":\"The topic of this meeting were non-linear partial differential and integro-differential equations (in particular kinetic equations and their macroscopic/fluid-dynamical limits) modeling the dynamics of many-particle systems with applications in physics, engineering, and mathematical biology. Typical questions of interest were the derivation of macro-models from micro-models, the mathematical analysis (well-posedness, stability, asymptotic behavior of solutions), and ``to a lesser extent'' numerical aspects of such equations. A highlight of this meeting was a mini-course on the recent mathematical theory of Landau damping.\",\"author\":[{\"firstnames\":[\"Robert\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Strain\"],\"suffixes\":[]}],\"booktitle\":\"Classical and quantum mechanical models of many-particle systems. Workshop held December 5 – December 11, 2010.\",\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-08 13:59:00 -0600\",\"doi\":\"10.4171/owr/2010/54\",\"editor\":[{\"firstnames\":[\"Anton\"],\"propositions\":[],\"lastnames\":[\"Arnold\"],\"suffixes\":[]},{\"firstnames\":[\"Eric\",\"A.\"],\"propositions\":[],\"lastnames\":[\"Carlen\"],\"suffixes\":[]},{\"firstnames\":[\"Laurent\"],\"propositions\":[],\"lastnames\":[\"Desvillettes\"],\"suffixes\":[]}],\"fjournal\":\"Oberwolfach Reports\",\"issn\":\"1660-8933; 1660-8941/e\",\"journal\":\"Oberwolfach Rep.\",\"keywords\":\"Boltzmann equation, Kinetic Theory, non-cutoff\",\"language\":\"English\",\"msc2010\":\"00B05 81-06 82-06 35Qxx 82Cxx 82B40 81S30\",\"number\":\"4\",\"pages\":\"3159–3236\",\"publisher\":\"European Mathematical Society (EMS) Publishing House, Zurich; Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach\",\"title\":\"Around the Boltzmann equation without angular cut-off\",\"urlpdf\":\"https://strain.math.upenn.edu/preprints/rms2010ow.pdf\",\"volume\":\"7\",\"year\":\"2010\",\"zblnumber\":\"1235.00027\",\"bdsk-url-1\":\"https://doi.org/10.4171/owr/2010/54\",\"bibtex\":\"@article{Arnold2010,\\n\\tabstract = {In this report, we will describe briefly several recent developments for the Boltzmann equation without the Grad angular cut-off assumption.},\\n\\tabstractow = {The topic of this meeting were non-linear partial differential and integro-differential equations (in particular kinetic equations and their macroscopic/fluid-dynamical limits) modeling the dynamics of many-particle systems with applications in physics, engineering, and mathematical biology. Typical questions of interest were the derivation of macro-models from micro-models, the mathematical analysis (well-posedness, stability, asymptotic behavior of solutions), and ``to a lesser extent'' numerical aspects of such equations. A highlight of this meeting was a mini-course on the recent mathematical theory of Landau damping.},\\n\\tauthor = {Robert M. Strain},\\n\\tbooktitle = {{Classical and quantum mechanical models of many-particle systems. Workshop held December 5 -- December 11, 2010.}},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-08 13:59:00 -0600},\\n\\tdoi = {10.4171/owr/2010/54},\\n\\teditor = {Anton {Arnold} and Eric A. {Carlen} and Laurent {Desvillettes}},\\n\\tfjournal = {{Oberwolfach Reports}},\\n\\tissn = {1660-8933; 1660-8941/e},\\n\\tjournal = {{Oberwolfach Rep.}},\\n\\tkeywords = {Boltzmann equation, Kinetic Theory, non-cutoff},\\n\\tlanguage = {English},\\n\\tmsc2010 = {00B05 81-06 82-06 35Qxx 82Cxx 82B40 81S30},\\n\\tnumber = {4},\\n\\tpages = {3159--3236},\\n\\tpublisher = {European Mathematical Society (EMS) Publishing House, Zurich; Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach},\\n\\ttitle = {Around the {B}oltzmann equation without angular cut-off},\\n\\turlpdf = {https://strain.math.upenn.edu/preprints/rms2010ow.pdf},\\n\\tvolume = {7},\\n\\tyear = {2010},\\n\\tzblnumber = {1235.00027},\\n\\tbdsk-url-1 = {https://doi.org/10.4171/owr/2010/54}}\\n\\n\",\"author_short\":[\"Strain, R. M.\"],\"editor_short\":[\"Arnold, A.\",\"Carlen, E. A.\",\"Desvillettes, L.\"],\"key\":\"Arnold2010\",\"id\":\"Arnold2010\",\"bibbaseid\":\"strain-aroundtheboltzmannequationwithoutangularcutoff-2010\",\"role\":\"author\",\"urls\":{\"Pdf\":\"https://strain.math.upenn.edu/preprints/rms2010ow.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Kinetic Theory\",\"non-cutoff\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":1,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"The relativistic Landau-Maxwell system is among the most fundamental and complete models for describing the dynamics of a dilute collisional plasma in which particles interact through Coulombic collisions and through their self-consistent electromagnetic field. We construct the first global in time classical solutions. Our solutions are constructed in a periodic box and near the relativistic Maxwellian, the Jüttner solution.\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Guo\"],\"firstnames\":[\"Yan\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 15:27:26 -0400\",\"date-modified\":\"2019-08-08 11:41:54 -0600\",\"doi\":\"10.1007/s00220-004-1151-2\",\"fjournal\":\"Communications in Mathematical Physics\",\"issn\":\"0010-3616\",\"journal\":\"Comm. Math. Phys.\",\"keywords\":\"Landau equation, relativistic Landau equation, Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems\",\"mrclass\":\"82D10 (35Q60 35Q75 76X05 76Y05)\",\"mrnumber\":\"2100057\",\"mrreviewer\":\"Cédric Villani\",\"number\":\"2\",\"pages\":\"263–320\",\"title\":\"Stability of the relativistic Maxwellian in a collisional plasma\",\"urlpdf\":\"https://strain.math.upenn.edu/preprints/2004SG.pdf\",\"volume\":\"251\",\"year\":\"2004\",\"zblnumber\":\"1113.82070\",\"bdsk-url-1\":\"https://doi.org/10.1007/s00220-004-1151-2\",\"bibtex\":\"@article{MR2100057,\\n\\tabstract = {The relativistic Landau-Maxwell system is among the most fundamental  and complete models for describing the dynamics of a dilute collisional plasma in which particles interact through Coulombic collisions and through their self-consistent electromagnetic field.   We construct the first global in time classical solutions.  Our solutions are constructed in a periodic box and near the relativistic Maxwellian, the J\\\\\\\"{u}ttner solution.},\\n\\tauthor = {Strain, Robert M. and Guo, Yan},\\n\\tdate-added = {2019-07-13 15:27:26 -0400},\\n\\tdate-modified = {2019-08-08 11:41:54 -0600},\\n\\tdoi = {10.1007/s00220-004-1151-2},\\n\\tfjournal = {Communications in Mathematical Physics},\\n\\tissn = {0010-3616},\\n\\tjournal = {Comm. Math. Phys.},\\n\\tkeywords = {Landau equation, relativistic Landau equation, Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},\\n\\tmrclass = {82D10 (35Q60 35Q75 76X05 76Y05)},\\n\\tmrnumber = {2100057},\\n\\tmrreviewer = {C\\\\'edric Villani},\\n\\tnumber = {2},\\n\\tpages = {263--320},\\n\\ttitle = {Stability of the relativistic {M}axwellian in a collisional plasma},\\n\\turlpdf = {https://strain.math.upenn.edu/preprints/2004SG.pdf},\\n\\tvolume = {251},\\n\\tyear = {2004},\\n\\tzblnumber = {1113.82070},\\n\\tbdsk-url-1 = {https://doi.org/10.1007/s00220-004-1151-2}}\\n\\n\",\"author_short\":[\"Strain, R. M.\",\"Guo, Y.\"],\"key\":\"MR2100057\",\"id\":\"MR2100057\",\"bibbaseid\":\"strain-guo-stabilityoftherelativisticmaxwellianinacollisionalplasma-2004\",\"role\":\"author\",\"urls\":{\"Pdf\":\"https://strain.math.upenn.edu/preprints/2004SG.pdf\"},\"keyword\":[\"Landau equation\",\"relativistic Landau equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\",\"Vlasov-Maxwell systems\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":6,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"abstract\":\"In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These $L^∞$ bounds have been known to be a challenging open problem in relativistic kinetic theory. To accomplish this, we establish two types of estimates for the gain part of the collision operator: first, we prove a potential type estimate and a relativistic hyper-surface integral estimate. We then combine those estimates using the relativistic counter-part of the Carleman representation to derive uniform control of the gain term for the relativistic collision operator. This allows us to prove the desired propagation of the uniform bounds of the solution. We further present two applications of the propagation of the uniform upper bounds: first we give another proof of the Boltzmann $H$-theorem, and second we prove the asymptotic convergence of solutions to the relativistic Maxwellian equilibrium.\",\"author\":[{\"firstnames\":[\"Jin\",\"Woo\"],\"propositions\":[],\"lastnames\":[\"Jang\"],\"suffixes\":[]},{\"firstnames\":[\"Robert\",\"M.\"],\"propositions\":[],\"lastnames\":[\"Strain\"],\"suffixes\":[]},{\"firstnames\":[\"Seok-Bae\"],\"propositions\":[],\"lastnames\":[\"Yun\"],\"suffixes\":[]}],\"doi\":\"10.1007/s00205-021-01649-0\",\"eprint\":\"1907.05784\",\"fjournal\":\"Archive for Rational Mechanics and Analysis\",\"journal\":\"Arch. Ration. Mech. Anal.\",\"keywords\":\"relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory\",\"note\":\"published online April 15, 2021\",\"pages\":\"149–186\",\"title\":\"Propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation\",\"url_arxiv\":\"https://arxiv.org/abs/1907.05784\",\"volume\":\"241\",\"year\":\"2021\",\"bdsk-url-1\":\"https://doi.org/10.1007/s00205-021-01649-0\",\"bibtex\":\"@article{1907.05784,\\n\\tabstract = {In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These $L^\\\\infty$ bounds have been known to be a challenging open problem in relativistic kinetic theory. To accomplish this, we establish two types of estimates for the gain part of the collision operator: first, we prove a potential type estimate and a relativistic hyper-surface integral estimate. We then combine those estimates using the relativistic counter-part of the Carleman representation to derive uniform control of the gain term for the relativistic collision operator. This allows us to prove the desired propagation of the uniform bounds of the solution. We further present two applications of the propagation of the uniform upper bounds: first we give another proof of the Boltzmann $H$-theorem, and second we prove the asymptotic convergence of solutions to the relativistic Maxwellian equilibrium.},\\n\\tauthor = {Jin Woo Jang and Robert M. Strain and Seok-Bae Yun},\\n\\tdoi = {10.1007/s00205-021-01649-0},\\n\\teprint = {1907.05784},\\n\\tfjournal = {Archive for Rational Mechanics and Analysis},\\n\\tjournal = {Arch. Ration. Mech. Anal.},\\n\\tkeywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\\n\\tnote = {published online April 15, 2021},\\n\\tpages = {149--186},\\n\\ttitle = {{Propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation}},\\n\\turl_arxiv = {https://arxiv.org/abs/1907.05784},\\n\\tvolume = {241},\\n\\tyear = {2021},\\n\\tbdsk-url-1 = {https://doi.org/10.1007/s00205-021-01649-0}}\\n\\n\",\"author_short\":[\"Jang, J. W.\",\"Strain, R. M.\",\"Yun, S.\"],\"key\":\"1907.05784\",\"id\":\"1907.05784\",\"bibbaseid\":\"jang-strain-yun-propagationofuniformupperboundsforthespatiallyhomogeneousrelativisticboltzmannequation-2021\",\"role\":\"author\",\"urls\":{\" arxiv\":\"https://arxiv.org/abs/1907.05784\"},\"keyword\":[\"relativistic Boltzmann equation\",\"Kinetic Theory\",\"relativistic Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":9,\"html\":\"\"},{\"bibtype\":\"phdthesis\",\"type\":\"phdthesis\",\"abstract\":\"The collisional Kinetic Equations we study are all of the form $$∂_t F + v· ∇_x F+V(t,x)· ∇_v F=Q(F,F).$$ Here $F=F(t,x,v)$ is a probabilistic density function (of time $t\\\\ge 0$, space $x∈Ω$ and velocity $v∈ℝ^3$) for a particle taken chosen randomly from a gas or plasma. $V(t,x)$ is a field term which usually represents Maxwell's theory of electricity and magnetism, sometimes this term is neglected. $Q(F,F)$ is the collision operator which models the interaction between colliding particles. We consider both the Boltzmann and Landau collision operators. We prove existence, uniqueness and regularity of close to equilibrium solutions to the relativistic Landau-Maxwell system in the first part of this thesis. Our main tool is an energy method. In the second part, we prove arbitrarily high polynomial time decay rates to equilibrium for four kinetic equations. These are cutoff soft potential Boltzmann and Landau equations, but also the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system. The main technique used here is interpolation. In the third part, we prove exponential decay for the cutoff soft potential Boltzmann and Landau equations. The main point here is to show that exponential decay of the initial data is propagated by a solution. In the fourth and final part of this thesis, we write down a few important calculations in the relativistic Boltzmann theory which are scattered around the literature. We also calculate a few Lorentz transformations which maybe useful in relativistic transport theory. We use these calculations to comment about extending the results in this thesis to the relativistic Boltzmann equation. \",\"annote\":\"(https://search.proquest.com)\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Strain\"],\"firstnames\":[\"Robert\",\"M.\"],\"suffixes\":[]}],\"date-added\":\"2019-07-13 17:21:46 -0400\",\"date-modified\":\"2019-08-08 11:28:11 -0600\",\"isbn\":\"978-0542-12875-2\",\"journal\":\"ProQuest Dissertations and Theses\",\"keywords\":\"Boltzmann equation, Landau equation, Balescu-Lenard equation, Kinetic Theory, Vlasov-Maxwell systems, relativistic Kinetic Theory\",\"language\":\"English\",\"mrclass\":\"Thesis\",\"mrnumber\":\"2707256\",\"note\":\"(ProQuest Document ID 305028444)\",\"pages\":\"1–200\",\"publisher\":\"ProQuest LLC, Ann Arbor, MI\",\"school\":\"Brown University\",\"title\":\"Some applications of an energy method in collisional Kinetic theory\",\"url_proquest\":\"http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3174679\",\"urlpdf\":\"https://strain.math.upenn.edu/preprints/2005PHDthesisS.pdf\",\"year\":\"2005\",\"bdsk-url-1\":\"http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3174679\",\"bibtex\":\"@phdthesis{MR2707256,\\n\\tabstract = {The collisional Kinetic Equations we study are all of the form\\n$$\\\\partial_t F + v\\\\cdot \\\\nabla_x F+V(t,x)\\\\cdot \\\\nabla_v F=Q(F,F).$$\\nHere $F=F(t,x,v)$ is a probabilistic density function (of time $t\\\\ge 0$, space $x\\\\in\\\\Omega$ and velocity $v\\\\in\\\\mathbb{R}^3$) for a particle taken chosen randomly from a gas or plasma. $V(t,x)$ is a field term which usually represents Maxwell's theory of electricity and magnetism, sometimes this term is neglected.   $Q(F,F)$ is the collision operator which models the interaction between colliding particles.  We consider both the Boltzmann and Landau collision operators.  \\n\\n\\nWe prove existence, uniqueness and regularity of close to equilibrium solutions to the relativistic Landau-Maxwell system in the first part of this thesis.  Our main tool is an energy method.  \\n\\nIn the second part, we prove arbitrarily high polynomial time decay rates to equilibrium for four kinetic equations.  These are cutoff soft potential Boltzmann and Landau equations, but also the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system.  The main technique used here is interpolation.  \\n\\nIn the third part, we prove exponential decay for the cutoff soft potential Boltzmann and Landau equations.  The main point here is to show that exponential decay of the initial data is propagated by a solution.  \\n\\nIn the fourth and final part of this thesis, we write down a few important calculations in the relativistic Boltzmann theory which are scattered around the literature.  We also calculate a few Lorentz transformations which maybe useful in relativistic transport theory.  We use these calculations to comment about extending the results in this thesis to the relativistic Boltzmann equation.\\n},\\n\\tannote = {(https://search.proquest.com)},\\n\\tauthor = {Strain, Robert M.},\\n\\tdate-added = {2019-07-13 17:21:46 -0400},\\n\\tdate-modified = {2019-08-08 11:28:11 -0600},\\n\\tisbn = {978-0542-12875-2},\\n\\tjournal = {ProQuest Dissertations and Theses},\\n\\tkeywords = {Boltzmann equation, Landau equation, Balescu-Lenard equation, Kinetic Theory, Vlasov-Maxwell systems, relativistic Kinetic Theory},\\n\\tlanguage = {English},\\n\\tmrclass = {Thesis},\\n\\tmrnumber = {2707256},\\n\\tnote = {(ProQuest Document ID 305028444)},\\n\\tpages = {1--200},\\n\\tpublisher = {ProQuest LLC, Ann Arbor, MI},\\n\\tschool = {Brown University},\\n\\ttitle = {Some applications of an energy method in collisional {K}inetic theory},\\n\\turl_proquest = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3174679},\\n\\turlpdf = {https://strain.math.upenn.edu/preprints/2005PHDthesisS.pdf},\\n\\tyear = {2005},\\n\\tbdsk-url-1 = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3174679}}\\n\",\"author_short\":[\"Strain, R. M.\"],\"key\":\"MR2707256\",\"id\":\"MR2707256\",\"bibbaseid\":\"strain-someapplicationsofanenergymethodincollisionalkinetictheory-2005\",\"role\":\"author\",\"urls\":{\" proquest\":\"http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3174679\",\"Pdf\":\"https://strain.math.upenn.edu/preprints/2005PHDthesisS.pdf\"},\"keyword\":[\"Boltzmann equation\",\"Landau equation\",\"Balescu-Lenard equation\",\"Kinetic Theory\",\"Vlasov-Maxwell systems\",\"relativistic Kinetic Theory\"],\"metadata\":{\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\"downloads\":12,\"html\":\"\"}],\n      metadata: {\"owner\":\"strain, r\",\"authorlinks\":{\"strain, r\":\"https://strain.math.upenn.edu/pubs.html\"}},\n      ownerID: \"qtada8FpZ25ps85An\"\n    };\n  </script>\n\n  <script type=\"text/javascript\">\n  var site$;\n  if (typeof $ != 'undefined') {\n    console.log('existing jquery: storing and then restoring');\n    site$ = $;\n    $ = null;\n  }\n  </script>\n\n  <script src=\"https://ajax.googleapis.com/ajax/libs/jquery/2.2.4/jquery.min.js\">\n  </script>\n  <script type=\"text/javascript\">\n  if (site$) {\n    console.log('existing jquery: avoiding conflict and restoring');\n    bb$ = $.noConflict(true);\n    $ = site$;\n  } else {\n    bb$ = $;\n  }\n  </script>\n\n  <script src=\"//bibbase.org/js/bibbase.min.js\"\n          type=\"text/javascript\">\n  </script>\n\n  <script type=\"text/javascript\"\n          src=\"https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML\">\n  </script>\n  <script type=\"text/x-mathjax-config\">\n    MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\\\(','\\\\)']]},\n                        skipTags: [\"body\"],\n                        processClass: \"bibbase_paper\"});\n  </script>\n\n  <style>\n  @media print {\n    .dontprint {\n        display: none !important;\n    }\n\n  .bibbase_group {\n    background-color: #E0E0E0;\n    font-style: italic;\n    font-size: larger;\n    text-shadow: 0px 0px 3px #FFFFFF;\n    border-radius: 4px 4px 4px 4px;\n    -moz-border-radius: 4px 4px 4px 4px;\n    -webkit-border-radius: 4px;\n    padding: 5px 40px 5px;\n    margin-top: 10px;\n    margin-bottom: 5px;\n    cursor: pointer;\n  }\n\n  .bibbase_paper {\n    margin-bottom: 5px;\n  }\n\n  }\n  </style>\n\n  \n  <div style=\"float: right; margin-right: 10px; margin-top: 10px; text-align: right; font-size: 12px\">\n    generated by\n    <a href=\"//bibbase.org\"\n       onclick=\"trackOutboundLink('bibbase', 'logo clicked', window.location.href, '//bibbase.org'); return false;\">\n      <img src=\"//bibbase.org/img/logo.svg\"\n           style=\"vertical-align: middle; margin-left: 3px; height: 16px;\"/\n           alt=\"bibbase.org\"\n           title=\"BibBase – The easiest way to maintain your publications page.\"\n        ></a>\n    \n  </div>\n  \n\n  <div id=\"bibbase_header\" class=\"dontprint\" style=\"\">\n\n    <ul class=\"nav nav-pills\" style=\"margin: 0px;\">\n\n      <li class=\"dropdown\" id=\"groupby_dropdown\">\n      <a class=\"dropdown-toggle\"\n         data-toggle=\"dropdown\"\n         href=\"#\">\n          Group by\n          <b class=\"caret\"></b>\n      </a>\n      <ul class=\"dropdown-menu\">\n        <li class=\"groupby year\"><a onclick=\"groupby('year')\">\n            Year</a>\n        </li>\n        <li class=\"groupby author\"><a onclick=\"groupby('author_short')\">\n            Author</a>\n        </li>\n        <li class=\"groupby type\"><a onclick=\"groupby('type')\">\n            Type</a>\n        </li>\n        <li class=\"groupby keyword\"><a onclick=\"groupby('keyword')\">\n            Keyword</a>\n        </li>\n        <li class=\"groupby downloads\"><a onclick=\"groupby('downloads')\">\n            Downloads</a>\n        </li>\n      </ul>\n      </li>\n\n\n      <li class=\"dropdown\" id=\"menu_dropdown\">\n      <a class=\"dropdown-toggle\"\n         data-toggle=\"dropdown\"\n         href=\"#\" alt=\"controls\">\n          <i class=\"fa fa-reorder\" alt=\"controls\"></i>\n          <b class=\"caret\"></b>\n      </a>\n      <ul class=\"dropdown-menu\">\n        <li>\n          <a onclick=\"toggleAll()\">\n            <i class=\"fa fa-chevron-down fa-fw\"></i> Expand/Collapse All\n          </a>\n        </li>\n        <li>\n          <a download=\"RMS3web.bib\" href=\"data:text/bibtex;base64,@article{161005271,
	abstract = {We prove time decay of solutions to the Muskat equation in 2D and in 3D. We consider the norm $\|f\|_{s}(t)$.  In this paper, for the 3D Muskat problem, given initial data $f_{0}\in H^{l}(\mathbb{R}^{2})$ for some $l\geq 3$ such that $\|f_{0}\|_{1} < k_{0}$ for a constant $k_{0} \approx 1/5$, we prove uniform in time bounds of $\|f\|_{s}(t)$ for $-d < s < l-1$ and assuming $\|f_{0}\|_{\nu} < \infty$ we prove time decay estimates of the form $\|f\|_{s}(t) \lesssim (1+t)^{-s+\nu}$ for $0 \leq s \leq l-1$ and $-d \leq \nu < s$.  These large time decay rates are the same as the optimal rate for the linear Muskat equation.  We also prove analogous results in 2D.},
	author = {Patel, Neel and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1080/03605302.2017.1321661},
	eprint = {1610.05271},
	fjournal = {Communications in Partial Differential Equations},
	issn = {0360-5302},
	journal = {Comm. Partial Differential Equations},
	keywords = {Fluid mechanics, Free boundary problems, Muskat problem},
	mrclass = {76D03 (35B40 35Q35 76S05)},
	mrnumber = {3683311},
	mrreviewer = {Youcef Amirat},
	number = {6},
	pages = {977--999},
	title = {Large time decay estimates for the {M}uskat equation},
	url_arxiv = {https://arxiv.org/abs/1610.05271},
	volume = {42},
	year = {2017},
	zblnumber = {1378.35245},
	bdsk-url-1 = {https://doi.org/10.1080/03605302.2017.1321661}}



@article{GancedoGarciaJuarezPatelStrain24,
	abstract = { In this paper we consider gravity-capillarity Muskat bubbles in 2D. We obtain
a new approach to improve our result in \cite{GancedoGarciaJuarezPatelStrain23}. Due to a new bubble-adapted
formulation, the improvement is two fold.  We significantly condense the proof
and we now obtain the global well-posedness result for Muskat bubbles in
critical regularity.},
	author = {Gancedo, Francisco and Garc\'{\i}a-Ju\'{a}rez, Eduardo and Patel, Neel and Strain, Robert},
	eprint = {2312.14323},
	keywords = {Fluid mechanics, Free boundary problems, critical regularity, Muskat problem},
	month = {Dec},
	note = {preprint},
	pages = {26 pages},
	title = {{On Nonlinear Stability of Muskat Bubbles}},
	url_arxiv = {https://arxiv.org/abs/2312.14323},
	year = {2023}}



@article{GancedoGarciaJuarezPatelStrain23,
	abstract = {In this paper, we study the dynamics of fluids in porous media governed by Darcy's law: the Muskat problem. We consider the setting of two immiscible fluids of different densities and viscosities under the influence of gravity in which one fluid is completely surrounded by the other. This setting is gravity unstable because along a portion of the interface, the denser fluid must be above the other. Surprisingly, even without capillarity, the circle-shaped bubble is a steady state solution moving with vertical constant velocity determined by the density jump between the fluids. Taking advantage of our discovery of this steady state, we are able to prove global in time existence and uniqueness of dynamic bubbles of nearly circular shapes under the influence of surface tension. We prove this global existence result for low regularity initial data. Moreover, we prove that these solutions are instantly analytic and decay exponentially fast in time to the circle.},
	author = {Gancedo, Francisco and Garc\'{\i}a-Ju\'{a}rez, Eduardo and Patel, Neel and Strain, Robert},
	doi = {10.1090/memo/1455},
	eprint = {1902.02318},
	fjournal = {Memoirs of the American Mathematical Society},
	issn = {0065-9266,1947-6221},
	journal = {Mem. Amer. Math. Soc.},
	keywords = {Fluid mechanics, Free boundary problems, Muskat problem},
	mrclass = {99-06},
	mrnumber = {4679708},
	number = {1455},
	pages = {87 pages},
	title = {Global {R}egularity for {G}ravity {U}nstable {M}uskat {B}ubbles},
	url = {https://doi.org/10.1090/memo/1455},
	url_arxiv = {https://arxiv.org/abs/1902.02318},
	volume = {292},
	year = {2023},
	bdsk-url-1 = {https://doi.org/10.1090/memo/1455}}



@article{2112.00692,
	abstract = {We study the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid.  This is known as the Stokes immersed boundary problem and also as the Peskin problem.   We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem.  In this case we prove local in time well-posedness for arbitrary initial data in the scaling critical Besov space $\dot{B}^{3/2}_{2,1}(\mathbb{T} ; \mathbb{R}^2)$.  We additionally prove the optimal higher order smoothing effects for the solution.  To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancellation structure.},
	author = {Cameron, Stephen and Strain, Robert M.},
	blurbreport = {This paper studies the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time well-posedness for arbitrary initial data in the scaling critical Besov space $\dot{B}^{3/2}_{2,1}(\mathbb{T} ; \mathbb{R}^2)$. We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancellation structure.},
	doi = {10.1002/cpa.22139},
	eprint = {2112.00692},
	fjournal = {Communications on Pure and Applied Mathematics},
	issn = {0010-3640,1097-0312},
	journal = {Comm. Pure Appl. Math.},
	keywords = {Fluid mechanics, Free boundary problems, Peskin problem, Fluid-Structure interface, critical regularity, immersed boundary problem},
	mrclass = {35Q35 (35C15 35R11 35R35 76D07)},
	mrnumber = {4673875},
	number = {2},
	pages = {901--989},
	title = {Critical local well-posedness for the fully nonlinear {P}eskin problem},
	url_arxiv = {https://arxiv.org/abs/2112.00692},
	volume = {77},
	year = {2024},
	bdsk-url-1 = {https://doi.org/10.1002/cpa.22139}}



@article{Peskin3D,
	abstract = {This paper introduces the {\em{3D Peskin problem}}: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow. We obtain the equations that model this free boundary problem and show that they admit a boundary integral reduction, providing an evolution equation for the elastic interface. We consider general nonlinear elastic laws, i.e., the fully nonlinear Peskin problem, and prove that the problem is well-posed in low-regularity H\"older spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time.},
	author = {Eduardo Garc{\'{i}}a-Ju{\'{a}}rez and Po-Chun Kuo and Yoichiro Mori and Robert M. Strain},
	eprint = {2301.12153},
	keywords = {Fluid mechanics, Free boundary problems, Peskin problem, Fluid-Structure interface, critical regularity, immersed boundary problem},
	month = {Jan},
	note = {preprint},
	pages = {93 pages},
	title = {{Well-Posedness of the 3D Peskin Problem}},
	url_arxiv = {https://arxiv.org/abs/2301.12153},
	year = {2023}}



@article{2103.15885,
	abstract = {This paper is concerned with the relativistic Boltzmann equation without angular cutoff.  We establish the global-in-time existence, uniqueness and asymptotic stability for solutions nearby the relativistic Maxwellian to the special relativistic Boltzmann equation without any angular cutoff.  We work in the case of a spatially periodic box. We assume the generic hard-interaction and  soft-interaction conditions on the collision kernel that were derived by Dudy\'nski and Ekiel-Je$\dot{\text{z}}$ewska (Comm. Math. Phys. \textbf{115}(4):607--629, 1988), and our assumptions include the case of Israel particles (J. Math. Phys. \textbf{4}:1163--1181, 1963).   In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator.   We further derive the relativistic analogue of the Carleman dual representation of Boltzmann collision operator.  This is the first global existence and stability result for relativistic Boltzmann equation without angular cutoff and this resolves the open question of perturbative global existence without the Grad's angular cut-off assumption.},
	author = {{J}in {W}oo {J}ang and {R}obert {M}. {S}train},
	date-modified = {2024-03-20 16:23:02 -0400},
	doi = {10.1007/s40818-022-00137-2},
	eprint = {2103.15885},
	fjournal = {Annals of PDE. Journal Dedicated to the Analysis of Problems from Physical Sciences},
	journal = {Ann. {PDE}},
	keywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},
	number = {2},
	pages = {167 pages},
	title = {{Asymptotic Stability of the Relativistic Boltzmann Equation without Angular Cut-off}},
	url_arxiv = {https://arxiv.org/abs/2103.15885},
	volume = {8},
	year = {2022},
	bdsk-url-1 = {https://doi.org/10.1007/s40818-022-00137-2}}



@article{2009.03360,
	abstract = {The Peskin problem models the dynamics of a closed elastic filament immersed
in an incompressible fluid. In this paper, we consider the case when the inner
and outer viscosities are possibly different. This viscosity contrast adds
further non-local effects to the system through the implicit non-local relation
between the net force and the free interface. We prove the first global
well-posedness result for the Peskin problem in this setting. The result
applies for medium size initial interfaces in critical spaces and shows instant
analytic smoothing. We carefully calculate the medium size constraint on the
initial data. These results are new even without viscosity contrast.},
	author = {Eduardo Garc{\'{i}}a-Ju{\'{a}}rez and Yoichiro Mori and Robert M. Strain},
	doi = {10.2140/apde.2023.16.785},
	eprint = {2009.03360},
	fjournal = {Analysis \& PDE},
	journal = {Anal. PDE},
	keywords = {Fluid mechanics, Free boundary problems, Peskin problem, Fluid-Structure interface, critical regularity, immersed boundary problem},
	number = {3},
	pages = {785--838},
	title = {{The Peskin Problem with Viscosity Contrast}},
	url_arxiv = {https://arxiv.org/abs/2009.03360},
	volume = {16},
	year = {2023},
	bdsk-url-1 = {https://doi.org/10.2140/apde.2023.16.785}}



@article{JangStrainWong2021,
	abstract = {Although the nuclear fusion process has received a great deal of attention in recent years, the amount of the mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the \textit{Vlasov-Maxwell} system in a two-dimensional annulus when a huge (\textit{but finite-in-time}) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. This work is the first step to a more generalized work on the three dimensional Tokamak structure. The highlight of this work is from the physical assumptions on the external magnetic potential well which remains finite \textit{within a finite time interval} and from that we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical-coordinate-forms of the \textit{Vlasov-Maxwell} system.},
	author = {{J}in {W}oo {J}ang and {R}obert {M}. {S}train and {T}ak {K}wong {W}ong},
	doi = {10.3934/krm.2021039},
	eprint = {2111.04583},
	fjournal = {Kinetic and Related Models},
	journal = {Kinet. Relat. Models},
	keywords = {Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},
	note = {published online November 25, 2021},
	pages = {36 pages},
	title = {{Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus}},
	url_arxiv = {https://arxiv.org/abs/2111.04583},
	year = {2021},
	bdsk-url-1 = {https://doi.org/10.3934/krm.2021039}}



@article{ChapmanJangStrain2020,
	abstract = {This article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in the center-of-momentum coordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular we mention the widely used cancellation lemma by Alexandre et al. (Arch. Ration. Mech. Anal. 152(4):327--355, 2000). In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentum $p$ to the post collisional momentum $p'$; specifically we calculate the determinant for $p\mapsto u = \theta p'+(1-\theta )p$ for $\theta \in [0,1]$. Afterwards we give an upper-bound for this determinant that has no singularity in both $p$ and $q$ variables. Next we give an example where we prove that the Jacobian goes to zero in a specific pointwise limit. We further explain the results of our numerical study which shows that the Jacobian determinant has a very large number of distinct points at which it is machine zero. This generalizes the work of Glassey-Strauss (Transport Theory Statist. Phys. 20(1):55--68, 1991) and Guo-Strain (Comm. Math. Phys. 310(3):649--673, 2012). These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates.},
	author = {Chapman, James and Jang, Jin Woo and Strain, Robert M.},
	doi = {10.1007/s00220-021-04101-2},
	eprint = {2006.02540},
	fjournal = {Communications in Mathematical Physics},
	journal = {Comm. Math. Phys.},
	keywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},
	pages = {1913--1943},
	published = {published online May 07, 2021},
	title = {{On the Determinant Problem for the Relativistic Boltzmann Equation}},
	url_arxiv = {https://arxiv.org/abs/2006.02540},
	volume = {384},
	year = {2021},
	bdsk-url-1 = {https://doi.org/10.1007/s00220-021-04101-2}}



@article{RIMSbessatsu2020,
	abstract = {This report succinctly summarizes results proved in the authors' recent work (2019) where the unique existence of solutions to the Boltzmann equation without angular cut-off and the Landau equation with Coulomb potential are studied in a perturbation framework. A major feature is the use of the Wiener space $A(\Omega)$, which can be expected to play a similar role to $L^\infty$. Compared to the $L^2$-based solution spaces that were employed for prior known results, this function space enables us to establish a new global existence theory. One further feature is that, not only an initial value problem, but also an initial boundary value problem whose boundary conditions can be regarded as physical boundaries in some simple situation, are considered for both equations. 
In addition to unique existence, large-time behavior of the solutions and propagation of spatial regularity are also proved. In the end of report, key ideas of the proof will be explained in a concise way.},
	author = {Renjun Duan and Shuangqian Liu and Shota Sakamoto and Robert M. Strain},
	fjournal = {RIMS K{{o}}ky{{u}}roku {B}essatsu},
	journal = {RIMS K{{o}}ky{{u}}roku {B}essatsu},
	keywords = {Boltzmann equation, Landau equation, Kinetic Theory, non-cutoff},
	note = {proceedings of symposium on ``Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations''},
	pages = {29--46},
	title = {{G}lobal solutions to the {B}oltzmann equation without angular cutoff and the {L}andau equation with {C}olomb potential},
	urlpdf = {http://hdl.handle.net/2433/260676},
	volume = {B82},
	year = {2020},
	bdsk-url-1 = {http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu.html}}



@article{1904.12086,
	abstract = {This paper proves the existence of small-amplitude global-in-time unique mild
solutions to both the Landau equation including the Coulomb potential and the
Boltzmann equation without angular cutoff. Since the well-known works (Guo,
2002) and (Gressman-Strain-2011, AMUXY-2012) on the construction of classical
solutions in smooth Sobolev spaces which in particular are regular in the
spatial variables, it still remains an open problem to obtain global solutions
in an $L^\infty_{x,v}$ framework, similar to that in (Guo-2010), for the
Boltzmann equation with cutoff in general bounded domains. One main difficulty
arises from the interaction between the transport operator and the
velocity-diffusion-type collision operator in the non-cutoff Boltzmann and
Landau equations; another major difficulty is the potential formation of
singularities for solutions to the boundary value problem. In the present work
we introduce a new function space with low regularity in the spatial variable
to treat the problem in cases when the spatial domain is either a torus, or a
finite channel with boundary. For the latter case, either the inflow boundary
condition or the specular reflection boundary condition is considered. An
important property of the function space is that the $L^\infty_T L^2_v$ norm,
in velocity and time, of the distribution function is in the Wiener algebra
$A(\Omega)$ in the spatial variables. Besides the construction of global
solutions in these function spaces, we additionally study the large-time
behavior of solutions for both hard and soft potentials, and we further justify
the property of propagation of regularity of solutions in the spatial
variables.},
	author = {Renjun Duan and Shuangqian Liu and Shota Sakamoto and Robert M. Strain},
	doi = {10.1002/cpa.21920},
	eprint = {1904.12086},
	fjournal = {Communications on Pure and Applied Mathematics},
	journal = {Comm. Pure Appl. Math.},
	keywords = {Boltzmann equation, Landau equation, Kinetic Theory, non-cutoff},
	number = {5},
	pages = {932--1020},
	title = {Global mild solutions of the {L}andau and non-cutoff {B}oltzmann equations},
	url_arxiv = {https://arxiv.org/abs/1904.12086},
	volume = {74},
	year = {2021},
	bdsk-url-1 = {https://arxiv.org/abs/1904.12086}}



@article{GGPS19,
	abstract = {The Muskat problem models the filtration of two incompressible immiscible
fluids of different characteristics in porous media. In this paper, we consider
both the 2D and 3D setting of two fluids of different constant densities and
different constant viscosities. In this situation, the related contour
equations are non-local, not only in the evolution system, but also in the
implicit relation between the amplitude of the vorticity and the free
interface. Among other extra difficulties, no maximum principles are available
for the amplitude and the slopes of the interface in $L^\infty$. We prove
global in time existence results for medium size initial stable data in
critical spaces. We also enhance previous methods by showing smoothing (instant
analyticity), improving the medium size constant in 3D, together with sharp
decay rates of analytic norms. The found technique is twofold, giving
ill-posedness in unstable situations for very low regular solutions.},
	author = {Francisco Gancedo and Eduardo Garc{\'{i}}a-Ju{\'{a}}rez and Neel Patel and Robert M. Strain},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1016/j.aim.2019.01.017},
	eprint = {1710.11604},
	fjournal = {Advances in Mathematics},
	issn = {0001-8708},
	journal = {Adv. Math.},
	keywords = {Fluid mechanics, Free boundary problems, Muskat problem},
	mrclass = {76S05},
	mrnumber = {3899970},
	pages = {552--597},
	title = {On the {M}uskat problem with viscosity jump: {G}lobal in time results},
	url_arxiv = {https://arxiv.org/abs/1710.11604},
	volume = {345},
	year = {2019},
	zblnumber = {07021548},
	bdsk-url-1 = {https://doi.org/10.1016/j.aim.2019.01.017}}



@article{LS2019,
	abstract = {In this paper we prove the global existence, uniqueness, optimal large time
decay rates, and uniform gain of analyticity for the exponential PDE
$h_t=\Delta e^{-\Delta h}$ in the whole space $\mathbb{R}^d_x$. We assume the
initial data is of medium size in the critical Wiener algebra $\Delta h \in
A(\mathbb{R}^d)$. This exponential PDE was derived in (Krug, Dobbs, and
Majaniemi in 1995) and more recently in (Marzuola and Weare 2013).},
	author = {Jian-Guo Liu and Robert M. Strain},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-08 14:21:35 -0600},
	doi = {10.4171/IFB/417},
	eprint = {1805.02246},
	fjournal = {Interfaces and Free Boundaries. Mathematical Analysis, Computation and Applications.},
	issn = {1463-9963},
	journal = {Interfaces Free Bound.},
	keywords = {Free boundary problems, Materials science},
	mrclass = {35K25 (35B40 35K55 35K65 74A50)},
	mrnumber = {3951578},
	number = {1},
	pages = {61--86},
	title = {Global stability for solutions to the exponential {PDE} describing epitaxial growth},
	url_arxiv = {https://arxiv.org/abs/1805.02246},
	volume = {21},
	year = {2019},
	zblnumber = {07084773},
	bdsk-url-1 = {https://doi.org/10.4171/IFB/417}}



@article{StrainTas2019,
	abstract = {In this paper we study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. Despite it's physical importance, this equation has not received a lot of mathematical attention we think due to the extreme complexity of the relativistic structure of the kernel of the collision operator. In this paper we first largely decompose the structure of the relativistic Landau collision operator. After that we prove the global Entropy dissipation estimate. Then we prove the propagation of any polynomial moment for a weak solution. Lastly we prove the existence of a true weak solution for a large class of initial data.},
	author = {Strain, Robert M. and Taskovi{\'{c}}, Maja},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1016/j.jfa.2019.04.007},
	eprint = {1806.08720},
	fjournal = {Journal of Functional Analysis},
	issn = {0022-1236},
	journal = {J. Funct. Anal.},
	keywords = {relativistic Landau equation, Landau equation, Kinetic Theory, relativistic Kinetic Theory},
	mrclass = {82D10 (35A01 35B45 35Q75)},
	mrnumber = {3959729},
	number = {4},
	pages = {1139--1201},
	title = {{E}ntropy dissipation estimates for the relativistic {L}andau equation, and applications},
	url_arxiv = {https://arxiv.org/abs/1806.08720},
	volume = {277},
	year = {2019},
	zblnumber = {07066836},
	bdsk-url-1 = {https://doi.org/10.1016/j.jfa.2019.04.007}}



@article{1903.05301,
	abstract = {We prove the uniqueness of weak solutions to the spatially homogeneous special relativistic Landau equation under the conditional assumption that the solution satisfies $(p^0)^7 F(t,p) \in L^1 ([0,T]; L^\infty)$.  The existence of standard weak solutions to the relativistic Landau equation has been shown recently in (Strain-Taskovi{\'{c}} 2019, 1806.08720).},
	author = {Robert M. Strain and Zhenfu Wang},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1090/qam/1545},
	eprint = {1903.05301},
	fjournal = {Quarterly of Applied Mathematics},
	journal = {Quart. Appl. Math.},
	keywords = {relativistic Landau equation, Landau equation, Kinetic Theory, relativistic Kinetic Theory},
	mrnumber = {4042221},
	pages = {107--145},
	title = {Uniqueness of Bounded Solutions for the Homogeneous Relativistic {L}andau Equation with {C}oulomb Interactions},
	url_arxiv = {https://arxiv.org/abs/1903.05301},
	volume = {78},
	year = {2019},
	zblnumber = {1427.82044},
	bdsk-url-1 = {https://arxiv.org/abs/1903.05301}}



@article{CCGRS16,
	abstract = {This paper considers the three dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an $L^2$ maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants.  Furthermore we refine the estimates from our prior paper to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in critical spaces, giving solutions with bounded slope and time integrable bounded curvature.},
	author = {Constantin, Peter and C\'{o}rdoba, Diego and Gancedo, Francisco and Rodr\'{i}guez-Piazza, Luis and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1353/ajm.2016.0044},
	eprint = {1310.0953},
	fjournal = {American Journal of Mathematics},
	issn = {0002-9327},
	journal = {Amer. J. Math.},
	keywords = {Fluid mechanics, Free boundary problems, Muskat problem},
	mrclass = {35Q35 (76S05)},
	mrnumber = {3595492},
	mrreviewer = {Maria Specovius-Neugebauer},
	number = {6},
	pages = {1455--1494},
	title = {On the {M}uskat problem: global in time results in 2{D} and 3{D}},
	url_arxiv = {https://arxiv.org/abs/1310.0953},
	volume = {138},
	year = {2016},
	zblnumber = {1369.35053},
	bdsk-url-1 = {https://doi.org/10.1353/ajm.2016.0044}}



@article{MR3437855,
	abstract = {We consider the relativistic Vlasov-Maxwell system with data of unrestricted size and without compact support in momentum space. In the two dimensional and the two-and-a-half dimensional cases, Glassey-Schaeffer proved (Commun Math Phys 185:257--284, 1997; Arch Ration Mech Anal 141:331--354, 1998; Arch Ration Mech Anal. 141:355--374, 1998) that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the 2D and the $2\frac 12$D cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the $L^\infty_x$ norms of the electromagnetic fields compared to (Commun Math Phys 185:257--284, 1997; Arch Ration Mech Anal 141:355--374, 1998).  In the three dimensional case, we apply Strichartz estimates and moment bounds to show that a regular solution can be extended as long as 
$\|(p^0)^{\theta} f \|_{L^q_xL^1_{p}}$ remains bounded for $\theta>{2/q}$ and $q \in (2, \infty]$. This improves previous results of Pallard (Indiana Univ Math J 54(5):1395--1409, 2005; Commun Math Sci 13(2):347--354, 2015).},
	author = {Luk, Jonathan and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-08 13:24:39 -0600},
	doi = {10.1007/s00205-015-0899-1},
	fjournal = {Archive for Rational Mechanics and Analysis},
	issn = {0003-9527},
	journal = {Arch. Ration. Mech. Anal.},
	keywords = {Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},
	mrclass = {35Q83 (35Q61 35Q75 35Q82 76Y05 82D10)},
	mrnumber = {3437855},
	mrreviewer = {\L ukasz P\l ociniczak},
	number = {1},
	pages = {445--552},
	title = {Strichartz estimates and moment bounds for the relativistic {V}lasov-{M}axwell system},
	url_arxiv1 = {https://arxiv.org/abs/1406.0168},
	url_arxiv2 = {https://arxiv.org/abs/1406.0169},
	volume = {219},
	year = {2016},
	zblnumber = {1337.35150},
	bdsk-url-1 = {https://doi.org/10.1007/s00205-015-0899-1}}



@article{MR3437599,
	abstract = {In this article we will introduce a new model to describe the leading order behavior of an ideal and axisymmetric fluid moving in a very narrow domain. After providing a formal derivation of the model, we will prove the well-posedness and provide a rigorous mathematical justification for the formal derivation under a new sign condition. Finally, a blowup result regarding this model will be discussed as well.},
	author = {Strain, Robert M. and Wong, Tak Kwong},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-08 13:17:31 -0600},
	doi = {10.1016/j.jde.2015.11.023},
	eprint = {1505.06281},
	fjournal = {Journal of Differential Equations},
	issn = {0022-0396},
	journal = {J. Differential Equations},
	keywords = {Fluid mechanics},
	mrclass = {35Q31 (35B30 35Q35 76B03)},
	mrnumber = {3437599},
	mrreviewer = {Franck Sueur},
	number = {5},
	pages = {4619--4656},
	title = {Axisymmetric flow of ideal fluid moving in a narrow domain: a study of the axisymmetric hydrostatic {E}uler equations},
	url_arxiv = {https://arxiv.org/abs/1505.06281},
	volume = {260},
	year = {2016},
	zblnumber = {1333.35170},
	bdsk-url-1 = {https://doi.org/10.1016/j.jde.2015.11.023}}



@article{MR3181769,
	abstract = {In this paper for either the sharp front Surface Quasi-Geostrophic equation or the Muskat problem we rule out the `'splash singularity'' blow-up scenario; in other words we prove that the contours evolving from either of these systems can not intersect at a single point while the free boundary remains smooth. Splash singularities have been shown to hold for the free boundary incompressible Euler equation in the form of the water waves contour evolution problem (Castro A, et al. (2012) Proc Natl Acad Sci USA 109:733-738). Our result confirms the numerical simulations in (C{\'o}rdoba D, et al. (2005) Proc Natl Acad Sci USA 102:5949-5952) where it is shown that the curvature blows up due to the contours collapsing at a point. Here we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse. Another conclusion that we provide is a better understanding of the work (C{\'o}rdoba D, Gancedo F (2010) Comm Math Phys 299:561-575) in which squirt singularities are ruled out; in this case a positive volume of fluid between the contours can not be ejected in finite time.},
	author = {Gancedo, Francisco and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1073/pnas.1320554111},
	eprint = {1309.4023},
	fjournal = {Proceedings of the National Academy of Sciences of the United States of America},
	issn = {0027-8424},
	journal = {Proc. Natl. Acad. Sci. USA},
	keywords = {Fluid mechanics, Free boundary problems, Muskat problem},
	mrclass = {76S05 (35B35 76D27 76Txx 86A05)},
	mrnumber = {3181769},
	mrreviewer = {Jos\'{e} Miguel Pacheco Castelao},
	number = {2},
	pages = {635--639},
	publisher = {Proceedings of the National Academy of Sciences},
	title = {Absence of splash singularities for surface quasi-geostrophic sharp fronts and the {M}uskat problem},
	url_arxiv = {https://arxiv.org/abs/1309.4023},
	volume = {111},
	year = {2014},
	zblnumber = {1355.76065},
	bdsk-url-1 = {https://www.pnas.org/content/111/2/635},
	bdsk-url-2 = {https://doi.org/10.1073/pnas.1320554111}}



@article{MR3248056,
	abstract = {The global existence of solutions to the relativistic Vlasov-Maxwell system given sufficiently regular finite energy initial data is a longstanding open problem.  The main result of Glassey-Strauss 1986 shows that a solution $(f, E, B)$ remains $C^1$ as long as the momentum support of $f$ remains bounded.  Alternate proofs were later given by Bouchut-Golse-Pallard 2003 and Klainerman-Staffilani 2002.  We show that only the boundedness of the momentum support of $f$ {\it after projecting to any two dimensional plane} is needed  for $(f, E, B)$ to remain $C^1$. },
	author = {Luk, Jonathan and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1007/s00220-014-2108-8},
	eprint = {1406.0165},
	fjournal = {Communications in Mathematical Physics},
	issn = {0010-3616},
	journal = {Comm. Math. Phys.},
	keywords = {Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},
	mrclass = {35Q75 (35Q83 81T20)},
	mrnumber = {3248056},
	mrreviewer = {Calvin Tadmon},
	number = {3},
	pages = {1005--1027},
	title = {A new continuation criterion for the relativistic {V}lasov-{M}axwell system},
	url_arxiv = {https://arxiv.org/abs/1406.0165},
	volume = {331},
	year = {2014},
	zblnumber = {1309.35174},
	bdsk-url-1 = {https://doi.org/10.1007/s00220-014-2108-8}}



@article{MR3213301,
	abstract = {We prove that $k$-th order derivatives of perturbative classical solutions to the hard and soft potential Boltzmann equation (without the angular cut-off assumption) in the whole space, ${\mathbb R}^{n}_x$ with ${n \ge 3}$, converge in large time to the global Maxwellian with the optimal decay rate of   $t^{-\frac{1}{2}(k+r+\frac{n}{2}-\frac{n}{r})}$
in the $L^r_x(L^2_{v})$-norm for any $2\leq r\leq \infty$.   These results hold for any $r \in (0, n/2]$ as long as initially $\| f_0\|_{\dot{B}^{-r,\infty}_2 L^2_{v}} < \infty$.   In the hard potential case, we prove faster decay results in the sense that if $\| \mathbf{P} f_0\|_{\dot{B}^{-r,\infty}_2 L^2_{v}} < \infty$ and 
$\| \{\mathbf{I} - \mathbf{P}\} f_0\|_{\dot{B}^{-r+1,\infty}_2 L^2_{v}} < \infty$ for $r \in (n/2, (n+2)/2]$  then the solution decays the global Maxwellian in $L^2_v(L^2_x)$ with the optimal large time decay rate of $t^{-\frac{1}{2} r}$.},
	author = {Sohinger, Vedran and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-15 15:47:31 -0400},
	doi = {10.1016/j.aim.2014.04.012},
	eprint = {1206.0027},
	fjournal = {Advances in Mathematics},
	issn = {0001-8708},
	journal = {Adv. Math.},
	keywords = {Boltzmann equation, non-cutoff, Kinetic Theory},
	mrclass = {35Q20 (35B40 35F25 76P05 82C40)},
	mrnumber = {3213301},
	mrreviewer = {Cecil Pompiliu Gr\"unfeld},
	msc2010 = {35Q20 35R11 76P05 82C40 35B65 26A33},
	pages = {274--332},
	publisher = {Elsevier (Academic Press), San Diego, CA},
	title = {The {B}oltzmann equation, {B}esov spaces, and optimal time decay rates in {$\mathbb{R}_x^n$}},
	url_arxiv = {https://arxiv.org/abs/1206.0027},
	volume = {261},
	year = {2014},
	zblnumber = {1293.35195},
	bdsk-url-1 = {https://doi.org/10.1016/j.aim.2014.04.012}}



@article{MR3166961,
	abstract = {The spatially homogeneous Boltzmann equation has been studied extensively in the Newtonian case, but not much is known for the special relativistic case. In this paper, we address several issues for the spatially homogeneous Boltzmann equation for relativistic particles. We first derive the relativistic version of the Povzner inequality. Using this, we study the Cauchy problem and investigate how the polynomial and exponential moments in $L^1$ are propagated. Several key differences between the relativistic and the Newtonian cases are confronted and discussed.},
	author = {Strain, Robert M. and Yun, Seok-Bae},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-08 11:57:02 -0600},
	doi = {10.1137/130923531},
	fjournal = {SIAM Journal on Mathematical Analysis},
	issn = {0036-1410},
	journal = {SIAM J. Math. Anal.},
	keywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},
	mrclass = {35Q20 (76P05 76Y05)},
	mrnumber = {3166961},
	mrreviewer = {Piotr Biler},
	number = {1},
	pages = {917--938},
	title = {Spatially homogeneous {B}oltzmann equation for relativistic particles},
	urlpdf = {https://strain.math.upenn.edu/preprints/92353.pdf},
	volume = {46},
	year = {2014},
	zblnumber = {1321.35127},
	bdsk-url-1 = {https://doi.org/10.1137/130923531}}



@article{CCGS13,
	abstract = {The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities.
In this work we prove three results.  First we prove an $L^2(\mathbb{R})$ maximum principle, in the form of a new ``log''  conservation law which is satisfied by the equation for the interface.  Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy 
$\|f_0\|_{L^\infty}<\infty$ and $\|\partial_x f_0\|_{L^\infty}<1$. We take advantage of the fact that the bound $\|\partial_x f_0\|_{L^\infty}<1$ is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.
Lastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance $\| f\|_1 \le 1/5$. Previous results of this sort used a small constant $\epsilon \ll1$ which was not explicit.},
	author = {Constantin, Peter and C\'{o}rdoba, Diego and Gancedo, Francisco and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.4171/JEMS/360},
	eprint = {1007.3744},
	fjournal = {Journal of the European Mathematical Society (JEMS)},
	issn = {1435-9855},
	journal = {J. Eur. Math. Soc. (JEMS)},
	keywords = {Fluid mechanics, Free boundary problems, Muskat problem},
	mrclass = {35Q35 (35B50 35D35 76S05)},
	mrnumber = {2998834},
	mrreviewer = {Weiran Sun},
	number = {1},
	pages = {201--227},
	title = {On the global existence for the {M}uskat problem},
	url_arxiv = {https://arxiv.org/abs/1007.3744},
	url_pdf = {https://strain.math.upenn.edu/preprints/ccgsMuskat.pdf},
	volume = {15},
	year = {2013},
	zblnumber = {1258.35002},
	bdsk-url-1 = {https://doi.org/10.4171/JEMS/360}}



@article{MR2982812,
	abstract = {We present the uniform $L^1$-stability estimate for the relativistic Boltzmann equation near vacuum. For this, we explicitly construct a relativistic counterpart of the nonlinear functional which is a linear combination of  $L^1$-distance and a collision potential. This functional measures the  $L^1$-distance between two continuous mild solutions. When the initial data is sufficiently small and decays exponentially fast, we show that the functional satisfies the uniform stability estimate leading to the uniform  $L^1$-stability estimate with respect to initial data.},
	author = {Ha, Seung-Yeal and Jeong, Eunhee and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-08 14:17:12 -0600},
	doi = {10.3934/cpaa.2013.12.1141},
	fjournal = {Communications on Pure and Applied Analysis},
	issn = {1534-0392},
	journal = {Commun. Pure Appl. Anal.},
	keywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},
	mrclass = {35Q20 (35B35 35Q75 82C40)},
	mrnumber = {2982812},
	mrreviewer = {Calvin Tadmon},
	number = {2},
	pages = {1141--1161},
	title = {Uniform {$L^1$}-stability of the relativistic {B}oltzmann equation near vacuum},
	urlpdf = {https://strain.math.upenn.edu/preprints/cpaa0494.pdf},
	volume = {12},
	year = {2013},
	zblnumber = {1267.35162},
	bdsk-url-1 = {https://doi.org/10.3934/cpaa.2013.12.1141}}



@article{MR3101794,
	abstract = {For the Landau-Poisson system with Coulomb interaction in $\mathbb{R}^3_x$, we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close.},
	author = {Strain, Robert M. and Zhu, Keya},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1007/s00205-013-0658-0},
	eprint = {1202.2471},
	fjournal = {Archive for Rational Mechanics and Analysis},
	issn = {0003-9527},
	journal = {Arch. Ration. Mech. Anal.},
	keywords = {Landau equation, Vlasov-Maxwell systems, Kinetic Theory},
	mrclass = {35Q83 (35A01 35A02 35B40)},
	mrnumber = {3101794},
	mrreviewer = {Jonathan Ben-Artzi},
	number = {2},
	pages = {615--671},
	title = {The {V}lasov-{P}oisson-{L}andau system in {$\mathbb{R}^3_x$}},
	url_arxiv = {https://arxiv.org/abs/1202.2471},
	url_pdf = {https://strain.math.upenn.edu/preprints/2012szLandauP.pdf},
	volume = {210},
	year = {2013},
	zblnumber = {1294.35168},
	bdsk-url-1 = {https://doi.org/10.1007/s00205-013-0658-0}}



@article{MR3129108,
	abstract = {Consider steady-state weak solutions to the incompressible
Navier-Stokes equations in six spatial dimensions. We prove that the 2D
Hausdorff measure of the set of singular points is equal to zero.  This problem was mentioned in 1988 by Struwe, during his study of the five dimensional case.},
	author = {Dong, Hongjie and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1512/iumj.2012.61.4765},
	eprint = {1101.5580},
	fjournal = {Indiana University Mathematics Journal},
	issn = {0022-2518},
	journal = {Indiana Univ. Math. J.},
	keywords = {Navier-Stokes equations, Fluid mechanics},
	mrclass = {35Q30 (35B65 76D03 76D05)},
	mrnumber = {3129108},
	mrreviewer = {Kazuo Yamazaki},
	number = {6},
	pages = {2211--2229},
	title = {On partial regularity of steady-state solutions to the 6{D} {N}avier-{S}tokes equations},
	url_arxiv = {https://arxiv.org/abs/1101.5580},
	url_pdf = {https://strain.math.upenn.edu/preprints/ds6DssNSE.pdf},
	volume = {61},
	year = {2012},
	zblnumber = {1286.35193},
	bdsk-url-1 = {https://doi.org/10.1512/iumj.2012.61.4765}}



@incollection{MR3050180,
	abstract = {In this paper, we present a new approach of studying the full dissipative property of the Vlasov-Poisson-Boltzmann system over the whole space. The key part of this approach is to design the interactive functional to capture the dissipation of the system along the degenerate components. The developed approach is generally applicable to other relevant models arising from plasma physics both at the kinetic and fluid levels.},
	address = {Singapore},
	annote = {Proceedings of the HYP2010 conference in Beijing},
	author = {Renjun Duan and Robert M. Strain},
	booktitle = {Hyperbolic problems---theory, numerics and applications. {V}olume 2},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-08 14:12:11 -0600},
	doi = {10/c74v},
	editor = {Tatsien Li and Song Jiang},
	keywords = {Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory},
	mrnumber = {3050180},
	pages = {398--405},
	publisher = {World Scientific Publishing and Higher Education Press},
	series = {Ser. Contemp. Appl. Math. CAM},
	title = {On the {F}ull {D}issipative {P}roperty of the {V}lasov-{P}oisson-{B}oltzmann {S}ystem},
	urlpdf = {https://strain.math.upenn.edu/preprints/2012DuanStrainHype.pdf},
	volume = {17},
	year = {2012},
	zblnumber = {1293.35339},
	bdsk-url-1 = {https://doi.org/10.1142/9789814417099_0037}}



@article{MR2914961,
	abstract = {In this article we prove a collection of new non-linear and non-local integral inequalities.  We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions.  Specifically, we include all $\alpha\in (0, \frac{74}{75})$.},
	author = {Gressman, Philip T. and Krieger, Joachim and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1016/j.aim.2012.02.017},
	eprint = {1202.4088},
	fjournal = {Advances in Mathematics},
	issn = {0001-8708},
	journal = {Adv. Math.},
	keywords = {Landau equation, Kinetic Theory},
	mrclass = {35K55 (35A01 35B65 35R11)},
	mrnumber = {2914961},
	mrreviewer = {M. Pilar Velasco},
	number = {2},
	pages = {642--648},
	title = {A non-local inequality and global existence},
	url_arxiv = {https://arxiv.org/abs/1202.4088},
	url_pdf = {https://strain.math.upenn.edu/preprints/2012gksINEQ.pdf},
	volume = {230},
	year = {2012},
	zblnumber = {1248.35005},
	bdsk-url-1 = {https://doi.org/10.1016/j.aim.2012.02.017}}



@article{MR2891870,
	abstract = {In the study of solutions to the relativistic Boltzmann equation, their
regularity with respect to the momentum variables has been an outstanding
question, even local in time, due to the initially unexpected growth in the
post-collisional momentum variables which was discovered in 1991 by Glassey
\& Strauss. We establish momentum regularity within energy
spaces via a new splitting technique and interplay between the
Glassey-Strauss frame and the center of mass frame of the relativistic collision
operator. In a periodic box, these new momentum regularity estimates lead to
a proof of global existence of classical solutions to the two-species
relativistic Vlasov-Maxwell-Boltzmann system for charged particles near
Maxwellian with hard ball interaction.},
	author = {Guo, Yan and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1007/s00220-012-1417-z},
	eprint = {1012.1158},
	fjournal = {Communications in Mathematical Physics},
	issn = {0010-3616},
	journal = {Comm. Math. Phys.},
	keywords = {Boltzmann equation, relativistic Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory, relativistic Kinetic Theory},
	mrclass = {82D10 (35B35 35B65 35Q83 82C24 82C40)},
	mrnumber = {2891870},
	mrreviewer = {Stephen Wollman},
	number = {3},
	pages = {649--673},
	title = {Momentum regularity and stability of the relativistic {V}lasov-{M}axwell-{B}oltzmann system},
	url_arxiv = {https://arxiv.org/abs/1012.1158},
	url_pdf = {https://strain.math.upenn.edu/preprints/gsRVMB.pdf},
	volume = {310},
	year = {2012},
	zblnumber = {1245.35130},
	bdsk-url-1 = {https://doi.org/10.1007/s00220-012-1417-z}}



@article{MR2901061,
	abstract = {In this paper we prove the global in time well-posedness of the following non-local diffusion equation with $\alpha \in (0,2/3)$:
$\partial_t u = ((-\Delta)^{-1}u)\Delta u + \alpha u^2$.  The initial condition $u_0$ is positive, radial, and non-increasing with
$u_0\in L^1\cap L^{2+\delta}({\mathbb R}^3)$ for some small $\delta >0$.  There is no size restriction on $u_0$.
This model problem appears of interest due to its structural similarity with Landau's equation from plasma physics, and moreover its radically different behavior from the semi-linear Heat equation: $u_t = \Delta u + \alpha u^2$. },
	author = {Krieger, Joachim and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1080/03605302.2011.643437},
	eprint = {1012.2890},
	fjournal = {Communications in Partial Differential Equations},
	issn = {0360-5302},
	journal = {Comm. Partial Differential Equations},
	keywords = {Landau equation, Kinetic Theory},
	mrclass = {35K55 (35B30 35B65 35D35 35Q20)},
	mrnumber = {2901061},
	mrreviewer = {Jana Kopfova},
	number = {4},
	pages = {647--689},
	title = {Global solutions to a non-local diffusion equation with quadratic non-linearity},
	url_arxiv = {https://arxiv.org/abs/1012.2890},
	url_pdf = {https://strain.math.upenn.edu/preprints/03605302.2011.pdf},
	volume = {37},
	year = {2012},
	zblnumber = {1247.35087},
	bdsk-url-1 = {https://doi.org/10.1080/03605302.2011.643437}}



@article{MR2911100,
	abstract = {For the relativistic Boltzmann equation in $R^3_x$, this work proves the global existence, uniqueness, positivity, and optimal time convergence rates to the relativistic Maxwellian for solutions which start out sufficiently close under the general physical soft potential assumption proposed in 1988.},
	author = {Strain, Robert M. and Zhu, Keya},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-15 15:47:40 -0400},
	doi = {10.3934/krm.2012.5.383},
	eprint = {1106.1579},
	fjournal = {Kinetic and Related Models},
	issn = {1937-5093},
	journal = {Kinet. Relat. Models},
	keywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory},
	mrclass = {82C05 (76P05 76Y05)},
	mrnumber = {2911100},
	mrreviewer = {Mark Thompson},
	number = {2},
	pages = {383--415},
	title = {Large-time decay of the soft potential relativistic {B}oltzmann equation in {$\mathbb R^3_x$}},
	url_arxiv = {https://arxiv.org/abs/1106.1579},
	url_pdf = {https://strain.math.upenn.edu/preprints/szRBwhole.pdf},
	volume = {5},
	year = {2012},
	zblnumber = {1247.76071},
	bdsk-url-1 = {https://doi.org/10.3934/krm.2012.5.383}}



@article{MR2972454,
	abstract = {In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space ${\mathbb R}^{n}_x$ with $n \ge 3$.    We use the existence theory of global in time nearby Maxwellian solutions from previous work.  
It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption.  For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of  $O(t^{-\frac{n}{2}+\frac{n}{2r}})$ in the $L^2_v(L^r_x)$-norm for any $2\leq r\leq \infty$. },
	author = {Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.3934/krm.2012.5.583},
	eprint = {1011.5561},
	fjournal = {Kinetic and Related Models},
	issn = {1937-5093},
	journal = {Kinet. Relat. Models},
	keywords = {Boltzmann equation, Kinetic Theory, non-cutoff},
	mrclass = {76P05 (26A33 35F20 82Cxx)},
	mrnumber = {2972454},
	number = {3},
	pages = {583--613},
	title = {Optimal time decay of the non cut-off {B}oltzmann equation in the whole space},
	url_arxiv = {https://arxiv.org/abs/1011.5561},
	url_pdf = {https://strain.math.upenn.edu/preprints/SdecaySOFT.pdf},
	volume = {5},
	year = {2012},
	zblnumber = {1383.76414},
	bdsk-url-1 = {https://doi.org/10.3934/krm.2012.5.583}}



@article{MR2832167,
	abstract = {In this paper we study the large-time behavior of classical
solutions to the two-species Vlasov-Maxwell-Boltzmann system in the
whole space $R^3_x$.    The existence of  global in time nearby
Maxwellian solutions is known from the work of Strain in 2006.  However
the asymptotic behavior of these solutions  has been a challenging
open problem.  Building on our previous work on time
decay for the simpler Vlasov-Poisson-Boltzmann system, we prove that
these solutions converge to the global Maxwellian with the optimal
decay rate of  $O(t^{-\frac{3}{2}+\frac{3}{2r}})$ in
$L^2_\xi(L^r_x)$-norm for any $2\leq r\leq \infty$ if initial
perturbation is smooth enough and decays in space-velocity fast
enough at infinity. Moreover, some explicit rates for the
electromagnetic field tending to zero are also provided.},
	author = {Duan, Renjun and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1002/cpa.20381},
	eprint = {1006.3605},
	fjournal = {Communications on Pure and Applied Mathematics},
	issn = {0010-3640},
	journal = {Comm. Pure Appl. Math.},
	keywords = {Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory},
	mrclass = {82C40 (35B40 35Q20 35Q83 76W05)},
	mrnumber = {2832167},
	mrreviewer = {Stephen Wollman},
	number = {11},
	pages = {1497--1546},
	title = {Optimal large-time behavior of the {V}lasov-{M}axwell-{B}oltzmann system in the whole space},
	url_arxiv = {https://arxiv.org/abs/1006.3605},
	url_pdf = {https://strain.math.upenn.edu/preprints/dsVMB.pdf},
	volume = {64},
	year = {2011},
	zblnumber = {1244.35010},
	bdsk-url-1 = {https://doi.org/10.1002/cpa.20381}}



@article{MR2754344,
	abstract = {The Vlasov-Poisson-Boltzmann System governs the time evolution of
the distribution function for the dilute charged particles in the
presence of a self-consistent electric potential force through the
Poisson equation. In this paper, we are concerned with the rate of
convergence of solutions to equilibrium for this system over $R^3_x$.
It is shown that the electric field which is indeed responsible for
the lowest-order part in the energy space reduces the speed of
convergence and hence the dispersion of this system over the full
space is slower than that of the Boltzmann equation without forces,
where the exact difference between both power indices in the
algebraic rates of convergence is $1/4$. For the proof, in the
linearized case with a given non-homogeneous source,  Fourier
analysis is employed to obtain time-decay properties of the solution
operator. In the nonlinear case, the combination of the linearized
results and the nonlinear energy estimates with the help of the
proper Lyapunov-type inequalities leads to the optimal time-decay
rate of perturbed solutions under some conditions on initial data.},
	author = {Duan, Renjun and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1007/s00205-010-0318-6},
	eprint = {0912.1742},
	fjournal = {Archive for Rational Mechanics and Analysis},
	issn = {0003-9527},
	journal = {Arch. Ration. Mech. Anal.},
	keywords = {Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory},
	mrclass = {35Q83 (35B40 35Q20 76P05 76X05 82C40 82D10)},
	mrnumber = {2754344},
	number = {1},
	pages = {291--328},
	title = {Optimal time decay of the {V}lasov-{P}oisson-{B}oltzmann system in {$\mathbb{R}^3$}},
	url_arxiv = {https://arxiv.org/abs/0912.1742},
	url_pdf = {https://strain.math.upenn.edu/preprints/dsVPB.pdf},
	volume = {199},
	year = {2011},
	zblnumber = {1232.35169},
	bdsk-url-1 = {https://doi.org/10.1007/s00205-010-0318-6}}



@article{MR2784329,
	abstract = {This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials, $r^{-(p-1)}$ with $p>2$,
for initial perturbations of the Maxwellian equilibrium states.  
We  more generally cover collision kernels with parameters $s\in (0,1)$ and $\gamma$ satisfying 
$\gamma  > -n$ in arbitrary dimensions $\mathbb{T}^n \times \mathbb{R}^n$ with $n\ge 2$.  
Moreover, we prove rapid convergence as predicted by the celebrated Boltzmann $H$-theorem.
When $\gamma \ge -2s$, we have  exponential time decay to the Maxwellian equilibrium states.  When $\gamma <-2s$, our solutions decay polynomially fast in time with any rate.  
These results are completely constructive.  Additionally, we prove  sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $\gamma  \ge -2s$, as conjectured in  Mouhot-Strain.  It will be observed that this fundamental equation, derived by both Boltzmann  and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world.  Our methods provide a new  understanding of the grazing collisions in the Boltzmann theory. },
	author = {Gressman, Philip T. and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1090/S0894-0347-2011-00697-8},
	eprint = {1011.5441},
	fjournal = {Journal of the American Mathematical Society},
	issn = {0894-0347},
	journal = {J. Amer. Math. Soc.},
	keywords = {Boltzmann equation, Kinetic Theory, non-cutoff},
	mrclass = {82C40 (35H20 35Q20 35R11 76P05)},
	mrnumber = {2784329},
	mrreviewer = {Laurent Desvillettes},
	number = {3},
	pages = {771--847},
	title = {Global classical solutions of the {B}oltzmann equation without angular cut-off},
	url_arxiv = {https://arxiv.org/abs/1011.5441},
	url_pdf = {https://strain.math.upenn.edu/preprints/gsNonCut.pdf},
	volume = {24},
	year = {2011},
	zblnumber = {1248.35140},
	bdsk-url-1 = {https://doi.org/10.1090/S0894-0347-2011-00697-8}}



@article{MR2807092,
	abstract = {This article provides sharp constructive upper and lower bound estimates for the  
Boltzmann collision operator 
with the full range of physical non cut-off collision kernels ($\gamma > -n$ and $s\in (0,1)$)
 in the trilinear $L^2(\mathbb{R}^n)$ energy $( \mathcal{Q}(g,f), f )$.  These new estimates prove that, for a very general class of $g(v)$, the  global diffusive behavior (on $f$) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works.  We further prove new global entropy production estimates with the same anisotropic semi-norm.  
 This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non cut-off Boltzmann collision operator in the energy space 
$L^2(\mathbb{R}^n)$.},
	author = {Gressman, Philip T. and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1016/j.aim.2011.05.005},
	eprint = {1007.1276},
	fjournal = {Advances in Mathematics},
	issn = {0001-8708},
	journal = {Adv. Math.},
	keywords = {Boltzmann equation, Kinetic Theory, non-cutoff},
	mrclass = {35Q20 (35B65 35R11 82C40)},
	mrnumber = {2807092},
	mrreviewer = {Francesco Salvarani},
	number = {6},
	pages = {2349--2384},
	title = {Sharp anisotropic estimates for the {B}oltzmann collision operator and its entropy production},
	url_arxiv = {https://arxiv.org/abs/1007.1276},
	url_pdf = {https://strain.math.upenn.edu/preprints/gsNonCutEst.pdf},
	volume = {227},
	year = {2011},
	zblnumber = {1234.35173},
	bdsk-url-1 = {https://doi.org/10.1016/j.aim.2011.05.005}}



@article{MR2793935,
	abstract = {We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. 
More specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellian constructed from a class of solutions to the relativistic Euler equations that includes a large 
subclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Euler equations. },
	author = {Speck, Jared and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1007/s00220-011-1207-z},
	eprint = {1009.5033},
	fjournal = {Communications in Mathematical Physics},
	issn = {0010-3616},
	journal = {Comm. Math. Phys.},
	keywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory, Fluid mechanics},
	mrclass = {82C40 (35Q20 76P05 76Y05)},
	mrnumber = {2793935},
	mrreviewer = {Stephen Wollman},
	number = {1},
	pages = {229--280},
	title = {Hilbert expansion from the {B}oltzmann equation to relativistic fluids},
	url_arxiv = {https://arxiv.org/abs/1009.5033},
	url_pdf = {https://strain.math.upenn.edu/preprints/ssHilbertExp.pdf},
	volume = {304},
	year = {2011},
	zblnumber = {1221.35271},
	bdsk-url-1 = {https://doi.org/10.1007/s00220-011-1207-z}}



@article{MR2765751,
	abstract = {It is often the case in mathematical analysis that solving an open problem can be facilitated by finding a new set of coordinates  which may illuminate the known difficulties. In this article, we illustrate how to derive an assortment coordinates in which to represent the relativistic Boltzmann collision operator.  We show the equivalence between some known representations, and others which seem to be new.  One of these representations has been used recently to solve several open problems.},
	author = {Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.3934/krm.2011.4.345},
	eprint = {1011.5093},
	fjournal = {Kinetic and Related Models},
	issn = {1937-5093},
	journal = {Kinet. Relat. Models},
	keywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},
	mrclass = {82C40 (76P05 83A05)},
	mrnumber = {2765751},
	mrreviewer = {Stephen Wollman},
	number = {1},
	pages = {345--359},
	title = {Coordinates in the relativistic {B}oltzmann theory},
	url_arxiv = {https://arxiv.org/abs/1011.5093},
	url_pdf = {https://strain.math.upenn.edu/preprints/sCCkrm.pdf},
	volume = {4},
	year = {2011},
	zblnumber = {05869610},
	bdsk-url-1 = {https://doi.org/10.3934/krm.2011.4.345}}



@article{MR2629879,
	abstract = {This is a brief announcement of our recent proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation without any angular cutoff, that is, for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential $r^{-(p-1)}$ with $p > 2$, and more generally. We present here a mathematical framework for unique global in time solutions for all of these potentials. We consider it remarkable that this equation, derived by Boltzmann (1) in 1872 and Maxwell (2) in 1867, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects due to grazing collisions.},
	author = {Gressman, Philip T. and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-08 11:55:21 -0600},
	doi = {10.1073/pnas.1001185107},
	fjournal = {Proceedings of the National Academy of Sciences of the United States of America},
	issn = {1091-6490},
	journal = {Proc. Natl. Acad. Sci. USA},
	keywords = {Boltzmann equation, Kinetic Theory, non-cutoff},
	mrclass = {82C40 (35Q20)},
	mrnumber = {2629879},
	number = {13},
	pages = {5744--5749},
	title = {Global classical solutions of the {B}oltzmann equation with long-range interactions},
	urlpdf = {https://strain.math.upenn.edu/preprints/gsPNAS2010.pdf},
	volume = {107},
	year = {2010},
	zblnumber = {1205.82120},
	bdsk-url-1 = {https://doi.org/10.1073/pnas.1001185107}}



@article{MR2679588,
	abstract = {We study the Cauchy Problem for the relativistic Boltzmann equation with near Vacuum initial data. Unique global in time mild solutions are obtained uniformly in the speed of light parameter $c \ge 1$.  We furthermore prove that solutions to the relativistic Boltzmann equation converge to solutions of  the Newtonian Boltzmann equation in the limit as $c\to\infty$ on arbitrary time intervals $[0,T]$, with  convergence rate $1/c^{2-\epsilon}$ for any $\epsilon \in(0,2)$.  This may be the first proof of unique global in time validity of the Newtonian limit for a Kinetic equation.},
	author = {Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1137/090762695},
	eprint = {1004.5407},
	fjournal = {SIAM Journal on Mathematical Analysis},
	issn = {0036-1410},
	journal = {SIAM J. Math. Anal.},
	keywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},
	mrclass = {82C40 (35Q20 35Q75 76P05)},
	mrnumber = {2679588},
	mrreviewer = {Silvia Lorenzani},
	number = {4},
	pages = {1568--1601},
	title = {Global {N}ewtonian limit for the relativistic {B}oltzmann equation near vacuum},
	url_arxiv = {https://arxiv.org/abs/1004.5407},
	url_pdf = {https://strain.math.upenn.edu/preprints/rBnewt.pdf},
	volume = {42},
	year = {2010},
	zblnumber = {05894999},
	bdsk-url-1 = {https://doi.org/10.1137/090762695}}



@article{MR2728733,
	abstract = {In this paper it is  shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in $L^\infty_\ell$.  If the initial data  are continuous then so is the corresponding solution.  We work in the case of a spatially periodic box.  Conditions on the collision kernel are generic; this resolves the open question of global existence for the soft potentials.},
	author = {Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1007/s00220-010-1129-1},
	eprint = {1003.4893},
	fjournal = {Communications in Mathematical Physics},
	issn = {0010-3616},
	journal = {Comm. Math. Phys.},
	keywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},
	mrclass = {82C40 (35Q20)},
	mrnumber = {2728733},
	mrreviewer = {Stephen Wollman},
	number = {2},
	pages = {529--597},
	title = {Asymptotic stability of the relativistic {B}oltzmann equation for the soft potentials},
	url_arxiv = {https://arxiv.org/abs/1003.4893},
	url_pdf = {https://strain.math.upenn.edu/preprints/sRBsoft.pdf},
	volume = {300},
	year = {2010},
	zblnumber = {1214.35072},
	bdsk-url-1 = {https://doi.org/10.1007/s00220-010-1129-1}}



@article{MR2512859,
	abstract = {Consider axisymmetric strong solutions of the incompressible
Navier-Stokes equations in $\mathbb{R}^3$ with non-trivial swirl.  Let $z$
denote the axis of symmetry and $r$ measure the distance to the
$z$-axis.  Suppose the solution satisfies,for some $0 \le \varepsilon \le 1$ that $|v (x,t)| \le C_* r^{-1+\varepsilon } |t|^{-\varepsilon /2}$ for $-T_0\le t < 0$ and a positive finite constant $C_*$ which is  allowed to be large,  we then prove that $v$ is regular at time zero. },
	author = {Chen, Chiun-Chuan and Strain, Robert M. and Tsai, Tai-Peng and Yau, Horng-Tzer},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1080/03605300902793956},
	eprint = {0709.4230},
	fjournal = {Communications in Partial Differential Equations},
	issn = {0360-5302},
	journal = {Comm. Partial Differential Equations},
	keywords = {Navier-Stokes equations, Fluid mechanics},
	mrclass = {35Q30 (35B40 35B44 35B65 76D05)},
	mrnumber = {2512859},
	mrreviewer = {Thierry Goudon},
	number = {1-3},
	pages = {203--232},
	title = {Lower bounds on the blow-up rate of the axisymmetric {N}avier-{S}tokes equations. {II}},
	url_arxiv = {https://arxiv.org/abs/0709.4230},
	url_pdf = {https://strain.math.upenn.edu/preprints/2009ChenStrainTsaiYau.pdf},
	volume = {34},
	year = {2009},
	zblnumber = {1173.35095},
	bdsk-url-1 = {https://doi.org/10.1080/03605300902793956}}



@article{MR2429247,
	abstract = {Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations  in $R^3$ with non-trivial swirl.  Such solutions are not known to be globally defined, but it is shown by Caffarelli-Kohn-Nirenberg in 1982 that they could only blow up on the axis of symmetry.  Let $z$ denote the axis of symmetry and $r$ measure the distance 
to the $z$-axis.   Suppose the solution satisfies the pointwise scale invariant bound $|v (x,t)| \le C_*{(r^2 -t)^{-1/2}} $ for $-T_0\le t < 0$ and a positive finite constant $C_*$ which is  allowed to be large,  we then prove that $v$ is regular at time zero. },
	author = {Chen, Chiun-Chuan and Strain, Robert M. and Yau, Horng-Tzer and Tsai, Tai-Peng},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1093/imrn/rnn016},
	eprint = {math/0701796},
	fjournal = {International Mathematics Research Notices. IMRN},
	issn = {1073-7928},
	journal = {Int. Math. Res. Not. IMRN},
	keywords = {Navier-Stokes equations, Fluid mechanics},
	mrclass = {35Q30 (35B40 76D03 76D05)},
	mrnumber = {2429247},
	mrreviewer = {Thierry Goudon},
	number = {9},
	pages = {Art. ID rnn016, 31},
	title = {Lower bound on the blow-up rate of the axisymmetric {N}avier-{S}tokes equations},
	url_arxiv = {https://arxiv.org/abs/math/0701796},
	url_pdf = {https://strain.math.upenn.edu/preprints/2008rnn016.pdf},
	year = {2008},
	zblnumber = {1154.35068},
	bdsk-url-1 = {https://doi.org/10.1093/imrn/rnn016}}



@article{MR2366140,
	abstract = {Consider both soft potentials with angular cutoff and Landau collision kernels in the Boltzmann theory inside a periodic box. We prove that any smooth perturbation near a given Maxwellian approaches to zero at the rate of $e^{-\lambda t^{p}}$ for some $\lambda >0$ and  $p\in (0,1)$. Our method is based on an unified energy estimate with appropriate exponential velocity
weight. Our results extend the classical Caflisch 1980 result to the case of very soft potential and Coulomb interactions.},
	author = {Strain, Robert M. and Guo, Yan},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-08 11:40:38 -0600},
	doi = {10.1007/s00205-007-0067-3},
	fjournal = {Archive for Rational Mechanics and Analysis},
	issn = {0003-9527},
	journal = {Arch. Ration. Mech. Anal.},
	keywords = {Boltzmann equation, Landau equation, Kinetic Theory},
	mrclass = {82B05 (35B45 35F25)},
	mrnumber = {2366140},
	mrreviewer = {Simone Calogero},
	number = {2},
	pages = {287--339},
	read = {0},
	title = {Exponential decay for soft potentials near {M}axwellian},
	urlpdf = {https://strain.math.upenn.edu/preprints/2005SGed.pdf},
	volume = {187},
	year = {2008},
	zblnumber = {1130.76069},
	bdsk-url-1 = {https://doi.org/10.1007/s00205-007-0067-3}}



@article{MR2322149,
	abstract = {In this paper we prove new constructive coercivity estimates for the Boltzmann collision operator without cutoff, that is for {\em long-range} interactions. 
In particular we give a generalized sufficient condition for the existence of a spectral gap which involves both the growth behavior of the collision kernel at large relative velocities and its singular behavior at grazing and frontal collisions. It provides in particular existence of a spectral gap and estimates on it for interactions deriving from the hard potentials $\phi(r)=r^{-(s-1)}$, $s \ge 5$ or the so-called moderately soft potentials $\phi(r)=r^{-(s-1)}$, $3< s < 5$, (without angular cutoff). 
In particular this paper recovers (by constructive means), improves and extends previous results of Pao 1974. We also obtain constructive coercivity estimates for 
the Landau collision operator for the optimal coercivity norm pointed out in Guo 2002 and we formulate a conjecture about a unified necessary and sufficient condition 
for the existence of a spectral gap for Boltzmann and Landau linearized collision operators. },
	author = {Mouhot, Cl{\'{e}}ment and Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1016/j.matpur.2007.03.003},
	eprint = {math/0607495},
	fjournal = {Journal de Math{\'{e}}matiques Pures et Appliqu{\'{e}}es. Neuvi{\`e}me S{\'{e}}rie},
	issn = {0021-7824},
	journal = {J. Math. Pures Appl. (9)},
	keywords = {Boltzmann equation, Landau equation, Kinetic Theory, non-cutoff},
	mrclass = {82C40 (47G20 76P05)},
	mrnumber = {2322149},
	mrreviewer = {C\'{e}dric Villani},
	number = {5},
	pages = {515--535},
	title = {Spectral gap and coercivity estimates for linearized {B}oltzmann collision operators without angular cutoff},
	url_arxiv = {https://arxiv.org/abs/math/0607495},
	url_pdf = {https://strain.math.upenn.edu/preprints/2007mouhotStrain.pdf},
	volume = {87},
	year = {2007},
	zblnumber = {1388.76338},
	bdsk-url-1 = {https://doi.org/10.1016/j.matpur.2007.03.003}}



@article{MR2372479,
	abstract = {The Balescu-Lenard equation  from  plasma physics is widely considered to include a highly accurate correction to Landau's fundamental collision operator.   
Yet so far it has seen very little mathematical study. We perform an extensive linearized analysis of this equation, 
which includes determining the asymptotic behavior of the new components of the linearized operator and establishing time decay rates for the linearized equation.  },
	author = {Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1080/03605300601088609},
	eprint = {math/0603490},
	fjournal = {Communications in Partial Differential Equations},
	issn = {0360-5302},
	journal = {Comm. Partial Differential Equations},
	keywords = {Kinetic Theory, Balescu-Lenard equation},
	mrclass = {82D10 (76P05 82C40)},
	mrnumber = {2372479},
	mrreviewer = {Stephen Wollman},
	number = {10-12},
	pages = {1551--1586},
	title = {On the linearized {B}alescu-{L}enard equation},
	url_pdf = {https://strain.math.upenn.edu/preprints/2007BalescuLenard.pdf},
	volume = {32},
	year = {2007},
	zblnumber = {1128.76068},
	bdsk-url-1 = {https://doi.org/10.1080/03605300601088609}}



@article{MR2209761,
	abstract = {By direct interpolation of a family of smooth energy estimates for solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann type equations, we show convergence to Maxwellian with any polynomial rate in time. Our results not only resolve the important open problem for both the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system for charged particles, but also lead to a simpler alternative proof of recent decay results for soft potentials as well as the Coulombic interaction, with precise decay rate depending on the initial conditions.
},
	author = {Strain, Robert M. and Guo, Yan},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-08 11:41:10 -0600},
	doi = {10.1080/03605300500361545},
	fjournal = {Communications in Partial Differential Equations},
	issn = {0360-5302},
	journal = {Comm. Partial Differential Equations},
	keywords = {Boltzmann equation, Landau equation, relativistic Landau equation, Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},
	mrclass = {82B40 (82C40 82D05 82D10)},
	mrnumber = {2209761},
	mrreviewer = {Hai-Liang Li},
	number = {1-3},
	pages = {417--429},
	title = {Almost exponential decay near {M}axwellian},
	urlpdf = {https://strain.math.upenn.edu/preprints/2005SGaed.pdf},
	volume = {31},
	year = {2006},
	zblnumber = {1096.82010},
	bdsk-url-1 = {https://doi.org/10.1080/03605300500361545}}



@article{MR2259206,
	abstract = {The Vlasov-Maxwell-Boltzmann system is a fundamental model to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians.},
	author = {Strain, Robert M.},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-07-13 15:27:26 -0400},
	doi = {10.1007/s00220-006-0109-y},
	eprint = {math/0512002},
	fjournal = {Communications in Mathematical Physics},
	issn = {0010-3616},
	journal = {Comm. Math. Phys.},
	keywords = {Boltzmann equation, Kinetic Theory, Vlasov-Maxwell systems},
	mrclass = {82D10 (35A05 35F20 35Q60 76P05 76X05)},
	mrnumber = {2259206},
	mrreviewer = {Simone Calogero},
	number = {2},
	pages = {543--567},
	title = {The {V}lasov-{M}axwell-{B}oltzmann system in the whole space},
	url_pdf = {https://strain.math.upenn.edu/preprints/2005Sws.pdf},
	volume = {268},
	year = {2006},
	zblnumber = {1129.35022},
	bdsk-url-1 = {https://doi.org/10.1007/s00220-006-0109-y}}



@article{Arnold2006,
	abstract = {We discuss recent work proving exponential time decay rates to equilibrium for Boltzmann equations such as the soft potentials, Landau's equation and the lin- earized Balescu-Lenard model. We also mention a proof of existence and unique- ness of solutions near Maxwellian to the Vlasov-Maxwell-Boltzmann system in the whole space. Some of these projects are joint work with Yan Guo.},
	abstractow = {The Oberwolfach meeting described here aimed at presenting the latest mathematical results in the field of kinetic theory (both classical and quantum). There were 50 participants, among which 15 young participants (PhD students, post-docs or young assistant professors). Two of them (M.-P. Gualdani and R. Strain) were invited within the program "US Junior Oberwolfach Fellows": they are promising young researchers working in the US. The program of the meeting was made in such a way that a lot of time remained for people to meet informally and discuss about scientific issues. It also ensured that almost everybody attended all (or most of) the scheduled talks. The program was structured in the following way: three subtopics were defined (relationships between micro/meso/macroscopic models; kinetic theory for complex particles: granular media, coagulation/fragmentation, chemotaxis and sprays; quantum mechanical kinetic theory). For each subtopic, there were a few (2 or 3) longer talks (about 40 minutes) by senior participants. For those talks, a specific effort of clarity was asked to the speakers, and the subject had to be rather broad. Then, shorter talks of about 20 minutes were planned, on more specialized issues. Finally, a special (much longer) talk in two parts was presented by one of the participants (C. Villani), in order to describe in a very didactical way an emerging link between kinetic theory, optimal transport, and Riemannian geometry. Some participants did not give a talk within this program, but organized an informal discussion around a poster, with an audience composed of specially interested people.},
	author = {Robert M. Strain},
	booktitle = {{Classical and quantum mechanical models of many-particle systems. Workshop held December 3--9, 2006.}},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-08 13:59:27 -0600},
	doi = {10.4171/owr/2006/54},
	editor = {Anton {Arnold} and Carlo {Cercignani} and Laurent {Desvillettes}},
	fjournal = {{Oberwolfach Reports}},
	issn = {1660-8933; 1660-8941/e},
	journal = {{Oberwolfach Rep.}},
	keywords = {Boltzmann equation, Landau equation, relativistic Landau equation, Kinetic Theory, relativistic Kinetic Theory},
	language = {English},
	msc2010 = {82B40 81-06 82-06 81S30 00B05},
	number = {4},
	pages = {3189--3258},
	publisher = {European Mathematical Society (EMS) Publishing House, Zurich; Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach},
	title = {{R}ecent results on existence, uniqueness and asymptotic decay rates for collisional kinetic models},
	urlpdf = {https://strain.math.upenn.edu/preprints/rms2006ow.pdf},
	volume = {3},
	year = {2006},
	zblnumber = {1177.82057},
	bdsk-url-1 = {https://doi.org/10.4171/owr/2006/54}}



@article{Arnold2010,
	abstract = {In this report, we will describe briefly several recent developments for the Boltzmann equation without the Grad angular cut-off assumption.},
	abstractow = {The topic of this meeting were non-linear partial differential and integro-differential equations (in particular kinetic equations and their macroscopic/fluid-dynamical limits) modeling the dynamics of many-particle systems with applications in physics, engineering, and mathematical biology. Typical questions of interest were the derivation of macro-models from micro-models, the mathematical analysis (well-posedness, stability, asymptotic behavior of solutions), and ``to a lesser extent'' numerical aspects of such equations. A highlight of this meeting was a mini-course on the recent mathematical theory of Landau damping.},
	author = {Robert M. Strain},
	booktitle = {{Classical and quantum mechanical models of many-particle systems. Workshop held December 5 -- December 11, 2010.}},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-08 13:59:00 -0600},
	doi = {10.4171/owr/2010/54},
	editor = {Anton {Arnold} and Eric A. {Carlen} and Laurent {Desvillettes}},
	fjournal = {{Oberwolfach Reports}},
	issn = {1660-8933; 1660-8941/e},
	journal = {{Oberwolfach Rep.}},
	keywords = {Boltzmann equation, Kinetic Theory, non-cutoff},
	language = {English},
	msc2010 = {00B05 81-06 82-06 35Qxx 82Cxx 82B40 81S30},
	number = {4},
	pages = {3159--3236},
	publisher = {European Mathematical Society (EMS) Publishing House, Zurich; Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach},
	title = {Around the {B}oltzmann equation without angular cut-off},
	urlpdf = {https://strain.math.upenn.edu/preprints/rms2010ow.pdf},
	volume = {7},
	year = {2010},
	zblnumber = {1235.00027},
	bdsk-url-1 = {https://doi.org/10.4171/owr/2010/54}}



@article{MR2100057,
	abstract = {The relativistic Landau-Maxwell system is among the most fundamental  and complete models for describing the dynamics of a dilute collisional plasma in which particles interact through Coulombic collisions and through their self-consistent electromagnetic field.   We construct the first global in time classical solutions.  Our solutions are constructed in a periodic box and near the relativistic Maxwellian, the J\"{u}ttner solution.},
	author = {Strain, Robert M. and Guo, Yan},
	date-added = {2019-07-13 15:27:26 -0400},
	date-modified = {2019-08-08 11:41:54 -0600},
	doi = {10.1007/s00220-004-1151-2},
	fjournal = {Communications in Mathematical Physics},
	issn = {0010-3616},
	journal = {Comm. Math. Phys.},
	keywords = {Landau equation, relativistic Landau equation, Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},
	mrclass = {82D10 (35Q60 35Q75 76X05 76Y05)},
	mrnumber = {2100057},
	mrreviewer = {C\'edric Villani},
	number = {2},
	pages = {263--320},
	title = {Stability of the relativistic {M}axwellian in a collisional plasma},
	urlpdf = {https://strain.math.upenn.edu/preprints/2004SG.pdf},
	volume = {251},
	year = {2004},
	zblnumber = {1113.82070},
	bdsk-url-1 = {https://doi.org/10.1007/s00220-004-1151-2}}



@article{1907.05784,
	abstract = {In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These $L^\infty$ bounds have been known to be a challenging open problem in relativistic kinetic theory. To accomplish this, we establish two types of estimates for the gain part of the collision operator: first, we prove a potential type estimate and a relativistic hyper-surface integral estimate. We then combine those estimates using the relativistic counter-part of the Carleman representation to derive uniform control of the gain term for the relativistic collision operator. This allows us to prove the desired propagation of the uniform bounds of the solution. We further present two applications of the propagation of the uniform upper bounds: first we give another proof of the Boltzmann $H$-theorem, and second we prove the asymptotic convergence of solutions to the relativistic Maxwellian equilibrium.},
	author = {Jin Woo Jang and Robert M. Strain and Seok-Bae Yun},
	doi = {10.1007/s00205-021-01649-0},
	eprint = {1907.05784},
	fjournal = {Archive for Rational Mechanics and Analysis},
	journal = {Arch. Ration. Mech. Anal.},
	keywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},
	note = {published online April 15, 2021},
	pages = {149--186},
	title = {{Propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation}},
	url_arxiv = {https://arxiv.org/abs/1907.05784},
	volume = {241},
	year = {2021},
	bdsk-url-1 = {https://doi.org/10.1007/s00205-021-01649-0}}



@phdthesis{MR2707256,
	abstract = {The collisional Kinetic Equations we study are all of the form
$$\partial_t F + v\cdot \nabla_x F+V(t,x)\cdot \nabla_v F=Q(F,F).$$
Here $F=F(t,x,v)$ is a probabilistic density function (of time $t\ge 0$, space $x\in\Omega$ and velocity $v\in\mathbb{R}^3$) for a particle taken chosen randomly from a gas or plasma. $V(t,x)$ is a field term which usually represents Maxwell's theory of electricity and magnetism, sometimes this term is neglected.   $Q(F,F)$ is the collision operator which models the interaction between colliding particles.  We consider both the Boltzmann and Landau collision operators.  


We prove existence, uniqueness and regularity of close to equilibrium solutions to the relativistic Landau-Maxwell system in the first part of this thesis.  Our main tool is an energy method.  

In the second part, we prove arbitrarily high polynomial time decay rates to equilibrium for four kinetic equations.  These are cutoff soft potential Boltzmann and Landau equations, but also the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system.  The main technique used here is interpolation.  

In the third part, we prove exponential decay for the cutoff soft potential Boltzmann and Landau equations.  The main point here is to show that exponential decay of the initial data is propagated by a solution.  

In the fourth and final part of this thesis, we write down a few important calculations in the relativistic Boltzmann theory which are scattered around the literature.  We also calculate a few Lorentz transformations which maybe useful in relativistic transport theory.  We use these calculations to comment about extending the results in this thesis to the relativistic Boltzmann equation.
},
	annote = {(https://search.proquest.com)},
	author = {Strain, Robert M.},
	date-added = {2019-07-13 17:21:46 -0400},
	date-modified = {2019-08-08 11:28:11 -0600},
	isbn = {978-0542-12875-2},
	journal = {ProQuest Dissertations and Theses},
	keywords = {Boltzmann equation, Landau equation, Balescu-Lenard equation, Kinetic Theory, Vlasov-Maxwell systems, relativistic Kinetic Theory},
	language = {English},
	mrclass = {Thesis},
	mrnumber = {2707256},
	note = {(ProQuest Document ID 305028444)},
	pages = {1--200},
	publisher = {ProQuest LLC, Ann Arbor, MI},
	school = {Brown University},
	title = {Some applications of an energy method in collisional {K}inetic theory},
	url_proquest = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3174679},
	urlpdf = {https://strain.math.upenn.edu/preprints/2005PHDthesisS.pdf},
	year = {2005},
	bdsk-url-1 = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3174679}}
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In order to keep using Mendeley with BibBase past April\n        14th, you need to:\n        <ol>\n          <li>renew the authorization for BibBase on Mendeley, and</li>\n          <li>update the BibBase URL\n            in your page the same way you did when you initially set up\n            this page.\n          </li>\n        </ol>\n      </p>\n\n      <p>\n        <a href=\"http://bibbase.org/cgi-bin/mendeley2/get.py\">\n          <i class=\"fa fa-angle-double-right\"></i>\n          Fix it now</a>\n      </p>\n    </div>\n\n  </div>\n\n\n  <div id=\"bibbase_body\">\n    \n  \n  <div class=\"bibbase_group_whole\" id=\"group_2024\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2024')\"\n      onkeypress=\"toggleGroup('group_2024')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2024</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"cameron-strain-criticallocalwellposednessforthefullynonlinearpeskinproblem-2024\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Stephen Cameron;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2112.00692\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'cameron-strain-criticallocalwellposednessforthefullynonlinearpeskinproblem-2024', 'https://arxiv.org/abs/2112.00692', event);\">\n    Critical local well-posedness for the fully nonlinear Peskin problem.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Pure Appl. Math.</i>, 77(2): 901–989.  2024.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2112.00692\"\n     onclick=\"log_download('cameron-strain-criticallocalwellposednessforthefullynonlinearpeskinproblem-2024', 'https://arxiv.org/abs/2112.00692', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Critical local well-posedness for the fully nonlinear Peskin problem [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1002/cpa.22139\"\n     onclick=\"log_download('cameron-strain-criticallocalwellposednessforthefullynonlinearpeskinproblem-2024', 'http://doi.org/10.1002/cpa.22139', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/cameron-strain-criticallocalwellposednessforthefullynonlinearpeskinproblem-2024\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>4 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('cameron-strain-criticallocalwellposednessforthefullynonlinearpeskinproblem-2024')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{2112.00692,\n\tabstract = {We study the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid.  This is known as the Stokes immersed boundary problem and also as the Peskin problem.   We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem.  In this case we prove local in time well-posedness for arbitrary initial data in the scaling critical Besov space $\\dot{B}^{3/2}_{2,1}(\\mathbb{T} ; \\mathbb{R}^2)$.  We additionally prove the optimal higher order smoothing effects for the solution.  To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancellation structure.},\n\tauthor = {Cameron, Stephen and Strain, Robert M.},\n\tblurbreport = {This paper studies the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time well-posedness for arbitrary initial data in the scaling critical Besov space $\\dot{B}^{3/2}_{2,1}(\\mathbb{T} ; \\mathbb{R}^2)$. We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancellation structure.},\n\tdoi = {10.1002/cpa.22139},\n\teprint = {2112.00692},\n\tfjournal = {Communications on Pure and Applied Mathematics},\n\tissn = {0010-3640,1097-0312},\n\tjournal = {Comm. Pure Appl. Math.},\n\tkeywords = {Fluid mechanics, Free boundary problems, Peskin problem, Fluid-Structure interface, critical regularity, immersed boundary problem},\n\tmrclass = {35Q35 (35C15 35R11 35R35 76D07)},\n\tmrnumber = {4673875},\n\tnumber = {2},\n\tpages = {901--989},\n\ttitle = {Critical local well-posedness for the fully nonlinear {P}eskin problem},\n\turl_arxiv = {https://arxiv.org/abs/2112.00692},\n\tvolume = {77},\n\tyear = {2024},\n\tbdsk-url-1 = {https://doi.org/10.1002/cpa.22139}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We study the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time well-posedness for arbitrary initial data in the scaling critical Besov space $Ḃ^{3/2}_{2,1}(\\mathbb{T} ; ℝ^2)$. We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancellation structure.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2023\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2023')\"\n      onkeypress=\"toggleGroup('group_2023')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2023</span>\n      <span class=\"bibbase_group_count\">\n        \n        (4)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"gancedo-garcajurez-patel-strain-onnonlinearstabilityofmuskatbubbles-2023\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Francisco Gancedo;  Eduardo García-Juárez;  Neel Patel;  and  Robert Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2312.14323\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'gancedo-garcajurez-patel-strain-onnonlinearstabilityofmuskatbubbles-2023', 'https://arxiv.org/abs/2312.14323', event);\">\n    On Nonlinear Stability of Muskat Bubbles.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  ,26 pages. Dec 2023.\n  <span class=\"note\">preprint</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2312.14323\"\n     onclick=\"log_download('gancedo-garcajurez-patel-strain-onnonlinearstabilityofmuskatbubbles-2023', 'https://arxiv.org/abs/2312.14323', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"On Nonlinear Stability of Muskat Bubbles [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/gancedo-garcajurez-patel-strain-onnonlinearstabilityofmuskatbubbles-2023\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>1 download</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('gancedo-garcajurez-patel-strain-onnonlinearstabilityofmuskatbubbles-2023')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{GancedoGarciaJuarezPatelStrain24,\n\tabstract = { In this paper we consider gravity-capillarity Muskat bubbles in 2D. We obtain\na new approach to improve our result in \\cite{GancedoGarciaJuarezPatelStrain23}. Due to a new bubble-adapted\nformulation, the improvement is two fold.  We significantly condense the proof\nand we now obtain the global well-posedness result for Muskat bubbles in\ncritical regularity.},\n\tauthor = {Gancedo, Francisco and Garc\\&#x27;{\\i}a-Ju\\&#x27;{a}rez, Eduardo and Patel, Neel and Strain, Robert},\n\teprint = {2312.14323},\n\tkeywords = {Fluid mechanics, Free boundary problems, critical regularity, Muskat problem},\n\tmonth = {Dec},\n\tnote = {preprint},\n\tpages = {26 pages},\n\ttitle = {{On Nonlinear Stability of Muskat Bubbles}},\n\turl_arxiv = {https://arxiv.org/abs/2312.14323},\n\tyear = {2023}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper we consider gravity-capillarity Muskat bubbles in 2D. We obtain a new approach to improve our result in i̧teGancedoGarciaJuarezPatelStrain23. Due to a new bubble-adapted formulation, the improvement is two fold. We significantly condense the proof and we now obtain the global well-posedness result for Muskat bubbles in critical regularity.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"gancedo-garcajurez-patel-strain-globalregularityforgravityunstablemuskatbubbles-2023\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Francisco Gancedo;  Eduardo García-Juárez;  Neel Patel;  and  Robert Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://doi.org/10.1090/memo/1455\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'gancedo-garcajurez-patel-strain-globalregularityforgravityunstablemuskatbubbles-2023', 'https://doi.org/10.1090/memo/1455', event);\">\n    Global Regularity for Gravity Unstable Muskat Bubbles.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Mem. Amer. Math. Soc.</i>, 292(1455): 87 pages.  2023.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://doi.org/10.1090/memo/1455\"\n     onclick=\"log_download('gancedo-garcajurez-patel-strain-globalregularityforgravityunstablemuskatbubbles-2023', 'https://doi.org/10.1090/memo/1455', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Global Regularity for Gravity Unstable Muskat Bubbles [link]\"\n       title=\"Paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Paper</span></a>\n  &nbsp;\n  \n  <a href=\"https://arxiv.org/abs/1902.02318\"\n     onclick=\"log_download('gancedo-garcajurez-patel-strain-globalregularityforgravityunstablemuskatbubbles-2023', 'https://arxiv.org/abs/1902.02318', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Global Regularity for Gravity Unstable Muskat Bubbles [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1090/memo/1455\"\n     onclick=\"log_download('gancedo-garcajurez-patel-strain-globalregularityforgravityunstablemuskatbubbles-2023', 'http://doi.org/10.1090/memo/1455', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/gancedo-garcajurez-patel-strain-globalregularityforgravityunstablemuskatbubbles-2023\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('gancedo-garcajurez-patel-strain-globalregularityforgravityunstablemuskatbubbles-2023')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{GancedoGarciaJuarezPatelStrain23,\n\tabstract = {In this paper, we study the dynamics of fluids in porous media governed by Darcy&#x27;s law: the Muskat problem. We consider the setting of two immiscible fluids of different densities and viscosities under the influence of gravity in which one fluid is completely surrounded by the other. This setting is gravity unstable because along a portion of the interface, the denser fluid must be above the other. Surprisingly, even without capillarity, the circle-shaped bubble is a steady state solution moving with vertical constant velocity determined by the density jump between the fluids. Taking advantage of our discovery of this steady state, we are able to prove global in time existence and uniqueness of dynamic bubbles of nearly circular shapes under the influence of surface tension. We prove this global existence result for low regularity initial data. Moreover, we prove that these solutions are instantly analytic and decay exponentially fast in time to the circle.},\n\tauthor = {Gancedo, Francisco and Garc\\&#x27;{\\i}a-Ju\\&#x27;{a}rez, Eduardo and Patel, Neel and Strain, Robert},\n\tdoi = {10.1090/memo/1455},\n\teprint = {1902.02318},\n\tfjournal = {Memoirs of the American Mathematical Society},\n\tissn = {0065-9266,1947-6221},\n\tjournal = {Mem. Amer. Math. Soc.},\n\tkeywords = {Fluid mechanics, Free boundary problems, Muskat problem},\n\tmrclass = {99-06},\n\tmrnumber = {4679708},\n\tnumber = {1455},\n\tpages = {87 pages},\n\ttitle = {Global {R}egularity for {G}ravity {U}nstable {M}uskat {B}ubbles},\n\turl = {https://doi.org/10.1090/memo/1455},\n\turl_arxiv = {https://arxiv.org/abs/1902.02318},\n\tvolume = {292},\n\tyear = {2023},\n\tbdsk-url-1 = {https://doi.org/10.1090/memo/1455}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper, we study the dynamics of fluids in porous media governed by Darcy's law: the Muskat problem. We consider the setting of two immiscible fluids of different densities and viscosities under the influence of gravity in which one fluid is completely surrounded by the other. This setting is gravity unstable because along a portion of the interface, the denser fluid must be above the other. Surprisingly, even without capillarity, the circle-shaped bubble is a steady state solution moving with vertical constant velocity determined by the density jump between the fluids. Taking advantage of our discovery of this steady state, we are able to prove global in time existence and uniqueness of dynamic bubbles of nearly circular shapes under the influence of surface tension. We prove this global existence result for low regularity initial data. Moreover, we prove that these solutions are instantly analytic and decay exponentially fast in time to the circle.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"garciajurez-kuo-mori-strain-wellposednessofthe3dpeskinproblem-2023\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Eduardo García-Juárez;  Po-Chun Kuo;  Yoichiro Mori;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2301.12153\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'garciajurez-kuo-mori-strain-wellposednessofthe3dpeskinproblem-2023', 'https://arxiv.org/abs/2301.12153', event);\">\n    Well-Posedness of the 3D Peskin Problem.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  ,93 pages. Jan 2023.\n  <span class=\"note\">preprint</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2301.12153\"\n     onclick=\"log_download('garciajurez-kuo-mori-strain-wellposednessofthe3dpeskinproblem-2023', 'https://arxiv.org/abs/2301.12153', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Well-Posedness of the 3D Peskin Problem [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/garciajurez-kuo-mori-strain-wellposednessofthe3dpeskinproblem-2023\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>14 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('garciajurez-kuo-mori-strain-wellposednessofthe3dpeskinproblem-2023')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{Peskin3D,\n\tabstract = {This paper introduces the {\\em{3D Peskin problem}}: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow. We obtain the equations that model this free boundary problem and show that they admit a boundary integral reduction, providing an evolution equation for the elastic interface. We consider general nonlinear elastic laws, i.e., the fully nonlinear Peskin problem, and prove that the problem is well-posed in low-regularity H\\&quot;older spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time.},\n\tauthor = {Eduardo Garc{\\&#x27;{i}}a-Ju{\\&#x27;{a}}rez and Po-Chun Kuo and Yoichiro Mori and Robert M. Strain},\n\teprint = {2301.12153},\n\tkeywords = {Fluid mechanics, Free boundary problems, Peskin problem, Fluid-Structure interface, critical regularity, immersed boundary problem},\n\tmonth = {Jan},\n\tnote = {preprint},\n\tpages = {93 pages},\n\ttitle = {{Well-Posedness of the 3D Peskin Problem}},\n\turl_arxiv = {https://arxiv.org/abs/2301.12153},\n\tyear = {2023}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  This paper introduces the \\em3D Peskin problem: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow. We obtain the equations that model this free boundary problem and show that they admit a boundary integral reduction, providing an evolution equation for the elastic interface. We consider general nonlinear elastic laws, i.e., the fully nonlinear Peskin problem, and prove that the problem is well-posed in low-regularity Hölder spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"garciajurez-mori-strain-thepeskinproblemwithviscositycontrast-2023\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Eduardo García-Juárez;  Yoichiro Mori;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2009.03360\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'garciajurez-mori-strain-thepeskinproblemwithviscositycontrast-2023', 'https://arxiv.org/abs/2009.03360', event);\">\n    The Peskin Problem with Viscosity Contrast.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Anal. PDE</i>, 16(3): 785–838.  2023.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2009.03360\"\n     onclick=\"log_download('garciajurez-mori-strain-thepeskinproblemwithviscositycontrast-2023', 'https://arxiv.org/abs/2009.03360', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"The Peskin Problem with Viscosity Contrast [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.2140/apde.2023.16.785\"\n     onclick=\"log_download('garciajurez-mori-strain-thepeskinproblemwithviscositycontrast-2023', 'http://doi.org/10.2140/apde.2023.16.785', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/garciajurez-mori-strain-thepeskinproblemwithviscositycontrast-2023\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>6 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('garciajurez-mori-strain-thepeskinproblemwithviscositycontrast-2023')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{2009.03360,\n\tabstract = {The Peskin problem models the dynamics of a closed elastic filament immersed\nin an incompressible fluid. In this paper, we consider the case when the inner\nand outer viscosities are possibly different. This viscosity contrast adds\nfurther non-local effects to the system through the implicit non-local relation\nbetween the net force and the free interface. We prove the first global\nwell-posedness result for the Peskin problem in this setting. The result\napplies for medium size initial interfaces in critical spaces and shows instant\nanalytic smoothing. We carefully calculate the medium size constraint on the\ninitial data. These results are new even without viscosity contrast.},\n\tauthor = {Eduardo Garc{\\&#x27;{i}}a-Ju{\\&#x27;{a}}rez and Yoichiro Mori and Robert M. Strain},\n\tdoi = {10.2140/apde.2023.16.785},\n\teprint = {2009.03360},\n\tfjournal = {Analysis \\&amp; PDE},\n\tjournal = {Anal. PDE},\n\tkeywords = {Fluid mechanics, Free boundary problems, Peskin problem, Fluid-Structure interface, critical regularity, immersed boundary problem},\n\tnumber = {3},\n\tpages = {785--838},\n\ttitle = {{The Peskin Problem with Viscosity Contrast}},\n\turl_arxiv = {https://arxiv.org/abs/2009.03360},\n\tvolume = {16},\n\tyear = {2023},\n\tbdsk-url-1 = {https://doi.org/10.2140/apde.2023.16.785}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The Peskin problem models the dynamics of a closed elastic filament immersed in an incompressible fluid. In this paper, we consider the case when the inner and outer viscosities are possibly different. This viscosity contrast adds further non-local effects to the system through the implicit non-local relation between the net force and the free interface. We prove the first global well-posedness result for the Peskin problem in this setting. The result applies for medium size initial interfaces in critical spaces and shows instant analytic smoothing. We carefully calculate the medium size constraint on the initial data. These results are new even without viscosity contrast.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2022\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2022')\"\n      onkeypress=\"toggleGroup('group_2022')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2022</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"jang-strain-asymptoticstabilityoftherelativisticboltzmannequationwithoutangularcutoff-2022\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Jin Woo Jang;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2103.15885\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'jang-strain-asymptoticstabilityoftherelativisticboltzmannequationwithoutangularcutoff-2022', 'https://arxiv.org/abs/2103.15885', event);\">\n    Asymptotic Stability of the Relativistic Boltzmann Equation without Angular Cut-off.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Ann. PDE</i>, 8(2): 167 pages.  2022.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2103.15885\"\n     onclick=\"log_download('jang-strain-asymptoticstabilityoftherelativisticboltzmannequationwithoutangularcutoff-2022', 'https://arxiv.org/abs/2103.15885', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Asymptotic Stability of the Relativistic Boltzmann Equation without Angular Cut-off [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s40818-022-00137-2\"\n     onclick=\"log_download('jang-strain-asymptoticstabilityoftherelativisticboltzmannequationwithoutangularcutoff-2022', 'http://doi.org/10.1007/s40818-022-00137-2', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/jang-strain-asymptoticstabilityoftherelativisticboltzmannequationwithoutangularcutoff-2022\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>5 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('jang-strain-asymptoticstabilityoftherelativisticboltzmannequationwithoutangularcutoff-2022')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{2103.15885,\n\tabstract = {This paper is concerned with the relativistic Boltzmann equation without angular cutoff.  We establish the global-in-time existence, uniqueness and asymptotic stability for solutions nearby the relativistic Maxwellian to the special relativistic Boltzmann equation without any angular cutoff.  We work in the case of a spatially periodic box. We assume the generic hard-interaction and  soft-interaction conditions on the collision kernel that were derived by Dudy\\&#x27;nski and Ekiel-Je$\\dot{\\text{z}}$ewska (Comm. Math. Phys. \\textbf{115}(4):607--629, 1988), and our assumptions include the case of Israel particles (J. Math. Phys. \\textbf{4}:1163--1181, 1963).   In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator.   We further derive the relativistic analogue of the Carleman dual representation of Boltzmann collision operator.  This is the first global existence and stability result for relativistic Boltzmann equation without angular cutoff and this resolves the open question of perturbative global existence without the Grad&#x27;s angular cut-off assumption.},\n\tauthor = {{J}in {W}oo {J}ang and {R}obert {M}. {S}train},\n\tdate-modified = {2024-03-20 16:23:02 -0400},\n\tdoi = {10.1007/s40818-022-00137-2},\n\teprint = {2103.15885},\n\tfjournal = {Annals of PDE. Journal Dedicated to the Analysis of Problems from Physical Sciences},\n\tjournal = {Ann. {PDE}},\n\tkeywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\n\tnumber = {2},\n\tpages = {167 pages},\n\ttitle = {{Asymptotic Stability of the Relativistic Boltzmann Equation without Angular Cut-off}},\n\turl_arxiv = {https://arxiv.org/abs/2103.15885},\n\tvolume = {8},\n\tyear = {2022},\n\tbdsk-url-1 = {https://doi.org/10.1007/s40818-022-00137-2}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  This paper is concerned with the relativistic Boltzmann equation without angular cutoff. We establish the global-in-time existence, uniqueness and asymptotic stability for solutions nearby the relativistic Maxwellian to the special relativistic Boltzmann equation without any angular cutoff. We work in the case of a spatially periodic box. We assume the generic hard-interaction and soft-interaction conditions on the collision kernel that were derived by Dudyński and Ekiel-Je$\\̇text{z}}$ewska (Comm. Math. Phys. \\textbf115(4):607–629, 1988), and our assumptions include the case of Israel particles (J. Math. Phys. \\textbf4:1163–1181, 1963). In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. We further derive the relativistic analogue of the Carleman dual representation of Boltzmann collision operator. This is the first global existence and stability result for relativistic Boltzmann equation without angular cutoff and this resolves the open question of perturbative global existence without the Grad's angular cut-off assumption.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2021\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2021')\"\n      onkeypress=\"toggleGroup('group_2021')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2021</span>\n      <span class=\"bibbase_group_count\">\n        \n        (4)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"jang-strain-wong-magneticconfinementforthe2daxisymmetricrelativisticvlasovmaxwellsysteminanannulus-2021\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Jin Woo Jang;  Robert M. Strain;  and  Tak Kwong Wong.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2111.04583\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'jang-strain-wong-magneticconfinementforthe2daxisymmetricrelativisticvlasovmaxwellsysteminanannulus-2021', 'https://arxiv.org/abs/2111.04583', event);\">\n    Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Kinet. Relat. Models</i>,36 pages.  2021.\n  <span class=\"note\">published online November 25, 2021</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2111.04583\"\n     onclick=\"log_download('jang-strain-wong-magneticconfinementforthe2daxisymmetricrelativisticvlasovmaxwellsysteminanannulus-2021', 'https://arxiv.org/abs/2111.04583', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.3934/krm.2021039\"\n     onclick=\"log_download('jang-strain-wong-magneticconfinementforthe2daxisymmetricrelativisticvlasovmaxwellsysteminanannulus-2021', 'http://doi.org/10.3934/krm.2021039', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/jang-strain-wong-magneticconfinementforthe2daxisymmetricrelativisticvlasovmaxwellsysteminanannulus-2021\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>5 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('jang-strain-wong-magneticconfinementforthe2daxisymmetricrelativisticvlasovmaxwellsysteminanannulus-2021')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{JangStrainWong2021,\n\tabstract = {Although the nuclear fusion process has received a great deal of attention in recent years, the amount of the mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the \\textit{Vlasov-Maxwell} system in a two-dimensional annulus when a huge (\\textit{but finite-in-time}) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. This work is the first step to a more generalized work on the three dimensional Tokamak structure. The highlight of this work is from the physical assumptions on the external magnetic potential well which remains finite \\textit{within a finite time interval} and from that we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical-coordinate-forms of the \\textit{Vlasov-Maxwell} system.},\n\tauthor = {{J}in {W}oo {J}ang and {R}obert {M}. {S}train and {T}ak {K}wong {W}ong},\n\tdoi = {10.3934/krm.2021039},\n\teprint = {2111.04583},\n\tfjournal = {Kinetic and Related Models},\n\tjournal = {Kinet. Relat. Models},\n\tkeywords = {Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},\n\tnote = {published online November 25, 2021},\n\tpages = {36 pages},\n\ttitle = {{Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus}},\n\turl_arxiv = {https://arxiv.org/abs/2111.04583},\n\tyear = {2021},\n\tbdsk-url-1 = {https://doi.org/10.3934/krm.2021039}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Although the nuclear fusion process has received a great deal of attention in recent years, the amount of the mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the <i>Vlasov-Maxwell</i> system in a two-dimensional annulus when a huge (<i>but finite-in-time</i>) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. This work is the first step to a more generalized work on the three dimensional Tokamak structure. The highlight of this work is from the physical assumptions on the external magnetic potential well which remains finite <i>within a finite time interval</i> and from that we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical-coordinate-forms of the <i>Vlasov-Maxwell</i> system.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"chapman-jang-strain-onthedeterminantproblemfortherelativisticboltzmannequation-2021\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   James Chapman;  Jin Woo Jang;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/2006.02540\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'chapman-jang-strain-onthedeterminantproblemfortherelativisticboltzmannequation-2021', 'https://arxiv.org/abs/2006.02540', event);\">\n    On the Determinant Problem for the Relativistic Boltzmann Equation.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Math. Phys.</i>, 384: 1913–1943.  2021.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/2006.02540\"\n     onclick=\"log_download('chapman-jang-strain-onthedeterminantproblemfortherelativisticboltzmannequation-2021', 'https://arxiv.org/abs/2006.02540', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"On the Determinant Problem for the Relativistic Boltzmann Equation [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00220-021-04101-2\"\n     onclick=\"log_download('chapman-jang-strain-onthedeterminantproblemfortherelativisticboltzmannequation-2021', 'http://doi.org/10.1007/s00220-021-04101-2', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/chapman-jang-strain-onthedeterminantproblemfortherelativisticboltzmannequation-2021\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>11 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('chapman-jang-strain-onthedeterminantproblemfortherelativisticboltzmannequation-2021')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{ChapmanJangStrain2020,\n\tabstract = {This article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in the center-of-momentum coordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular we mention the widely used cancellation lemma by Alexandre et al. (Arch. Ration. Mech. Anal. 152(4):327--355, 2000). In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentum $p$ to the post collisional momentum $p&#x27;$; specifically we calculate the determinant for $p\\mapsto u = \\theta p&#x27;+(1-\\theta )p$ for $\\theta \\in [0,1]$. Afterwards we give an upper-bound for this determinant that has no singularity in both $p$ and $q$ variables. Next we give an example where we prove that the Jacobian goes to zero in a specific pointwise limit. We further explain the results of our numerical study which shows that the Jacobian determinant has a very large number of distinct points at which it is machine zero. This generalizes the work of Glassey-Strauss (Transport Theory Statist. Phys. 20(1):55--68, 1991) and Guo-Strain (Comm. Math. Phys. 310(3):649--673, 2012). These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates.},\n\tauthor = {Chapman, James and Jang, Jin Woo and Strain, Robert M.},\n\tdoi = {10.1007/s00220-021-04101-2},\n\teprint = {2006.02540},\n\tfjournal = {Communications in Mathematical Physics},\n\tjournal = {Comm. Math. Phys.},\n\tkeywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\n\tpages = {1913--1943},\n\tpublished = {published online May 07, 2021},\n\ttitle = {{On the Determinant Problem for the Relativistic Boltzmann Equation}},\n\turl_arxiv = {https://arxiv.org/abs/2006.02540},\n\tvolume = {384},\n\tyear = {2021},\n\tbdsk-url-1 = {https://doi.org/10.1007/s00220-021-04101-2}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  This article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in the center-of-momentum coordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular we mention the widely used cancellation lemma by Alexandre et al. (Arch. Ration. Mech. Anal. 152(4):327–355, 2000). In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentum $p$ to the post collisional momentum $p'$; specifically we calculate the determinant for $p↦ u = θ p'+(1-θ )p$ for $θ ∈ [0,1]$. Afterwards we give an upper-bound for this determinant that has no singularity in both $p$ and $q$ variables. Next we give an example where we prove that the Jacobian goes to zero in a specific pointwise limit. We further explain the results of our numerical study which shows that the Jacobian determinant has a very large number of distinct points at which it is machine zero. This generalizes the work of Glassey-Strauss (Transport Theory Statist. Phys. 20(1):55–68, 1991) and Guo-Strain (Comm. Math. Phys. 310(3):649–673, 2012). These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"duan-liu-sakamoto-strain-globalmildsolutionsofthelandauandnoncutoffboltzmannequations-2021\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Renjun Duan;  Shuangqian Liu;  Shota Sakamoto;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1904.12086\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'duan-liu-sakamoto-strain-globalmildsolutionsofthelandauandnoncutoffboltzmannequations-2021', 'https://arxiv.org/abs/1904.12086', event);\">\n    Global mild solutions of the Landau and non-cutoff Boltzmann equations.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Pure Appl. Math.</i>, 74(5): 932–1020.  2021.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1904.12086\"\n     onclick=\"log_download('duan-liu-sakamoto-strain-globalmildsolutionsofthelandauandnoncutoffboltzmannequations-2021', 'https://arxiv.org/abs/1904.12086', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Global mild solutions of the Landau and non-cutoff Boltzmann equations [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1002/cpa.21920\"\n     onclick=\"log_download('duan-liu-sakamoto-strain-globalmildsolutionsofthelandauandnoncutoffboltzmannequations-2021', 'http://doi.org/10.1002/cpa.21920', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/duan-liu-sakamoto-strain-globalmildsolutionsofthelandauandnoncutoffboltzmannequations-2021\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>10 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('duan-liu-sakamoto-strain-globalmildsolutionsofthelandauandnoncutoffboltzmannequations-2021')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{1904.12086,\n\tabstract = {This paper proves the existence of small-amplitude global-in-time unique mild\nsolutions to both the Landau equation including the Coulomb potential and the\nBoltzmann equation without angular cutoff. Since the well-known works (Guo,\n2002) and (Gressman-Strain-2011, AMUXY-2012) on the construction of classical\nsolutions in smooth Sobolev spaces which in particular are regular in the\nspatial variables, it still remains an open problem to obtain global solutions\nin an $L^\\infty_{x,v}$ framework, similar to that in (Guo-2010), for the\nBoltzmann equation with cutoff in general bounded domains. One main difficulty\narises from the interaction between the transport operator and the\nvelocity-diffusion-type collision operator in the non-cutoff Boltzmann and\nLandau equations; another major difficulty is the potential formation of\nsingularities for solutions to the boundary value problem. In the present work\nwe introduce a new function space with low regularity in the spatial variable\nto treat the problem in cases when the spatial domain is either a torus, or a\nfinite channel with boundary. For the latter case, either the inflow boundary\ncondition or the specular reflection boundary condition is considered. An\nimportant property of the function space is that the $L^\\infty_T L^2_v$ norm,\nin velocity and time, of the distribution function is in the Wiener algebra\n$A(\\Omega)$ in the spatial variables. Besides the construction of global\nsolutions in these function spaces, we additionally study the large-time\nbehavior of solutions for both hard and soft potentials, and we further justify\nthe property of propagation of regularity of solutions in the spatial\nvariables.},\n\tauthor = {Renjun Duan and Shuangqian Liu and Shota Sakamoto and Robert M. Strain},\n\tdoi = {10.1002/cpa.21920},\n\teprint = {1904.12086},\n\tfjournal = {Communications on Pure and Applied Mathematics},\n\tjournal = {Comm. Pure Appl. Math.},\n\tkeywords = {Boltzmann equation, Landau equation, Kinetic Theory, non-cutoff},\n\tnumber = {5},\n\tpages = {932--1020},\n\ttitle = {Global mild solutions of the {L}andau and non-cutoff {B}oltzmann equations},\n\turl_arxiv = {https://arxiv.org/abs/1904.12086},\n\tvolume = {74},\n\tyear = {2021},\n\tbdsk-url-1 = {https://arxiv.org/abs/1904.12086}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  This paper proves the existence of small-amplitude global-in-time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well-known works (Guo, 2002) and (Gressman-Strain-2011, AMUXY-2012) on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem to obtain global solutions in an $L^∞_{x,v}$ framework, similar to that in (Guo-2010), for the Boltzmann equation with cutoff in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity-diffusion-type collision operator in the non-cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem. In the present work we introduce a new function space with low regularity in the spatial variable to treat the problem in cases when the spatial domain is either a torus, or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the $L^∞_T L^2_v$ norm, in velocity and time, of the distribution function is in the Wiener algebra $A(Ω)$ in the spatial variables. Besides the construction of global solutions in these function spaces, we additionally study the large-time behavior of solutions for both hard and soft potentials, and we further justify the property of propagation of regularity of solutions in the spatial variables.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"jang-strain-yun-propagationofuniformupperboundsforthespatiallyhomogeneousrelativisticboltzmannequation-2021\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Jin Woo Jang;  Robert M. Strain;  and  Seok-Bae Yun.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1907.05784\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'jang-strain-yun-propagationofuniformupperboundsforthespatiallyhomogeneousrelativisticboltzmannequation-2021', 'https://arxiv.org/abs/1907.05784', event);\">\n    Propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Arch. Ration. Mech. Anal.</i>, 241: 149–186.  2021.\n  <span class=\"note\">published online April 15, 2021</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1907.05784\"\n     onclick=\"log_download('jang-strain-yun-propagationofuniformupperboundsforthespatiallyhomogeneousrelativisticboltzmannequation-2021', 'https://arxiv.org/abs/1907.05784', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00205-021-01649-0\"\n     onclick=\"log_download('jang-strain-yun-propagationofuniformupperboundsforthespatiallyhomogeneousrelativisticboltzmannequation-2021', 'http://doi.org/10.1007/s00205-021-01649-0', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/jang-strain-yun-propagationofuniformupperboundsforthespatiallyhomogeneousrelativisticboltzmannequation-2021\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>9 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('jang-strain-yun-propagationofuniformupperboundsforthespatiallyhomogeneousrelativisticboltzmannequation-2021')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{1907.05784,\n\tabstract = {In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These $L^\\infty$ bounds have been known to be a challenging open problem in relativistic kinetic theory. To accomplish this, we establish two types of estimates for the gain part of the collision operator: first, we prove a potential type estimate and a relativistic hyper-surface integral estimate. We then combine those estimates using the relativistic counter-part of the Carleman representation to derive uniform control of the gain term for the relativistic collision operator. This allows us to prove the desired propagation of the uniform bounds of the solution. We further present two applications of the propagation of the uniform upper bounds: first we give another proof of the Boltzmann $H$-theorem, and second we prove the asymptotic convergence of solutions to the relativistic Maxwellian equilibrium.},\n\tauthor = {Jin Woo Jang and Robert M. Strain and Seok-Bae Yun},\n\tdoi = {10.1007/s00205-021-01649-0},\n\teprint = {1907.05784},\n\tfjournal = {Archive for Rational Mechanics and Analysis},\n\tjournal = {Arch. Ration. Mech. Anal.},\n\tkeywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\n\tnote = {published online April 15, 2021},\n\tpages = {149--186},\n\ttitle = {{Propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation}},\n\turl_arxiv = {https://arxiv.org/abs/1907.05784},\n\tvolume = {241},\n\tyear = {2021},\n\tbdsk-url-1 = {https://doi.org/10.1007/s00205-021-01649-0}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These $L^∞$ bounds have been known to be a challenging open problem in relativistic kinetic theory. To accomplish this, we establish two types of estimates for the gain part of the collision operator: first, we prove a potential type estimate and a relativistic hyper-surface integral estimate. We then combine those estimates using the relativistic counter-part of the Carleman representation to derive uniform control of the gain term for the relativistic collision operator. This allows us to prove the desired propagation of the uniform bounds of the solution. We further present two applications of the propagation of the uniform upper bounds: first we give another proof of the Boltzmann $H$-theorem, and second we prove the asymptotic convergence of solutions to the relativistic Maxwellian equilibrium.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2020\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2020')\"\n      onkeypress=\"toggleGroup('group_2020')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2020</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"duan-liu-sakamoto-strain-globalsolutionstotheboltzmannequationwithoutangularcutoffandthelandauequationwithcolombpotential-2020\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Renjun Duan;  Shuangqian Liu;  Shota Sakamoto;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"http://hdl.handle.net/2433/260676\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'duan-liu-sakamoto-strain-globalsolutionstotheboltzmannequationwithoutangularcutoffandthelandauequationwithcolombpotential-2020', 'http://hdl.handle.net/2433/260676', event);\">\n    Global solutions to the Boltzmann equation without angular cutoff and the Landau equation with Colomb potential.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>RIMS Kokyuroku Bessatsu</i>, B82: 29–46.  2020.\n  <span class=\"note\">proceedings of symposium on ``Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations''</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://hdl.handle.net/2433/260676\"\n     onclick=\"log_download('duan-liu-sakamoto-strain-globalsolutionstotheboltzmannequationwithoutangularcutoffandthelandauequationwithcolombpotential-2020', 'http://hdl.handle.net/2433/260676', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Global solutions to the Boltzmann equation without angular cutoff and the Landau equation with Colomb potential [link]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/duan-liu-sakamoto-strain-globalsolutionstotheboltzmannequationwithoutangularcutoffandthelandauequationwithcolombpotential-2020\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>6 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('duan-liu-sakamoto-strain-globalsolutionstotheboltzmannequationwithoutangularcutoffandthelandauequationwithcolombpotential-2020')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{RIMSbessatsu2020,\n\tabstract = {This report succinctly summarizes results proved in the authors&#x27; recent work (2019) where the unique existence of solutions to the Boltzmann equation without angular cut-off and the Landau equation with Coulomb potential are studied in a perturbation framework. A major feature is the use of the Wiener space $A(\\Omega)$, which can be expected to play a similar role to $L^\\infty$. Compared to the $L^2$-based solution spaces that were employed for prior known results, this function space enables us to establish a new global existence theory. One further feature is that, not only an initial value problem, but also an initial boundary value problem whose boundary conditions can be regarded as physical boundaries in some simple situation, are considered for both equations. \nIn addition to unique existence, large-time behavior of the solutions and propagation of spatial regularity are also proved. In the end of report, key ideas of the proof will be explained in a concise way.},\n\tauthor = {Renjun Duan and Shuangqian Liu and Shota Sakamoto and Robert M. Strain},\n\tfjournal = {RIMS K{{o}}ky{{u}}roku {B}essatsu},\n\tjournal = {RIMS K{{o}}ky{{u}}roku {B}essatsu},\n\tkeywords = {Boltzmann equation, Landau equation, Kinetic Theory, non-cutoff},\n\tnote = {proceedings of symposium on &#x60;&#x60;Regularity and Asymptotic Analysis for Critical Cases of Partial Differential Equations&#x27;&#x27;},\n\tpages = {29--46},\n\ttitle = {{G}lobal solutions to the {B}oltzmann equation without angular cutoff and the {L}andau equation with {C}olomb potential},\n\turlpdf = {http://hdl.handle.net/2433/260676},\n\tvolume = {B82},\n\tyear = {2020},\n\tbdsk-url-1 = {http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu.html}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  This report succinctly summarizes results proved in the authors' recent work (2019) where the unique existence of solutions to the Boltzmann equation without angular cut-off and the Landau equation with Coulomb potential are studied in a perturbation framework. A major feature is the use of the Wiener space $A(Ω)$, which can be expected to play a similar role to $L^∞$. Compared to the $L^2$-based solution spaces that were employed for prior known results, this function space enables us to establish a new global existence theory. One further feature is that, not only an initial value problem, but also an initial boundary value problem whose boundary conditions can be regarded as physical boundaries in some simple situation, are considered for both equations. In addition to unique existence, large-time behavior of the solutions and propagation of spatial regularity are also proved. In the end of report, key ideas of the proof will be explained in a concise way.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2019\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2019')\"\n      onkeypress=\"toggleGroup('group_2019')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2019</span>\n      <span class=\"bibbase_group_count\">\n        \n        (4)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"gancedo-garciajurez-patel-strain-onthemuskatproblemwithviscosityjumpglobalintimeresults-2019\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Francisco Gancedo;  Eduardo García-Juárez;  Neel Patel;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1710.11604\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'gancedo-garciajurez-patel-strain-onthemuskatproblemwithviscosityjumpglobalintimeresults-2019', 'https://arxiv.org/abs/1710.11604', event);\">\n    On the Muskat problem with viscosity jump: Global in time results.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Adv. Math.</i>, 345: 552–597.  2019.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1710.11604\"\n     onclick=\"log_download('gancedo-garciajurez-patel-strain-onthemuskatproblemwithviscosityjumpglobalintimeresults-2019', 'https://arxiv.org/abs/1710.11604', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"On the Muskat problem with viscosity jump: Global in time results [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1016/j.aim.2019.01.017\"\n     onclick=\"log_download('gancedo-garciajurez-patel-strain-onthemuskatproblemwithviscosityjumpglobalintimeresults-2019', 'http://doi.org/10.1016/j.aim.2019.01.017', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/gancedo-garciajurez-patel-strain-onthemuskatproblemwithviscosityjumpglobalintimeresults-2019\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>4 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('gancedo-garciajurez-patel-strain-onthemuskatproblemwithviscosityjumpglobalintimeresults-2019')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{GGPS19,\n\tabstract = {The Muskat problem models the filtration of two incompressible immiscible\nfluids of different characteristics in porous media. In this paper, we consider\nboth the 2D and 3D setting of two fluids of different constant densities and\ndifferent constant viscosities. In this situation, the related contour\nequations are non-local, not only in the evolution system, but also in the\nimplicit relation between the amplitude of the vorticity and the free\ninterface. Among other extra difficulties, no maximum principles are available\nfor the amplitude and the slopes of the interface in $L^\\infty$. We prove\nglobal in time existence results for medium size initial stable data in\ncritical spaces. We also enhance previous methods by showing smoothing (instant\nanalyticity), improving the medium size constant in 3D, together with sharp\ndecay rates of analytic norms. The found technique is twofold, giving\nill-posedness in unstable situations for very low regular solutions.},\n\tauthor = {Francisco Gancedo and Eduardo Garc{\\&#x27;{i}}a-Ju{\\&#x27;{a}}rez and Neel Patel and Robert M. Strain},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1016/j.aim.2019.01.017},\n\teprint = {1710.11604},\n\tfjournal = {Advances in Mathematics},\n\tissn = {0001-8708},\n\tjournal = {Adv. Math.},\n\tkeywords = {Fluid mechanics, Free boundary problems, Muskat problem},\n\tmrclass = {76S05},\n\tmrnumber = {3899970},\n\tpages = {552--597},\n\ttitle = {On the {M}uskat problem with viscosity jump: {G}lobal in time results},\n\turl_arxiv = {https://arxiv.org/abs/1710.11604},\n\tvolume = {345},\n\tyear = {2019},\n\tzblnumber = {07021548},\n\tbdsk-url-1 = {https://doi.org/10.1016/j.aim.2019.01.017}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The Muskat problem models the filtration of two incompressible immiscible fluids of different characteristics in porous media. In this paper, we consider both the 2D and 3D setting of two fluids of different constant densities and different constant viscosities. In this situation, the related contour equations are non-local, not only in the evolution system, but also in the implicit relation between the amplitude of the vorticity and the free interface. Among other extra difficulties, no maximum principles are available for the amplitude and the slopes of the interface in $L^∞$. We prove global in time existence results for medium size initial stable data in critical spaces. We also enhance previous methods by showing smoothing (instant analyticity), improving the medium size constant in 3D, together with sharp decay rates of analytic norms. The found technique is twofold, giving ill-posedness in unstable situations for very low regular solutions.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"liu-strain-globalstabilityforsolutionstotheexponentialpdedescribingepitaxialgrowth-2019\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Jian-Guo Liu;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1805.02246\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'liu-strain-globalstabilityforsolutionstotheexponentialpdedescribingepitaxialgrowth-2019', 'https://arxiv.org/abs/1805.02246', event);\">\n    Global stability for solutions to the exponential PDE describing epitaxial growth.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Interfaces Free Bound.</i>, 21(1): 61–86.  2019.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1805.02246\"\n     onclick=\"log_download('liu-strain-globalstabilityforsolutionstotheexponentialpdedescribingepitaxialgrowth-2019', 'https://arxiv.org/abs/1805.02246', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Global stability for solutions to the exponential PDE describing epitaxial growth [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.4171/IFB/417\"\n     onclick=\"log_download('liu-strain-globalstabilityforsolutionstotheexponentialpdedescribingepitaxialgrowth-2019', 'http://doi.org/10.4171/IFB/417', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/liu-strain-globalstabilityforsolutionstotheexponentialpdedescribingepitaxialgrowth-2019\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('liu-strain-globalstabilityforsolutionstotheexponentialpdedescribingepitaxialgrowth-2019')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{LS2019,\n\tabstract = {In this paper we prove the global existence, uniqueness, optimal large time\ndecay rates, and uniform gain of analyticity for the exponential PDE\n$h_t=\\Delta e^{-\\Delta h}$ in the whole space $\\mathbb{R}^d_x$. We assume the\ninitial data is of medium size in the critical Wiener algebra $\\Delta h \\in\nA(\\mathbb{R}^d)$. This exponential PDE was derived in (Krug, Dobbs, and\nMajaniemi in 1995) and more recently in (Marzuola and Weare 2013).},\n\tauthor = {Jian-Guo Liu and Robert M. Strain},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-08 14:21:35 -0600},\n\tdoi = {10.4171/IFB/417},\n\teprint = {1805.02246},\n\tfjournal = {Interfaces and Free Boundaries. Mathematical Analysis, Computation and Applications.},\n\tissn = {1463-9963},\n\tjournal = {Interfaces Free Bound.},\n\tkeywords = {Free boundary problems, Materials science},\n\tmrclass = {35K25 (35B40 35K55 35K65 74A50)},\n\tmrnumber = {3951578},\n\tnumber = {1},\n\tpages = {61--86},\n\ttitle = {Global stability for solutions to the exponential {PDE} describing epitaxial growth},\n\turl_arxiv = {https://arxiv.org/abs/1805.02246},\n\tvolume = {21},\n\tyear = {2019},\n\tzblnumber = {07084773},\n\tbdsk-url-1 = {https://doi.org/10.4171/IFB/417}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper we prove the global existence, uniqueness, optimal large time decay rates, and uniform gain of analyticity for the exponential PDE $h_t=Δ e^{-Δ h}$ in the whole space $ℝ^d_x$. We assume the initial data is of medium size in the critical Wiener algebra $Δ h ∈ A(ℝ^d)$. This exponential PDE was derived in (Krug, Dobbs, and Majaniemi in 1995) and more recently in (Marzuola and Weare 2013).\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-taskovi-entropydissipationestimatesfortherelativisticlandauequationandapplications-2019\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain;  and  Maja Tasković.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1806.08720\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-taskovi-entropydissipationestimatesfortherelativisticlandauequationandapplications-2019', 'https://arxiv.org/abs/1806.08720', event);\">\n    Entropy dissipation estimates for the relativistic Landau equation, and applications.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>J. Funct. Anal.</i>, 277(4): 1139–1201.  2019.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1806.08720\"\n     onclick=\"log_download('strain-taskovi-entropydissipationestimatesfortherelativisticlandauequationandapplications-2019', 'https://arxiv.org/abs/1806.08720', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Entropy dissipation estimates for the relativistic Landau equation, and applications [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1016/j.jfa.2019.04.007\"\n     onclick=\"log_download('strain-taskovi-entropydissipationestimatesfortherelativisticlandauequationandapplications-2019', 'http://doi.org/10.1016/j.jfa.2019.04.007', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-taskovi-entropydissipationestimatesfortherelativisticlandauequationandapplications-2019\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>6 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-taskovi-entropydissipationestimatesfortherelativisticlandauequationandapplications-2019')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{StrainTas2019,\n\tabstract = {In this paper we study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. Despite it&#x27;s physical importance, this equation has not received a lot of mathematical attention we think due to the extreme complexity of the relativistic structure of the kernel of the collision operator. In this paper we first largely decompose the structure of the relativistic Landau collision operator. After that we prove the global Entropy dissipation estimate. Then we prove the propagation of any polynomial moment for a weak solution. Lastly we prove the existence of a true weak solution for a large class of initial data.},\n\tauthor = {Strain, Robert M. and Taskovi{\\&#x27;{c}}, Maja},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1016/j.jfa.2019.04.007},\n\teprint = {1806.08720},\n\tfjournal = {Journal of Functional Analysis},\n\tissn = {0022-1236},\n\tjournal = {J. Funct. Anal.},\n\tkeywords = {relativistic Landau equation, Landau equation, Kinetic Theory, relativistic Kinetic Theory},\n\tmrclass = {82D10 (35A01 35B45 35Q75)},\n\tmrnumber = {3959729},\n\tnumber = {4},\n\tpages = {1139--1201},\n\ttitle = {{E}ntropy dissipation estimates for the relativistic {L}andau equation, and applications},\n\turl_arxiv = {https://arxiv.org/abs/1806.08720},\n\tvolume = {277},\n\tyear = {2019},\n\tzblnumber = {07066836},\n\tbdsk-url-1 = {https://doi.org/10.1016/j.jfa.2019.04.007}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper we study the Cauchy problem for the spatially homogeneous relativistic Landau equation with Coulomb interactions. Despite it's physical importance, this equation has not received a lot of mathematical attention we think due to the extreme complexity of the relativistic structure of the kernel of the collision operator. In this paper we first largely decompose the structure of the relativistic Landau collision operator. After that we prove the global Entropy dissipation estimate. Then we prove the propagation of any polynomial moment for a weak solution. Lastly we prove the existence of a true weak solution for a large class of initial data.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-wang-uniquenessofboundedsolutionsforthehomogeneousrelativisticlandauequationwithcoulombinteractions-2019\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain;  and  Zhenfu Wang.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1903.05301\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-wang-uniquenessofboundedsolutionsforthehomogeneousrelativisticlandauequationwithcoulombinteractions-2019', 'https://arxiv.org/abs/1903.05301', event);\">\n    Uniqueness of Bounded Solutions for the Homogeneous Relativistic Landau Equation with Coulomb Interactions.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Quart. Appl. Math.</i>, 78: 107–145.  2019.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1903.05301\"\n     onclick=\"log_download('strain-wang-uniquenessofboundedsolutionsforthehomogeneousrelativisticlandauequationwithcoulombinteractions-2019', 'https://arxiv.org/abs/1903.05301', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Uniqueness of Bounded Solutions for the Homogeneous Relativistic Landau Equation with Coulomb Interactions [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1090/qam/1545\"\n     onclick=\"log_download('strain-wang-uniquenessofboundedsolutionsforthehomogeneousrelativisticlandauequationwithcoulombinteractions-2019', 'http://doi.org/10.1090/qam/1545', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-wang-uniquenessofboundedsolutionsforthehomogeneousrelativisticlandauequationwithcoulombinteractions-2019\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>1 download</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-wang-uniquenessofboundedsolutionsforthehomogeneousrelativisticlandauequationwithcoulombinteractions-2019')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{1903.05301,\n\tabstract = {We prove the uniqueness of weak solutions to the spatially homogeneous special relativistic Landau equation under the conditional assumption that the solution satisfies $(p^0)^7 F(t,p) \\in L^1 ([0,T]; L^\\infty)$.  The existence of standard weak solutions to the relativistic Landau equation has been shown recently in (Strain-Taskovi{\\&#x27;{c}} 2019, 1806.08720).},\n\tauthor = {Robert M. Strain and Zhenfu Wang},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1090/qam/1545},\n\teprint = {1903.05301},\n\tfjournal = {Quarterly of Applied Mathematics},\n\tjournal = {Quart. Appl. Math.},\n\tkeywords = {relativistic Landau equation, Landau equation, Kinetic Theory, relativistic Kinetic Theory},\n\tmrnumber = {4042221},\n\tpages = {107--145},\n\ttitle = {Uniqueness of Bounded Solutions for the Homogeneous Relativistic {L}andau Equation with {C}oulomb Interactions},\n\turl_arxiv = {https://arxiv.org/abs/1903.05301},\n\tvolume = {78},\n\tyear = {2019},\n\tzblnumber = {1427.82044},\n\tbdsk-url-1 = {https://arxiv.org/abs/1903.05301}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We prove the uniqueness of weak solutions to the spatially homogeneous special relativistic Landau equation under the conditional assumption that the solution satisfies $(p^0)^7 F(t,p) ∈ L^1 ([0,T]; L^∞)$. The existence of standard weak solutions to the relativistic Landau equation has been shown recently in (Strain-Tasković 2019, 1806.08720).\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2017\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2017')\"\n      onkeypress=\"toggleGroup('group_2017')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2017</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"patel-strain-largetimedecayestimatesforthemuskatequation-2017\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Neel Patel;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1610.05271\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'patel-strain-largetimedecayestimatesforthemuskatequation-2017', 'https://arxiv.org/abs/1610.05271', event);\">\n    Large time decay estimates for the Muskat equation.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Partial Differential Equations</i>, 42(6): 977–999.  2017.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1610.05271\"\n     onclick=\"log_download('patel-strain-largetimedecayestimatesforthemuskatequation-2017', 'https://arxiv.org/abs/1610.05271', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Large time decay estimates for the Muskat equation [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1080/03605302.2017.1321661\"\n     onclick=\"log_download('patel-strain-largetimedecayestimatesforthemuskatequation-2017', 'http://doi.org/10.1080/03605302.2017.1321661', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/patel-strain-largetimedecayestimatesforthemuskatequation-2017\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>4 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('patel-strain-largetimedecayestimatesforthemuskatequation-2017')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{161005271,\n\tabstract = {We prove time decay of solutions to the Muskat equation in 2D and in 3D. We consider the norm $\\|f\\|_{s}(t)$.  In this paper, for the 3D Muskat problem, given initial data $f_{0}\\in H^{l}(\\mathbb{R}^{2})$ for some $l\\geq 3$ such that $\\|f_{0}\\|_{1} &lt; k_{0}$ for a constant $k_{0} \\approx 1/5$, we prove uniform in time bounds of $\\|f\\|_{s}(t)$ for $-d &lt; s &lt; l-1$ and assuming $\\|f_{0}\\|_{\\nu} &lt; \\infty$ we prove time decay estimates of the form $\\|f\\|_{s}(t) \\lesssim (1+t)^{-s+\\nu}$ for $0 \\leq s \\leq l-1$ and $-d \\leq \\nu &lt; s$.  These large time decay rates are the same as the optimal rate for the linear Muskat equation.  We also prove analogous results in 2D.},\n\tauthor = {Patel, Neel and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1080/03605302.2017.1321661},\n\teprint = {1610.05271},\n\tfjournal = {Communications in Partial Differential Equations},\n\tissn = {0360-5302},\n\tjournal = {Comm. Partial Differential Equations},\n\tkeywords = {Fluid mechanics, Free boundary problems, Muskat problem},\n\tmrclass = {76D03 (35B40 35Q35 76S05)},\n\tmrnumber = {3683311},\n\tmrreviewer = {Youcef Amirat},\n\tnumber = {6},\n\tpages = {977--999},\n\ttitle = {Large time decay estimates for the {M}uskat equation},\n\turl_arxiv = {https://arxiv.org/abs/1610.05271},\n\tvolume = {42},\n\tyear = {2017},\n\tzblnumber = {1378.35245},\n\tbdsk-url-1 = {https://doi.org/10.1080/03605302.2017.1321661}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We prove time decay of solutions to the Muskat equation in 2D and in 3D. We consider the norm $\\|f\\|_{s}(t)$. In this paper, for the 3D Muskat problem, given initial data $f_{0}∈ H^{l}(ℝ^{2})$ for some $l≥ 3$ such that $\\|f_{0}\\|_{1} < k_{0}$ for a constant $k_{0} ≈ 1/5$, we prove uniform in time bounds of $\\|f\\|_{s}(t)$ for $-d < s < l-1$ and assuming $\\|f_{0}\\|_{ν} < ∞$ we prove time decay estimates of the form $\\|f\\|_{s}(t) łesssim (1+t)^{-s+ν}$ for $0 ≤ s ≤ l-1$ and $-d ≤ ν < s$. These large time decay rates are the same as the optimal rate for the linear Muskat equation. We also prove analogous results in 2D.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2016\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2016')\"\n      onkeypress=\"toggleGroup('group_2016')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2016</span>\n      <span class=\"bibbase_group_count\">\n        \n        (3)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"constantin-crdoba-gancedo-rodriguezpiazza-strain-onthemuskatproblemglobalintimeresultsin2dand3d-2016\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Peter Constantin;  Diego Córdoba;  Francisco Gancedo;  Luis Rodríguez-Piazza;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1310.0953\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'constantin-crdoba-gancedo-rodriguezpiazza-strain-onthemuskatproblemglobalintimeresultsin2dand3d-2016', 'https://arxiv.org/abs/1310.0953', event);\">\n    On the Muskat problem: global in time results in 2D and 3D.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Amer. J. Math.</i>, 138(6): 1455–1494.  2016.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1310.0953\"\n     onclick=\"log_download('constantin-crdoba-gancedo-rodriguezpiazza-strain-onthemuskatproblemglobalintimeresultsin2dand3d-2016', 'https://arxiv.org/abs/1310.0953', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"On the Muskat problem: global in time results in 2D and 3D [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1353/ajm.2016.0044\"\n     onclick=\"log_download('constantin-crdoba-gancedo-rodriguezpiazza-strain-onthemuskatproblemglobalintimeresultsin2dand3d-2016', 'http://doi.org/10.1353/ajm.2016.0044', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/constantin-crdoba-gancedo-rodriguezpiazza-strain-onthemuskatproblemglobalintimeresultsin2dand3d-2016\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('constantin-crdoba-gancedo-rodriguezpiazza-strain-onthemuskatproblemglobalintimeresultsin2dand3d-2016')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{CCGRS16,\n\tabstract = {This paper considers the three dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an $L^2$ maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants.  Furthermore we refine the estimates from our prior paper to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in critical spaces, giving solutions with bounded slope and time integrable bounded curvature.},\n\tauthor = {Constantin, Peter and C\\&#x27;{o}rdoba, Diego and Gancedo, Francisco and Rodr\\&#x27;{i}guez-Piazza, Luis and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1353/ajm.2016.0044},\n\teprint = {1310.0953},\n\tfjournal = {American Journal of Mathematics},\n\tissn = {0002-9327},\n\tjournal = {Amer. J. Math.},\n\tkeywords = {Fluid mechanics, Free boundary problems, Muskat problem},\n\tmrclass = {35Q35 (76S05)},\n\tmrnumber = {3595492},\n\tmrreviewer = {Maria Specovius-Neugebauer},\n\tnumber = {6},\n\tpages = {1455--1494},\n\ttitle = {On the {M}uskat problem: global in time results in 2{D} and 3{D}},\n\turl_arxiv = {https://arxiv.org/abs/1310.0953},\n\tvolume = {138},\n\tyear = {2016},\n\tzblnumber = {1369.35053},\n\tbdsk-url-1 = {https://doi.org/10.1353/ajm.2016.0044}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  This paper considers the three dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an $L^2$ maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants. Furthermore we refine the estimates from our prior paper to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in critical spaces, giving solutions with bounded slope and time integrable bounded curvature.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"luk-strain-strichartzestimatesandmomentboundsfortherelativisticvlasovmaxwellsystem-2016\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Jonathan Luk;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1406.0168\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'luk-strain-strichartzestimatesandmomentboundsfortherelativisticvlasovmaxwellsystem-2016', 'https://arxiv.org/abs/1406.0168', event);\">\n    Strichartz estimates and moment bounds for the relativistic Vlasov-Maxwell system.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Arch. Ration. Mech. Anal.</i>, 219(1): 445–552.  2016.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1406.0168\"\n     onclick=\"log_download('luk-strain-strichartzestimatesandmomentboundsfortherelativisticvlasovmaxwellsystem-2016', 'https://arxiv.org/abs/1406.0168', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Strichartz estimates and moment bounds for the relativistic Vlasov-Maxwell system [link]\"\n       title=\" arxiv1\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv1</span></a>\n  &nbsp;\n  \n  <a href=\"https://arxiv.org/abs/1406.0169\"\n     onclick=\"log_download('luk-strain-strichartzestimatesandmomentboundsfortherelativisticvlasovmaxwellsystem-2016', 'https://arxiv.org/abs/1406.0169', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Strichartz estimates and moment bounds for the relativistic Vlasov-Maxwell system [link]\"\n       title=\" arxiv2\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv2</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00205-015-0899-1\"\n     onclick=\"log_download('luk-strain-strichartzestimatesandmomentboundsfortherelativisticvlasovmaxwellsystem-2016', 'http://doi.org/10.1007/s00205-015-0899-1', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/luk-strain-strichartzestimatesandmomentboundsfortherelativisticvlasovmaxwellsystem-2016\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>1 download</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('luk-strain-strichartzestimatesandmomentboundsfortherelativisticvlasovmaxwellsystem-2016')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR3437855,\n\tabstract = {We consider the relativistic Vlasov-Maxwell system with data of unrestricted size and without compact support in momentum space. In the two dimensional and the two-and-a-half dimensional cases, Glassey-Schaeffer proved (Commun Math Phys 185:257--284, 1997; Arch Ration Mech Anal 141:331--354, 1998; Arch Ration Mech Anal. 141:355--374, 1998) that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the 2D and the $2\\frac 12$D cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the $L^\\infty_x$ norms of the electromagnetic fields compared to (Commun Math Phys 185:257--284, 1997; Arch Ration Mech Anal 141:355--374, 1998).  In the three dimensional case, we apply Strichartz estimates and moment bounds to show that a regular solution can be extended as long as \n$\\|(p^0)^{\\theta} f \\|_{L^q_xL^1_{p}}$ remains bounded for $\\theta&gt;{2/q}$ and $q \\in (2, \\infty]$. This improves previous results of Pallard (Indiana Univ Math J 54(5):1395--1409, 2005; Commun Math Sci 13(2):347--354, 2015).},\n\tauthor = {Luk, Jonathan and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-08 13:24:39 -0600},\n\tdoi = {10.1007/s00205-015-0899-1},\n\tfjournal = {Archive for Rational Mechanics and Analysis},\n\tissn = {0003-9527},\n\tjournal = {Arch. Ration. Mech. Anal.},\n\tkeywords = {Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},\n\tmrclass = {35Q83 (35Q61 35Q75 35Q82 76Y05 82D10)},\n\tmrnumber = {3437855},\n\tmrreviewer = {\\L ukasz P\\l ociniczak},\n\tnumber = {1},\n\tpages = {445--552},\n\ttitle = {Strichartz estimates and moment bounds for the relativistic {V}lasov-{M}axwell system},\n\turl_arxiv1 = {https://arxiv.org/abs/1406.0168},\n\turl_arxiv2 = {https://arxiv.org/abs/1406.0169},\n\tvolume = {219},\n\tyear = {2016},\n\tzblnumber = {1337.35150},\n\tbdsk-url-1 = {https://doi.org/10.1007/s00205-015-0899-1}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We consider the relativistic Vlasov-Maxwell system with data of unrestricted size and without compact support in momentum space. In the two dimensional and the two-and-a-half dimensional cases, Glassey-Schaeffer proved (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:331–354, 1998; Arch Ration Mech Anal. 141:355–374, 1998) that for regular initial data with compact momentum support this system has unique global in time classical solutions. In this work we do not assume compact momentum support for the initial data and instead require only that the data have polynomial decay in momentum space. In the 2D and the $2\\frac 12$D cases, we prove the global existence, uniqueness and regularity for solutions arising from this class of initial data. To this end we use Strichartz estimates and prove that suitable moments of the solution remain bounded. Moreover, we obtain a slight improvement of the temporal growth of the $L^∞_x$ norms of the electromagnetic fields compared to (Commun Math Phys 185:257–284, 1997; Arch Ration Mech Anal 141:355–374, 1998). In the three dimensional case, we apply Strichartz estimates and moment bounds to show that a regular solution can be extended as long as $\\|(p^0)^{θ} f \\|_{L^q_xL^1_{p}}$ remains bounded for $θ>{2/q}$ and $q ∈ (2, ∞]$. This improves previous results of Pallard (Indiana Univ Math J 54(5):1395–1409, 2005; Commun Math Sci 13(2):347–354, 2015).\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-wong-axisymmetricflowofidealfluidmovinginanarrowdomainastudyoftheaxisymmetrichydrostaticeulerequations-2016\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain;  and  Tak Kwong Wong.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1505.06281\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-wong-axisymmetricflowofidealfluidmovinginanarrowdomainastudyoftheaxisymmetrichydrostaticeulerequations-2016', 'https://arxiv.org/abs/1505.06281', event);\">\n    Axisymmetric flow of ideal fluid moving in a narrow domain: a study of the axisymmetric hydrostatic Euler equations.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>J. Differential Equations</i>, 260(5): 4619–4656.  2016.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1505.06281\"\n     onclick=\"log_download('strain-wong-axisymmetricflowofidealfluidmovinginanarrowdomainastudyoftheaxisymmetrichydrostaticeulerequations-2016', 'https://arxiv.org/abs/1505.06281', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Axisymmetric flow of ideal fluid moving in a narrow domain: a study of the axisymmetric hydrostatic Euler equations [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1016/j.jde.2015.11.023\"\n     onclick=\"log_download('strain-wong-axisymmetricflowofidealfluidmovinginanarrowdomainastudyoftheaxisymmetrichydrostaticeulerequations-2016', 'http://doi.org/10.1016/j.jde.2015.11.023', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-wong-axisymmetricflowofidealfluidmovinginanarrowdomainastudyoftheaxisymmetrichydrostaticeulerequations-2016\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-wong-axisymmetricflowofidealfluidmovinginanarrowdomainastudyoftheaxisymmetrichydrostaticeulerequations-2016')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR3437599,\n\tabstract = {In this article we will introduce a new model to describe the leading order behavior of an ideal and axisymmetric fluid moving in a very narrow domain. After providing a formal derivation of the model, we will prove the well-posedness and provide a rigorous mathematical justification for the formal derivation under a new sign condition. Finally, a blowup result regarding this model will be discussed as well.},\n\tauthor = {Strain, Robert M. and Wong, Tak Kwong},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-08 13:17:31 -0600},\n\tdoi = {10.1016/j.jde.2015.11.023},\n\teprint = {1505.06281},\n\tfjournal = {Journal of Differential Equations},\n\tissn = {0022-0396},\n\tjournal = {J. Differential Equations},\n\tkeywords = {Fluid mechanics},\n\tmrclass = {35Q31 (35B30 35Q35 76B03)},\n\tmrnumber = {3437599},\n\tmrreviewer = {Franck Sueur},\n\tnumber = {5},\n\tpages = {4619--4656},\n\ttitle = {Axisymmetric flow of ideal fluid moving in a narrow domain: a study of the axisymmetric hydrostatic {E}uler equations},\n\turl_arxiv = {https://arxiv.org/abs/1505.06281},\n\tvolume = {260},\n\tyear = {2016},\n\tzblnumber = {1333.35170},\n\tbdsk-url-1 = {https://doi.org/10.1016/j.jde.2015.11.023}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this article we will introduce a new model to describe the leading order behavior of an ideal and axisymmetric fluid moving in a very narrow domain. After providing a formal derivation of the model, we will prove the well-posedness and provide a rigorous mathematical justification for the formal derivation under a new sign condition. Finally, a blowup result regarding this model will be discussed as well.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2014\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2014')\"\n      onkeypress=\"toggleGroup('group_2014')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2014</span>\n      <span class=\"bibbase_group_count\">\n        \n        (4)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"gancedo-strain-absenceofsplashsingularitiesforsurfacequasigeostrophicsharpfrontsandthemuskatproblem-2014\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Francisco Gancedo;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1309.4023\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'gancedo-strain-absenceofsplashsingularitiesforsurfacequasigeostrophicsharpfrontsandthemuskatproblem-2014', 'https://arxiv.org/abs/1309.4023', event);\">\n    Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Proc. Natl. Acad. Sci. USA</i>, 111(2): 635–639.  2014.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1309.4023\"\n     onclick=\"log_download('gancedo-strain-absenceofsplashsingularitiesforsurfacequasigeostrophicsharpfrontsandthemuskatproblem-2014', 'https://arxiv.org/abs/1309.4023', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1073/pnas.1320554111\"\n     onclick=\"log_download('gancedo-strain-absenceofsplashsingularitiesforsurfacequasigeostrophicsharpfrontsandthemuskatproblem-2014', 'http://doi.org/10.1073/pnas.1320554111', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/gancedo-strain-absenceofsplashsingularitiesforsurfacequasigeostrophicsharpfrontsandthemuskatproblem-2014\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('gancedo-strain-absenceofsplashsingularitiesforsurfacequasigeostrophicsharpfrontsandthemuskatproblem-2014')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR3181769,\n\tabstract = {In this paper for either the sharp front Surface Quasi-Geostrophic equation or the Muskat problem we rule out the &#x60;&#x27;splash singularity&#x27;&#x27; blow-up scenario; in other words we prove that the contours evolving from either of these systems can not intersect at a single point while the free boundary remains smooth. Splash singularities have been shown to hold for the free boundary incompressible Euler equation in the form of the water waves contour evolution problem (Castro A, et al. (2012) Proc Natl Acad Sci USA 109:733-738). Our result confirms the numerical simulations in (C{\\&#x27;o}rdoba D, et al. (2005) Proc Natl Acad Sci USA 102:5949-5952) where it is shown that the curvature blows up due to the contours collapsing at a point. Here we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse. Another conclusion that we provide is a better understanding of the work (C{\\&#x27;o}rdoba D, Gancedo F (2010) Comm Math Phys 299:561-575) in which squirt singularities are ruled out; in this case a positive volume of fluid between the contours can not be ejected in finite time.},\n\tauthor = {Gancedo, Francisco and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1073/pnas.1320554111},\n\teprint = {1309.4023},\n\tfjournal = {Proceedings of the National Academy of Sciences of the United States of America},\n\tissn = {0027-8424},\n\tjournal = {Proc. Natl. Acad. Sci. USA},\n\tkeywords = {Fluid mechanics, Free boundary problems, Muskat problem},\n\tmrclass = {76S05 (35B35 76D27 76Txx 86A05)},\n\tmrnumber = {3181769},\n\tmrreviewer = {Jos\\&#x27;{e} Miguel Pacheco Castelao},\n\tnumber = {2},\n\tpages = {635--639},\n\tpublisher = {Proceedings of the National Academy of Sciences},\n\ttitle = {Absence of splash singularities for surface quasi-geostrophic sharp fronts and the {M}uskat problem},\n\turl_arxiv = {https://arxiv.org/abs/1309.4023},\n\tvolume = {111},\n\tyear = {2014},\n\tzblnumber = {1355.76065},\n\tbdsk-url-1 = {https://www.pnas.org/content/111/2/635},\n\tbdsk-url-2 = {https://doi.org/10.1073/pnas.1320554111}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper for either the sharp front Surface Quasi-Geostrophic equation or the Muskat problem we rule out the `'splash singularity'' blow-up scenario; in other words we prove that the contours evolving from either of these systems can not intersect at a single point while the free boundary remains smooth. Splash singularities have been shown to hold for the free boundary incompressible Euler equation in the form of the water waves contour evolution problem (Castro A, et al. (2012) Proc Natl Acad Sci USA 109:733-738). Our result confirms the numerical simulations in (Córdoba D, et al. (2005) Proc Natl Acad Sci USA 102:5949-5952) where it is shown that the curvature blows up due to the contours collapsing at a point. Here we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse. Another conclusion that we provide is a better understanding of the work (Córdoba D, Gancedo F (2010) Comm Math Phys 299:561-575) in which squirt singularities are ruled out; in this case a positive volume of fluid between the contours can not be ejected in finite time.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"luk-strain-anewcontinuationcriterionfortherelativisticvlasovmaxwellsystem-2014\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Jonathan Luk;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1406.0165\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'luk-strain-anewcontinuationcriterionfortherelativisticvlasovmaxwellsystem-2014', 'https://arxiv.org/abs/1406.0165', event);\">\n    A new continuation criterion for the relativistic Vlasov-Maxwell system.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Math. Phys.</i>, 331(3): 1005–1027.  2014.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1406.0165\"\n     onclick=\"log_download('luk-strain-anewcontinuationcriterionfortherelativisticvlasovmaxwellsystem-2014', 'https://arxiv.org/abs/1406.0165', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A new continuation criterion for the relativistic Vlasov-Maxwell system [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00220-014-2108-8\"\n     onclick=\"log_download('luk-strain-anewcontinuationcriterionfortherelativisticvlasovmaxwellsystem-2014', 'http://doi.org/10.1007/s00220-014-2108-8', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/luk-strain-anewcontinuationcriterionfortherelativisticvlasovmaxwellsystem-2014\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>3 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('luk-strain-anewcontinuationcriterionfortherelativisticvlasovmaxwellsystem-2014')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR3248056,\n\tabstract = {The global existence of solutions to the relativistic Vlasov-Maxwell system given sufficiently regular finite energy initial data is a longstanding open problem.  The main result of Glassey-Strauss 1986 shows that a solution $(f, E, B)$ remains $C^1$ as long as the momentum support of $f$ remains bounded.  Alternate proofs were later given by Bouchut-Golse-Pallard 2003 and Klainerman-Staffilani 2002.  We show that only the boundedness of the momentum support of $f$ {\\it after projecting to any two dimensional plane} is needed  for $(f, E, B)$ to remain $C^1$. },\n\tauthor = {Luk, Jonathan and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1007/s00220-014-2108-8},\n\teprint = {1406.0165},\n\tfjournal = {Communications in Mathematical Physics},\n\tissn = {0010-3616},\n\tjournal = {Comm. Math. Phys.},\n\tkeywords = {Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},\n\tmrclass = {35Q75 (35Q83 81T20)},\n\tmrnumber = {3248056},\n\tmrreviewer = {Calvin Tadmon},\n\tnumber = {3},\n\tpages = {1005--1027},\n\ttitle = {A new continuation criterion for the relativistic {V}lasov-{M}axwell system},\n\turl_arxiv = {https://arxiv.org/abs/1406.0165},\n\tvolume = {331},\n\tyear = {2014},\n\tzblnumber = {1309.35174},\n\tbdsk-url-1 = {https://doi.org/10.1007/s00220-014-2108-8}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The global existence of solutions to the relativistic Vlasov-Maxwell system given sufficiently regular finite energy initial data is a longstanding open problem. The main result of Glassey-Strauss 1986 shows that a solution $(f, E, B)$ remains $C^1$ as long as the momentum support of $f$ remains bounded. Alternate proofs were later given by Bouchut-Golse-Pallard 2003 and Klainerman-Staffilani 2002. We show that only the boundedness of the momentum support of $f$ ıt after projecting to any two dimensional plane is needed for $(f, E, B)$ to remain $C^1$. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"sohinger-strain-theboltzmannequationbesovspacesandoptimaltimedecayratesinxn-2014\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Vedran Sohinger;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1206.0027\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'sohinger-strain-theboltzmannequationbesovspacesandoptimaltimedecayratesinxn-2014', 'https://arxiv.org/abs/1206.0027', event);\">\n    The Boltzmann equation, Besov spaces, and optimal time decay rates in $ℝ_x^n$.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Adv. Math.</i>, 261: 274–332.  2014.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1206.0027\"\n     onclick=\"log_download('sohinger-strain-theboltzmannequationbesovspacesandoptimaltimedecayratesinxn-2014', 'https://arxiv.org/abs/1206.0027', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"The Boltzmann equation, Besov spaces, and optimal time decay rates in $ℝ_x^n$ [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1016/j.aim.2014.04.012\"\n     onclick=\"log_download('sohinger-strain-theboltzmannequationbesovspacesandoptimaltimedecayratesinxn-2014', 'http://doi.org/10.1016/j.aim.2014.04.012', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/sohinger-strain-theboltzmannequationbesovspacesandoptimaltimedecayratesinxn-2014\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('sohinger-strain-theboltzmannequationbesovspacesandoptimaltimedecayratesinxn-2014')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR3213301,\n\tabstract = {We prove that $k$-th order derivatives of perturbative classical solutions to the hard and soft potential Boltzmann equation (without the angular cut-off assumption) in the whole space, ${\\mathbb R}^{n}_x$ with ${n \\ge 3}$, converge in large time to the global Maxwellian with the optimal decay rate of   $t^{-\\frac{1}{2}(k+r+\\frac{n}{2}-\\frac{n}{r})}$\nin the $L^r_x(L^2_{v})$-norm for any $2\\leq r\\leq \\infty$.   These results hold for any $r \\in (0, n/2]$ as long as initially $\\| f_0\\|_{\\dot{B}^{-r,\\infty}_2 L^2_{v}} &lt; \\infty$.   In the hard potential case, we prove faster decay results in the sense that if $\\| \\mathbf{P} f_0\\|_{\\dot{B}^{-r,\\infty}_2 L^2_{v}} &lt; \\infty$ and \n$\\| \\{\\mathbf{I} - \\mathbf{P}\\} f_0\\|_{\\dot{B}^{-r+1,\\infty}_2 L^2_{v}} &lt; \\infty$ for $r \\in (n/2, (n+2)/2]$  then the solution decays the global Maxwellian in $L^2_v(L^2_x)$ with the optimal large time decay rate of $t^{-\\frac{1}{2} r}$.},\n\tauthor = {Sohinger, Vedran and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-15 15:47:31 -0400},\n\tdoi = {10.1016/j.aim.2014.04.012},\n\teprint = {1206.0027},\n\tfjournal = {Advances in Mathematics},\n\tissn = {0001-8708},\n\tjournal = {Adv. Math.},\n\tkeywords = {Boltzmann equation, non-cutoff, Kinetic Theory},\n\tmrclass = {35Q20 (35B40 35F25 76P05 82C40)},\n\tmrnumber = {3213301},\n\tmrreviewer = {Cecil Pompiliu Gr\\&quot;unfeld},\n\tmsc2010 = {35Q20 35R11 76P05 82C40 35B65 26A33},\n\tpages = {274--332},\n\tpublisher = {Elsevier (Academic Press), San Diego, CA},\n\ttitle = {The {B}oltzmann equation, {B}esov spaces, and optimal time decay rates in {$\\mathbb{R}_x^n$}},\n\turl_arxiv = {https://arxiv.org/abs/1206.0027},\n\tvolume = {261},\n\tyear = {2014},\n\tzblnumber = {1293.35195},\n\tbdsk-url-1 = {https://doi.org/10.1016/j.aim.2014.04.012}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We prove that $k$-th order derivatives of perturbative classical solutions to the hard and soft potential Boltzmann equation (without the angular cut-off assumption) in the whole space, ${\\mathbb R}^{n}_x$ with ${n \\ge 3}$, converge in large time to the global Maxwellian with the optimal decay rate of $t^{-\\frac{1}{2}(k+r+\\frac{n}{2}-\\frac{n}{r})}$ in the $L^r_x(L^2_{v})$-norm for any $2≤ r≤ ∞$. These results hold for any $r ∈ (0, n/2]$ as long as initially $\\| f_0\\|_{Ḃ^{-r,∞}_2 L^2_{v}} < ∞$. In the hard potential case, we prove faster decay results in the sense that if $\\| \\mathbf{P} f_0\\|_{Ḃ^{-r,∞}_2 L^2_{v}} < ∞$ and $\\| \\{\\mathbf{I} - \\mathbf{P}\\} f_0\\|_{Ḃ^{-r+1,∞}_2 L^2_{v}} < ∞$ for $r ∈ (n/2, (n+2)/2]$ then the solution decays the global Maxwellian in $L^2_v(L^2_x)$ with the optimal large time decay rate of $t^{-\\frac{1}{2} r}$.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-yun-spatiallyhomogeneousboltzmannequationforrelativisticparticles-2014\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain;  and  Seok-Bae Yun.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/92353.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-yun-spatiallyhomogeneousboltzmannequationforrelativisticparticles-2014', 'https://strain.math.upenn.edu/preprints/92353.pdf', event);\">\n    Spatially homogeneous Boltzmann equation for relativistic particles.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>SIAM J. Math. Anal.</i>, 46(1): 917–938.  2014.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/92353.pdf\"\n     onclick=\"log_download('strain-yun-spatiallyhomogeneousboltzmannequationforrelativisticparticles-2014', 'https://strain.math.upenn.edu/preprints/92353.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Spatially homogeneous Boltzmann equation for relativistic particles [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1137/130923531\"\n     onclick=\"log_download('strain-yun-spatiallyhomogeneousboltzmannequationforrelativisticparticles-2014', 'http://doi.org/10.1137/130923531', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-yun-spatiallyhomogeneousboltzmannequationforrelativisticparticles-2014\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>1 download</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-yun-spatiallyhomogeneousboltzmannequationforrelativisticparticles-2014')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR3166961,\n\tabstract = {The spatially homogeneous Boltzmann equation has been studied extensively in the Newtonian case, but not much is known for the special relativistic case. In this paper, we address several issues for the spatially homogeneous Boltzmann equation for relativistic particles. We first derive the relativistic version of the Povzner inequality. Using this, we study the Cauchy problem and investigate how the polynomial and exponential moments in $L^1$ are propagated. Several key differences between the relativistic and the Newtonian cases are confronted and discussed.},\n\tauthor = {Strain, Robert M. and Yun, Seok-Bae},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-08 11:57:02 -0600},\n\tdoi = {10.1137/130923531},\n\tfjournal = {SIAM Journal on Mathematical Analysis},\n\tissn = {0036-1410},\n\tjournal = {SIAM J. Math. Anal.},\n\tkeywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\n\tmrclass = {35Q20 (76P05 76Y05)},\n\tmrnumber = {3166961},\n\tmrreviewer = {Piotr Biler},\n\tnumber = {1},\n\tpages = {917--938},\n\ttitle = {Spatially homogeneous {B}oltzmann equation for relativistic particles},\n\turlpdf = {https://strain.math.upenn.edu/preprints/92353.pdf},\n\tvolume = {46},\n\tyear = {2014},\n\tzblnumber = {1321.35127},\n\tbdsk-url-1 = {https://doi.org/10.1137/130923531}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The spatially homogeneous Boltzmann equation has been studied extensively in the Newtonian case, but not much is known for the special relativistic case. In this paper, we address several issues for the spatially homogeneous Boltzmann equation for relativistic particles. We first derive the relativistic version of the Povzner inequality. Using this, we study the Cauchy problem and investigate how the polynomial and exponential moments in $L^1$ are propagated. Several key differences between the relativistic and the Newtonian cases are confronted and discussed.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2013\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2013')\"\n      onkeypress=\"toggleGroup('group_2013')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2013</span>\n      <span class=\"bibbase_group_count\">\n        \n        (3)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"constantin-crdoba-gancedo-strain-ontheglobalexistenceforthemuskatproblem-2013\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Peter Constantin;  Diego Córdoba;  Francisco Gancedo;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/ccgsMuskat.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'constantin-crdoba-gancedo-strain-ontheglobalexistenceforthemuskatproblem-2013', 'https://strain.math.upenn.edu/preprints/ccgsMuskat.pdf', event);\">\n    On the global existence for the Muskat problem.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>J. Eur. Math. Soc. (JEMS)</i>, 15(1): 201–227.  2013.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1007.3744\"\n     onclick=\"log_download('constantin-crdoba-gancedo-strain-ontheglobalexistenceforthemuskatproblem-2013', 'https://arxiv.org/abs/1007.3744', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"On the global existence for the Muskat problem [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/ccgsMuskat.pdf\"\n     onclick=\"log_download('constantin-crdoba-gancedo-strain-ontheglobalexistenceforthemuskatproblem-2013', 'https://strain.math.upenn.edu/preprints/ccgsMuskat.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"On the global existence for the Muskat problem [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.4171/JEMS/360\"\n     onclick=\"log_download('constantin-crdoba-gancedo-strain-ontheglobalexistenceforthemuskatproblem-2013', 'http://doi.org/10.4171/JEMS/360', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/constantin-crdoba-gancedo-strain-ontheglobalexistenceforthemuskatproblem-2013\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>4 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('constantin-crdoba-gancedo-strain-ontheglobalexistenceforthemuskatproblem-2013')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{CCGS13,\n\tabstract = {The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities.\nIn this work we prove three results.  First we prove an $L^2(\\mathbb{R})$ maximum principle, in the form of a new &#x60;&#x60;log&#x27;&#x27;  conservation law which is satisfied by the equation for the interface.  Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy \n$\\|f_0\\|_{L^\\infty}&lt;\\infty$ and $\\|\\partial_x f_0\\|_{L^\\infty}&lt;1$. We take advantage of the fact that the bound $\\|\\partial_x f_0\\|_{L^\\infty}&lt;1$ is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.\nLastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance $\\| f\\|_1 \\le 1/5$. Previous results of this sort used a small constant $\\epsilon \\ll1$ which was not explicit.},\n\tauthor = {Constantin, Peter and C\\&#x27;{o}rdoba, Diego and Gancedo, Francisco and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.4171/JEMS/360},\n\teprint = {1007.3744},\n\tfjournal = {Journal of the European Mathematical Society (JEMS)},\n\tissn = {1435-9855},\n\tjournal = {J. Eur. Math. Soc. (JEMS)},\n\tkeywords = {Fluid mechanics, Free boundary problems, Muskat problem},\n\tmrclass = {35Q35 (35B50 35D35 76S05)},\n\tmrnumber = {2998834},\n\tmrreviewer = {Weiran Sun},\n\tnumber = {1},\n\tpages = {201--227},\n\ttitle = {On the global existence for the {M}uskat problem},\n\turl_arxiv = {https://arxiv.org/abs/1007.3744},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/ccgsMuskat.pdf},\n\tvolume = {15},\n\tyear = {2013},\n\tzblnumber = {1258.35002},\n\tbdsk-url-1 = {https://doi.org/10.4171/JEMS/360}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2(ℝ)$ maximum principle, in the form of a new ``log'' conservation law which is satisfied by the equation for the interface. Our second result is a proof of global existence of Lipschitz continuous solutions for initial data that satisfy $\\|f_0\\|_{L^∞}<∞$ and $\\|∂_x f_0\\|_{L^∞}<1$. We take advantage of the fact that the bound $\\|∂_x f_0\\|_{L^∞}<1$ is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law. Lastly, we prove a global existence result for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance $\\| f\\|_1 łe 1/5$. Previous results of this sort used a small constant $ε ≪1$ which was not explicit.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"ha-jeong-strain-uniforml1stabilityoftherelativisticboltzmannequationnearvacuum-2013\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Seung-Yeal Ha;  Eunhee Jeong;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/cpaa0494.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'ha-jeong-strain-uniforml1stabilityoftherelativisticboltzmannequationnearvacuum-2013', 'https://strain.math.upenn.edu/preprints/cpaa0494.pdf', event);\">\n    Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Commun. Pure Appl. Anal.</i>, 12(2): 1141–1161.  2013.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/cpaa0494.pdf\"\n     onclick=\"log_download('ha-jeong-strain-uniforml1stabilityoftherelativisticboltzmannequationnearvacuum-2013', 'https://strain.math.upenn.edu/preprints/cpaa0494.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.3934/cpaa.2013.12.1141\"\n     onclick=\"log_download('ha-jeong-strain-uniforml1stabilityoftherelativisticboltzmannequationnearvacuum-2013', 'http://doi.org/10.3934/cpaa.2013.12.1141', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/ha-jeong-strain-uniforml1stabilityoftherelativisticboltzmannequationnearvacuum-2013\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('ha-jeong-strain-uniforml1stabilityoftherelativisticboltzmannequationnearvacuum-2013')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2982812,\n\tabstract = {We present the uniform $L^1$-stability estimate for the relativistic Boltzmann equation near vacuum. For this, we explicitly construct a relativistic counterpart of the nonlinear functional which is a linear combination of  $L^1$-distance and a collision potential. This functional measures the  $L^1$-distance between two continuous mild solutions. When the initial data is sufficiently small and decays exponentially fast, we show that the functional satisfies the uniform stability estimate leading to the uniform  $L^1$-stability estimate with respect to initial data.},\n\tauthor = {Ha, Seung-Yeal and Jeong, Eunhee and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-08 14:17:12 -0600},\n\tdoi = {10.3934/cpaa.2013.12.1141},\n\tfjournal = {Communications on Pure and Applied Analysis},\n\tissn = {1534-0392},\n\tjournal = {Commun. Pure Appl. Anal.},\n\tkeywords = {relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\n\tmrclass = {35Q20 (35B35 35Q75 82C40)},\n\tmrnumber = {2982812},\n\tmrreviewer = {Calvin Tadmon},\n\tnumber = {2},\n\tpages = {1141--1161},\n\ttitle = {Uniform {$L^1$}-stability of the relativistic {B}oltzmann equation near vacuum},\n\turlpdf = {https://strain.math.upenn.edu/preprints/cpaa0494.pdf},\n\tvolume = {12},\n\tyear = {2013},\n\tzblnumber = {1267.35162},\n\tbdsk-url-1 = {https://doi.org/10.3934/cpaa.2013.12.1141}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We present the uniform $L^1$-stability estimate for the relativistic Boltzmann equation near vacuum. For this, we explicitly construct a relativistic counterpart of the nonlinear functional which is a linear combination of $L^1$-distance and a collision potential. This functional measures the $L^1$-distance between two continuous mild solutions. When the initial data is sufficiently small and decays exponentially fast, we show that the functional satisfies the uniform stability estimate leading to the uniform $L^1$-stability estimate with respect to initial data.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-zhu-thevlasovpoissonlandausystemin3x-2013\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain;  and  Keya Zhu.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2012szLandauP.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-zhu-thevlasovpoissonlandausystemin3x-2013', 'https://strain.math.upenn.edu/preprints/2012szLandauP.pdf', event);\">\n    The Vlasov-Poisson-Landau system in $ℝ^3_x$.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Arch. Ration. Mech. Anal.</i>, 210(2): 615–671.  2013.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1202.2471\"\n     onclick=\"log_download('strain-zhu-thevlasovpoissonlandausystemin3x-2013', 'https://arxiv.org/abs/1202.2471', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"The Vlasov-Poisson-Landau system in $ℝ^3_x$ [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2012szLandauP.pdf\"\n     onclick=\"log_download('strain-zhu-thevlasovpoissonlandausystemin3x-2013', 'https://strain.math.upenn.edu/preprints/2012szLandauP.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"The Vlasov-Poisson-Landau system in $ℝ^3_x$ [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00205-013-0658-0\"\n     onclick=\"log_download('strain-zhu-thevlasovpoissonlandausystemin3x-2013', 'http://doi.org/10.1007/s00205-013-0658-0', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-zhu-thevlasovpoissonlandausystemin3x-2013\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-zhu-thevlasovpoissonlandausystemin3x-2013')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR3101794,\n\tabstract = {For the Landau-Poisson system with Coulomb interaction in $\\mathbb{R}^3_x$, we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close.},\n\tauthor = {Strain, Robert M. and Zhu, Keya},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1007/s00205-013-0658-0},\n\teprint = {1202.2471},\n\tfjournal = {Archive for Rational Mechanics and Analysis},\n\tissn = {0003-9527},\n\tjournal = {Arch. Ration. Mech. Anal.},\n\tkeywords = {Landau equation, Vlasov-Maxwell systems, Kinetic Theory},\n\tmrclass = {35Q83 (35A01 35A02 35B40)},\n\tmrnumber = {3101794},\n\tmrreviewer = {Jonathan Ben-Artzi},\n\tnumber = {2},\n\tpages = {615--671},\n\ttitle = {The {V}lasov-{P}oisson-{L}andau system in {$\\mathbb{R}^3_x$}},\n\turl_arxiv = {https://arxiv.org/abs/1202.2471},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/2012szLandauP.pdf},\n\tvolume = {210},\n\tyear = {2013},\n\tzblnumber = {1294.35168},\n\tbdsk-url-1 = {https://doi.org/10.1007/s00205-013-0658-0}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  For the Landau-Poisson system with Coulomb interaction in $ℝ^3_x$, we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2012\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2012')\"\n      onkeypress=\"toggleGroup('group_2012')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2012</span>\n      <span class=\"bibbase_group_count\">\n        \n        (7)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"dong-strain-onpartialregularityofsteadystatesolutionstothe6dnavierstokesequations-2012\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Hongjie Dong;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/ds6DssNSE.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'dong-strain-onpartialregularityofsteadystatesolutionstothe6dnavierstokesequations-2012', 'https://strain.math.upenn.edu/preprints/ds6DssNSE.pdf', event);\">\n    On partial regularity of steady-state solutions to the 6D Navier-Stokes equations.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Indiana Univ. Math. J.</i>, 61(6): 2211–2229.  2012.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1101.5580\"\n     onclick=\"log_download('dong-strain-onpartialregularityofsteadystatesolutionstothe6dnavierstokesequations-2012', 'https://arxiv.org/abs/1101.5580', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"On partial regularity of steady-state solutions to the 6D Navier-Stokes equations [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/ds6DssNSE.pdf\"\n     onclick=\"log_download('dong-strain-onpartialregularityofsteadystatesolutionstothe6dnavierstokesequations-2012', 'https://strain.math.upenn.edu/preprints/ds6DssNSE.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"On partial regularity of steady-state solutions to the 6D Navier-Stokes equations [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1512/iumj.2012.61.4765\"\n     onclick=\"log_download('dong-strain-onpartialregularityofsteadystatesolutionstothe6dnavierstokesequations-2012', 'http://doi.org/10.1512/iumj.2012.61.4765', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/dong-strain-onpartialregularityofsteadystatesolutionstothe6dnavierstokesequations-2012\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('dong-strain-onpartialregularityofsteadystatesolutionstothe6dnavierstokesequations-2012')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR3129108,\n\tabstract = {Consider steady-state weak solutions to the incompressible\nNavier-Stokes equations in six spatial dimensions. We prove that the 2D\nHausdorff measure of the set of singular points is equal to zero.  This problem was mentioned in 1988 by Struwe, during his study of the five dimensional case.},\n\tauthor = {Dong, Hongjie and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1512/iumj.2012.61.4765},\n\teprint = {1101.5580},\n\tfjournal = {Indiana University Mathematics Journal},\n\tissn = {0022-2518},\n\tjournal = {Indiana Univ. Math. J.},\n\tkeywords = {Navier-Stokes equations, Fluid mechanics},\n\tmrclass = {35Q30 (35B65 76D03 76D05)},\n\tmrnumber = {3129108},\n\tmrreviewer = {Kazuo Yamazaki},\n\tnumber = {6},\n\tpages = {2211--2229},\n\ttitle = {On partial regularity of steady-state solutions to the 6{D} {N}avier-{S}tokes equations},\n\turl_arxiv = {https://arxiv.org/abs/1101.5580},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/ds6DssNSE.pdf},\n\tvolume = {61},\n\tyear = {2012},\n\tzblnumber = {1286.35193},\n\tbdsk-url-1 = {https://doi.org/10.1512/iumj.2012.61.4765}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Consider steady-state weak solutions to the incompressible Navier-Stokes equations in six spatial dimensions. We prove that the 2D Hausdorff measure of the set of singular points is equal to zero. This problem was mentioned in 1988 by Struwe, during his study of the five dimensional case.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"duan-strain-onthefulldissipativepropertyofthevlasovpoissonboltzmannsystem-2012\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Renjun Duan;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2012DuanStrainHype.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'duan-strain-onthefulldissipativepropertyofthevlasovpoissonboltzmannsystem-2012', 'https://strain.math.upenn.edu/preprints/2012DuanStrainHype.pdf', event);\">\n    On the Full Dissipative Property of the Vlasov-Poisson-Boltzmann System.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  In  Tatsien Li;  and  Song Jiang., editor(s), <i>Hyperbolic problems—theory, numerics and applications. Volume 2</i>, volume 17, of Ser. Contemp. Appl. Math. CAM, pages 398–405. World Scientific Publishing and Higher Education Press, Singapore,  2012.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2012DuanStrainHype.pdf\"\n     onclick=\"log_download('duan-strain-onthefulldissipativepropertyofthevlasovpoissonboltzmannsystem-2012', 'https://strain.math.upenn.edu/preprints/2012DuanStrainHype.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"On the Full Dissipative Property of the Vlasov-Poisson-Boltzmann System [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10/c74v\"\n     onclick=\"log_download('duan-strain-onthefulldissipativepropertyofthevlasovpoissonboltzmannsystem-2012', 'http://doi.org/10/c74v', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/duan-strain-onthefulldissipativepropertyofthevlasovpoissonboltzmannsystem-2012\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>1 download</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('duan-strain-onthefulldissipativepropertyofthevlasovpoissonboltzmannsystem-2012')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@incollection{MR3050180,\n\tabstract = {In this paper, we present a new approach of studying the full dissipative property of the Vlasov-Poisson-Boltzmann system over the whole space. The key part of this approach is to design the interactive functional to capture the dissipation of the system along the degenerate components. The developed approach is generally applicable to other relevant models arising from plasma physics both at the kinetic and fluid levels.},\n\taddress = {Singapore},\n\tannote = {Proceedings of the HYP2010 conference in Beijing},\n\tauthor = {Renjun Duan and Robert M. Strain},\n\tbooktitle = {Hyperbolic problems---theory, numerics and applications. {V}olume 2},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-08 14:12:11 -0600},\n\tdoi = {10/c74v},\n\teditor = {Tatsien Li and Song Jiang},\n\tkeywords = {Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory},\n\tmrnumber = {3050180},\n\tpages = {398--405},\n\tpublisher = {World Scientific Publishing and Higher Education Press},\n\tseries = {Ser. Contemp. Appl. Math. CAM},\n\ttitle = {On the {F}ull {D}issipative {P}roperty of the {V}lasov-{P}oisson-{B}oltzmann {S}ystem},\n\turlpdf = {https://strain.math.upenn.edu/preprints/2012DuanStrainHype.pdf},\n\tvolume = {17},\n\tyear = {2012},\n\tzblnumber = {1293.35339},\n\tbdsk-url-1 = {https://doi.org/10.1142/9789814417099_0037}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper, we present a new approach of studying the full dissipative property of the Vlasov-Poisson-Boltzmann system over the whole space. The key part of this approach is to design the interactive functional to capture the dissipation of the system along the degenerate components. The developed approach is generally applicable to other relevant models arising from plasma physics both at the kinetic and fluid levels.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"gressman-krieger-strain-anonlocalinequalityandglobalexistence-2012\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Philip T. Gressman;  Joachim Krieger;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2012gksINEQ.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'gressman-krieger-strain-anonlocalinequalityandglobalexistence-2012', 'https://strain.math.upenn.edu/preprints/2012gksINEQ.pdf', event);\">\n    A non-local inequality and global existence.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Adv. Math.</i>, 230(2): 642–648.  2012.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1202.4088\"\n     onclick=\"log_download('gressman-krieger-strain-anonlocalinequalityandglobalexistence-2012', 'https://arxiv.org/abs/1202.4088', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A non-local inequality and global existence [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2012gksINEQ.pdf\"\n     onclick=\"log_download('gressman-krieger-strain-anonlocalinequalityandglobalexistence-2012', 'https://strain.math.upenn.edu/preprints/2012gksINEQ.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"A non-local inequality and global existence [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1016/j.aim.2012.02.017\"\n     onclick=\"log_download('gressman-krieger-strain-anonlocalinequalityandglobalexistence-2012', 'http://doi.org/10.1016/j.aim.2012.02.017', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/gressman-krieger-strain-anonlocalinequalityandglobalexistence-2012\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>6 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('gressman-krieger-strain-anonlocalinequalityandglobalexistence-2012')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2914961,\n\tabstract = {In this article we prove a collection of new non-linear and non-local integral inequalities.  We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions.  Specifically, we include all $\\alpha\\in (0, \\frac{74}{75})$.},\n\tauthor = {Gressman, Philip T. and Krieger, Joachim and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1016/j.aim.2012.02.017},\n\teprint = {1202.4088},\n\tfjournal = {Advances in Mathematics},\n\tissn = {0001-8708},\n\tjournal = {Adv. Math.},\n\tkeywords = {Landau equation, Kinetic Theory},\n\tmrclass = {35K55 (35A01 35B65 35R11)},\n\tmrnumber = {2914961},\n\tmrreviewer = {M. Pilar Velasco},\n\tnumber = {2},\n\tpages = {642--648},\n\ttitle = {A non-local inequality and global existence},\n\turl_arxiv = {https://arxiv.org/abs/1202.4088},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/2012gksINEQ.pdf},\n\tvolume = {230},\n\tyear = {2012},\n\tzblnumber = {1248.35005},\n\tbdsk-url-1 = {https://doi.org/10.1016/j.aim.2012.02.017}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this article we prove a collection of new non-linear and non-local integral inequalities. We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions. Specifically, we include all $α∈ (0, \\frac{74}{75})$.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"guo-strain-momentumregularityandstabilityoftherelativisticvlasovmaxwellboltzmannsystem-2012\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Yan Guo;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/gsRVMB.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'guo-strain-momentumregularityandstabilityoftherelativisticvlasovmaxwellboltzmannsystem-2012', 'https://strain.math.upenn.edu/preprints/gsRVMB.pdf', event);\">\n    Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Math. Phys.</i>, 310(3): 649–673.  2012.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1012.1158\"\n     onclick=\"log_download('guo-strain-momentumregularityandstabilityoftherelativisticvlasovmaxwellboltzmannsystem-2012', 'https://arxiv.org/abs/1012.1158', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/gsRVMB.pdf\"\n     onclick=\"log_download('guo-strain-momentumregularityandstabilityoftherelativisticvlasovmaxwellboltzmannsystem-2012', 'https://strain.math.upenn.edu/preprints/gsRVMB.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00220-012-1417-z\"\n     onclick=\"log_download('guo-strain-momentumregularityandstabilityoftherelativisticvlasovmaxwellboltzmannsystem-2012', 'http://doi.org/10.1007/s00220-012-1417-z', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/guo-strain-momentumregularityandstabilityoftherelativisticvlasovmaxwellboltzmannsystem-2012\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('guo-strain-momentumregularityandstabilityoftherelativisticvlasovmaxwellboltzmannsystem-2012')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2891870,\n\tabstract = {In the study of solutions to the relativistic Boltzmann equation, their\nregularity with respect to the momentum variables has been an outstanding\nquestion, even local in time, due to the initially unexpected growth in the\npost-collisional momentum variables which was discovered in 1991 by Glassey\n\\&amp; Strauss. We establish momentum regularity within energy\nspaces via a new splitting technique and interplay between the\nGlassey-Strauss frame and the center of mass frame of the relativistic collision\noperator. In a periodic box, these new momentum regularity estimates lead to\na proof of global existence of classical solutions to the two-species\nrelativistic Vlasov-Maxwell-Boltzmann system for charged particles near\nMaxwellian with hard ball interaction.},\n\tauthor = {Guo, Yan and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1007/s00220-012-1417-z},\n\teprint = {1012.1158},\n\tfjournal = {Communications in Mathematical Physics},\n\tissn = {0010-3616},\n\tjournal = {Comm. Math. Phys.},\n\tkeywords = {Boltzmann equation, relativistic Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory, relativistic Kinetic Theory},\n\tmrclass = {82D10 (35B35 35B65 35Q83 82C24 82C40)},\n\tmrnumber = {2891870},\n\tmrreviewer = {Stephen Wollman},\n\tnumber = {3},\n\tpages = {649--673},\n\ttitle = {Momentum regularity and stability of the relativistic {V}lasov-{M}axwell-{B}oltzmann system},\n\turl_arxiv = {https://arxiv.org/abs/1012.1158},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/gsRVMB.pdf},\n\tvolume = {310},\n\tyear = {2012},\n\tzblnumber = {1245.35130},\n\tbdsk-url-1 = {https://doi.org/10.1007/s00220-012-1417-z}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In the study of solutions to the relativistic Boltzmann equation, their regularity with respect to the momentum variables has been an outstanding question, even local in time, due to the initially unexpected growth in the post-collisional momentum variables which was discovered in 1991 by Glassey & Strauss. We establish momentum regularity within energy spaces via a new splitting technique and interplay between the Glassey-Strauss frame and the center of mass frame of the relativistic collision operator. In a periodic box, these new momentum regularity estimates lead to a proof of global existence of classical solutions to the two-species relativistic Vlasov-Maxwell-Boltzmann system for charged particles near Maxwellian with hard ball interaction.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"krieger-strain-globalsolutionstoanonlocaldiffusionequationwithquadraticnonlinearity-2012\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Joachim Krieger;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/03605302.2011.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'krieger-strain-globalsolutionstoanonlocaldiffusionequationwithquadraticnonlinearity-2012', 'https://strain.math.upenn.edu/preprints/03605302.2011.pdf', event);\">\n    Global solutions to a non-local diffusion equation with quadratic non-linearity.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Partial Differential Equations</i>, 37(4): 647–689.  2012.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1012.2890\"\n     onclick=\"log_download('krieger-strain-globalsolutionstoanonlocaldiffusionequationwithquadraticnonlinearity-2012', 'https://arxiv.org/abs/1012.2890', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Global solutions to a non-local diffusion equation with quadratic non-linearity [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/03605302.2011.pdf\"\n     onclick=\"log_download('krieger-strain-globalsolutionstoanonlocaldiffusionequationwithquadraticnonlinearity-2012', 'https://strain.math.upenn.edu/preprints/03605302.2011.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Global solutions to a non-local diffusion equation with quadratic non-linearity [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1080/03605302.2011.643437\"\n     onclick=\"log_download('krieger-strain-globalsolutionstoanonlocaldiffusionequationwithquadraticnonlinearity-2012', 'http://doi.org/10.1080/03605302.2011.643437', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/krieger-strain-globalsolutionstoanonlocaldiffusionequationwithquadraticnonlinearity-2012\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>9 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('krieger-strain-globalsolutionstoanonlocaldiffusionequationwithquadraticnonlinearity-2012')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2901061,\n\tabstract = {In this paper we prove the global in time well-posedness of the following non-local diffusion equation with $\\alpha \\in (0,2/3)$:\n$\\partial_t u = ((-\\Delta)^{-1}u)\\Delta u + \\alpha u^2$.  The initial condition $u_0$ is positive, radial, and non-increasing with\n$u_0\\in L^1\\cap L^{2+\\delta}({\\mathbb R}^3)$ for some small $\\delta &gt;0$.  There is no size restriction on $u_0$.\nThis model problem appears of interest due to its structural similarity with Landau&#x27;s equation from plasma physics, and moreover its radically different behavior from the semi-linear Heat equation: $u_t = \\Delta u + \\alpha u^2$. },\n\tauthor = {Krieger, Joachim and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1080/03605302.2011.643437},\n\teprint = {1012.2890},\n\tfjournal = {Communications in Partial Differential Equations},\n\tissn = {0360-5302},\n\tjournal = {Comm. Partial Differential Equations},\n\tkeywords = {Landau equation, Kinetic Theory},\n\tmrclass = {35K55 (35B30 35B65 35D35 35Q20)},\n\tmrnumber = {2901061},\n\tmrreviewer = {Jana Kopfova},\n\tnumber = {4},\n\tpages = {647--689},\n\ttitle = {Global solutions to a non-local diffusion equation with quadratic non-linearity},\n\turl_arxiv = {https://arxiv.org/abs/1012.2890},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/03605302.2011.pdf},\n\tvolume = {37},\n\tyear = {2012},\n\tzblnumber = {1247.35087},\n\tbdsk-url-1 = {https://doi.org/10.1080/03605302.2011.643437}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper we prove the global in time well-posedness of the following non-local diffusion equation with $α ∈ (0,2/3)$: $∂_t u = ((-Δ)^{-1}u)Δ u + α u^2$. The initial condition $u_0$ is positive, radial, and non-increasing with $u_0∈ L^1∩ L^{2+δ}({\\mathbb R}^3)$ for some small $δ >0$. There is no size restriction on $u_0$. This model problem appears of interest due to its structural similarity with Landau's equation from plasma physics, and moreover its radically different behavior from the semi-linear Heat equation: $u_t = Δ u + α u^2$. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-zhu-largetimedecayofthesoftpotentialrelativisticboltzmannequationinmathbbr3x-2012\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain;  and  Keya Zhu.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/szRBwhole.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-zhu-largetimedecayofthesoftpotentialrelativisticboltzmannequationinmathbbr3x-2012', 'https://strain.math.upenn.edu/preprints/szRBwhole.pdf', event);\">\n    Large-time decay of the soft potential relativistic Boltzmann equation in $\\mathbb R^3_x$.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Kinet. Relat. Models</i>, 5(2): 383–415.  2012.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1106.1579\"\n     onclick=\"log_download('strain-zhu-largetimedecayofthesoftpotentialrelativisticboltzmannequationinmathbbr3x-2012', 'https://arxiv.org/abs/1106.1579', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Large-time decay of the soft potential relativistic Boltzmann equation in $\\mathbb R^3_x$ [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/szRBwhole.pdf\"\n     onclick=\"log_download('strain-zhu-largetimedecayofthesoftpotentialrelativisticboltzmannequationinmathbbr3x-2012', 'https://strain.math.upenn.edu/preprints/szRBwhole.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Large-time decay of the soft potential relativistic Boltzmann equation in $\\mathbb R^3_x$ [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.3934/krm.2012.5.383\"\n     onclick=\"log_download('strain-zhu-largetimedecayofthesoftpotentialrelativisticboltzmannequationinmathbbr3x-2012', 'http://doi.org/10.3934/krm.2012.5.383', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-zhu-largetimedecayofthesoftpotentialrelativisticboltzmannequationinmathbbr3x-2012\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-zhu-largetimedecayofthesoftpotentialrelativisticboltzmannequationinmathbbr3x-2012')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2911100,\n\tabstract = {For the relativistic Boltzmann equation in $R^3_x$, this work proves the global existence, uniqueness, positivity, and optimal time convergence rates to the relativistic Maxwellian for solutions which start out sufficiently close under the general physical soft potential assumption proposed in 1988.},\n\tauthor = {Strain, Robert M. and Zhu, Keya},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-15 15:47:40 -0400},\n\tdoi = {10.3934/krm.2012.5.383},\n\teprint = {1106.1579},\n\tfjournal = {Kinetic and Related Models},\n\tissn = {1937-5093},\n\tjournal = {Kinet. Relat. Models},\n\tkeywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory},\n\tmrclass = {82C05 (76P05 76Y05)},\n\tmrnumber = {2911100},\n\tmrreviewer = {Mark Thompson},\n\tnumber = {2},\n\tpages = {383--415},\n\ttitle = {Large-time decay of the soft potential relativistic {B}oltzmann equation in {$\\mathbb R^3_x$}},\n\turl_arxiv = {https://arxiv.org/abs/1106.1579},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/szRBwhole.pdf},\n\tvolume = {5},\n\tyear = {2012},\n\tzblnumber = {1247.76071},\n\tbdsk-url-1 = {https://doi.org/10.3934/krm.2012.5.383}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  For the relativistic Boltzmann equation in $R^3_x$, this work proves the global existence, uniqueness, positivity, and optimal time convergence rates to the relativistic Maxwellian for solutions which start out sufficiently close under the general physical soft potential assumption proposed in 1988.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-optimaltimedecayofthenoncutoffboltzmannequationinthewholespace-2012\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/SdecaySOFT.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-optimaltimedecayofthenoncutoffboltzmannequationinthewholespace-2012', 'https://strain.math.upenn.edu/preprints/SdecaySOFT.pdf', event);\">\n    Optimal time decay of the non cut-off Boltzmann equation in the whole space.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Kinet. Relat. Models</i>, 5(3): 583–613.  2012.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1011.5561\"\n     onclick=\"log_download('strain-optimaltimedecayofthenoncutoffboltzmannequationinthewholespace-2012', 'https://arxiv.org/abs/1011.5561', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Optimal time decay of the non cut-off Boltzmann equation in the whole space [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/SdecaySOFT.pdf\"\n     onclick=\"log_download('strain-optimaltimedecayofthenoncutoffboltzmannequationinthewholespace-2012', 'https://strain.math.upenn.edu/preprints/SdecaySOFT.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Optimal time decay of the non cut-off Boltzmann equation in the whole space [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.3934/krm.2012.5.583\"\n     onclick=\"log_download('strain-optimaltimedecayofthenoncutoffboltzmannequationinthewholespace-2012', 'http://doi.org/10.3934/krm.2012.5.583', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-optimaltimedecayofthenoncutoffboltzmannequationinthewholespace-2012\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>1 download</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-optimaltimedecayofthenoncutoffboltzmannequationinthewholespace-2012')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2972454,\n\tabstract = {In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space ${\\mathbb R}^{n}_x$ with $n \\ge 3$.    We use the existence theory of global in time nearby Maxwellian solutions from previous work.  \nIt has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption.  For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of  $O(t^{-\\frac{n}{2}+\\frac{n}{2r}})$ in the $L^2_v(L^r_x)$-norm for any $2\\leq r\\leq \\infty$. },\n\tauthor = {Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.3934/krm.2012.5.583},\n\teprint = {1011.5561},\n\tfjournal = {Kinetic and Related Models},\n\tissn = {1937-5093},\n\tjournal = {Kinet. Relat. Models},\n\tkeywords = {Boltzmann equation, Kinetic Theory, non-cutoff},\n\tmrclass = {76P05 (26A33 35F20 82Cxx)},\n\tmrnumber = {2972454},\n\tnumber = {3},\n\tpages = {583--613},\n\ttitle = {Optimal time decay of the non cut-off {B}oltzmann equation in the whole space},\n\turl_arxiv = {https://arxiv.org/abs/1011.5561},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/SdecaySOFT.pdf},\n\tvolume = {5},\n\tyear = {2012},\n\tzblnumber = {1383.76414},\n\tbdsk-url-1 = {https://doi.org/10.3934/krm.2012.5.583}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space ${\\mathbb R}^{n}_x$ with $n \\ge 3$. We use the existence theory of global in time nearby Maxwellian solutions from previous work. It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption. For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of $O(t^{-\\frac{n}{2}+\\frac{n}{2r}})$ in the $L^2_v(L^r_x)$-norm for any $2≤ r≤ ∞$. \n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2011\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2011')\"\n      onkeypress=\"toggleGroup('group_2011')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2011</span>\n      <span class=\"bibbase_group_count\">\n        \n        (6)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"duan-strain-optimallargetimebehaviorofthevlasovmaxwellboltzmannsysteminthewholespace-2011\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Renjun Duan;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/dsVMB.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'duan-strain-optimallargetimebehaviorofthevlasovmaxwellboltzmannsysteminthewholespace-2011', 'https://strain.math.upenn.edu/preprints/dsVMB.pdf', event);\">\n    Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Pure Appl. Math.</i>, 64(11): 1497–1546.  2011.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1006.3605\"\n     onclick=\"log_download('duan-strain-optimallargetimebehaviorofthevlasovmaxwellboltzmannsysteminthewholespace-2011', 'https://arxiv.org/abs/1006.3605', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/dsVMB.pdf\"\n     onclick=\"log_download('duan-strain-optimallargetimebehaviorofthevlasovmaxwellboltzmannsysteminthewholespace-2011', 'https://strain.math.upenn.edu/preprints/dsVMB.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1002/cpa.20381\"\n     onclick=\"log_download('duan-strain-optimallargetimebehaviorofthevlasovmaxwellboltzmannsysteminthewholespace-2011', 'http://doi.org/10.1002/cpa.20381', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/duan-strain-optimallargetimebehaviorofthevlasovmaxwellboltzmannsysteminthewholespace-2011\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('duan-strain-optimallargetimebehaviorofthevlasovmaxwellboltzmannsysteminthewholespace-2011')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2832167,\n\tabstract = {In this paper we study the large-time behavior of classical\nsolutions to the two-species Vlasov-Maxwell-Boltzmann system in the\nwhole space $R^3_x$.    The existence of  global in time nearby\nMaxwellian solutions is known from the work of Strain in 2006.  However\nthe asymptotic behavior of these solutions  has been a challenging\nopen problem.  Building on our previous work on time\ndecay for the simpler Vlasov-Poisson-Boltzmann system, we prove that\nthese solutions converge to the global Maxwellian with the optimal\ndecay rate of  $O(t^{-\\frac{3}{2}+\\frac{3}{2r}})$ in\n$L^2_\\xi(L^r_x)$-norm for any $2\\leq r\\leq \\infty$ if initial\nperturbation is smooth enough and decays in space-velocity fast\nenough at infinity. Moreover, some explicit rates for the\nelectromagnetic field tending to zero are also provided.},\n\tauthor = {Duan, Renjun and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1002/cpa.20381},\n\teprint = {1006.3605},\n\tfjournal = {Communications on Pure and Applied Mathematics},\n\tissn = {0010-3640},\n\tjournal = {Comm. Pure Appl. Math.},\n\tkeywords = {Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory},\n\tmrclass = {82C40 (35B40 35Q20 35Q83 76W05)},\n\tmrnumber = {2832167},\n\tmrreviewer = {Stephen Wollman},\n\tnumber = {11},\n\tpages = {1497--1546},\n\ttitle = {Optimal large-time behavior of the {V}lasov-{M}axwell-{B}oltzmann system in the whole space},\n\turl_arxiv = {https://arxiv.org/abs/1006.3605},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/dsVMB.pdf},\n\tvolume = {64},\n\tyear = {2011},\n\tzblnumber = {1244.35010},\n\tbdsk-url-1 = {https://doi.org/10.1002/cpa.20381}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space $R^3_x$. The existence of global in time nearby Maxwellian solutions is known from the work of Strain in 2006. However the asymptotic behavior of these solutions has been a challenging open problem. Building on our previous work on time decay for the simpler Vlasov-Poisson-Boltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of $O(t^{-\\frac{3}{2}+\\frac{3}{2r}})$ in $L^2_ξ(L^r_x)$-norm for any $2≤ r≤ ∞$ if initial perturbation is smooth enough and decays in space-velocity fast enough at infinity. Moreover, some explicit rates for the electromagnetic field tending to zero are also provided.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"duan-strain-optimaltimedecayofthevlasovpoissonboltzmannsystemin3-2011\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Renjun Duan;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/dsVPB.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'duan-strain-optimaltimedecayofthevlasovpoissonboltzmannsystemin3-2011', 'https://strain.math.upenn.edu/preprints/dsVPB.pdf', event);\">\n    Optimal time decay of the Vlasov-Poisson-Boltzmann system in $ℝ^3$.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Arch. Ration. Mech. Anal.</i>, 199(1): 291–328.  2011.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/0912.1742\"\n     onclick=\"log_download('duan-strain-optimaltimedecayofthevlasovpoissonboltzmannsystemin3-2011', 'https://arxiv.org/abs/0912.1742', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Optimal time decay of the Vlasov-Poisson-Boltzmann system in $ℝ^3$ [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/dsVPB.pdf\"\n     onclick=\"log_download('duan-strain-optimaltimedecayofthevlasovpoissonboltzmannsystemin3-2011', 'https://strain.math.upenn.edu/preprints/dsVPB.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Optimal time decay of the Vlasov-Poisson-Boltzmann system in $ℝ^3$ [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00205-010-0318-6\"\n     onclick=\"log_download('duan-strain-optimaltimedecayofthevlasovpoissonboltzmannsystemin3-2011', 'http://doi.org/10.1007/s00205-010-0318-6', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/duan-strain-optimaltimedecayofthevlasovpoissonboltzmannsystemin3-2011\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('duan-strain-optimaltimedecayofthevlasovpoissonboltzmannsystemin3-2011')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2754344,\n\tabstract = {The Vlasov-Poisson-Boltzmann System governs the time evolution of\nthe distribution function for the dilute charged particles in the\npresence of a self-consistent electric potential force through the\nPoisson equation. In this paper, we are concerned with the rate of\nconvergence of solutions to equilibrium for this system over $R^3_x$.\nIt is shown that the electric field which is indeed responsible for\nthe lowest-order part in the energy space reduces the speed of\nconvergence and hence the dispersion of this system over the full\nspace is slower than that of the Boltzmann equation without forces,\nwhere the exact difference between both power indices in the\nalgebraic rates of convergence is $1/4$. For the proof, in the\nlinearized case with a given non-homogeneous source,  Fourier\nanalysis is employed to obtain time-decay properties of the solution\noperator. In the nonlinear case, the combination of the linearized\nresults and the nonlinear energy estimates with the help of the\nproper Lyapunov-type inequalities leads to the optimal time-decay\nrate of perturbed solutions under some conditions on initial data.},\n\tauthor = {Duan, Renjun and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1007/s00205-010-0318-6},\n\teprint = {0912.1742},\n\tfjournal = {Archive for Rational Mechanics and Analysis},\n\tissn = {0003-9527},\n\tjournal = {Arch. Ration. Mech. Anal.},\n\tkeywords = {Boltzmann equation, Vlasov-Maxwell systems, Kinetic Theory},\n\tmrclass = {35Q83 (35B40 35Q20 76P05 76X05 82C40 82D10)},\n\tmrnumber = {2754344},\n\tnumber = {1},\n\tpages = {291--328},\n\ttitle = {Optimal time decay of the {V}lasov-{P}oisson-{B}oltzmann system in {$\\mathbb{R}^3$}},\n\turl_arxiv = {https://arxiv.org/abs/0912.1742},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/dsVPB.pdf},\n\tvolume = {199},\n\tyear = {2011},\n\tzblnumber = {1232.35169},\n\tbdsk-url-1 = {https://doi.org/10.1007/s00205-010-0318-6}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The Vlasov-Poisson-Boltzmann System governs the time evolution of the distribution function for the dilute charged particles in the presence of a self-consistent electric potential force through the Poisson equation. In this paper, we are concerned with the rate of convergence of solutions to equilibrium for this system over $R^3_x$. It is shown that the electric field which is indeed responsible for the lowest-order part in the energy space reduces the speed of convergence and hence the dispersion of this system over the full space is slower than that of the Boltzmann equation without forces, where the exact difference between both power indices in the algebraic rates of convergence is $1/4$. For the proof, in the linearized case with a given non-homogeneous source, Fourier analysis is employed to obtain time-decay properties of the solution operator. In the nonlinear case, the combination of the linearized results and the nonlinear energy estimates with the help of the proper Lyapunov-type inequalities leads to the optimal time-decay rate of perturbed solutions under some conditions on initial data.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithoutangularcutoff-2011\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Philip T. Gressman;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/gsNonCut.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithoutangularcutoff-2011', 'https://strain.math.upenn.edu/preprints/gsNonCut.pdf', event);\">\n    Global classical solutions of the Boltzmann equation without angular cut-off.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>J. Amer. Math. Soc.</i>, 24(3): 771–847.  2011.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1011.5441\"\n     onclick=\"log_download('gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithoutangularcutoff-2011', 'https://arxiv.org/abs/1011.5441', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Global classical solutions of the Boltzmann equation without angular cut-off [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/gsNonCut.pdf\"\n     onclick=\"log_download('gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithoutangularcutoff-2011', 'https://strain.math.upenn.edu/preprints/gsNonCut.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Global classical solutions of the Boltzmann equation without angular cut-off [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1090/S0894-0347-2011-00697-8\"\n     onclick=\"log_download('gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithoutangularcutoff-2011', 'http://doi.org/10.1090/S0894-0347-2011-00697-8', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithoutangularcutoff-2011\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>9 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithoutangularcutoff-2011')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2784329,\n\tabstract = {This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials, $r^{-(p-1)}$ with $p&gt;2$,\nfor initial perturbations of the Maxwellian equilibrium states.  \nWe  more generally cover collision kernels with parameters $s\\in (0,1)$ and $\\gamma$ satisfying \n$\\gamma  &gt; -n$ in arbitrary dimensions $\\mathbb{T}^n \\times \\mathbb{R}^n$ with $n\\ge 2$.  \nMoreover, we prove rapid convergence as predicted by the celebrated Boltzmann $H$-theorem.\nWhen $\\gamma \\ge -2s$, we have  exponential time decay to the Maxwellian equilibrium states.  When $\\gamma &lt;-2s$, our solutions decay polynomially fast in time with any rate.  \nThese results are completely constructive.  Additionally, we prove  sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $\\gamma  \\ge -2s$, as conjectured in  Mouhot-Strain.  It will be observed that this fundamental equation, derived by both Boltzmann  and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world.  Our methods provide a new  understanding of the grazing collisions in the Boltzmann theory. },\n\tauthor = {Gressman, Philip T. and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1090/S0894-0347-2011-00697-8},\n\teprint = {1011.5441},\n\tfjournal = {Journal of the American Mathematical Society},\n\tissn = {0894-0347},\n\tjournal = {J. Amer. Math. Soc.},\n\tkeywords = {Boltzmann equation, Kinetic Theory, non-cutoff},\n\tmrclass = {82C40 (35H20 35Q20 35R11 76P05)},\n\tmrnumber = {2784329},\n\tmrreviewer = {Laurent Desvillettes},\n\tnumber = {3},\n\tpages = {771--847},\n\ttitle = {Global classical solutions of the {B}oltzmann equation without angular cut-off},\n\turl_arxiv = {https://arxiv.org/abs/1011.5441},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/gsNonCut.pdf},\n\tvolume = {24},\n\tyear = {2011},\n\tzblnumber = {1248.35140},\n\tbdsk-url-1 = {https://doi.org/10.1090/S0894-0347-2011-00697-8}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials, $r^{-(p-1)}$ with $p>2$, for initial perturbations of the Maxwellian equilibrium states. We more generally cover collision kernels with parameters $s∈ (0,1)$ and $γ$ satisfying $γ > -n$ in arbitrary dimensions $\\mathbb{T}^n × ℝ^n$ with $n\\ge 2$. Moreover, we prove rapid convergence as predicted by the celebrated Boltzmann $H$-theorem. When $γ \\ge -2s$, we have exponential time decay to the Maxwellian equilibrium states. When $γ <-2s$, our solutions decay polynomially fast in time with any rate. These results are completely constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $γ \\ge -2s$, as conjectured in Mouhot-Strain. It will be observed that this fundamental equation, derived by both Boltzmann and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the grazing collisions in the Boltzmann theory. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"gressman-strain-sharpanisotropicestimatesfortheboltzmanncollisionoperatoranditsentropyproduction-2011\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Philip T. Gressman;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/gsNonCutEst.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'gressman-strain-sharpanisotropicestimatesfortheboltzmanncollisionoperatoranditsentropyproduction-2011', 'https://strain.math.upenn.edu/preprints/gsNonCutEst.pdf', event);\">\n    Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Adv. Math.</i>, 227(6): 2349–2384.  2011.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1007.1276\"\n     onclick=\"log_download('gressman-strain-sharpanisotropicestimatesfortheboltzmanncollisionoperatoranditsentropyproduction-2011', 'https://arxiv.org/abs/1007.1276', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/gsNonCutEst.pdf\"\n     onclick=\"log_download('gressman-strain-sharpanisotropicestimatesfortheboltzmanncollisionoperatoranditsentropyproduction-2011', 'https://strain.math.upenn.edu/preprints/gsNonCutEst.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1016/j.aim.2011.05.005\"\n     onclick=\"log_download('gressman-strain-sharpanisotropicestimatesfortheboltzmanncollisionoperatoranditsentropyproduction-2011', 'http://doi.org/10.1016/j.aim.2011.05.005', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/gressman-strain-sharpanisotropicestimatesfortheboltzmanncollisionoperatoranditsentropyproduction-2011\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('gressman-strain-sharpanisotropicestimatesfortheboltzmanncollisionoperatoranditsentropyproduction-2011')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2807092,\n\tabstract = {This article provides sharp constructive upper and lower bound estimates for the  \nBoltzmann collision operator \nwith the full range of physical non cut-off collision kernels ($\\gamma &gt; -n$ and $s\\in (0,1)$)\n in the trilinear $L^2(\\mathbb{R}^n)$ energy $( \\mathcal{Q}(g,f), f )$.  These new estimates prove that, for a very general class of $g(v)$, the  global diffusive behavior (on $f$) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works.  We further prove new global entropy production estimates with the same anisotropic semi-norm.  \n This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non cut-off Boltzmann collision operator in the energy space \n$L^2(\\mathbb{R}^n)$.},\n\tauthor = {Gressman, Philip T. and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1016/j.aim.2011.05.005},\n\teprint = {1007.1276},\n\tfjournal = {Advances in Mathematics},\n\tissn = {0001-8708},\n\tjournal = {Adv. Math.},\n\tkeywords = {Boltzmann equation, Kinetic Theory, non-cutoff},\n\tmrclass = {35Q20 (35B65 35R11 82C40)},\n\tmrnumber = {2807092},\n\tmrreviewer = {Francesco Salvarani},\n\tnumber = {6},\n\tpages = {2349--2384},\n\ttitle = {Sharp anisotropic estimates for the {B}oltzmann collision operator and its entropy production},\n\turl_arxiv = {https://arxiv.org/abs/1007.1276},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/gsNonCutEst.pdf},\n\tvolume = {227},\n\tyear = {2011},\n\tzblnumber = {1234.35173},\n\tbdsk-url-1 = {https://doi.org/10.1016/j.aim.2011.05.005}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  This article provides sharp constructive upper and lower bound estimates for the Boltzmann collision operator with the full range of physical non cut-off collision kernels ($γ > -n$ and $s∈ (0,1)$) in the trilinear $L^2(ℝ^n)$ energy $( \\mathcal{Q}(g,f), f )$. These new estimates prove that, for a very general class of $g(v)$, the global diffusive behavior (on $f$) in the energy space is that of the geometric fractional derivative semi-norm identified in the linearized context in our earlier works. We further prove new global entropy production estimates with the same anisotropic semi-norm. This resolves the longstanding, widespread heuristic conjecture about the sharp diffusive nature of the non cut-off Boltzmann collision operator in the energy space $L^2(ℝ^n)$.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"speck-strain-hilbertexpansionfromtheboltzmannequationtorelativisticfluids-2011\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Jared Speck;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/ssHilbertExp.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'speck-strain-hilbertexpansionfromtheboltzmannequationtorelativisticfluids-2011', 'https://strain.math.upenn.edu/preprints/ssHilbertExp.pdf', event);\">\n    Hilbert expansion from the Boltzmann equation to relativistic fluids.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Math. Phys.</i>, 304(1): 229–280.  2011.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1009.5033\"\n     onclick=\"log_download('speck-strain-hilbertexpansionfromtheboltzmannequationtorelativisticfluids-2011', 'https://arxiv.org/abs/1009.5033', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Hilbert expansion from the Boltzmann equation to relativistic fluids [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/ssHilbertExp.pdf\"\n     onclick=\"log_download('speck-strain-hilbertexpansionfromtheboltzmannequationtorelativisticfluids-2011', 'https://strain.math.upenn.edu/preprints/ssHilbertExp.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Hilbert expansion from the Boltzmann equation to relativistic fluids [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00220-011-1207-z\"\n     onclick=\"log_download('speck-strain-hilbertexpansionfromtheboltzmannequationtorelativisticfluids-2011', 'http://doi.org/10.1007/s00220-011-1207-z', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/speck-strain-hilbertexpansionfromtheboltzmannequationtorelativisticfluids-2011\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>6 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('speck-strain-hilbertexpansionfromtheboltzmannequationtorelativisticfluids-2011')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2793935,\n\tabstract = {We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. \nMore specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellian constructed from a class of solutions to the relativistic Euler equations that includes a large \nsubclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Euler equations. },\n\tauthor = {Speck, Jared and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1007/s00220-011-1207-z},\n\teprint = {1009.5033},\n\tfjournal = {Communications in Mathematical Physics},\n\tissn = {0010-3616},\n\tjournal = {Comm. Math. Phys.},\n\tkeywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory, Fluid mechanics},\n\tmrclass = {82C40 (35Q20 76P05 76Y05)},\n\tmrnumber = {2793935},\n\tmrreviewer = {Stephen Wollman},\n\tnumber = {1},\n\tpages = {229--280},\n\ttitle = {Hilbert expansion from the {B}oltzmann equation to relativistic fluids},\n\turl_arxiv = {https://arxiv.org/abs/1009.5033},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/ssHilbertExp.pdf},\n\tvolume = {304},\n\tyear = {2011},\n\tzblnumber = {1221.35271},\n\tbdsk-url-1 = {https://doi.org/10.1007/s00220-011-1207-z}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We study the local-in-time hydrodynamic limit of the relativistic Boltzmann equation using a Hilbert expansion. More specifically, we prove the existence of local solutions to the relativistic Boltzmann equation that are nearby the local relativistic Maxwellian constructed from a class of solutions to the relativistic Euler equations that includes a large subclass of near-constant, non-vacuum fluid states. In particular, for small Knudsen number, these solutions to the relativistic Boltzmann equation have dynamics that are effectively captured by corresponding solutions to the relativistic Euler equations. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-coordinatesintherelativisticboltzmanntheory-2011\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/sCCkrm.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-coordinatesintherelativisticboltzmanntheory-2011', 'https://strain.math.upenn.edu/preprints/sCCkrm.pdf', event);\">\n    Coordinates in the relativistic Boltzmann theory.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Kinet. Relat. Models</i>, 4(1): 345–359.  2011.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1011.5093\"\n     onclick=\"log_download('strain-coordinatesintherelativisticboltzmanntheory-2011', 'https://arxiv.org/abs/1011.5093', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Coordinates in the relativistic Boltzmann theory [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/sCCkrm.pdf\"\n     onclick=\"log_download('strain-coordinatesintherelativisticboltzmanntheory-2011', 'https://strain.math.upenn.edu/preprints/sCCkrm.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Coordinates in the relativistic Boltzmann theory [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.3934/krm.2011.4.345\"\n     onclick=\"log_download('strain-coordinatesintherelativisticboltzmanntheory-2011', 'http://doi.org/10.3934/krm.2011.4.345', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-coordinatesintherelativisticboltzmanntheory-2011\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>1 download</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-coordinatesintherelativisticboltzmanntheory-2011')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2765751,\n\tabstract = {It is often the case in mathematical analysis that solving an open problem can be facilitated by finding a new set of coordinates  which may illuminate the known difficulties. In this article, we illustrate how to derive an assortment coordinates in which to represent the relativistic Boltzmann collision operator.  We show the equivalence between some known representations, and others which seem to be new.  One of these representations has been used recently to solve several open problems.},\n\tauthor = {Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.3934/krm.2011.4.345},\n\teprint = {1011.5093},\n\tfjournal = {Kinetic and Related Models},\n\tissn = {1937-5093},\n\tjournal = {Kinet. Relat. Models},\n\tkeywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\n\tmrclass = {82C40 (76P05 83A05)},\n\tmrnumber = {2765751},\n\tmrreviewer = {Stephen Wollman},\n\tnumber = {1},\n\tpages = {345--359},\n\ttitle = {Coordinates in the relativistic {B}oltzmann theory},\n\turl_arxiv = {https://arxiv.org/abs/1011.5093},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/sCCkrm.pdf},\n\tvolume = {4},\n\tyear = {2011},\n\tzblnumber = {05869610},\n\tbdsk-url-1 = {https://doi.org/10.3934/krm.2011.4.345}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  It is often the case in mathematical analysis that solving an open problem can be facilitated by finding a new set of coordinates which may illuminate the known difficulties. In this article, we illustrate how to derive an assortment coordinates in which to represent the relativistic Boltzmann collision operator. We show the equivalence between some known representations, and others which seem to be new. One of these representations has been used recently to solve several open problems.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2010\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2010')\"\n      onkeypress=\"toggleGroup('group_2010')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2010</span>\n      <span class=\"bibbase_group_count\">\n        \n        (4)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithlongrangeinteractions-2010\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Philip T. Gressman;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/gsPNAS2010.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithlongrangeinteractions-2010', 'https://strain.math.upenn.edu/preprints/gsPNAS2010.pdf', event);\">\n    Global classical solutions of the Boltzmann equation with long-range interactions.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Proc. Natl. Acad. Sci. USA</i>, 107(13): 5744–5749.  2010.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/gsPNAS2010.pdf\"\n     onclick=\"log_download('gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithlongrangeinteractions-2010', 'https://strain.math.upenn.edu/preprints/gsPNAS2010.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Global classical solutions of the Boltzmann equation with long-range interactions [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1073/pnas.1001185107\"\n     onclick=\"log_download('gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithlongrangeinteractions-2010', 'http://doi.org/10.1073/pnas.1001185107', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithlongrangeinteractions-2010\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('gressman-strain-globalclassicalsolutionsoftheboltzmannequationwithlongrangeinteractions-2010')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2629879,\n\tabstract = {This is a brief announcement of our recent proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation without any angular cutoff, that is, for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential $r^{-(p-1)}$ with $p &gt; 2$, and more generally. We present here a mathematical framework for unique global in time solutions for all of these potentials. We consider it remarkable that this equation, derived by Boltzmann (1) in 1872 and Maxwell (2) in 1867, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects due to grazing collisions.},\n\tauthor = {Gressman, Philip T. and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-08 11:55:21 -0600},\n\tdoi = {10.1073/pnas.1001185107},\n\tfjournal = {Proceedings of the National Academy of Sciences of the United States of America},\n\tissn = {1091-6490},\n\tjournal = {Proc. Natl. Acad. Sci. USA},\n\tkeywords = {Boltzmann equation, Kinetic Theory, non-cutoff},\n\tmrclass = {82C40 (35Q20)},\n\tmrnumber = {2629879},\n\tnumber = {13},\n\tpages = {5744--5749},\n\ttitle = {Global classical solutions of the {B}oltzmann equation with long-range interactions},\n\turlpdf = {https://strain.math.upenn.edu/preprints/gsPNAS2010.pdf},\n\tvolume = {107},\n\tyear = {2010},\n\tzblnumber = {1205.82120},\n\tbdsk-url-1 = {https://doi.org/10.1073/pnas.1001185107}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  This is a brief announcement of our recent proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation without any angular cutoff, that is, for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential $r^{-(p-1)}$ with $p > 2$, and more generally. We present here a mathematical framework for unique global in time solutions for all of these potentials. We consider it remarkable that this equation, derived by Boltzmann (1) in 1872 and Maxwell (2) in 1867, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects due to grazing collisions.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-globalnewtonianlimitfortherelativisticboltzmannequationnearvacuum-2010\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/rBnewt.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-globalnewtonianlimitfortherelativisticboltzmannequationnearvacuum-2010', 'https://strain.math.upenn.edu/preprints/rBnewt.pdf', event);\">\n    Global Newtonian limit for the relativistic Boltzmann equation near vacuum.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>SIAM J. Math. Anal.</i>, 42(4): 1568–1601.  2010.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1004.5407\"\n     onclick=\"log_download('strain-globalnewtonianlimitfortherelativisticboltzmannequationnearvacuum-2010', 'https://arxiv.org/abs/1004.5407', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Global Newtonian limit for the relativistic Boltzmann equation near vacuum [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/rBnewt.pdf\"\n     onclick=\"log_download('strain-globalnewtonianlimitfortherelativisticboltzmannequationnearvacuum-2010', 'https://strain.math.upenn.edu/preprints/rBnewt.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Global Newtonian limit for the relativistic Boltzmann equation near vacuum [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1137/090762695\"\n     onclick=\"log_download('strain-globalnewtonianlimitfortherelativisticboltzmannequationnearvacuum-2010', 'http://doi.org/10.1137/090762695', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-globalnewtonianlimitfortherelativisticboltzmannequationnearvacuum-2010\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>5 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-globalnewtonianlimitfortherelativisticboltzmannequationnearvacuum-2010')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2679588,\n\tabstract = {We study the Cauchy Problem for the relativistic Boltzmann equation with near Vacuum initial data. Unique global in time mild solutions are obtained uniformly in the speed of light parameter $c \\ge 1$.  We furthermore prove that solutions to the relativistic Boltzmann equation converge to solutions of  the Newtonian Boltzmann equation in the limit as $c\\to\\infty$ on arbitrary time intervals $[0,T]$, with  convergence rate $1/c^{2-\\epsilon}$ for any $\\epsilon \\in(0,2)$.  This may be the first proof of unique global in time validity of the Newtonian limit for a Kinetic equation.},\n\tauthor = {Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1137/090762695},\n\teprint = {1004.5407},\n\tfjournal = {SIAM Journal on Mathematical Analysis},\n\tissn = {0036-1410},\n\tjournal = {SIAM J. Math. Anal.},\n\tkeywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\n\tmrclass = {82C40 (35Q20 35Q75 76P05)},\n\tmrnumber = {2679588},\n\tmrreviewer = {Silvia Lorenzani},\n\tnumber = {4},\n\tpages = {1568--1601},\n\ttitle = {Global {N}ewtonian limit for the relativistic {B}oltzmann equation near vacuum},\n\turl_arxiv = {https://arxiv.org/abs/1004.5407},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/rBnewt.pdf},\n\tvolume = {42},\n\tyear = {2010},\n\tzblnumber = {05894999},\n\tbdsk-url-1 = {https://doi.org/10.1137/090762695}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We study the Cauchy Problem for the relativistic Boltzmann equation with near Vacuum initial data. Unique global in time mild solutions are obtained uniformly in the speed of light parameter $c \\ge 1$. We furthermore prove that solutions to the relativistic Boltzmann equation converge to solutions of the Newtonian Boltzmann equation in the limit as $c\\to∞$ on arbitrary time intervals $[0,T]$, with convergence rate $1/c^{2-ε}$ for any $ε ∈(0,2)$. This may be the first proof of unique global in time validity of the Newtonian limit for a Kinetic equation.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-asymptoticstabilityoftherelativisticboltzmannequationforthesoftpotentials-2010\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/sRBsoft.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-asymptoticstabilityoftherelativisticboltzmannequationforthesoftpotentials-2010', 'https://strain.math.upenn.edu/preprints/sRBsoft.pdf', event);\">\n    Asymptotic stability of the relativistic Boltzmann equation for the soft potentials.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Math. Phys.</i>, 300(2): 529–597.  2010.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1003.4893\"\n     onclick=\"log_download('strain-asymptoticstabilityoftherelativisticboltzmannequationforthesoftpotentials-2010', 'https://arxiv.org/abs/1003.4893', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Asymptotic stability of the relativistic Boltzmann equation for the soft potentials [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/sRBsoft.pdf\"\n     onclick=\"log_download('strain-asymptoticstabilityoftherelativisticboltzmannequationforthesoftpotentials-2010', 'https://strain.math.upenn.edu/preprints/sRBsoft.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Asymptotic stability of the relativistic Boltzmann equation for the soft potentials [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00220-010-1129-1\"\n     onclick=\"log_download('strain-asymptoticstabilityoftherelativisticboltzmannequationforthesoftpotentials-2010', 'http://doi.org/10.1007/s00220-010-1129-1', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-asymptoticstabilityoftherelativisticboltzmannequationforthesoftpotentials-2010\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-asymptoticstabilityoftherelativisticboltzmannequationforthesoftpotentials-2010')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2728733,\n\tabstract = {In this paper it is  shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in $L^\\infty_\\ell$.  If the initial data  are continuous then so is the corresponding solution.  We work in the case of a spatially periodic box.  Conditions on the collision kernel are generic; this resolves the open question of global existence for the soft potentials.},\n\tauthor = {Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1007/s00220-010-1129-1},\n\teprint = {1003.4893},\n\tfjournal = {Communications in Mathematical Physics},\n\tissn = {0010-3616},\n\tjournal = {Comm. Math. Phys.},\n\tkeywords = {Boltzmann equation, relativistic Boltzmann equation, Kinetic Theory, relativistic Kinetic Theory},\n\tmrclass = {82C40 (35Q20)},\n\tmrnumber = {2728733},\n\tmrreviewer = {Stephen Wollman},\n\tnumber = {2},\n\tpages = {529--597},\n\ttitle = {Asymptotic stability of the relativistic {B}oltzmann equation for the soft potentials},\n\turl_arxiv = {https://arxiv.org/abs/1003.4893},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/sRBsoft.pdf},\n\tvolume = {300},\n\tyear = {2010},\n\tzblnumber = {1214.35072},\n\tbdsk-url-1 = {https://doi.org/10.1007/s00220-010-1129-1}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in $L^∞_\\ell$. If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box. Conditions on the collision kernel are generic; this resolves the open question of global existence for the soft potentials.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-aroundtheboltzmannequationwithoutangularcutoff-2010\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/rms2010ow.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-aroundtheboltzmannequationwithoutangularcutoff-2010', 'https://strain.math.upenn.edu/preprints/rms2010ow.pdf', event);\">\n    Around the Boltzmann equation without angular cut-off.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Oberwolfach Rep.</i>, 7(4): 3159–3236.  2010.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/rms2010ow.pdf\"\n     onclick=\"log_download('strain-aroundtheboltzmannequationwithoutangularcutoff-2010', 'https://strain.math.upenn.edu/preprints/rms2010ow.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Around the Boltzmann equation without angular cut-off [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.4171/owr/2010/54\"\n     onclick=\"log_download('strain-aroundtheboltzmannequationwithoutangularcutoff-2010', 'http://doi.org/10.4171/owr/2010/54', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-aroundtheboltzmannequationwithoutangularcutoff-2010\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>1 download</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-aroundtheboltzmannequationwithoutangularcutoff-2010')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{Arnold2010,\n\tabstract = {In this report, we will describe briefly several recent developments for the Boltzmann equation without the Grad angular cut-off assumption.},\n\tabstractow = {The topic of this meeting were non-linear partial differential and integro-differential equations (in particular kinetic equations and their macroscopic/fluid-dynamical limits) modeling the dynamics of many-particle systems with applications in physics, engineering, and mathematical biology. Typical questions of interest were the derivation of macro-models from micro-models, the mathematical analysis (well-posedness, stability, asymptotic behavior of solutions), and &#x60;&#x60;to a lesser extent&#x27;&#x27; numerical aspects of such equations. A highlight of this meeting was a mini-course on the recent mathematical theory of Landau damping.},\n\tauthor = {Robert M. Strain},\n\tbooktitle = {{Classical and quantum mechanical models of many-particle systems. Workshop held December 5 -- December 11, 2010.}},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-08 13:59:00 -0600},\n\tdoi = {10.4171/owr/2010/54},\n\teditor = {Anton {Arnold} and Eric A. {Carlen} and Laurent {Desvillettes}},\n\tfjournal = {{Oberwolfach Reports}},\n\tissn = {1660-8933; 1660-8941/e},\n\tjournal = {{Oberwolfach Rep.}},\n\tkeywords = {Boltzmann equation, Kinetic Theory, non-cutoff},\n\tlanguage = {English},\n\tmsc2010 = {00B05 81-06 82-06 35Qxx 82Cxx 82B40 81S30},\n\tnumber = {4},\n\tpages = {3159--3236},\n\tpublisher = {European Mathematical Society (EMS) Publishing House, Zurich; Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach},\n\ttitle = {Around the {B}oltzmann equation without angular cut-off},\n\turlpdf = {https://strain.math.upenn.edu/preprints/rms2010ow.pdf},\n\tvolume = {7},\n\tyear = {2010},\n\tzblnumber = {1235.00027},\n\tbdsk-url-1 = {https://doi.org/10.4171/owr/2010/54}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this report, we will describe briefly several recent developments for the Boltzmann equation without the Grad angular cut-off assumption.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2009\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2009')\"\n      onkeypress=\"toggleGroup('group_2009')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2009</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"chen-strain-tsai-yau-lowerboundsontheblowuprateoftheaxisymmetricnavierstokesequationsii-2009\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Chiun-Chuan Chen;  Robert M. Strain;  Tai-Peng Tsai;  and  Horng-Tzer Yau.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2009ChenStrainTsaiYau.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'chen-strain-tsai-yau-lowerboundsontheblowuprateoftheaxisymmetricnavierstokesequationsii-2009', 'https://strain.math.upenn.edu/preprints/2009ChenStrainTsaiYau.pdf', event);\">\n    Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations. II.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Partial Differential Equations</i>, 34(1-3): 203–232.  2009.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/0709.4230\"\n     onclick=\"log_download('chen-strain-tsai-yau-lowerboundsontheblowuprateoftheaxisymmetricnavierstokesequationsii-2009', 'https://arxiv.org/abs/0709.4230', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations. II [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2009ChenStrainTsaiYau.pdf\"\n     onclick=\"log_download('chen-strain-tsai-yau-lowerboundsontheblowuprateoftheaxisymmetricnavierstokesequationsii-2009', 'https://strain.math.upenn.edu/preprints/2009ChenStrainTsaiYau.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations. II [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1080/03605300902793956\"\n     onclick=\"log_download('chen-strain-tsai-yau-lowerboundsontheblowuprateoftheaxisymmetricnavierstokesequationsii-2009', 'http://doi.org/10.1080/03605300902793956', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/chen-strain-tsai-yau-lowerboundsontheblowuprateoftheaxisymmetricnavierstokesequationsii-2009\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('chen-strain-tsai-yau-lowerboundsontheblowuprateoftheaxisymmetricnavierstokesequationsii-2009')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2512859,\n\tabstract = {Consider axisymmetric strong solutions of the incompressible\nNavier-Stokes equations in $\\mathbb{R}^3$ with non-trivial swirl.  Let $z$\ndenote the axis of symmetry and $r$ measure the distance to the\n$z$-axis.  Suppose the solution satisfies,for some $0 \\le \\varepsilon \\le 1$ that $|v (x,t)| \\le C_* r^{-1+\\varepsilon } |t|^{-\\varepsilon /2}$ for $-T_0\\le t &lt; 0$ and a positive finite constant $C_*$ which is  allowed to be large,  we then prove that $v$ is regular at time zero. },\n\tauthor = {Chen, Chiun-Chuan and Strain, Robert M. and Tsai, Tai-Peng and Yau, Horng-Tzer},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1080/03605300902793956},\n\teprint = {0709.4230},\n\tfjournal = {Communications in Partial Differential Equations},\n\tissn = {0360-5302},\n\tjournal = {Comm. Partial Differential Equations},\n\tkeywords = {Navier-Stokes equations, Fluid mechanics},\n\tmrclass = {35Q30 (35B40 35B44 35B65 76D05)},\n\tmrnumber = {2512859},\n\tmrreviewer = {Thierry Goudon},\n\tnumber = {1-3},\n\tpages = {203--232},\n\ttitle = {Lower bounds on the blow-up rate of the axisymmetric {N}avier-{S}tokes equations. {II}},\n\turl_arxiv = {https://arxiv.org/abs/0709.4230},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/2009ChenStrainTsaiYau.pdf},\n\tvolume = {34},\n\tyear = {2009},\n\tzblnumber = {1173.35095},\n\tbdsk-url-1 = {https://doi.org/10.1080/03605300902793956}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $ℝ^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the $z$-axis. Suppose the solution satisfies,for some $0 łe ɛ łe 1$ that $|v (x,t)| łe C_* r^{-1+ɛ } |t|^{-ɛ /2}$ for $-T_0łe t < 0$ and a positive finite constant $C_*$ which is allowed to be large, we then prove that $v$ is regular at time zero. \n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2008\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2008')\"\n      onkeypress=\"toggleGroup('group_2008')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2008</span>\n      <span class=\"bibbase_group_count\">\n        \n        (2)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"chen-strain-yau-tsai-lowerboundontheblowuprateoftheaxisymmetricnavierstokesequations-2008\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Chiun-Chuan Chen;  Robert M. Strain;  Horng-Tzer Yau;  and  Tai-Peng Tsai.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2008rnn016.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'chen-strain-yau-tsai-lowerboundontheblowuprateoftheaxisymmetricnavierstokesequations-2008', 'https://strain.math.upenn.edu/preprints/2008rnn016.pdf', event);\">\n    Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Int. Math. Res. Not. IMRN</i>, (9): Art. ID rnn016, 31.  2008.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/math/0701796\"\n     onclick=\"log_download('chen-strain-yau-tsai-lowerboundontheblowuprateoftheaxisymmetricnavierstokesequations-2008', 'https://arxiv.org/abs/math/0701796', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2008rnn016.pdf\"\n     onclick=\"log_download('chen-strain-yau-tsai-lowerboundontheblowuprateoftheaxisymmetricnavierstokesequations-2008', 'https://strain.math.upenn.edu/preprints/2008rnn016.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1093/imrn/rnn016\"\n     onclick=\"log_download('chen-strain-yau-tsai-lowerboundontheblowuprateoftheaxisymmetricnavierstokesequations-2008', 'http://doi.org/10.1093/imrn/rnn016', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/chen-strain-yau-tsai-lowerboundontheblowuprateoftheaxisymmetricnavierstokesequations-2008\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('chen-strain-yau-tsai-lowerboundontheblowuprateoftheaxisymmetricnavierstokesequations-2008')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2429247,\n\tabstract = {Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations  in $R^3$ with non-trivial swirl.  Such solutions are not known to be globally defined, but it is shown by Caffarelli-Kohn-Nirenberg in 1982 that they could only blow up on the axis of symmetry.  Let $z$ denote the axis of symmetry and $r$ measure the distance \nto the $z$-axis.   Suppose the solution satisfies the pointwise scale invariant bound $|v (x,t)| \\le C_*{(r^2 -t)^{-1/2}} $ for $-T_0\\le t &lt; 0$ and a positive finite constant $C_*$ which is  allowed to be large,  we then prove that $v$ is regular at time zero. },\n\tauthor = {Chen, Chiun-Chuan and Strain, Robert M. and Yau, Horng-Tzer and Tsai, Tai-Peng},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1093/imrn/rnn016},\n\teprint = {math/0701796},\n\tfjournal = {International Mathematics Research Notices. IMRN},\n\tissn = {1073-7928},\n\tjournal = {Int. Math. Res. Not. IMRN},\n\tkeywords = {Navier-Stokes equations, Fluid mechanics},\n\tmrclass = {35Q30 (35B40 76D03 76D05)},\n\tmrnumber = {2429247},\n\tmrreviewer = {Thierry Goudon},\n\tnumber = {9},\n\tpages = {Art. ID rnn016, 31},\n\ttitle = {Lower bound on the blow-up rate of the axisymmetric {N}avier-{S}tokes equations},\n\turl_arxiv = {https://arxiv.org/abs/math/0701796},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/2008rnn016.pdf},\n\tyear = {2008},\n\tzblnumber = {1154.35068},\n\tbdsk-url-1 = {https://doi.org/10.1093/imrn/rnn016}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $R^3$ with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown by Caffarelli-Kohn-Nirenberg in 1982 that they could only blow up on the axis of symmetry. Let $z$ denote the axis of symmetry and $r$ measure the distance to the $z$-axis. Suppose the solution satisfies the pointwise scale invariant bound $|v (x,t)| łe C_*{(r^2 -t)^{-1/2}} $ for $-T_0łe t < 0$ and a positive finite constant $C_*$ which is allowed to be large, we then prove that $v$ is regular at time zero. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-guo-exponentialdecayforsoftpotentialsnearmaxwellian-2008\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain;  and  Yan Guo.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2005SGed.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-guo-exponentialdecayforsoftpotentialsnearmaxwellian-2008', 'https://strain.math.upenn.edu/preprints/2005SGed.pdf', event);\">\n    Exponential decay for soft potentials near Maxwellian.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Arch. Ration. Mech. Anal.</i>, 187(2): 287–339.  2008.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2005SGed.pdf\"\n     onclick=\"log_download('strain-guo-exponentialdecayforsoftpotentialsnearmaxwellian-2008', 'https://strain.math.upenn.edu/preprints/2005SGed.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Exponential decay for soft potentials near Maxwellian [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00205-007-0067-3\"\n     onclick=\"log_download('strain-guo-exponentialdecayforsoftpotentialsnearmaxwellian-2008', 'http://doi.org/10.1007/s00205-007-0067-3', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-guo-exponentialdecayforsoftpotentialsnearmaxwellian-2008\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>5 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-guo-exponentialdecayforsoftpotentialsnearmaxwellian-2008')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2366140,\n\tabstract = {Consider both soft potentials with angular cutoff and Landau collision kernels in the Boltzmann theory inside a periodic box. We prove that any smooth perturbation near a given Maxwellian approaches to zero at the rate of $e^{-\\lambda t^{p}}$ for some $\\lambda &gt;0$ and  $p\\in (0,1)$. Our method is based on an unified energy estimate with appropriate exponential velocity\nweight. Our results extend the classical Caflisch 1980 result to the case of very soft potential and Coulomb interactions.},\n\tauthor = {Strain, Robert M. and Guo, Yan},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-08 11:40:38 -0600},\n\tdoi = {10.1007/s00205-007-0067-3},\n\tfjournal = {Archive for Rational Mechanics and Analysis},\n\tissn = {0003-9527},\n\tjournal = {Arch. Ration. Mech. Anal.},\n\tkeywords = {Boltzmann equation, Landau equation, Kinetic Theory},\n\tmrclass = {82B05 (35B45 35F25)},\n\tmrnumber = {2366140},\n\tmrreviewer = {Simone Calogero},\n\tnumber = {2},\n\tpages = {287--339},\n\tread = {0},\n\ttitle = {Exponential decay for soft potentials near {M}axwellian},\n\turlpdf = {https://strain.math.upenn.edu/preprints/2005SGed.pdf},\n\tvolume = {187},\n\tyear = {2008},\n\tzblnumber = {1130.76069},\n\tbdsk-url-1 = {https://doi.org/10.1007/s00205-007-0067-3}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Consider both soft potentials with angular cutoff and Landau collision kernels in the Boltzmann theory inside a periodic box. We prove that any smooth perturbation near a given Maxwellian approaches to zero at the rate of $e^{-λ t^{p}}$ for some $λ >0$ and $p∈ (0,1)$. Our method is based on an unified energy estimate with appropriate exponential velocity weight. Our results extend the classical Caflisch 1980 result to the case of very soft potential and Coulomb interactions.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2007\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2007')\"\n      onkeypress=\"toggleGroup('group_2007')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2007</span>\n      <span class=\"bibbase_group_count\">\n        \n        (2)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"mouhot-strain-spectralgapandcoercivityestimatesforlinearizedboltzmanncollisionoperatorswithoutangularcutoff-2007\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Clément Mouhot;  and  Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2007mouhotStrain.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'mouhot-strain-spectralgapandcoercivityestimatesforlinearizedboltzmanncollisionoperatorswithoutangularcutoff-2007', 'https://strain.math.upenn.edu/preprints/2007mouhotStrain.pdf', event);\">\n    Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>J. Math. Pures Appl. (9)</i>, 87(5): 515–535.  2007.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/math/0607495\"\n     onclick=\"log_download('mouhot-strain-spectralgapandcoercivityestimatesforlinearizedboltzmanncollisionoperatorswithoutangularcutoff-2007', 'https://arxiv.org/abs/math/0607495', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff [link]\"\n       title=\" arxiv\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> arxiv</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2007mouhotStrain.pdf\"\n     onclick=\"log_download('mouhot-strain-spectralgapandcoercivityestimatesforlinearizedboltzmanncollisionoperatorswithoutangularcutoff-2007', 'https://strain.math.upenn.edu/preprints/2007mouhotStrain.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1016/j.matpur.2007.03.003\"\n     onclick=\"log_download('mouhot-strain-spectralgapandcoercivityestimatesforlinearizedboltzmanncollisionoperatorswithoutangularcutoff-2007', 'http://doi.org/10.1016/j.matpur.2007.03.003', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/mouhot-strain-spectralgapandcoercivityestimatesforlinearizedboltzmanncollisionoperatorswithoutangularcutoff-2007\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('mouhot-strain-spectralgapandcoercivityestimatesforlinearizedboltzmanncollisionoperatorswithoutangularcutoff-2007')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2322149,\n\tabstract = {In this paper we prove new constructive coercivity estimates for the Boltzmann collision operator without cutoff, that is for {\\em long-range} interactions. \nIn particular we give a generalized sufficient condition for the existence of a spectral gap which involves both the growth behavior of the collision kernel at large relative velocities and its singular behavior at grazing and frontal collisions. It provides in particular existence of a spectral gap and estimates on it for interactions deriving from the hard potentials $\\phi(r)=r^{-(s-1)}$, $s \\ge 5$ or the so-called moderately soft potentials $\\phi(r)=r^{-(s-1)}$, $3&lt; s &lt; 5$, (without angular cutoff). \nIn particular this paper recovers (by constructive means), improves and extends previous results of Pao 1974. We also obtain constructive coercivity estimates for \nthe Landau collision operator for the optimal coercivity norm pointed out in Guo 2002 and we formulate a conjecture about a unified necessary and sufficient condition \nfor the existence of a spectral gap for Boltzmann and Landau linearized collision operators. },\n\tauthor = {Mouhot, Cl{\\&#x27;{e}}ment and Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1016/j.matpur.2007.03.003},\n\teprint = {math/0607495},\n\tfjournal = {Journal de Math{\\&#x27;{e}}matiques Pures et Appliqu{\\&#x27;{e}}es. Neuvi{\\&#x60;e}me S{\\&#x27;{e}}rie},\n\tissn = {0021-7824},\n\tjournal = {J. Math. Pures Appl. (9)},\n\tkeywords = {Boltzmann equation, Landau equation, Kinetic Theory, non-cutoff},\n\tmrclass = {82C40 (47G20 76P05)},\n\tmrnumber = {2322149},\n\tmrreviewer = {C\\&#x27;{e}dric Villani},\n\tnumber = {5},\n\tpages = {515--535},\n\ttitle = {Spectral gap and coercivity estimates for linearized {B}oltzmann collision operators without angular cutoff},\n\turl_arxiv = {https://arxiv.org/abs/math/0607495},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/2007mouhotStrain.pdf},\n\tvolume = {87},\n\tyear = {2007},\n\tzblnumber = {1388.76338},\n\tbdsk-url-1 = {https://doi.org/10.1016/j.matpur.2007.03.003}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  In this paper we prove new constructive coercivity estimates for the Boltzmann collision operator without cutoff, that is for \\em long-range interactions. In particular we give a generalized sufficient condition for the existence of a spectral gap which involves both the growth behavior of the collision kernel at large relative velocities and its singular behavior at grazing and frontal collisions. It provides in particular existence of a spectral gap and estimates on it for interactions deriving from the hard potentials $ϕ(r)=r^{-(s-1)}$, $s \\ge 5$ or the so-called moderately soft potentials $ϕ(r)=r^{-(s-1)}$, $3< s < 5$, (without angular cutoff). In particular this paper recovers (by constructive means), improves and extends previous results of Pao 1974. We also obtain constructive coercivity estimates for the Landau collision operator for the optimal coercivity norm pointed out in Guo 2002 and we formulate a conjecture about a unified necessary and sufficient condition for the existence of a spectral gap for Boltzmann and Landau linearized collision operators. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-onthelinearizedbalesculenardequation-2007\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2007BalescuLenard.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-onthelinearizedbalesculenardequation-2007', 'https://strain.math.upenn.edu/preprints/2007BalescuLenard.pdf', event);\">\n    On the linearized Balescu-Lenard equation.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Partial Differential Equations</i>, 32(10-12): 1551–1586.  2007.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2007BalescuLenard.pdf\"\n     onclick=\"log_download('strain-onthelinearizedbalesculenardequation-2007', 'https://strain.math.upenn.edu/preprints/2007BalescuLenard.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"On the linearized Balescu-Lenard equation [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1080/03605300601088609\"\n     onclick=\"log_download('strain-onthelinearizedbalesculenardequation-2007', 'http://doi.org/10.1080/03605300601088609', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-onthelinearizedbalesculenardequation-2007\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-onthelinearizedbalesculenardequation-2007')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2372479,\n\tabstract = {The Balescu-Lenard equation  from  plasma physics is widely considered to include a highly accurate correction to Landau&#x27;s fundamental collision operator.   \nYet so far it has seen very little mathematical study. We perform an extensive linearized analysis of this equation, \nwhich includes determining the asymptotic behavior of the new components of the linearized operator and establishing time decay rates for the linearized equation.  },\n\tauthor = {Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1080/03605300601088609},\n\teprint = {math/0603490},\n\tfjournal = {Communications in Partial Differential Equations},\n\tissn = {0360-5302},\n\tjournal = {Comm. Partial Differential Equations},\n\tkeywords = {Kinetic Theory, Balescu-Lenard equation},\n\tmrclass = {82D10 (76P05 82C40)},\n\tmrnumber = {2372479},\n\tmrreviewer = {Stephen Wollman},\n\tnumber = {10-12},\n\tpages = {1551--1586},\n\ttitle = {On the linearized {B}alescu-{L}enard equation},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/2007BalescuLenard.pdf},\n\tvolume = {32},\n\tyear = {2007},\n\tzblnumber = {1128.76068},\n\tbdsk-url-1 = {https://doi.org/10.1080/03605300601088609}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The Balescu-Lenard equation from plasma physics is widely considered to include a highly accurate correction to Landau's fundamental collision operator. Yet so far it has seen very little mathematical study. We perform an extensive linearized analysis of this equation, which includes determining the asymptotic behavior of the new components of the linearized operator and establishing time decay rates for the linearized equation. \n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2006\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2006')\"\n      onkeypress=\"toggleGroup('group_2006')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2006</span>\n      <span class=\"bibbase_group_count\">\n        \n        (3)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"strain-guo-almostexponentialdecaynearmaxwellian-2006\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain;  and  Yan Guo.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2005SGaed.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-guo-almostexponentialdecaynearmaxwellian-2006', 'https://strain.math.upenn.edu/preprints/2005SGaed.pdf', event);\">\n    Almost exponential decay near Maxwellian.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Partial Differential Equations</i>, 31(1-3): 417–429.  2006.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2005SGaed.pdf\"\n     onclick=\"log_download('strain-guo-almostexponentialdecaynearmaxwellian-2006', 'https://strain.math.upenn.edu/preprints/2005SGaed.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Almost exponential decay near Maxwellian [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1080/03605300500361545\"\n     onclick=\"log_download('strain-guo-almostexponentialdecaynearmaxwellian-2006', 'http://doi.org/10.1080/03605300500361545', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-guo-almostexponentialdecaynearmaxwellian-2006\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-guo-almostexponentialdecaynearmaxwellian-2006')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2209761,\n\tabstract = {By direct interpolation of a family of smooth energy estimates for solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann type equations, we show convergence to Maxwellian with any polynomial rate in time. Our results not only resolve the important open problem for both the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system for charged particles, but also lead to a simpler alternative proof of recent decay results for soft potentials as well as the Coulombic interaction, with precise decay rate depending on the initial conditions.\n},\n\tauthor = {Strain, Robert M. and Guo, Yan},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-08 11:41:10 -0600},\n\tdoi = {10.1080/03605300500361545},\n\tfjournal = {Communications in Partial Differential Equations},\n\tissn = {0360-5302},\n\tjournal = {Comm. Partial Differential Equations},\n\tkeywords = {Boltzmann equation, Landau equation, relativistic Landau equation, Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},\n\tmrclass = {82B40 (82C40 82D05 82D10)},\n\tmrnumber = {2209761},\n\tmrreviewer = {Hai-Liang Li},\n\tnumber = {1-3},\n\tpages = {417--429},\n\ttitle = {Almost exponential decay near {M}axwellian},\n\turlpdf = {https://strain.math.upenn.edu/preprints/2005SGaed.pdf},\n\tvolume = {31},\n\tyear = {2006},\n\tzblnumber = {1096.82010},\n\tbdsk-url-1 = {https://doi.org/10.1080/03605300500361545}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  By direct interpolation of a family of smooth energy estimates for solutions near Maxwellian equilibrium and in a periodic box to several Boltzmann type equations, we show convergence to Maxwellian with any polynomial rate in time. Our results not only resolve the important open problem for both the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system for charged particles, but also lead to a simpler alternative proof of recent decay results for soft potentials as well as the Coulombic interaction, with precise decay rate depending on the initial conditions. \n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-thevlasovmaxwellboltzmannsysteminthewholespace-2006\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2005Sws.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-thevlasovmaxwellboltzmannsysteminthewholespace-2006', 'https://strain.math.upenn.edu/preprints/2005Sws.pdf', event);\">\n    The Vlasov-Maxwell-Boltzmann system in the whole space.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Math. Phys.</i>, 268(2): 543–567.  2006.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2005Sws.pdf\"\n     onclick=\"log_download('strain-thevlasovmaxwellboltzmannsysteminthewholespace-2006', 'https://strain.math.upenn.edu/preprints/2005Sws.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"The Vlasov-Maxwell-Boltzmann system in the whole space [pdf]\"\n       title=\" pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00220-006-0109-y\"\n     onclick=\"log_download('strain-thevlasovmaxwellboltzmannsysteminthewholespace-2006', 'http://doi.org/10.1007/s00220-006-0109-y', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-thevlasovmaxwellboltzmannsysteminthewholespace-2006\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>3 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-thevlasovmaxwellboltzmannsysteminthewholespace-2006')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2259206,\n\tabstract = {The Vlasov-Maxwell-Boltzmann system is a fundamental model to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians.},\n\tauthor = {Strain, Robert M.},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-07-13 15:27:26 -0400},\n\tdoi = {10.1007/s00220-006-0109-y},\n\teprint = {math/0512002},\n\tfjournal = {Communications in Mathematical Physics},\n\tissn = {0010-3616},\n\tjournal = {Comm. Math. Phys.},\n\tkeywords = {Boltzmann equation, Kinetic Theory, Vlasov-Maxwell systems},\n\tmrclass = {82D10 (35A05 35F20 35Q60 76P05 76X05)},\n\tmrnumber = {2259206},\n\tmrreviewer = {Simone Calogero},\n\tnumber = {2},\n\tpages = {543--567},\n\ttitle = {The {V}lasov-{M}axwell-{B}oltzmann system in the whole space},\n\turl_pdf = {https://strain.math.upenn.edu/preprints/2005Sws.pdf},\n\tvolume = {268},\n\tyear = {2006},\n\tzblnumber = {1129.35022},\n\tbdsk-url-1 = {https://doi.org/10.1007/s00220-006-0109-y}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The Vlasov-Maxwell-Boltzmann system is a fundamental model to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"strain-recentresultsonexistenceuniquenessandasymptoticdecayratesforcollisionalkineticmodels-2006\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/rms2006ow.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-recentresultsonexistenceuniquenessandasymptoticdecayratesforcollisionalkineticmodels-2006', 'https://strain.math.upenn.edu/preprints/rms2006ow.pdf', event);\">\n    Recent results on existence, uniqueness and asymptotic decay rates for collisional kinetic models.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Oberwolfach Rep.</i>, 3(4): 3189–3258.  2006.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/rms2006ow.pdf\"\n     onclick=\"log_download('strain-recentresultsonexistenceuniquenessandasymptoticdecayratesforcollisionalkineticmodels-2006', 'https://strain.math.upenn.edu/preprints/rms2006ow.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Recent results on existence, uniqueness and asymptotic decay rates for collisional kinetic models [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.4171/owr/2006/54\"\n     onclick=\"log_download('strain-recentresultsonexistenceuniquenessandasymptoticdecayratesforcollisionalkineticmodels-2006', 'http://doi.org/10.4171/owr/2006/54', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-recentresultsonexistenceuniquenessandasymptoticdecayratesforcollisionalkineticmodels-2006\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>3 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-recentresultsonexistenceuniquenessandasymptoticdecayratesforcollisionalkineticmodels-2006')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{Arnold2006,\n\tabstract = {We discuss recent work proving exponential time decay rates to equilibrium for Boltzmann equations such as the soft potentials, Landau&#x27;s equation and the lin- earized Balescu-Lenard model. We also mention a proof of existence and unique- ness of solutions near Maxwellian to the Vlasov-Maxwell-Boltzmann system in the whole space. Some of these projects are joint work with Yan Guo.},\n\tabstractow = {The Oberwolfach meeting described here aimed at presenting the latest mathematical results in the field of kinetic theory (both classical and quantum). There were 50 participants, among which 15 young participants (PhD students, post-docs or young assistant professors). Two of them (M.-P. Gualdani and R. Strain) were invited within the program &quot;US Junior Oberwolfach Fellows&quot;: they are promising young researchers working in the US. The program of the meeting was made in such a way that a lot of time remained for people to meet informally and discuss about scientific issues. It also ensured that almost everybody attended all (or most of) the scheduled talks. The program was structured in the following way: three subtopics were defined (relationships between micro/meso/macroscopic models; kinetic theory for complex particles: granular media, coagulation/fragmentation, chemotaxis and sprays; quantum mechanical kinetic theory). For each subtopic, there were a few (2 or 3) longer talks (about 40 minutes) by senior participants. For those talks, a specific effort of clarity was asked to the speakers, and the subject had to be rather broad. Then, shorter talks of about 20 minutes were planned, on more specialized issues. Finally, a special (much longer) talk in two parts was presented by one of the participants (C. Villani), in order to describe in a very didactical way an emerging link between kinetic theory, optimal transport, and Riemannian geometry. Some participants did not give a talk within this program, but organized an informal discussion around a poster, with an audience composed of specially interested people.},\n\tauthor = {Robert M. Strain},\n\tbooktitle = {{Classical and quantum mechanical models of many-particle systems. Workshop held December 3--9, 2006.}},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-08 13:59:27 -0600},\n\tdoi = {10.4171/owr/2006/54},\n\teditor = {Anton {Arnold} and Carlo {Cercignani} and Laurent {Desvillettes}},\n\tfjournal = {{Oberwolfach Reports}},\n\tissn = {1660-8933; 1660-8941/e},\n\tjournal = {{Oberwolfach Rep.}},\n\tkeywords = {Boltzmann equation, Landau equation, relativistic Landau equation, Kinetic Theory, relativistic Kinetic Theory},\n\tlanguage = {English},\n\tmsc2010 = {82B40 81-06 82-06 81S30 00B05},\n\tnumber = {4},\n\tpages = {3189--3258},\n\tpublisher = {European Mathematical Society (EMS) Publishing House, Zurich; Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach},\n\ttitle = {{R}ecent results on existence, uniqueness and asymptotic decay rates for collisional kinetic models},\n\turlpdf = {https://strain.math.upenn.edu/preprints/rms2006ow.pdf},\n\tvolume = {3},\n\tyear = {2006},\n\tzblnumber = {1177.82057},\n\tbdsk-url-1 = {https://doi.org/10.4171/owr/2006/54}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We discuss recent work proving exponential time decay rates to equilibrium for Boltzmann equations such as the soft potentials, Landau's equation and the lin- earized Balescu-Lenard model. We also mention a proof of existence and unique- ness of solutions near Maxwellian to the Vlasov-Maxwell-Boltzmann system in the whole space. Some of these projects are joint work with Yan Guo.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2005\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2005')\"\n      onkeypress=\"toggleGroup('group_2005')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2005</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"strain-someapplicationsofanenergymethodincollisionalkinetictheory-2005\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2005PHDthesisS.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-someapplicationsofanenergymethodincollisionalkinetictheory-2005', 'https://strain.math.upenn.edu/preprints/2005PHDthesisS.pdf', event);\">\n    Some applications of an energy method in collisional Kinetic theory.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  Ph.D. Thesis, Brown University,  2005.\n  <span class=\"note\">(ProQuest Document ID 305028444)</span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&amp;rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&amp;res_dat=xri:pqdiss&amp;rft_dat=xri:pqdiss:3174679\"\n     onclick=\"log_download('strain-someapplicationsofanenergymethodincollisionalkinetictheory-2005', 'http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&amp;rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&amp;res_dat=xri:pqdiss&amp;rft_dat=xri:pqdiss:3174679', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Some applications of an energy method in collisional Kinetic theory [link]\"\n       title=\" proquest\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> proquest</span></a>\n  &nbsp;\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2005PHDthesisS.pdf\"\n     onclick=\"log_download('strain-someapplicationsofanenergymethodincollisionalkinetictheory-2005', 'https://strain.math.upenn.edu/preprints/2005PHDthesisS.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Some applications of an energy method in collisional Kinetic theory [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-someapplicationsofanenergymethodincollisionalkinetictheory-2005\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>12 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-someapplicationsofanenergymethodincollisionalkinetictheory-2005')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@phdthesis{MR2707256,\n\tabstract = {The collisional Kinetic Equations we study are all of the form\n$$\\partial_t F + v\\cdot \\nabla_x F+V(t,x)\\cdot \\nabla_v F=Q(F,F).$$\nHere $F=F(t,x,v)$ is a probabilistic density function (of time $t\\ge 0$, space $x\\in\\Omega$ and velocity $v\\in\\mathbb{R}^3$) for a particle taken chosen randomly from a gas or plasma. $V(t,x)$ is a field term which usually represents Maxwell&#x27;s theory of electricity and magnetism, sometimes this term is neglected.   $Q(F,F)$ is the collision operator which models the interaction between colliding particles.  We consider both the Boltzmann and Landau collision operators.  \n\n\nWe prove existence, uniqueness and regularity of close to equilibrium solutions to the relativistic Landau-Maxwell system in the first part of this thesis.  Our main tool is an energy method.  \n\nIn the second part, we prove arbitrarily high polynomial time decay rates to equilibrium for four kinetic equations.  These are cutoff soft potential Boltzmann and Landau equations, but also the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system.  The main technique used here is interpolation.  \n\nIn the third part, we prove exponential decay for the cutoff soft potential Boltzmann and Landau equations.  The main point here is to show that exponential decay of the initial data is propagated by a solution.  \n\nIn the fourth and final part of this thesis, we write down a few important calculations in the relativistic Boltzmann theory which are scattered around the literature.  We also calculate a few Lorentz transformations which maybe useful in relativistic transport theory.  We use these calculations to comment about extending the results in this thesis to the relativistic Boltzmann equation.\n},\n\tannote = {(https://search.proquest.com)},\n\tauthor = {Strain, Robert M.},\n\tdate-added = {2019-07-13 17:21:46 -0400},\n\tdate-modified = {2019-08-08 11:28:11 -0600},\n\tisbn = {978-0542-12875-2},\n\tjournal = {ProQuest Dissertations and Theses},\n\tkeywords = {Boltzmann equation, Landau equation, Balescu-Lenard equation, Kinetic Theory, Vlasov-Maxwell systems, relativistic Kinetic Theory},\n\tlanguage = {English},\n\tmrclass = {Thesis},\n\tmrnumber = {2707256},\n\tnote = {(ProQuest Document ID 305028444)},\n\tpages = {1--200},\n\tpublisher = {ProQuest LLC, Ann Arbor, MI},\n\tschool = {Brown University},\n\ttitle = {Some applications of an energy method in collisional {K}inetic theory},\n\turl_proquest = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&amp;rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&amp;res_dat=xri:pqdiss&amp;rft_dat=xri:pqdiss:3174679},\n\turlpdf = {https://strain.math.upenn.edu/preprints/2005PHDthesisS.pdf},\n\tyear = {2005},\n\tbdsk-url-1 = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&amp;rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&amp;res_dat=xri:pqdiss&amp;rft_dat=xri:pqdiss:3174679}}\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The collisional Kinetic Equations we study are all of the form $$∂_t F + v· ∇_x F+V(t,x)· ∇_v F=Q(F,F).$$ Here $F=F(t,x,v)$ is a probabilistic density function (of time $t\\ge 0$, space $x∈Ω$ and velocity $v∈ℝ^3$) for a particle taken chosen randomly from a gas or plasma. $V(t,x)$ is a field term which usually represents Maxwell's theory of electricity and magnetism, sometimes this term is neglected. $Q(F,F)$ is the collision operator which models the interaction between colliding particles. We consider both the Boltzmann and Landau collision operators. We prove existence, uniqueness and regularity of close to equilibrium solutions to the relativistic Landau-Maxwell system in the first part of this thesis. Our main tool is an energy method. In the second part, we prove arbitrarily high polynomial time decay rates to equilibrium for four kinetic equations. These are cutoff soft potential Boltzmann and Landau equations, but also the Vlasov-Maxwell-Boltzmann system and the relativistic Landau-Maxwell system. The main technique used here is interpolation. In the third part, we prove exponential decay for the cutoff soft potential Boltzmann and Landau equations. The main point here is to show that exponential decay of the initial data is propagated by a solution. In the fourth and final part of this thesis, we write down a few important calculations in the relativistic Boltzmann theory which are scattered around the literature. We also calculate a few Lorentz transformations which maybe useful in relativistic transport theory. We use these calculations to comment about extending the results in this thesis to the relativistic Boltzmann equation. \n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2004\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2004')\"\n      onkeypress=\"toggleGroup('group_2004')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2004</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"strain-guo-stabilityoftherelativisticmaxwellianinacollisionalplasma-2004\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n   Robert M. Strain;  and  Yan Guo.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2004SG.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'strain-guo-stabilityoftherelativisticmaxwellianinacollisionalplasma-2004', 'https://strain.math.upenn.edu/preprints/2004SG.pdf', event);\">\n    Stability of the relativistic Maxwellian in a collisional plasma.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Comm. Math. Phys.</i>, 251(2): 263–320.  2004.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://strain.math.upenn.edu/preprints/2004SG.pdf\"\n     onclick=\"log_download('strain-guo-stabilityoftherelativisticmaxwellianinacollisionalplasma-2004', 'https://strain.math.upenn.edu/preprints/2004SG.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Stability of the relativistic Maxwellian in a collisional plasma [pdf]\"\n       title=\"Pdf\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\">Pdf</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"http://doi.org/10.1007/s00220-004-1151-2\"\n     onclick=\"log_download('strain-guo-stabilityoftherelativisticmaxwellianinacollisionalplasma-2004', 'http://doi.org/10.1007/s00220-004-1151-2', event.ctrlKey)\"\n     class=\"bibbase doi link\">\n    <span>doi</span></a>\n  &nbsp;\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/strain-guo-stabilityoftherelativisticmaxwellianinacollisionalplasma-2004\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>6 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('strain-guo-stabilityoftherelativisticmaxwellianinacollisionalplasma-2004')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{MR2100057,\n\tabstract = {The relativistic Landau-Maxwell system is among the most fundamental  and complete models for describing the dynamics of a dilute collisional plasma in which particles interact through Coulombic collisions and through their self-consistent electromagnetic field.   We construct the first global in time classical solutions.  Our solutions are constructed in a periodic box and near the relativistic Maxwellian, the J\\&quot;{u}ttner solution.},\n\tauthor = {Strain, Robert M. and Guo, Yan},\n\tdate-added = {2019-07-13 15:27:26 -0400},\n\tdate-modified = {2019-08-08 11:41:54 -0600},\n\tdoi = {10.1007/s00220-004-1151-2},\n\tfjournal = {Communications in Mathematical Physics},\n\tissn = {0010-3616},\n\tjournal = {Comm. Math. Phys.},\n\tkeywords = {Landau equation, relativistic Landau equation, Kinetic Theory, relativistic Kinetic Theory, Vlasov-Maxwell systems},\n\tmrclass = {82D10 (35Q60 35Q75 76X05 76Y05)},\n\tmrnumber = {2100057},\n\tmrreviewer = {C\\&#x27;edric Villani},\n\tnumber = {2},\n\tpages = {263--320},\n\ttitle = {Stability of the relativistic {M}axwellian in a collisional plasma},\n\turlpdf = {https://strain.math.upenn.edu/preprints/2004SG.pdf},\n\tvolume = {251},\n\tyear = {2004},\n\tzblnumber = {1113.82070},\n\tbdsk-url-1 = {https://doi.org/10.1007/s00220-004-1151-2}}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The relativistic Landau-Maxwell system is among the most fundamental and complete models for describing the dynamics of a dilute collisional plasma in which particles interact through Coulombic collisions and through their self-consistent electromagnetic field. We construct the first global in time classical solutions. Our solutions are constructed in a periodic box and near the relativistic Maxwellian, the Jüttner solution.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n\n\n\n  </div>\n\n\n  <!-- Modal -->\n\n  <!-- <iframe id=\"bibbase_network_frame\" -->\n  <!--         src=\"//bibbase.org/network/control\" -->\n  <!--         style=\"display: none;\"> -->\n  <!-- </iframe> -->\n\n</div>\n"}; document.write(bibbase_data.data);