Arxiv doi abstract bibtex

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.

@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}}

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