On the Determinant Problem for the Relativistic Boltzmann Equation. Chapman, J., Jang, J. W., & Strain, R. M. Comm. Math. Phys., 384:1913–1943, 2021. ![On the Determinant Problem for the Relativistic Boltzmann Equation [link]](https://bibbase.org/img/filetypes/link.svg "link")
Arxiv doi abstract bibtex 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.
@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}}
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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). 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