تقرير
On the Determinant Problem for the Relativistic Boltzmann Equation
| العنوان: | On the Determinant Problem for the Relativistic Boltzmann Equation |
|---|---|
| المؤلفون: | Chapman, James, Jang, Jin Woo, Strain, Robert M. |
| سنة النشر: | 2020 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Analysis of PDEs |
| الوصف: | 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 $p\mapsto u = \theta p'+\left(1-\theta\right)p$ 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. Comment: 27 pages, 2 figures |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1007/s00220-021-04101-2 |
| URL الوصول: | http://arxiv.org/abs/2006.02540 |
| رقم الأكسشن: | edsarx.2006.02540 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.1007/s00220-021-04101-2 |
|---|