On the Determinant Problem for the Relativistic Boltzmann Equation
On the Determinant Problem for the Relativistic Boltzmann Equation
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DOI：
10.1007/s00220-021-04101-2
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发表时间：
2020-06
影响因子：
2.4
通讯作者：
James L. Chapman;Jin Woo Jang;Robert M. Strain
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文献类型：
--
作者：
James L. Chapman;Jin Woo Jang;Robert M. Strain
本文考虑动量中心坐标下相对论玻尔兹曼方程的雅可比行列式的一个长期悬而未决的问题。对于牛顿玻尔兹曼方程，动量中心坐标在牛顿非截止玻尔兹曼方程的研究中起到了很大的作用，特别是我们提到了广泛使用的消去引理。在本文中，我们具体计算了从动量到碰撞后动量的相对论碰撞映象的十个变量中非常复杂的雅可比行列式；具体地说，我们计算了以下元素的行列式：\secumentclass[12pt]{minimum}\usepackage{amsath}\usepackage{wa ysym}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathsfs}\usepackage{upgreek}\setlong{\oddsidemargin}{-69pt}\Begin{Document}$$p\mapsto u=\theta p‘+\Left(1\theta\right)p$\\end然后，我们给出了这个行列式的一个上界，它在两个变元中都没有奇性。接下来，我们给出一个例子，在这里我们证明了雅可比在一个特定的逐点极限下趋于零。我们进一步解释了我们的数值研究的结果，它表明雅可比行列式有非常多的不同的点，在那里它是机器零。这推广了Glassey-Strauss(1991)和Guo-Strand(2012)的工作。这些结论使得我们很难想象动量中心坐标下牛顿抵消引理的直接相对论类比。
This article considers a long-outstanding open question regarding the Jacobian determinant for the relativistic Boltzmann equation in thecenter-of-momentumcoordinates. For the Newtonian Boltzmann equation, the center-of-momentum coordinates have played a large role in the study of the Newtonian non-cutoff Boltzmann equation, in particular we mention the widely used cancellation lemma . In this article we calculate specifically the very complicated Jacobian determinant, in ten variables, for the relativistic collision map from the momentumpto the post collisional momentum; specifically we calculate the determinant for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\mapsto u = \theta p'+\left( 1-\theta \right) p$$\end{document} for. Afterwards we give an upper-bound for this determinant that has no singularity in bothpandqvariables. 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 (1991) and Guo-Strain (2012) . These conclusions make it difficult to envision a direct relativistic analog of the Newtonian cancellation lemma in the center-of-momentum coordinates.