论文标题
关于相对论鲍尔茨曼方程的决定性问题
On the Determinant Problem for the Relativistic Boltzmann Equation
论文作者
论文摘要
本文考虑了关于摩托米中心坐标中相对论玻尔兹曼方程的雅各布决定因素的长期以来的公开问题。对于牛顿玻尔兹曼方程，摩amentum的坐标在研究牛顿非切割玻尔兹曼方程的研究中发挥了重要作用，我们提到了Alexandre等人的广泛使用的取消列。 （Arch。Mation。Mech。Anal。152（4）：327-355，2000）。在本文中，我们特别计算出非常复杂的雅各布决定因素，即十个变量，用于相对论碰撞地图，从动量$ p $到产后动量$ p'$;具体来说，我们计算$ p \ mapsto u =θp'+\ left（1-θ\右）p $的决定因素，in [0,1] $。之后，我们为这个决定因素提供了上限，在$ p $和$ q $变量中都没有奇异性。接下来，我们举一个示例，我们证明雅各布在特定的限制中以零为零。我们进一步解释了我们的数值研究结果，该结果表明，雅各布的决定因素具有大量的不同点，即机器为零。这概括了格拉西·斯特劳斯（Glassey-Strauss）的工作（传输理论统计学家，物理学20（1）：55-68，1991）和guo-strain（Comm。Math。310Phys。310（3）：649-673，2012）。这些结论使得很难想象牛顿取消引理中的直接相对论类似物。
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 $p\mapsto u = θp'+\left(1-θ\right)p$ for $θ\in [0,1]$. 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.
