Regularity for the Boltzmann equation conditional to macroscopic bounds
Regularity for the Boltzmann equation conditional to macroscopic bounds
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DOI：
10.4171/emss/37
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发表时间：
2020-05
期刊：
影响因子：
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通讯作者：
C. Imbert;L. Silvestre
中科院分区：
文献类型：
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作者：
C. Imbert;L. Silvestre
玻尔兹曼方程是一种在统计力学中起核心作用的非线性偏微分方程。从数学的角度看，任意初始数据的全局光滑解的存在性是一个突出的开放问题。在这篇文章中，我们回顾了一个专注于非截止情况的程序，该程序致力于仅根据物理上有意义的条件来推导以$C^\inty$为单位的先验估计。我们证明，只要它的质量、能量和熵密度保持有界，并且远离真空，解将保持一致光滑。
The Boltzmann equation is a nonlinear partial differential equation that plays a central role in statistical mechanics. From the mathematical point of view, the existence of global smooth solutions for arbitrary initial data is an outstanding open problem. In the present article, we review a program focused on the non-cutoff case and dedicated to the derivation of a priori estimates in $C^\infty$, depending only on physically meaningful conditions. We prove that the solution will stay uniformly smooth provided that its mass, energy and entropy densities remain bounded, and away from vacuum.