Gaussian Lower Bounds for the Boltzmann Equation without Cutoff
Gaussian Lower Bounds for the Boltzmann Equation without Cutoff
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DOI：
10.1137/19m1252375
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发表时间：
2019-03
期刊：
影响因子：
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通讯作者：
C. Imbert;C. Mouhot;L. Silvestre
中科院分区：
文献类型：
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作者：
C. Imbert;C. Mouhot;L. Silvestre
玻尔兹曼方程解正性的研究可以追溯到Carleman (1933)， Carleman的初始论点是由第二作者ulvirenti- wennberg（1997）和Briant（2015）提出的。然而，高斯衰减下界的出现对于远程相互作用（所谓的非截止碰撞核）仍然是一个悬而未决的问题。我们回答了这个问题，并在空间周期条件下，在流体动力量（局部质量、局部能量和局部熵密度）保持有界的唯一假设下，建立了无截断的玻尔兹曼方程解的高斯下界。本文除了第三作者（2016）建立的解的统一上界外，大部分是自成体系的。
The study of positivity of solutions to the Boltzmann equation goes back to Carleman (1933), and the initial argument of Carleman was developed byPulvirenti-Wennberg (1997), the second author and Briant (2015). The appearance of a lower bound with Gaussian decay had however remained an open question for long-range interactions (the so-called non-cutoff collision kernels). We answer this question and establish such Gaussian lower bound for solutions to the Boltzmann equation without cutoff, in the case of hard and moderately soft potentials, with spatial periodic conditions, and under the sole assumption that hydrodynamic quantities (local mass, local energy and local entropy density) remain bounded. The paper is mostly self-contained, apart from the uniform upper bound on the solution established by the third author (2016).