Hölder regularity of the Boltzmann equation past an obstacle

IF 4.3 3区材料科学 Q1 ENGINEERING, ELECTRICAL & ELECTRONIC

ACS Applied Electronic Materials Pub Date : 2023-10-06 DOI:10.1002/cpa.22167

Chanwoo Kim, Donghyun Lee

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Abstract

Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in $C_{x, v}^{0, \frac{1}{2} -}$ for the Boltzmann equation of the hard-sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle. In particular, this Hölder regularity result is a stark contrast to the case of other physical boundary conditions (such as the diffuse reflection boundary condition and in-flow boundary condition), for which the solutions of the Boltzmann equation develop discontinuity in a codimension 1 subset (Kim [Comm. Math. Phys. 308 (2011)]), and therefore the best possible regularity is BV, which has been proved by Guo et al. [Arch. Rational Mech. Anal. 220 (2016)].

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Boltzmann方程越过障碍物的Hölder正则性

根据域形状的解的正则性和奇异性是玻尔兹曼理论中一个具有挑战性的研究主题。本文证明了Cx，v0,12−$C^｛0，\frac｛1｝中的一个Hölder正则性{2}-}_对于硬球分子的玻尔兹曼方程，{x，v}$，其在分子间碰撞和与凸障碍物边界的接触中经历弹性反射。特别是，这个Hölder正则性结果与其他物理边界条件（如漫反射边界条件和流中边界条件）的情况形成了鲜明对比，在其他物理边界情况下，玻尔兹曼方程的解在余维1的子集中产生了不连续性（Kim[Comm.Math.Phys.308（2011）]），因此最佳可能的正则性是BV，郭等人[Arch.RrationalMech.Anal.220（2016）]已经证明了这一点。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

ACS Applied Electronic Materials Multiple-

CiteScore

7.20

自引率

4.30%

发文量

567