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 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)].

中文翻译：

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玻尔兹曼方程越过障碍物的霍尔德正则性

根据域形状的解的规律性和奇异性是玻尔兹曼理论中具有挑战性的研究主题。在本文中，我们证明了 Hölder 正则为硬球分子的玻尔兹曼方程，其在分子间碰撞和与凸障碍物边界接触时发生弹性反射。特别是，这种霍尔德正则性结果与其他物理边界条件（例如漫反射边界条件和流入边界条件）的情况形成鲜明对比，对于其他物理边界条件，玻尔兹曼方程的解在余维 1 子集中出现不连续性（Kim [Comm. Math. Phys. 308 (2011)]），因此最好的可能规律是 BV，这已被Guo 等人证明。 [拱。理性机械。肛门。 220（2016）]。