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arXiv · 1212.1694

Regularity of the Boltzmann Equation in Convex Domains

Yan Guo·Chanwoo Kim·Daniela Tonon·Ariane Trescases
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Abstract

A basic question about regularity of Boltzmann solutions in the presence of physical boundary conditions has been open due to characteristic nature of the boundary as well as the non-local mixing of the collision operator. Consider the Boltzmann equation in a strictly convex domain with the specular, bounce-back and diffuse boundary condition. With the aid of a distance function toward the grazing set, we construct weighted classical $C^{1}$ solutions away from the grazing set for all boundary conditions. For the diffuse boundary condition, we construct $W^{1,p}$ solutions for $1< p<2$ and weighted $W^{1,p}$ solutions for $2\leq p\leq \infty $ as well. On the other hand, we show second derivatives do not exist up to the boundary in general by constructing counterexamples for all boundary conditions.

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Yan Guo, Chanwoo Kim, Daniela Tonon, Ariane Trescases. 2013-12-22. Regularity of the Boltzmann Equation in Convex Domains. https://doi.org/10.1007/s00222-016-0670-8

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