Regularity of the Boltzmann Equation in Convex Domains

TL;DR

This paper investigates the regularity of solutions to the Boltzmann equation in convex domains, establishing existence of certain smooth solutions away from the boundary and demonstrating the non-existence of second derivatives at the boundary.

Contribution

It constructs weighted classical and Sobolev solutions for the Boltzmann equation with various boundary conditions and provides counterexamples showing the non-existence of second derivatives at the boundary.

Findings

Weighted $C^{1}$ solutions exist away from the grazing set.

Existence of $W^{1,p}$ solutions for $1< p<2$ and weighted $W^{1,p}$ solutions for $2\, ext{to}\, ext{infinity}$.

Counterexamples show second derivatives generally do not exist up to the boundary.

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.