Regularity of Stationary Boltzmann equation in Convex Domains

TL;DR

This paper establishes the existence of $C^{1,eta}$ regularity for solutions to the stationary Boltzmann equation in convex domains, overcoming previous regularity limitations at the boundary.

Contribution

It proves the existence of $C^{1,eta}$ solutions away from the grazing boundary for the stationary Boltzmann equation in convex domains with small temperature fluctuations.

Findings

Constructed $C^{1,eta}$ solutions away from the boundary.

Achieved regularity results for non-isothermal diffuse boundary conditions.

Extended regularity theory for Boltzmann in convex domains.

Abstract

Higher regularity estimate has been a challenging question for the Boltzmann equation in bounded domains. Indeed, it is well-known to have "the non-existence of a second order derivative at the boundary" in [15] even for symmetric convex domains such as a disk or sphere. In this paper we answer this question in the affirmative by constructing the $C^{1,\beta}$ solutions away from the grazing boundary, for any $\beta<1$, to the stationary Boltzmann equation with the non-isothermal diffuse boundary condition in strictly convex domains, as long as a smooth wall temperature has small fluctuation pointwisely.