Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

TL;DR

This paper establishes pointwise regularity estimates for solutions to the stationary linearized Boltzmann equation with diffuse reflection boundary conditions in convex domains, extending velocity averaging techniques.

Contribution

It provides the first pointwise derivative estimates for solutions under diffuse reflection boundary conditions in convex domains, linking stationary and velocity averaging results.

Findings

Derived pointwise estimates for first derivatives of solutions.

Extended velocity averaging lemma to stationary boundary problems.

Applicable to hard sphere, cutoff hard potential, and Maxwellian gases.

Abstract

We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma.