Rarefied gas dynamics with external fields under specular reflection boundary condition

TL;DR

This paper studies the Boltzmann equation with external fields in convex domains, establishing the existence of classical solutions with specular reflection boundary conditions under specific regularity and positivity assumptions on the external field.

Contribution

It constructs classical solutions to the Boltzmann equation with external fields and specular reflection in convex domains, under new regularity and positivity conditions.

Findings

Existence of classical $C^1$ solutions away from grazing set.

Solutions constructed under $C^2$ regularity of external field.

Positivity of the normal derivative of the external field is crucial.

Abstract

We consider the Boltzmann equation with external fields in strictly convex domains with the specular reflection boundary condition. We construct classical $C^1$ solutions away from the grazing set under the assumption that the external field is $C^2$ and the normal derivative of the field is positive and bounded away from $0$.