BV-regularity of the Boltzmann equation in Non-Convex Domains

TL;DR

This paper proves optimal BV regularity estimates for solutions to the Boltzmann equation in non-convex domains with diffuse boundary conditions, using new trace estimates and neighborhood constructions.

Contribution

It introduces a novel $W^{1,1}$-trace estimate and a specialized neighborhood construction to achieve BV regularity in non-convex domains.

Findings

Established optimal BV estimates for Boltzmann solutions in non-convex domains.

Developed a new trace estimate for diffuse boundary conditions.

Created a delicate construction of an $ ext{epsilon}$-tubular neighborhood of the singular set.

Abstract

Consider the Boltzmann equation in a general non-convex domain with the diffuse boundary condition. We establish optimal BV estimates for such solutions. Our method consists of a new $W^{1,1}-$trace estimate for the diffuse boundary condition and a delicate construction of and an $\varepsilon-$tubular neighborhood of the singular set.