Quantitative lower bound for solutions to the Boltzmann equation in non-convex domains

TL;DR

This paper establishes lower bounds for solutions to the Boltzmann equation in bounded, non-convex domains, extending previous results that required convexity, under both cutoff and non-cutoff assumptions.

Contribution

It provides the first known Maxwellian lower bound in non-convex domains for the cutoff case and a weaker bound for the non-cutoff case, removing the convexity restriction.

Findings

Maxwellian lower bound in cutoff case for non-convex domains

Weaker lower bound in non-cutoff case for non-convex domains

Extension of previous convex-domain results to non-convex settings

Abstract

In this article, we study the continuous mild solutions to the Boltzmann equation in a bounded spatial domain, under either angular cutoff assumption or non-cutoff assumption. Without assuming convexity of the spatial domain, we establish a Maxwellian lower bound in the cutoff case, and a weaker-than-Maxwellian lower bound for the non-cutoff case. This extends the results of \cite{Bri1,Bri2}, where the convexity of the domain was required.