Asymptotic Stability of the Boltzmann Equation with Maxwell Boundary Conditions

TL;DR

This paper establishes the asymptotic stability of the Boltzmann equation with Maxwell boundary conditions in general domains, proving existence, uniqueness, and exponential convergence to equilibrium without contradiction methods.

Contribution

It introduces a new perturbative Cauchy theory for the Boltzmann equation with Maxwell boundary conditions, covering a broad class of weights and boundary parameters.

Findings

Proves existence and uniqueness of solutions in weighted spaces.

Demonstrates exponential convergence to equilibrium.

Analyzes the threshold for the accommodation coefficient .

Abstract

In a general $C^1$ domain, we study the perturbative Cauchy theory for the Boltzmann equation with Maxwell boundary conditions with an accommodation coefficient $\alpha$ in $(\sqrt{2/3},1]$, and discuss this threshold. We consider polynomial or stretched exponential weights $m(v)$ and prove existence, uniqueness and exponential trend to equilibrium around a global Maxwellian in $L^\infty_{x,v}(m)$. Of important note is the fact that the methods do not involve contradiction arguments.