On the large amplitude solution of the Boltzmann equation with large external potential and boundary effects

Abstract

The Boltzmann equation is a fundamental equation in kinetic theory that describes the motion of rarefied gases. In this study, we examine the Boltzmann equation within a $C^1$ bounded domain, subject to a large external potential $\Phi(x)$ and diffuse reflection boundary conditions. Initially, we prove the asymptotic stability of small perturbations near the local Maxwellian $\mu_E (x,v)$. Subsequently, we demonstrate the asymptotic stability of large amplitude solutions with initial data that is arbitrarily large in (weighted) $L^\infty$, but sufficiently small in the sense of relative entropy. Specifically, we extend the results for large amplitude solutions of the Boltzmann equation (with or without external potential) [10, 11, 12, 23] to scenarios involving significant external potentials [19, 28] under diffuse reflection boundary conditions.