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.