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

TL;DR

This paper establishes the stability of large amplitude solutions to the Boltzmann equation in a bounded domain with significant external potential and boundary effects, extending previous results to more complex scenarios.

Contribution

It proves the asymptotic stability of large amplitude solutions of the Boltzmann equation with large external potential and boundary effects, a significant extension of prior work.

Findings

Proved stability of small perturbations near local Maxwellian.

Demonstrated stability of large amplitude solutions with initial data large in weighted L-infinity.

Extended stability results to scenarios with significant external potentials and boundary conditions.

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.