The Non-cutoff Boltzmann Equation in Bounded Domains

TL;DR

This paper investigates the stability and long-time behavior of the non-cutoff Boltzmann equation in bounded domains, establishing global existence and exponential decay without angular cutoff assumptions, using advanced analytical techniques.

Contribution

It provides the first global existence and decay results for the non-cutoff Boltzmann equation in general bounded domains with physical boundary conditions.

Findings

Global-in-time existence of solutions

Exponential decay towards Maxwellian equilibrium

Applicable to both hard and soft potentials

Abstract

The initial-boundary value problem for the inhomogeneous non-cutoff Boltzmann equation is a challenging open problem. In this paper, we study the stability and long-time dynamics of the Boltzmann equation near a global Maxwellian without angular cutoff assumption in a general $C^3$ bounded domain $\Omega$ (including convex and non-convex cases) with physical boundary conditions: inflow boundary and Maxwell-reflection boundary with accommodation coefficient $\al\in(0,1)$. We obtain the global-in-time existence, which has an exponential decay rate towards the global Maxwellian for both hard and soft potentials. The crucial methods are the forward-backward extension of the boundary problem to the whole space by Vlasov-type equations, a level-function trace lemma, an improved velocity averaging lemma with less regularity but without cutoff in velocity, and an extra damping provided by the advection operator, followed by the De Giorgi iteration and the $L^2$--$L^\infty$ energy method.