Global solutions in $L^{p}_{v}L^{\infty}_{x}$ for the Boltzmann equation in bounded domains

TL;DR

This paper establishes the existence and exponential convergence of global solutions to the Boltzmann equation in bounded domains within relaxed function spaces, expanding the understanding of solution behavior beyond traditional bounded classes.

Contribution

It introduces a novel approach to solving the Boltzmann equation in $L^{p}_{v}L^inity_{x}$ spaces with diffuse reflection, allowing for large initial data and providing new insights into convergence to equilibrium.

Findings

Existence of unique global solutions for large initial data.

Exponential convergence to the Maxwellian equilibrium.

A new pointwise estimate for the gain term in the Boltzmann collision operator.

Abstract

The existence theory for solutions to the Boltzmann equation in bounded domains has primarily been developed within uniformly bounded function classes, such as $L^{\infty}_{x,v}$, as in [Duan-Huang-Wang-Yang,2017], [Duan-Wang,2019], [Guo,2010]. In this paper, we investigate solutions in relaxed function spaces $L^{p}_{v}L^\infty_{x}$ for the initial-boundary value problem of the Boltzmann equation in bounded domains. We consider the case of hard potential under diffuse reflection boundary conditions and assume cutoff model. For large initial data in a weighted $L^{p}_{v}L^\infty_{x}$ space with small relative entropy, we construct unique global-in-time mild solution that converge exponentially to the global Maxwellian. A pointwise estimate for the gain term, bounded in terms of $L^p_v$ and $L^2_v$ norms, is essential to prove our main results. Relative to [Gualdani-Mischler-Mouhot,2017], our work provides an alternative perspective on convergence to equilibrium in the presence of boundary conditions.