The Boltzmann equation with weakly inhomogeneous data in bounded domain

TL;DR

This paper establishes the existence and long-term behavior of solutions to the Boltzmann equation with specular reflection boundary conditions in bounded domains, for initial data near radially symmetric states.

Contribution

It extends previous results on the Boltzmann equation to include specular reflection boundary conditions in bounded domains, providing a global solution and asymptotic analysis.

Findings

Unique global solution constructed

Large time asymptotic behavior characterized

Extension of previous Cauchy problem results

Abstract

This paper is concerned with the Boltzmann equation with specular reflection boundary condition. We construct a unique global solution and obtain its large time asymptotic behavior in the case that the initial data is close enough to a radially symmetric homogeneous datum. The result extends the case of Cauchy problem considered by Arkeryd-Esposito-Pulvirenti [Comm. Math. Phys. 111(3): 393-407 (1987)] to the specular reflection boundary value problem in bounded domain.