Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions

TL;DR

This paper develops a perturbative approach to the Boltzmann equation in bounded domains with different boundary conditions, proving existence and uniqueness of solutions under less restrictive velocity weights.

Contribution

It extends Guo's work by establishing existence, uniqueness, and positivity for solutions with polynomial and exponential weights, using constructive methods.

Findings

Proves existence and uniqueness of solutions with polynomial weights.

Establishes positivity and continuity of solutions.

Provides constructive methods for solutions in bounded domains.

Abstract

We study the Boltzmann equation near a global Maxwellian in the case of bounded domains. We consider the boundary conditions to be either specular reflections or Maxwellian diffusion. Starting from the reference work of Guo in $L^\infty_{x,v}\left(\left(1+|v|\right)^\beta e^{|v|^2/4}\right)$, we prove existence, uniqueness, continuity and positivity of solutions for less restrictive weights in the velocity variable; namely, polynomials and stretch exponentials. The methods developed here are constructive.