Regularity of Boltzmann equation with Cercignani-Lampis boundary in convex domain

TL;DR

This paper investigates the regularity properties of the Boltzmann equation with Cercignani-Lampis boundary conditions in convex domains, constructing solutions under specific assumptions on boundary parameters.

Contribution

It introduces a method to construct local weighted $C^1$ solutions for both dynamical and steady Boltzmann equations with C-L boundary conditions in convex domains.

Findings

Constructed local weighted $C^1$ dynamical solutions.

Established steady solutions under small fluctuation assumptions.

Analyzed regularity in the context of intermediate boundary reflection laws.

Abstract

The Boltzmann equation is a fundamental kinetic equation that describes the dynamics of dilute gas. In this paper we study the regularity of both dynamical and steady Boltzmann equation in strictly convex domain with the Cercignani-Lampis (C-L) boundary condition. The C-L boundary condition describes the intermediate reflection law between diffuse reflection and specular reflection via two accommodation coefficients. We construct local weighted $C^1$ dynamical solution using repeated interaction through the characteristic. When we assume small fluctuation to the wall temperature and accommodation coefficients, we construct weighted $C^1$ steady solution.