# Cercignani-Lampis boundary in the Boltzmann theory

## TL;DR

This paper establishes local well-posedness of the Boltzmann equation with Cercignani-Lampis boundary conditions, introducing a new boundary term decomposition and constructing a unique steady solution under specific constraints.

## Contribution

It provides the first proof of local well-posedness for the Boltzmann equation with Cercignani-Lampis boundary conditions using a novel boundary decomposition method.

## Key findings

- Proved local-in-time well-posedness of the Boltzmann equation with Cercignani-Lampis boundary.
- Developed a new boundary term decomposition technique.
- Constructed a unique steady solution under wall temperature and accommodation constraints.

## Abstract

The Boltzmann equation is a fundamental kinetic equation that describes the dynamics of dilute gas. In this paper we study the local well-posedness of the Boltzmann equation in bounded domain with the Cercignani-Lampis boundary condition, which describes the intermediate reflection law between diffuse reflection and specular reflection via two accommodation coefficients. We prove the local-in-time well-posedness of the equation by establishing an $L^\infty$ estimate. In particular, for the $L^\infty$ bound we develop a new decomposition on the boundary term combining with repeated interaction through the characteristic. Via this method, we construct a unique steady solution of the Boltzmann equation with constraints on the wall temperature and the accommodation coefficient.