Macroscopic estimate of the linear Boltzmann and Landau equations with Specular reflection boundary

TL;DR

This paper extends the test function method to establish $L^6$ control of the macroscopic part of linear Boltzmann and Landau equations with specular reflection boundary conditions, using Korn's inequality and symmetric Poisson systems.

Contribution

It introduces an extension of the test function method to handle specular reflection boundaries in kinetic equations, incorporating Korn's inequality and symmetric Poisson equations.

Findings

Proves $L^6$ control of macroscopic parts for linear Boltzmann and Landau equations.

Extends existing methods to boundary conditions involving specular reflection.

Utilizes Korn's inequality and symmetric Poisson systems for the analysis.

Abstract

In this short note, we prove an $L^6$-control of the macroscopic part of the linear Boltzmann and Landau equations. This result is an extension of the test function method of Esposito-Guo-Kim-Marra~\cite{EGKM}\cite{EGKM2} to the specular reflection boundary condition, in which we crucially used the Korn's inequality \cite{DV2} and the system of symmetric Poisson equations \cite{Bernou}.