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

TL;DR

The paper proves an $L^6$-control of the macroscopic part of the linear Boltzmann and Landau equations under specular reflection boundary conditions. It extends the Esposito-Guo-Kim-Marra test function method to the specular boundary case. The method crucially uses Korn's inequality and a system of symmetric Poisson equations to derive the macroscopic estimate. This macroscopic bound advances the analysis of kinetic equations with reflecting boundaries by providing a robust a priori control that informs stability and long-time behavior.

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}.