Communications in Mathematical Sciences

Volume 22 (2024)

Number 1

Energy method for the Boltzmann equation of monatomic gaseous mixtures

Pages: 137 – 166

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n1.a6

Authors

Laurent Boudin (Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions, Paris, France)

Bérénice Grec (MAP5, CNRS UMR 8145, Université Paris Cité, Paris, France)

Milana Pavić-Čolić (Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Serbia)

Srboljub Simić (Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Serbia)

Abstract

In this paper, we present an energy method for the system of Boltzmann equations in the multicomponent mixture case, based on a micro-macro decomposition. More precisely, the perturbation of a solution to the Boltzmann equation around a global equilibrium is decomposed into the sum of a macroscopic and a microscopic part, for which we obtain a priori estimates at both lower and higher orders. These estimates are obtained under a suitable smallness assumption. The assumption can be justified a posteriori in the higher-order case, leading to the closure of the corresponding estimate.

Keywords

multicomponent gas mixture, energy method, micro-macro decomposition, conservation laws, smallness assumptions

2010 Mathematics Subject Classification

35Q20, 35Q35, 82C40

Received 13 October 2021

Received revised 30 August 2022

Accepted 9 May 2023

Published 7 December 2023