Energy method for the Boltzmann equation of monatomic gaseous mixtures

TL;DR

This paper develops an energy method using micro-macro decomposition to analyze the Boltzmann equations for monatomic gaseous mixtures, providing a priori estimates for solutions near equilibrium.

Contribution

It introduces a novel energy approach for multicomponent Boltzmann systems based on micro-macro decomposition, enabling rigorous a priori estimates.

Findings

Establishment of a priori estimates for solutions near equilibrium.

Justification of smallness assumptions for higher-order estimates.

Closure of estimates leading to potential stability results.

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.