Acoustic limit of Boltzmann equations for gas mixture

TL;DR

This paper rigorously derives the hydrodynamic and acoustic limits of Boltzmann equations for a two-species gas mixture with different particle masses, using a vector-valued framework and Hilbert expansion, addressing the loss of symmetry in the collision operator.

Contribution

It provides a novel framework for analyzing the limits of Boltzmann equations for gas mixtures with different masses, including the acoustic limit, by handling the asymmetry in the collision operator.

Findings

Hydrodynamic limit established for small Knudsen number.

Acoustic limit rigorously justified under initial data assumptions.

Analysis accounts for mass disparity in particle species.

Abstract

In this paper, we study the hydrodynamic and acoustic limit from Boltzmann equations for two species gas mixture with potential $\gamma \in \left(-3, 1\right]$. % in the whole space $(x \in \mathbb{R}^3)$.Here the particle masses are different which derives to the loss of symmetry to the linearized collision operator. %This paper resolves it precisely by using a framework based on vector-valued functions. We construct the hydrodynamic limit for two species based on the Hilbert expansion method when the Knudsen number is small. The key observation is the precise properties of the linearized collision operators, including the extra operators due to the different particle masses $(m^A \neq m^B)$. In additional, the acoustic limit of the Boltzmann equations for gas mixtures is rigorously justified by assuming the strength of the initial data depends on the Knudsen number.