Perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system

TL;DR

This paper develops a perturbative Cauchy theory for a flux-incompressible Maxwell-Stefan system, proving global existence, uniqueness, and exponential convergence of solutions near equilibrium states.

Contribution

It introduces a novel perturbative analysis of the Maxwell-Stefan system around non-constant equilibria, which was previously unexplored in the literature.

Findings

Proves global existence and uniqueness of strong solutions.

Demonstrates exponential convergence to equilibrium.

Shows the natural emergence of the equimolar diffusion condition.

Abstract

Recently, the authors proved [2] that the Maxwell-Stefan system with an incompressibility-like condition on the total flux can be rigorously derived from the multi-species Boltzmann equation. Similar cross-diffusion models have been widely investigated, but the particular case of a perturbative incompressible setting around a non constant equilibrium state of the mixture (needed in [2]) seems absent of the literature. We thus establish a quantitative perturbative Cauchy theory in Sobolev spaces for it. More precisely, by reducing the analysis of the Maxwell-Stefan system to the study of a quasilinear parabolic equation on the sole concentrations and with the use of a suitable anisotropic norm, we prove global existence and uniqueness of strong solutions and their exponential trend to equilibrium in a perturbative regime around any macroscopic equilibrium state of the mixture. As a by-product, we show that the equimolar diffusion condition naturally appears from this perturbative incompressible setting.