The non-isothermal Maxwell-Stefan asymptotics of the multi-species Boltzmann equations

TL;DR

This paper rigorously derives the non-isothermal Maxwell-Stefan system from the multi-species Boltzmann equations, establishing global well-posedness, spectral gap properties, and uniform solutions in the diffusive scaling, extending previous isothermal results.

Contribution

It provides the first rigorous derivation of the non-isothermal Maxwell-Stefan asymptotics from the Boltzmann equations, including well-posedness and spectral analysis.

Findings

Established global-in-time well-posedness of the Maxwell-Stefan system.

Proved uniform solutions to Boltzmann equations in the diffusive limit.

Extended previous isothermal results to the non-isothermal case.

Abstract

We study the convergence from the multi-species Boltzmann equations to the non-isothermal Maxwell-Stefan system. The global-in-time well-posedness of the Maxwell-Stefan system is first established. The solution is utilized as the fluid quantities to construct a local Maxwellian vector. The Maxwell-Stefan system can be derived from the multi-species Boltzmann equations under diffusive scaling by adding a relation on the total concentration. Different with the classical hydrodynamic limits of the Boltzmann equations, the Maxwellian based on the Maxwell-Stefan system is not a local equilibrium for the mixtures due to cross-interactions. A local coercivity property for the operator linearized around the local Maxwellian is established, based on the explicit spectral gap of the operator linearized around the global equilibrium. The global-in-time solution to the multi-species Boltzmann equations uniform in Knudsen number $\varepsilon$ is established in this scaling, thus the first non-isothermal Maxwell-Stefan asymptotics is rigorously justified. This generalizes Bondesan and Briant's work \cite{briant2021stability} from isothermal to non-isothermal case.