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

Andrea Bondesan; Marc Briant

doi:10.3934/dcds.2021210

Article Contents

2022, Volume 42, Issue 6: 2747-2773. Doi: 10.3934/dcds.2021210

This issue Previous Article A free boundary problem of nonlinear diffusion equation with positive bistable nonlinearity in high space dimensions I : Classification of asymptotic behavior Next Article On

$ n $

-tuplewise IP-sensitivity and thick sensitivity

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

Andrea Bondesan^1, and
Marc Briant^2, ,

1.
University of Graz, Institute of Mathematics and Scientific Computing, 8010 Graz, Austria
2.
Université de Paris, Laboratoire MAP5, UMR CNRS 8145, F-75006 Paris, France

^*Corresponding author
^*Corresponding author

Received: January 2021

Revised: October 2021

Early access: January 2022

Published: June 2022

The authors would like to thank Laurent Boudin and Bérénice Grec for fruitful discussions on the theory of gaseous and fluid mixtures

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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.

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Mathematics Subject Classification: Primary: 35A01, 76B03, 35Q35; Secondary: 35K51, 35K55.

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