Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions

Andrea Bondesan; Laurent Boudin; Marc Briant; Bérénice Grec

doi:10.3934/cpaa.2020112

Article Contents

2020, Volume 19, Issue 5: 2549-2573. Doi: 10.3934/cpaa.2020112

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Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions

1.
Université de Paris, Université Paris Descartes, Laboratoire MAP5, CNRS UMR 8145, F-75006 Paris, FRANCE
2.
Sorbonne Université, Université Paris–Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions, Équipe Reo, F-75005 Paris, FRANCE

^*Corresponding author
^*Corresponding author

Received: October 31, 2018

Revised: July 31, 2019

Published: March 2020

This work was partially funded by the French ANR-13-BS01-0004 project Kibord headed by L. Desvillettes. The first and fourth authors have been partially funded by Université Sorbonne Paris Cité, in the framework of the "Investissements d'Avenir", convention ANR-11-IDEX-0005

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Abstract

We consider the Boltzmann operator for mixtures with cutoff Maxwellian, hard potential, or hard-sphere collision kernels. In a perturbative regime around the global Maxwellian equilibrium, the linearized Boltzmann multi-species operator $ \mathbf{L} $ is known to possess an explicit spectral gap $ \lambda_{ \mathbf{L}} $, in the global equilibrium weighted $ L^2 $ space. We study a new operator $ \mathbf{ L^{\varepsilon}} $ obtained by linearizing the Boltzmann operator for mixtures around local Maxwellian distributions, where all the species evolve with different small macroscopic velocities of order $ \varepsilon $, $ \varepsilon >0 $. This is a non-equilibrium state for the mixture. We establish a quasi-stability property for the Dirichlet form of $ \mathbf{ L^{\varepsilon}} $ in the global equilibrium weighted $ L^2 $ space. More precisely, we consider the explicit upper bound that has been proved for the entropy production functional associated to $ \mathbf{L} $ and we show that the same estimate holds for the entropy production functional associated to $ \mathbf{ L^{\varepsilon}} $, up to a correction of order $ \varepsilon $.

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Mathematics Subject Classification: Primary: 76P05, 82B40; Secondary: 35P15.

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