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The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium

Abstract : We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the 3-dimensional torus. The ultimate aim of this work is to obtain existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in L 1 v L ∞ x (m), where m ∼ (1 + |v| k) is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an L 2 − L ∞ theoryàtheoryà la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (e.g. Carleman representation , Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the L 1 v L ∞ x framework is dealt with for any k > k 0 , recovering the optimal physical threshold of finite energy k 0 = 2 in the particular case of a multi-species hard spheres mixture with same masses.
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https://hal.archives-ouvertes.fr/hal-01492048
Contributor : Marc Briant Connect in order to contact the contributor
Submitted on : Friday, March 17, 2017 - 6:40:02 PM
Last modification on : Wednesday, October 13, 2021 - 7:58:04 PM
Long-term archiving on: : Sunday, June 18, 2017 - 1:52:17 PM

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Marc Briant, Esther Daus. The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium. Archive for Rational Mechanics and Analysis, Springer Verlag, 2016, 222 (3), pp.1367-1443. ⟨10.1007/s00205-016-1023-x⟩. ⟨hal-01492048⟩

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