Hypocoercivity for kinetic equations with linear relaxation terms

Jean Dolbeault; Clément Mouhot; Christian Schmeiser

doi:10.1016/j.crma.2009.02.025

Partial Differential Equations/Mathematical Physics

Hypocoercivity for kinetic equations with linear relaxation terms
[Hypocoercivité pour des équations cinétiques avec termes de relaxation linéaires]

Présenté par : Pierre-Louis Lions

Jean Dolbeault ¹ ; Clément Mouhot ¹ ; Christian Schmeiser ²

¹ Ceremade (UMR CNRS no. 7534), Université Paris-Dauphine, place de-Lattre-de-Tassigny, 75775 Paris cedex 16, France
² Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria

Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 511-516.

Résumé
Abstract

Cette Note est consacrée à une méthode simple pour démontrer l'hypocoercivité associée à une équation cinétique contenant un opérateur de relaxation linéaire ; il s'agit de construire une fonctionnelle de Lyapunov adaptée vérifiant une inégalité de type Gronwall. La méthode distingue clairement la coercivité au niveau microscopique, qui provient directement des propriétés de l'opérateur de relaxation, et une inégalité de trou spectral pour la densité spatiale, qui est reliée à la limite de diffusion. Elle améliore les résultats antérieurs. Notre approche est illustrée par le modèle de BGK linéaire et par un opérateur de relaxation qui correspond, au niveau macroscopique, à la diffusion rapide linéarisée.

This Note is devoted to a simple method for proving the hypocoercivity associated to a kinetic equation involving a linear time relaxation operator. It is based on the construction of an adapted Lyapunov functional satisfying a Gronwall-type inequality. The method clearly distinguishes the coercivity at microscopic level, which directly arises from the properties of the relaxation operator, and a spectral gap inequality at the macroscopic level for the spatial density, which is connected to the diffusion limit. It improves on previously known results. Our approach is illustrated by the linear BGK model and a relaxation operator which corresponds at macroscopic level to the linearized fast diffusion.

Reçu le : 2008-10-20
Accepté le : 2009-02-16
Publié le : 2009-04-02

DOI : 10.1016/j.crma.2009.02.025

Affiliations des auteurs :

Jean Dolbeault ¹ ; Clément Mouhot ¹ ; Christian Schmeiser ²

¹ Ceremade (UMR CNRS no. 7534), Université Paris-Dauphine, place de-Lattre-de-Tassigny, 75775 Paris cedex 16, France
² Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria

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     title = {Hypocoercivity for kinetic equations with linear relaxation terms},
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Jean Dolbeault; Clément Mouhot; Christian Schmeiser. Hypocoercivity for kinetic equations with linear relaxation terms. Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 511-516. doi : 10.1016/j.crma.2009.02.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.02.025/

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Jean Dolbeault; Clément Mouhot; Christian Schmeiser Hypocoercivity for kinetic equations with linear relaxation terms, Comptes Rendus. Mathématique. Académie des Sciences, Paris, Volume 347 (2009) no. 9-10, pp. 511-516 | DOI:10.1016/j.crma.2009.02.025 | Zbl:1177.35054

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