[Hypocoercivité pour des équations cinétiques avec termes de relaxation linéaires]
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.
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@article{CRMATH_2009__347_9-10_511_0,
author = {Dolbeault, Jean and Mouhot, Cl\'ement and Schmeiser, Christian},
title = {Hypocoercivity for kinetic equations with linear relaxation terms},
journal = {Comptes Rendus. Math\'ematique},
pages = {511--516},
publisher = {Elsevier},
volume = {347},
number = {9-10},
year = {2009},
doi = {10.1016/j.crma.2009.02.025},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.02.025/}
}
TY - JOUR AU - Dolbeault, Jean AU - Mouhot, Clément AU - Schmeiser, Christian TI - Hypocoercivity for kinetic equations with linear relaxation terms JO - Comptes Rendus. Mathématique PY - 2009 SP - 511 EP - 516 VL - 347 IS - 9-10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.02.025/ DO - 10.1016/j.crma.2009.02.025 LA - en ID - CRMATH_2009__347_9-10_511_0 ER -
%0 Journal Article %A Dolbeault, Jean %A Mouhot, Clément %A Schmeiser, Christian %T Hypocoercivity for kinetic equations with linear relaxation terms %J Comptes Rendus. Mathématique %D 2009 %P 511-516 %V 347 %N 9-10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.02.025/ %R 10.1016/j.crma.2009.02.025 %G en %F CRMATH_2009__347_9-10_511_0
Dolbeault, Jean; Mouhot, Clément; Schmeiser, Christian. Hypocoercivity for kinetic equations with linear relaxation terms. Comptes Rendus. Mathématique, Tome 347 (2009) no. 9-10, pp. 511-516. doi : 10.1016/j.crma.2009.02.025. http://www.numdam.org/articles/10.1016/j.crma.2009.02.025/
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