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Elsevier, Comptes Rendus Mathématique, 9-10(347), p. 511-516

DOI: 10.1016/j.crma.2009.02.025

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Hypocoercivity for kinetic equations with linear relaxation terms

Journal article published in 2009 by Jean Dolbeault ORCID, Clément Mouhot, Christian Schmeiser
This paper is available in a repository.
This paper is available in a repository.

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Abstract

This note is devoted to a simple method for proving hypocoercivity of the solutions of a kinetic equation involving a linear time relaxation operator, i.e. 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.