Hypocoercivity for kinetic equations with linear relaxation terms
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Peer-reviewed
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Authors
Dolbeault, Jean
Mouhot, Clément
Schmeiser, Christian
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.
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Keywords
math.AP, math.AP, 82C40; 35B40; 35F10; 35H10; 35H99; 76P05
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1
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0764-4442
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0
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Elsevier
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Partially supported by the French-Austrian Amadeus project no. 13785UA, the ANR project IFO, the Austrian Science Fund (project no. W8) and the European network DEASE. The authors thank an anonymous referee for his valuable comments and suggestions.