Hypocoercive relaxation to equilibrium for some kinetic models

Pierre Monmarché

doi:10.3934/krm.2014.7.341

Article Contents

2014, Volume 7, Issue 2: 341-360. Doi: 10.3934/krm.2014.7.341

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Hypocoercive relaxation to equilibrium for some kinetic models

Pierre Monmarché^1,

1.
Laboratoire de Statistiques et Probabilités, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse cedex

Received: June 2013

Revised: February 2014

Published: June 2014

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Abstract

This paper deals with the study of some particular kinetic models, where the randomness acts only on the velocity variable level. Usually, the Markovian generator cannot satisfy any Poincaré's inequality. Hence, no Gronwall's lemma can easily lead to the exponential decay of $F_t$ (the $L^2$ norm of a test function along the semi-group). Nevertheless for the kinetic Fokker-Planck dynamics and for a piecewise deterministic evolution we show that $F_t$ satisfies a third order differential inequality which gives an explicit rate of convergence to equilibrium.

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Mathematics Subject Classification: Primary: 60J99; Secondary: 35B40.

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