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Hypocoercivity for linear kinetic equations conserving mass


Authors: Jean Dolbeault, Clément Mouhot and Christian Schmeiser
Journal: Trans. Amer. Math. Soc. 367 (2015), 3807-3828
MSC (2010): Primary 82C40; Secondary 35B40, 35F10, 35H10, 35H99, 76P05
DOI: https://doi.org/10.1090/S0002-9947-2015-06012-7
Published electronically: February 3, 2015
MathSciNet review: 3324910
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Abstract: We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $ L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed.


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Additional Information

Jean Dolbeault
Affiliation: Ceremade (UMR CNRS no. 7534), Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16, France
Email: dolbeaul@ceremade.dauphine.fr

Clément Mouhot
Affiliation: Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
Email: C.Mouhot@dpmms.cam.ac.uk

Christian Schmeiser
Affiliation: Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Wien, Austria
Email: Christian.Schmeiser@univie.ac.at

DOI: https://doi.org/10.1090/S0002-9947-2015-06012-7
Keywords: Kinetic equations, hypocoercivity, Boltzmann, BGK, relaxation, diffusion limit, nonlinear diffusion, Fokker-Planck, confinement, spectral gap, Poincar\'e inequality, Hardy-Poincar\'e inequality
Received by editor(s): May 10, 2010
Received by editor(s) in revised form: November 11, 2012
Published electronically: February 3, 2015
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.