Vol. 2, No. 2, 2020

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Hypocoercivity without confinement

Emeric Bouin, Jean Dolbeault, Stéphane Mischler, Clément Mouhot and Christian Schmeiser

Vol. 2 (2020), No. 2, 203–232
Abstract

Hypocoercivity methods are applied to linear kinetic equations with mass conservation and without confinement in order to prove that the solutions have an algebraic decay rate in the long-time range, which the same as the rate of the heat equation. Two alternative approaches are developed: an analysis based on decoupled Fourier modes and a direct approach where, instead of the Poincaré inequality for the Dirichlet form, Nash’s inequality is employed. The first approach is also used to provide a simple proof of exponential decay to equilibrium on the flat torus. The results are obtained on a space with exponential weights and then extended to larger function spaces by a factorization method. The optimality of the rates is discussed. Algebraic rates of decay on the whole space are improved when the initial datum has moment cancellations.

Keywords
hypocoercivity, linear kinetic equations, Fokker–Planck operator, scattering operator, transport operator, Fourier mode decomposition, Nash's inequality, factorization method, Green's function, micro-/macrodecomposition, diffusion limit
Mathematical Subject Classification 2010
Primary: 82C40
Secondary: 76P05, 35H10, 35K65, 35P15, 35Q84
Milestones
Received: 7 November 2018
Revised: 20 September 2019
Accepted: 24 November 2019
Published: 22 May 2020
Authors
Emeric Bouin
Ceremade (CNRS UMR 7534)
PSL University
Université Paris-Dauphine
Paris
France
Jean Dolbeault
Ceremade (CNRS UMR 7534)
PSL University
Université Paris-Dauphine
Paris
France
Stéphane Mischler
Ceremade (CNRS UMR 7534)
PSL University
Université Paris-Dauphine
Paris
France
Clément Mouhot
DPMMS
Centre for Mathematical Sciences
University of Cambridge
Cambridge
United Kingdom
Christian Schmeiser
Fakultät für Mathematik
Universität Wien
Wien
Austria