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

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.

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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
Received: 7 November 2018
Revised: 20 September 2019
Accepted: 24 November 2019
Published: 22 May 2020
Emeric Bouin
Ceremade (CNRS UMR 7534)
PSL University
Université Paris-Dauphine
Jean Dolbeault
Ceremade (CNRS UMR 7534)
PSL University
Université Paris-Dauphine
Stéphane Mischler
Ceremade (CNRS UMR 7534)
PSL University
Université Paris-Dauphine
Clément Mouhot
Centre for Mathematical Sciences
University of Cambridge
United Kingdom
Christian Schmeiser
Fakultät für Mathematik
Universität Wien