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Hypocoercivity without confinement

Bouin, Emeric; Dolbeault, Jean; Mischler, Stéphane; Mouhot, Clément; Schmeiser, Christian (2020), Hypocoercivity without confinement, Pure and Applied Analysis, 2, 2, p. 203-232. 10.2140/paa.2020.2.203

Type
Article accepté pour publication ou publié
Date
2020
Journal name
Pure and Applied Analysis
Volume
2
Number
2
Pages
203-232
Publication identifier
Metadata
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Author(s)
Bouin, Emeric
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Dolbeault, Jean cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Mischler, Stéphane
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Mouhot, Clément
Unité de Mathématiques Pures et Appliquées [UMPA-ENSL]
Schmeiser, Christian
Fakultät für Mathematik [Wien]
Abstract (EN)
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.
Subjects / Keywords
diffusion limit; micro/macro decomposition; Hypocoercivity; linear kinetic equations; Fokker-Planck operator; scattering operator; transport operator; Fourier mode decomposition; Nash's inequality; factorization method; Green's function