Hypocoercivity without confinement

Emeric Bouin (1) , Jean Dolbeault (1) , Stéphane Mischler (1) , Clément Mouhot (2) , Christian Schmeiser (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 DPMMS - Department of Pure Mathematics and Mathematical Statistics
3 Fakultät für Mathematik [Wien]

In this paper, hypocoercivity methods are applied to linear kinetic equations with mass conservation and without confinement, in order to prove that the long-time behavior has algebraic decay as in the case 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 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 also improved when the initial datum has zero average.

Domaines: Equations aux dérivées partielles [math.AP]

Emeric Bouin, Jean Dolbeault, Stéphane Mischler, Clément Mouhot, Christian Schmeiser. Hypocoercivity without confinement. 2017. ⟨hal-01575501v1⟩