Young Researchers Workshop: Kinetic models in biology and social sciences

Hypocoercivity without confinement

Emeric Bouin

Paris Dauphine University

In this talk, 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 Poincar{\'e} 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.