Abstract:
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. |