Landau damping studies relaxation without dissipation of the electric field in a collisionless plasma in the large time (similar to inviscid damping found in an ideal fluid).

The talk aims to give an overview of this topics for both confined and unconfined plasmas, including linear and nonlinear damping results, based on several recent joint works with E. Grenier (ENS Lyon), I. Rodnianski (Princeton) and with D. Han-Kwan (Polytechnique), F. Rousset (Paris-Sud). In particular, the talk ends with an elementary proof of the nonlinear Landau damping for analytic and Gevrey data on torus, and a construction of echo solutions which are arbitrarily large in Sobolev spaces but damped in the large time.