Landau Damping and Plasma Echoes

This review article provides an overview and the state of the art of the study of Landau damping and plasma echoes in the classical Vlasov–Poisson system near spatially homogeneous equilibria that describes the dynamics of a plasma confined on a torus or in the whole space. In particular, the article reproduces an analyticity framework developed for nonspecialists in [E. Grenier, T. T. Nguyen, and I. Rodnianski, SIAM J. Math. Anal., 54 (2022), pp. 940–953] that constructs plasma echo solutions consisting of infinitely many interacting elementary waves with arbitrarily high frequencies. The echo solutions allow a much larger class of data than those in analytic or Gevrey spaces studied by Mouhot and Villani, now including certain data with arbitrarily large Sobolev norms; nonetheless, these solutions do exhibit Landau damping in the large time.