I will discuss the topic of linear and nonlinear stability of homogeneous equilibria among solutions of the Vlasov-Poisson system, both in the confined case of periodic domains and in the unconfined case of the Euclidean space. The Euclidean problem differs significantly from the classical work on Landau damping in the periodic setting, in several ways. Most importantly, the linearized problem cannot satisfy a “Penrose condition” so the system contains resonances (small divisors) and the electric field can decay at most polynomially in time.
This is joint work with Benoit Pausader, Xuecheng Wang, and Klaus Widmayer.