On nonlinear Landau damping and Gevrey regularity

In this article we study the problem of nonlinear Landau damping for the Vlasov-Poisson equations on the torus. As our main result we show that for perturbations initially of size $\epsilon > 0$ and time intervals $(0, \epsilon−N)$ one obtains nonlinear stability in regularity classes larger than Gevrey 3, uniformly in ǫ. As a complementary result we construct families of Sobolev regular initial data which exhibit nonlinear Landau damping. Our proof is based on the methods of Grenier, Nguyen and Rodnianski [GNR20].