Landau damping for analytic and Gevrey data

In this paper, we give an elementary proof of the nonlinear Landau damping for the Vlasov-Poisson system near Penrose stable equilibria on the torus Tᵈ × Rᵈ that was first obtained by Mouhot and Villani in {MV} for analytic data and subsequently extended by Bedrossian, Masmoudi, and Mouhot {BMM} for Gevrey-γ data, γ∈(13,1]. Our proof relies on simple pointwise resolvent estimates and a standard nonlinear bootstrap analysis, using an ad-hoc family of analytic and Gevrey-γ norms.