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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- data, (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- norms.
Grenier et al. (Fri,) studied this question.
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