Going beyond the linearized study has been a longstanding problem in the theory of the Landau damping. In this paper we establish Landau damping for the nonlinear Vlasov equation, for any interaction potential less singular than Coulomb and for analytic perturbation. We point out the (a priori unexpected) critical nature of the Coulomb potential and analytic regularity, which can be seen only at the nonlinear level; in this case we derive Landau damping over finite but exponentially long times. Physical implications are discussed.
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