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 $\mathbb{T}^d \times \mathbb{R}^d$ that was first obtained by Mouhot and Villani in [9] for analytic data and subsequently extended by Bedrossian, Masmoudi, and Mouhot [2] for Gevrey‑$\gamma$ data, $\gamma \in (\frac{1}{3},1]$. Our proof relies on simple pointwise resolvent estimates and a standard nonlinear bootstrap analysis, using an ad-hoc family of analytic and Gevrey‑$\gamma$ norms.

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T.N. was a Visiting Fellow at Department of Mathematics, Princeton University, and partly supported by the NSF under grant DMS-1764119, an AMS Centennial fellowship, and a Simons fellowship.

I.R. is partially supported by the NSF grant DMS #1709270 and a Simons Investigator Award.

Received 4 July 2020

Accepted 14 December 2020

Published 29 August 2022