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# Mathematical Research Letters

## Volume 28 (2021)

### Number 6

### Landau damping for analytic and Gevrey data

Pages: 1679 – 1702

DOI: https://dx.doi.org/10.4310/MRL.2021.v28.n6.a3

#### Authors

#### Abstract

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.

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