Mathematical Research Letters

Volume 28 (2021)

Number 6

Landau damping for analytic and Gevrey data

Pages: 1679 – 1702



Emmanuel Grenier (CNRS et École Normale Supérieure de Lyon, UMR 5669, Lyon, France)

Toan T. Nguyen (Department of Mathematics, Pennsylvania State University, State College, Pa., U.S.A.)

Igor Rodnianski (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)


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