On nonlinear Landau damping and Gevrey regularity

TL;DR

This paper investigates nonlinear Landau damping for the Vlasov-Poisson equations on a torus, demonstrating stability in certain regularity classes and constructing initial data that exhibit damping.

Contribution

It establishes nonlinear stability in Gevrey classes larger than 3 and constructs Sobolev regular initial data showing Landau damping, extending previous results.

Findings

Nonlinear stability holds for perturbations in classes larger than Gevrey 3.

Existence of Sobolev regular initial data exhibiting Landau damping.

Stability persists over time intervals up to ext{epsilon}^{-N}.

Abstract

In this article we study the problem of nonlinear Landau damping for the Vlasov-Poisson equations on the torus. As our main result we show that for perturbations initially of size $\epsilon>0$ and time intervals $(0,\epsilon^{-N})$ one obtains nonlinear stability in regularity classes larger than Gevrey $3$, uniformly in $\epsilon$. As a complementary result we construct families of Sobolev regular initial data which exhibit nonlinear Landau damping. Our proof is based on the methods of Grenier, Nguyen and Rodnianski.