On nonlinear Landau damping and Gevrey regularity

In this article we study the problem of nonlinear Landau damping for the Vlasov–Poisson equations on the torus. We introduce anisotropic Gevrey spaces as a new tool to capture the time- and frequency-dependence of resonances. In particular, we show that for small initial data of size $0<ϵ<{ϵ}_0\left(N\right)$ and time intervals $(0,{ϵ}^{-N})$ with $N\in N$ arbitrary but fixed, nonlinear stability holds in regularity classes larger than Gevrey-3, uniformly in $ϵ$. 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 (Grenier et al., 2021).