Scattering problem for Vlasov-type equations on the $d$-dimensional torus with Gevrey data

Abstract

In this article, we consider Vlasov-type equations describing the evolution of single-species-type plasmas, such as those composed of electrons (Vlasov–Poisson) or ions (screened Vlasov–Poisson/Vlasov–Poisson with massless electrons). We solve the final data problem on the torus ${\mathbb{T}}^d$, $d\ge 1$, by considering asymptotic states of regularity Gevrey-$\frac{1}{\gamma }$ with $\gamma >\frac{1}{3}$, small perturbations of homogeneous equilibria satisfying the Penrose stability condition. This extends to the Gevrey perturbative case, and to higher dimensions, the scattering result in analytic regularity obtained by Caglioti and Maffei [J. Statist. Phys. 92 (1998), 301–323] and answers an open question raised in Bedrossian (2022).