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

TL;DR

This paper proves scattering results for Vlasov-type equations on the torus with Gevrey regularity, extending previous analytic results to a broader Gevrey class and higher dimensions, and addresses an open question in the field.

Contribution

It extends scattering results for Vlasov equations from analytic to Gevrey regularity and higher dimensions, solving an open problem in the literature.

Findings

Established scattering for Gevrey-$rac{1}{eta}$ perturbations on the torus.

Extended previous analytic regularity results to Gevrey classes.

Addressed an open question by proving higher-dimensional scattering results.

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 $\mathbb{T}^d$, $d \geq 1$, by considering asymptotic states of regularity Gevrey-$\frac{1}{\gamma}$ with $\gamma>\frac13$, small perturbations of homogeneous equilibria satisfying the Penrose stability condition. This extends to the Gevrey perturbative case, and to higher dimension, the scattering result in analytic regularity obtained by E. Caglioti and C. Maffei in [14], and answers an open question raised by J. Bedrossian in arXiv:2211.13707.