On the stability of vacuum in the screened Vlasov-Poisson equation

TL;DR

This paper investigates the long-term behavior of small solutions to the screened Vlasov-Poisson equation near vacuum, demonstrating scattering in higher dimensions and long-time existence in one dimension under specific regularity conditions.

Contribution

It provides the first analysis of asymptotic stability and scattering for the screened Vlasov-Poisson equation in multiple dimensions, and establishes long-time existence results in one dimension.

Findings

Solutions scatter freely in dimensions d ≥ 2.

Long time existence for solutions in dimension d=1.

Requires mild localization and regularity assumptions.

Abstract

We study the asymptotic behavior of small data solutions to the screened Vlasov-Poisson equation on $\mathbb{R}^d\times\mathbb{R}^d$ near vacuum. We show that for dimensions $d\geq 2$, under mild assumptions on localization (in terms of spatial moments) and regularity (in terms of at most three Sobolev derivatives) solutions scatter freely. In dimension $d=1$, we obtain a long time existence result in analytic regularity.