Sharp estimates for screened Vlasov-Poisson system around Penrose-stable equilibria in $\mathbb{R}^d $, $ d\geq3$

TL;DR

This paper establishes sharp decay estimates for the density in the screened Vlasov-Poisson system around Penrose-stable equilibria in higher dimensions, improving previous results by handling less regular initial data and nonlinear effects.

Contribution

It provides the first sharp decay estimates for the density with Hölder continuous initial data and extends the analysis to systems with nonlinear Poisson equations.

Findings

Sharp decay estimates for density similar to free transport

Improved results for lower and higher derivatives of density

Extension to nonlinear Poisson systems with massless electrons/ions

Abstract

In this paper, we study the asymptotic stability of Penrose-stable equilibria among solutions of the screened Vlasov-Poisson system in $\mathbb{R}^d$ with $d\geq 3$ that was first established by Bedrossian, Masmoudi, and Mouhot in \cite{JBedrossian2018} with smooth initial data. More precisely, we prove the sharp decay estimates for the density of the perturbed system, exactly like the free transport with only H\"older (i.e., $C^{a}$ for $0<a<1$) perturbed initial data. This improves the recent works in \cite{HanKwanD2021} by Han-Kwan, Nguyen, and Rousset for lower derivatives of the density and in \cite{NguyenTT2020} by T. Nguyen for higher derivatives with a logarithmic correction in time. Furthermore, we establish new estimates and cancellations of the kernel to the linearized problem to obtain this result. Moreover, we also prove this result for the Vlasov-Poisson system in which the electric field obeys a general nonlinear Poisson equation containing massless electrons/ions case.