Stability estimates for the Vlasov-Poisson system in $p$-kinetic Wasserstein distances

TL;DR

This paper generalizes stability estimates for the Vlasov-Poisson system from the classical $L^2$ and Wasserstein distances to a broader class of $L^p$ and kinetic Wasserstein distances, enhancing understanding of system stability.

Contribution

It extends Loeper's $L^2$-estimate to $L^p$ spaces and generalizes stability estimates to kinetic Wasserstein distances of order $p$, broadening the analytical framework.

Findings

Extended Loeper's $L^2$-estimate to $L^p$ spaces.

Generalized stability estimates to kinetic Wasserstein distances for $p > 1$.

Provided new tools for analyzing the stability of the Vlasov-Poisson system.

Abstract

We extend Loeper's $L^2$-estimate relating the electric fields to the densities for the Vlasov-Poisson system to $L^p$, with $1 < p < +\infty$, based on the Helmholtz-Weyl decomposition. This allows us to generalize both the classical Loeper's $2$-Wasserstein stability estimate and the recent stability estimate by the first author relying on the newly introduced kinetic Wasserstein distance to kinetic Wasserstein distances of order $1 < p < +\infty$.