Scattering of the Vlasov-Riesz system in the three dimensions

TL;DR

This paper studies the long-term behavior of solutions to the Vlasov-Riesz system in three dimensions, proving scattering results for certain interaction orders and establishing the existence of wave operators, marking a novel contribution in kinetic models.

Contribution

It provides the first proof of modified scattering with polynomial correction for the Vlasov-Riesz system in 3D, including existence of wave operators for specific interaction ranges.

Findings

Proves small data scattering for 1/2<α<1.

Establishes modified scattering for 1<α<1+δ.

Shows existence of (modified) wave operators.

Abstract

We consider an asymptotic behavior of solutions to the Vlasov-Riesz system of order $\alpha$ in $\mathbb{R}^3$ which is a kinetic model induced by Riesz interactions. We prove small data scattering when $1/2<\alpha<1$ and modified scattering when $1<\alpha<1+\delta$ for some $\delta>0$. Moreover, we show the existence of (modified) wave operators for such a regime. To the best of our knowledge, this is the first result on the existence of modified scattering with polynomial correction in kinetic models.