Modified scattering for the Vlasov-Riesz system with long-range interactions

TL;DR

This paper investigates the long-term behavior of small solutions to the 3D Vlasov-Riesz system with long-range interactions, establishing modified scattering and extending previous results to a broader parameter range.

Contribution

It introduces a Lagrangian approach with wave operators to prove modified scattering for the Vlasov-Riesz system in a wider regime of the interaction parameter.

Findings

Established modified scattering for $0<\,\alpha<1$

Extended previous results to $\frac{1}{2}<\alpha<1$

Developed a robust Lagrangian proof technique

Abstract

We study the long-time asymptotic behavior of small-data solutions to the three-dimensional Vlasov--Riesz system with the inverse power-law potential $\lambda |x|^{-\alpha}$ in the strictly long-range regime ($0 < \alpha < 1$). By introducing finite- and infinite-time modified wave operators for the characteristic flows, we describe the asymptotic dynamics via convergence to an effective profile along a suitably modified reference flow, and establish modified scattering of solutions. Our proof relies mainly on ODE techniques for the characteristic flows, while also using PDE methods for weighted $W^{1,\infty}$-bounds. Compared with the earlier result (of Huang and Kwon), our Lagrangian approach extends modified scattering to the broader regime $\frac{1}{2}<\alpha<1$ and provides a distinct and more robust argument.