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

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 $\hat{I}»|x{|}^{-\hat{I}\pm }$ in the strictly long-range regime ($0<\hat{I}\pm <1$). 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 $\frac{1}{2}<\hat{I}\pm <1$ and provides a distinct and more robust argument.

32 pages