Scattering of the Vlasov-Riesz system in the three dimensions

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.

38 pages; v3. Many typos were fixed, and Theorem 1.3 was changed;