Propagation of moments and uniqueness of weak solutions to the Vlasov-Poisson-Fokker-Planck system

TL;DR

This paper establishes the uniqueness of weak solutions to the Vlasov-Poisson-Fokker-Planck system under certain conditions and demonstrates the propagation of velocity moments of all orders greater than two.

Contribution

It proves the uniqueness of weak solutions with specific initial data and shows moments with respect to velocity propagate for any order above two.

Findings

Uniqueness of weak solutions in $C([0,T]; L^p)$ under bounded density assumptions.

Propagation of velocity moments of order higher than 2.

Corollary establishing uniqueness for initial data with finite energy.

Abstract

In this paper, we prove the uniqueness of weak solutions to the Vlasov-Poisson-Fokker-Planck system in $C([0,T]; L^p)$, by assuming the solution has a local bounded density which tends to infinite with a "reasonable" rate as $t\to 0$. And particularly as a corollary, we get the uniqueness of weak solutions with initial data $f_0$ satisfying $f_0|v|^2\in L^1$, which solves the uniqueness of weak solutions with finite energy. In addition, we prove that the moments with respect to the velocity propagate for any order higher than 2.