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

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.