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

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.

arXiv commentThere is a gap in the proof, we have to improve the estimates. We thank the refree for pointing out this to us. Moreover the assumption of "reasonable rate" does not so make sense in physics. Hence, we decide to withdraw this paper