Uniqueness of the solution to the Vlasov-Poisson system with bounded density

In this paper, we show uniqueness of weak solutions to the Vlasov-Poisson system on the only condition that the macroscopic density rho defined by rho(t, x) = integral(Rd) f (t, x, xi) d xi is bounded in L-infinity. Our proof is based on optimal transportation. (C) 2006 Elsevier SAS. All rights reserved.