Existence and Uniqueness of a Global Smooth Solution for the Vlasov-Poisson-Fokker-Planck System in Three Dimensions

Studying precisely the regularity of the force field created by the solution of the linear Vlasov-Fokker-Planck equation, we prove existence and uniqueness of a smooth solution to the Vlasov-Poisson-Fokker-Planck system in three dimensions. The attractive and repulsive cases are treated. The method involves an analysis of the effect of the Fokker-Planck kernel, without any force field, on non-linear expressions coming from the Vlasov term.