Smoothing Effect for the Non-linear Vlasov-Poisson-Fokker-Planck System

We study a smoothing effect of a non-linear degenerate parabolic problem, the Vlasov-Poisson-Fokker-Planck system, in three dimensions. We prove that a solution gives rise actually to very smooth macroscopic density and force field for positive time. This is obtained by analyzing the effect of the Fokker-Planck kernel on the force term in the Vlasov equation and using classical convolution inequalities.