ON THE INITIAL-VALUE PROBLEM FOR THE VLASOV-POISSON-FOKKER-PLANCK SYSTEM WITH INITIAL DATA IN L(P) SPACES

In this paper the global existence of weak solutions for the Vlasov-Po isson-Fokker-Planck equations in three dimensions is proved with an L( 1) boolean AND L(p) initial data. Also, the global existence of weak s olutions in four dimensions with small initial data is studied. A conv ergence of the solutions is obtained to those built by E. Horst and R. Hunze when the Fokker-Planck term vanishes. In order to obtain the a priori necessary estimates a sequence of approximate problems is intro duced. This sequence is obtained starting from a non-linear regulariza tion of the problem together with a linearization via a time retarded mollification of the non-linear term. The a priori bounds are reached by means of the control of the kinetic energy in the approximate seque nce of problems. Then, the proof is completed obtaining the equicontin uity properties which allow to pass to the limit.