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On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in $$L^ p$$ spaces. (English) Zbl 0829.35096
Summary: The global existence of weak solutions for the Vlasov-Poisson-Fokker- Planck equations in three dimensions is proved with an $$L^1 \cap L^p$$ initial data. Also, the global existence of weak solutions in four dimensions with small initial data is studied. A convergence 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 introduced. This sequence is obtained starting from a nonlinear regularization of the problem together with a linearization via a time retarded mollification of the nonlinear term. The a priori bounds are reached by means of the control of the kinetic energy in the approximate sequence of problems. Then, the proof is completed obtaining the equicontinuity properties which allow to pass to the limit.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76X05 Ionized gas flow in electromagnetic fields; plasmic flow 35D05 Existence of generalized solutions of PDE (MSC2000) 85A05 Galactic and stellar dynamics
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