Global weak solutions for the initial-boundary-value problems to the Vlasov-Poisson-Fokker-Planck system

Abstract

This work is devoted to prove the existence of weak solutions of the kinctic Vlassorv-Poisson-Fokker-Planck system in bounded domains for attractive of repulsive forces. Absorbing and reflection-type boundary conditions are considered for the kinetic eqution and zero values for the potential on the boundary. The existence of weak solutions is proved for bounded and integrable initial and boundary data with finite energy. The main difficulty of this problem is to obtain an existence theory for the liner equation. This fact is analysed using a variational technique and the theory of elliptic-parabolic equations of second order. The proof of existence for the initial-boundary value problem is carried out following a procedure of regularization and linearization of the problem. © 1998 B. G. Teubner Stuttgart - John Wiley & Sons, Ltd.