Global Weak Solutions for the Initial–Boundary-Value Problems to the Vlasov–Poisson–Fokker–Planck System

This work is devoted to prove the existence of weak solutions of the kinetic Vlasov—Poisson— Fokker—Planck system in bounded domains for attractive or repulsive forces. Absorbing and reflectiontype boundary conditions are considered for the kinetic equation 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 linear 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 problems is carried out following a procedure of regularization and linearization of the problem. ( 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.