Green's Function and the Pointwise Behaviors of the Vlasov-Poisson-Fokker-Planck System

The pointwise space-time behaviors of the Green's function and the global solution to the Vlasov-Poisson- Fokker-Planck (VPFP) system in three dimensional space are studied in this paper. It is shown that  the Green's function  consists of  the  diffusion waves decaying exponentially in time but algebraically in space, and the singular kinetic waves which become smooth for all $(t,x,v)$ when $t>0.$ Furthermore, we establish the pointwise space-time  behaviors of the global solution to the nonlinear VPFP system when the initial data is not necessarily smooth in terms of the Green's function.