Green’s Function and Pointwise Space-time Behaviors of the Vlasov-Poisson-Boltzmann System

The pointwise space-time behaviors of the Green’s function and the global solution to the Vlasov-Poisson-Boltzmann (VPB) system in spatial three dimension are studied in this paper. It is shown that, due to the influence of electrostatic potential governed by the Poisson equation, the Green’s function admits only the macroscopic nonlinear diffusive waves, the singular kinetic waves, and the remainder term decaying exponentially in time but algebraically in space. These behaviors have an essential difference from the Boltzmann equation, namely, the Huygen’s type sound wave propagation and the space-time exponential decay of remainder term for Boltzmann equation (Liu and Yu in Commun Pure Appl Math 57:1543–1608, 2004; Bull Inst Math Acad Sin (NS) 1(1): 1–78, 2006) cannot be observed for VPB system. Furthermore, we establish the pointwise space-time nonlinear diffusive behaviors of the global solution to the nonlinear VPB system in terms of the Green’s function. Some new strategies are introduced to deal with the difficulties caused by the electric fields.