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

TL;DR

This paper analyzes the Green's function and global solutions of the 3D Vlasov-Poisson-Fokker-Planck system, revealing decay properties and wave behaviors that inform the system's long-term dynamics.

Contribution

It provides a detailed pointwise analysis of the Green's function and global solutions for the nonlinear VPFP system in three dimensions, including decay rates and wave smoothing effects.

Findings

Green's function includes diffusion and kinetic waves with specific decay properties.

The global solution exhibits algebraic decay in space and exponential decay in time.

Singular kinetic waves become smooth for all positive times.

Abstract

The pointwise space-time behaviors of the Green's function and the global solution to the Vlasov-Poisson-Fokker-Planck (VPFP) system in spatial three dimension 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.