Green's Function and Pointwise Space-time Behaviors of the three-Dimensional modified Vlasov-Poisson-Boltzmann System

TL;DR

This paper analyzes the Green's function for the 3D modified Vlasov-Poisson-Boltzmann system, revealing its wave decomposition and establishing pointwise estimates for solutions.

Contribution

It provides a detailed decomposition of the Green's function into diffusive, Huygens, and kinetic waves, and derives pointwise estimates for the nonlinear system's solutions.

Findings

Green's function decomposes into macroscopic diffusive and Huygens waves at low frequency.

Singular kinetic wave decays exponentially in space and time.

Pointwise estimates for the global solution are established based on the Green's function.

Abstract

The pointwise space-time behavior of the Green's function of the three-dimensional modified Vlasov-Poisson-Boltzmann system is studied in this paper. It is shown that the Green's function has a decomposition of the macroscopic diffusive waves and Huygens waves with the speed $\sqrt{\frac{8}{3}}$ at low-frequency, the singular kinetic wave and the remainder term decaying exponentially in space and time. In addition, we establish the pointwise space-time estimate of the global solution to the nonlinear modified Vlasov-Poisson-Boltzmann system based on the Green's function.