Green's function and Pointwise Behavior of the One-Dimensional Vlasov-Maxwell-Boltzmann System

TL;DR

This paper analyzes the Green's function of the one-dimensional Vlasov-Maxwell-Boltzmann system, revealing new hyperbolic waves and establishing pointwise estimates for solutions, advancing understanding of plasma dynamics with electromagnetic effects.

Contribution

It introduces the first detailed analysis of high-frequency hyperbolic waves in the VMB system and develops new methods to handle coupling effects in plasma models.

Findings

Identification of macroscopic diffusive and Huygens waves at low-frequency

Discovery of new high-frequency hyperbolic waves unique to VMB

Establishment of pointwise estimates for nonlinear VMB solutions

Abstract

The pointwise space-time behavior of the Green's function of the one-dimensional Vlasov-Maxwell-Boltzmann (VMB) system is studied in this paper. It is shown that the Green's function consists of the macroscopic diffusive waves and Huygens waves with the speed $\pm \sqrt{5/3}$ at low-frequency, the hyperbolic waves with the speed $\pm 1$ at high-frequency, the singular kinetic and leading short waves, and the remaining term decaying exponentially in space and time. Note that these high-frequency hyperbolic waves are completely new and can not be observed for the Boltzmann equation and the Vlasov-Poisson-Boltzmann system. In addition, we establish the pointwise space-time estimate of the global solution to the nonlinear VMB system based on the Green's function. Compared to the Boltzmann equation and the Vlasov-Poisson-Boltzmann system, some new ideas are introduced to overcome the difficulties caused by the coupling effects of the transport of particles and the rotating of electro-magnetic fields, and investigate the new hyperbolic waves and singular leading short waves.