Green’s Function and Pointwise Behavior of the One-Dimensional Vlasov–Maxwell–Boltzmann System

The pointwise space-time behavior of theGreen’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 Huygenswaves with the speed ±√5/3 at low-frequency, the hyperbolic waves with the speed ±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 cannot 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.