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

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 ± 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 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.

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

“…We now outline the main idea and make some comments on the proof of the above theorems. The results in Theorem 1.1 on the pointwise behavior of the Green's function to the VPFP system is proved based on the spectral analysis [13] and the ideas inspired by [14,15]. Indeed, we first decompose the Green function G into the lower frequency part G L and the high frequency part G H .…”

“…By the virtue of the spectrum analysis of the VPFP system, the lower frequency part G L decays at e −Ct , and in particular decays at e −Ct/2 (1 + |x| 2 ) −k in the region |x| ≤ e Ct/(2k) . Furthermore, we apply the Picard's iteration as the VPB system [14] to estimate G L outside the region |x| ≤ e Ct/(2k) . To be more precisely, we apply the macro-micro decomposition to construct the approximate sequence ( Îk , Ĵk ) of (P 2 ĜL , P 3 ĜL ) such that Îk (t, ξ) and Ĵk (t, ξ) are the solutions to the Navier-Stokes-Poisson system with damping and the microscopic VPFP system respectively, and Îk (t, ξ) and Ĵk (t, ξ) are smooth and compact supported functions in ξ and satisfy for any…”

“…For the linear Fokker-Planck equation, the structure of the Green's function was investigated in [12,15,25,26]. Recently, Li, Yang and Zhong established the Green's function and pointwise space-time behavior of the unipolar Vlasov-Poisson-Boltzmann system in [14].…”

Global existence and time decay rates for the 3D bipolar compressible Navier-Stokes-Poisson system with unequal viscosities

“…These two kinds of waves will be further introduced hereinbelow. After that, a series of research on the space-time behaviors for the other fluid models were established, for instance, the isentropic and non-isentropic Navier-Stokes equations in [5,6,17,22], damped Euler equations in [29,34], Boltzmann equations and thermal non-equilibrium flows [18,24,39,41].…”

Finite Speed of Propagation of the Relativistic Landau and Boltzmann Equations

“…Finally, we want to emphasis that this theorem reflects the difference in essence between the relativistic kinetic equations and the classical kinetic equations. For the classical kinetic equations (Boltzmann, Landau or Fokker-Planck), note that the speed of the transport part is v, which reflects the infinite speed of propagation of the solution (see for instance the Fokker-Planck equation with potentials [16], classical Boltzmann equation with hard sphere [19], for the linear problems regarding the classical Boltzmann equation with hard potentials and soft potentials [12,15,18], and for the Vlasov-Poisson-Boltzmann system [14]). Precisely, if imposing the initial data compactly supported in the x variable, then we only have spatial decay (exponentially, subexponentially or algebraically) of the solution to the classical kinetic equations.…”