Spectrum Structure and Behaviors of the Vlasov--Maxwell--Boltzmann Systems

Abstract

The spectrum structures and behaviors of the Vlasov-Maxwell-Boltzmann (VMB) systems for both two species and one species are studied in this paper. The analysis shows the effect of the Lorentz force induced by the electro-magnetic field leads to some different structure of spectrum from the classical Boltzmann equation and the closely related Vlasov-Poisson-Boltzmann system. And the significant difference between the twospecies VMB model and one-species VMB model are given. The structure in high frequency illustrates the hyperbolic structure of the Maxwell equation. Furthermore, the long time behaviors and the optimal convergence rates to the equilibrium of the Vlasov-Maxwell-Boltzmann systems for both two species and one species are established based on the spectrum analysis, and in particular the phenomena of the electric field dominating and magnetic field dominating are observed for the one-species Vlasov-Maxwell-Boltzmann system.

On the Vlasov-Poisson-Boltzmann limit of the Vlasov-Maxwell-Boltzmann system

“…Such a method is based on the so-called macro-microscopic decompositions of the solution of the Boltzmann equation and the equation itself and it has been proved to be effective to be used to yield the global solvability of some complex kinetic equations, cf. [5,7,9,12,17,20,24,27,32,33] for the Vlasov-Maxwell-Boltzmann system (1.1), (1.2) and [6,8,10,11,15,26,35,36,37,38,39,40,41,42] for the Vlasov-Poisson-Boltzmann system (1.4), (1.5), etc. Since the main purpose of this paper is to give a rigorous mathematical justification of the global-in-time limit from the Vlasov-Maxwell-Boltzmann system (1.1), (1.2) to the Vlasov-Poisson-Boltzmann system (1.4), we need to deduce certain a priori estimates on the solution f + (t, x, v), f − (t, x, v), E (t, x), B (t, x) of the Cauchy problem of (2.3), (2.4), (2.5) which are independent of the parameter = 1 c .…”

Large-time behavior for fluid and kinetic plasmas with collisions

“…To answer this question has to be based on the spectral analysis as in Ukai [38] for the pure Boltzmann equation without any force. The case with self-consistent forces has been recently done by Li-Yang-Zhong [26].…”

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

“…The Vlasov-Maxwell-Boltzmann system has been intensively studied and important progress has been made, cf. [6,7,10,11,12,16,20] and the references therein. For instance, the global existence of unique strong solution with initial data near the normalized global Maxwellian was obtained in spatial period domain [10] and in spatial 3D whole space [20] for hard sphere collision, and then in [4,5] for general collision kernels with or without angular cut-off assumption.…”

“…The long time behavior of the global solution near a global Maxwellian was studied in [6,7]. The spectrum analysis and the optimal decay rate of the global solution to the VMB systems for both one-spices and two-spices were studied in [16]. The fluid dynamic limit for the VMB system was investigated in [11,12,9].…”

“…In this paper, we study the pointwise space-time behaviors of the Green's function and the global solution to the VMB system (1.1) based on the spectral analysis [16].…”