Stability of Nonlinear Wave Patterns to the Bipolar Vlasov–Poisson–Boltzmann System

Abstract

Abstract. The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction wave for bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, motivated by the micro-macro decomposition to the Boltzmann equation in [21,23], we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as the applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave patterns, viscous shock waves and rarefaction wave, are proved for the 1D bipolar Vlasov-PoissonBoltzmann system. More precisely, it is first proved that the linear superposition of two Boltzmann shock profiles in the first and third characteristic fields is nonlinearly stable to the 1D bipolar VPB system up to some suitable shifts without the zero macroscopic mass conditions on the initial perturbations. Then the time-asymptotic stability of rarefaction wave fan to compressible Euler equations is proved to 1D bipolar VPB system. These two results are concerned with the nonlinear stability of wave patterns for Boltzmann equation coupled with additional (electric) forces, which together with spectral analysis made in [18] sheds light on understanding the complicated dynamic behaviors around the wave patterns in the transportation of charged particles under the binary collisions, mutual interactions, and the effect of the electrostatic potential forces.

Stability of the Superposition of a Viscous Contact Wave with Two Rarefaction Waves to the Bipolar Vlasov--Poisson--Boltzmann System

“…Remark that the fluid-system (2.8) is the compressible Navier-Stokes type system strongly coupled with microscopic terms determined by (2.9) and electric field terms in (1.4) 3 , and the system (2.8), (2.9), (1.4) 3 is not self-closed and the new decomposition to F 2 -equation (1.4) 2 around the local Maxellian is needed (cf. [26]):…”

“…For the stability of single basic wave patterns, Duan-Liu proved the stability of rarefaction wave to a unipolar VPB system [5], which can be viewed as an approximation of bipolar VPB system (1.1) when the electron density is very rarefied and reaches a local equilibrium state with small electron mass compared with the ion. Recently, Li-Wang-Yang-Zhong [26] gave a unified micro-macro decomposition to the bipolar VPB system and proved the stability of rarefaction wave and viscous shock profile to the bipolar VPB system by this new decomposition. Then, Duan-Liu [6] generalized the result in [5] to the bipolar VPB system with the disparate mass and Huang-Liu [24] showed the stability of a single viscous contact wave case.…”

“…Note that the above fluid variables (ρ, u, θ) and the equilibrium Maxwellian M [ρ,u,θ] related to the density distribution functions F A and F B are defined in (2.1) and (2.3), respectively. As in [26], set the total mass by F 1 and the neutrality by F 2 of the distribution functions as follows…”

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We investigate the nonlinear stability of the superposition of a viscous contact wave and two rarefaction waves for one-dimensional bipolar Vlasov-Poisson-Boltzmann (VPB) system, which can be used to describe the transportation of charged particles under the additional electrostatic potential force. Based on a new micromacro type decomposition around the local Maxwellian related to the bipolar VPB system in our previous work [26], we prove that the superposition of a viscous contact wave and two rarefaction waves is time-asymptotically stable to 1D bipolar VPB system under some smallness conditions on the initial perturbations and wave strength, which implies that this typical composite wave pattern is nonlinearly stable under the combined effects of the binary collisions, the electrostatic potential force, and the mutual interactions of different charged particles. Note that this is the first result about the nonlinear stability of the combination of two different wave patterns for the Vlasov-Poisson-Boltzmann system.

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Asymptotics toward viscous contact waves for solutions of the Landau equation

“…The nonlinear stability of rarefaction waves to the Boltzmann equation was studied in [35,51,52]. In addition, the stability of nonlinear wave patterns to the Boltzmann equation with a self-consistent electric field for the hard sphere model has been considered in [12,31] and references therein.…”