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

Abstract

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.

Asymptotic stability of the superposition of viscous contact wave with rarefaction waves for the compressible Navier-Stokes-Maxwell equations

“…This new technique was needed by Zeng in [41] to obtain the stability of the superposition of shock waves with contact discontinuities for systems of viscous conservation laws, and was also used by Huang-Wang in [11] for the global stability of the same wave patterns and system as in [7]. In addition, some results about the asymptotic stability of the composite wave as in [7] were also shown for some more complex models, and we refer interested readers to [36], [5], [24], [1], [28], and references therein. Due to the frameworks of viscous shock wave and rarefaction wave are not compatible with each other, the timeasymptotic stability of the combination of viscous shock wave and rarefaction wave is still an interesting and challenging question for the Navier-Stokes equations!…”

Green's Function and Pointwise Behaviors of the One-Dimensional modified Vlasov-Poisson-Boltzmann System

“…The pointwise space-time behaviors of the Green's function and the global solution to the one-species VPB system was studied in [16]. Moreover, the wave phenomena is observed for one-dimensional VPB system [7,14,15], such as the shock profile, rarefaction wave and viscous contact wave.…”

Asymptotic stability of the combination of a viscous contact wave with two rarefaction waves for 1-D Navier-Stokes equations under periodic perturbations

“…Recently, by utilizing the theory of a-contraction with shifts, Kang-Vasseur-Wang [13] investigated the time-asymptotic stability for the composite waves of a viscous shock and a rarefaction wave for the 1-D compressible barotropic Navier-Stokes equations. More results on combination waves for other significant systems refer to [16,21,28] and the references therein.…”