Stability of the Rarefaction Wave of the Vlasov--Poisson--Boltzmann System

Abstract

This paper is devoted to the study of the nonlinear stability of the rarefaction waves of the Vlasov--Poisson--Boltzmann system with slab symmetry in the case where the electron background density satisfies an analogue of the Boltzmann relation. We allow that the electric potential may take distinct constant states at both far fields. The rarefaction wave is constructed by the quasineutral Euler equations through the zero-order fluid dynamic approximation, and the wave strength is not necessarily small. We prove that the local Maxwellian with macroscopic quantities determined by the quasineutral rarefaction wave is time-asymptotically stable under small perturbations for the corresponding Cauchy problem. The main analytical tool is the combination of techniques we developed in [R.-J. Duan and S.-Q. Liu, J. Differential Equations, 258 (2015), pp. 2495--2530] for the viscous compressible fluid with the self-consistent electric field and the refined energy method based on the macro-micro decomposition of the Boltzmann equation around a local Maxwellian. Both the time decay property of the rarefaction waves and the structure of the system play a key role in the proof.