Global stability of the rarefaction wave of the Vlasov-Poisson-Boltzmann system

TL;DR

This paper proves the nonlinear stability of rarefaction waves in the Vlasov-Poisson-Boltzmann system with slab symmetry, allowing for large wave strength and varying electric potential states, using advanced analytical techniques.

Contribution

It constructs and proves the stability of non-small amplitude rarefaction waves in the kinetic system with electric fields, extending previous fluid dynamic results.

Findings

Time-asymptotic stability of the Maxwellian with rarefaction wave macroscopic quantities.

Decay properties of the rarefaction waves are established.

The structure of the Poisson equation is crucial in the analysis.

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 allows that the electric potential may take distinct constant states at both far-fields. The rarefaction wave whose strength is not necessarily small is constructed through the quasineutral Euler equations coming from the zero-order fluid dynamic approximation of the kinetic system. 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 on the Vlasov-Poisson-Boltzmann system. The main analytical tool is the combination of techniques we developed in [10] for the viscous compressible fluid with the self-consistent electric field and the reciprocal 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 Poisson equation play a key role in the analysis.