Stability of Rarefaction Waves for the Non-cutoff Vlasov-Poisson-Boltzmann System with Physical Boundary

TL;DR

This paper proves the stability of rarefaction wave solutions in the non-cutoff Vlasov-Poisson-Boltzmann system with physical boundaries, extending analysis to realistic boundary conditions and collision kernels.

Contribution

It introduces the first stability analysis of rarefaction waves for the non-cutoff VPB system with physical boundary conditions, including specular reflection.

Findings

Established time-asymptotic stability of rarefaction waves

Handled non-cutoff collision kernels in stability analysis

Extended results to Boltzmann equation as a simplified model

Abstract

In this paper, we are concerned with the Vlasov-Poisson-Boltzmann (VPB) system in three-dimensional spatial space without angular cutoff in a rectangular duct with or without physical boundary conditions. Near a local Maxwellian with macroscopic quantities given by rarefaction wave solution of one-dimensional compressible Euler equations, we establish the time-asymptotic stability of planar rarefaction wave solutions for the Cauchy problem to VPB system with periodic or specular-reflection boundary condition. In particular, we successfully introduce physical boundaries, namely, specular-reflection boundary, to the models describing wave patterns of kinetic equations. Moreover, we treat the non-cutoff collision kernel instead of the cutoff one. As a simplified model, we also consider the stability and large time behavior of the rarefaction wave solution for the Boltzmann equation.