Stability of Rarefaction Waves of the Compressible Navier--Stokes--Poisson System with Large Initial Perturbation
Lan Zhang, Huijiang Zhao, Qingsong Zhao
Abstract
We show in this paper that rarefaction waves of the one-dimensional compressible Navier--Stokes--Poisson system are time-asymptotically nonlinear stable both for the two-fluid case of charged particles consisting of ions and electrons and for the one-fluid case of single ions under the Boltzmann relation. The self-consistent electrostatic potential force is allowed to take different far-fields and the initial perturbation can be chosen arbitrarily large. Our analysis is based on the nonlinear energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid, and the crucial step is to derive the desired positive upper and lower bounds of the density for both ions and/or electrons which are uniform in time and space. Moreover, the desired uniform estimate on the electric potential together with its large time behavior are also obtained for the one-fluid case. Our main idea is to use the smallness of the strength of the rarefaction waves to control the possible growth of the solutions of the compressible Navier--Stokes--Poisson system induced by the nonlinearities of the system and by the interactions between the rarefaction waves and the solutions.