Stability of the superposition of boundary layer and rarefaction wave for outflow problem on the two-fluid Navier-Stokes-Poisson system

TL;DR

This paper proves the nonlinear stability of a superposition of boundary layer and rarefaction wave for the two-fluid Navier-Stokes-Poisson system on a half line, using elementary energy methods under small boundary layer strength.

Contribution

It constructs and proves the stability of a complex wave superposition in the Navier-Stokes-Poisson system, allowing large rarefaction wave strength and small boundary layer strength.

Findings

Stability of superposed boundary layer and rarefaction wave established.

Elementary $L^2$ energy method used for proofs.

Stability holds under small boundary layer strength, arbitrary large rarefaction wave strength.

Abstract

This paper is concerned with the study of nonlinear stability of superposition of boundary layer and rarefaction wave on the two-fluid Navier-Stokes-Poisson system in the half line $\mathbb{R}_{+}=:(0,+\infty)$. On account of the quasineutral assumption and the absence of the electric field for the large time behavior, we successfully construct the boundary layer and rarefaction wave, and then we give the rigorous proofs of the stability theorems on the superposition of boundary layer and rarefaction wave under small perturbations for the corresponding initial boundary value problem of the Navier-Stokes-Poisson system, only provided the strength of boundary layer is small while the strength of rarefaction wave can be arbitrarily large. The complexity of nonlinear composite wave leads to many complicated terms in the course of establishing the {\it a priori} estimates. The proofs are given by an elementary $L^2$ energy method.