Asymptotic stability of composite waves of shock profile and rarefaction for the Navier-Stokes-Poisson system

TL;DR

This paper proves the asymptotic stability of combined shock and rarefaction waves in the one-dimensional Navier-Stokes-Poisson system, showing solutions converge to these waves over time if initially close.

Contribution

It extends the $a$-contraction method with shifts to the NSP system, demonstrating stability of composite waves involving shock and rarefaction profiles.

Findings

Solutions converge to composite waves over time

Stability proven for initial data close in $H^2$ norm

Method adapted from previous shock stability work

Abstract

We study the stability of composite waves consisting of a shock profile and a rarefaction wave for the one-dimensional isothermal Navier--Stokes--Poisson (NSP) system, which describes the ion dynamics in a collision-dominated plasma. More precisely, we prove that if the initial data are sufficiently close in the $H^2$ norm to the Riemann data corresponding to a solution consisting of a shock and a rarefaction wave of the associated quasi-neutral Euler system, then the solution to the Cauchy problem for the NSP system converges, up to a dynamical shift, to a superposition of the corresponding shock profile and the rarefaction wave as time tends to infinity. Our proof is based on the method of $a$-contraction with shifts, which has recently been applied to the Navier--Stokes equations to establish the asymptotic stability of composite waves. To adapt this method to the NSP system, we employ a modulated relative functional introduced in our previous work on the stability of single shock profiles.