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

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 for which the associated quasi-neutral Euler system admits a Riemann solution consisting of a shock and a rarefaction wave, 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. The proof is based on the method of $a$-contraction with shifts, which has recently been applied to establish the asymptotic stability of composite waves for the Navier--Stokes equations. To adapt this method to our problem, we employ a modulated relative functional introduced in our previous work on the stability of single shock profiles for the NSP system.