Stability of rarefaction waves of the Navier–Stokes–Poisson system

Abstract

In the paper we are concerned with the large time behavior of solutions to the one-dimensional Navier–Stokes–Poisson system in the case when the potential function of the self-consistent electric field may take distinct constant states at x=±∞. Precisely, it is shown that if initial data are close to a constant state with asymptotic values at far fields chosen such that the Riemann problem on the corresponding quasineutral Euler system admits a rarefaction wave whose strength is not necessarily small, then the solution exists for all time and tends to the rarefaction wave as t→+∞. The construction of the nontrivial large-time profile of the potential basing on the quasineutral assumption plays a key role in the stability analysis. The proof is based on the energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.