Asymptotic stability of shock profiles and rarefaction waves to the Navier-Stokes-Poisson system under space-periodic perturbations

TL;DR

This paper studies the long-term stability of shock profiles and rarefaction waves in the one-dimensional Navier-Stokes-Poisson system under space-periodic perturbations, proving convergence to background waves with detailed analysis of interactions.

Contribution

It introduces a novel energy method and quadratic ansatzes to analyze the asymptotic behavior of solutions with space-periodic perturbations in a coupled physical system.

Findings

Viscous shock solutions tend to a shifted shock profile over time.

Rarefaction wave solutions tend to the background rarefaction wave.

The method handles complex interactions between waves and periodic perturbations.

Abstract

This paper concerns with the large-time behaviors of the viscous shock profile and rarefaction wave under initial perturbations which tend to space-periodic functions at infinities for the one-dimensional compressible Navier-Stokes-Poisson equations. It is proved that: (1) for the viscous shock with small strength, if the initial perturbation is suitably small and satisfies a zero-mass type condition, then the solution tends to background viscous shock with a constant shift as time tends to the infinity, and the shift depends on both the mass of the localized perturbation, and the space-periodic perturbation; (2) for the rarefaction wave, if the initial perturbation is suitably small, then the solution tends to background rarefaction wave as time tends to infinity. The proof is based on the delicate constructions of the quadratic ansatzes, which capture the infinitely many interactions between the background waves and the periodic perturbations, and the energy method in Eulerian coordinates involving the effect of self-consistent electric field. Moreover, an abstract lemma is established to distinguish the non-decaying terms and good-decaying terms from the error terms of the equations of the quadratic ansatzes, which will be benefit to constructing the ansatzes and simplifying calculations for other non-localized perturbation problems, especially those with complicatedly coupling physical effects.