Long-Time Behavior towards Shock Profiles for the Navier-Stokes-Poisson System

TL;DR

This paper proves that solutions to the one-dimensional Navier-Stokes-Poisson system with initial data near a shock profile will asymptotically approach a family of such profiles over time, demonstrating stability in the long-term behavior.

Contribution

It establishes the nonlinear stability of shock profiles for the NSP system without requiring the zero mass condition, using the method of $a$-contraction with shifts.

Findings

Solutions tend to the shock profile manifold as time approaches infinity.

The stability result holds for initial data close in $H^2$-norm.

The method used does not require the zero mass condition.

Abstract

We study the stability of shock profiles in one spatial dimension for the isothermal Navier-Stokes-Poisson (NSP) system, which describes the dynamics of ions in a collision-dominated plasma. The NSP system admits a one-parameter family of smooth traveling waves, called shock profiles, for a given far-field condition satisfying the Lax entropy condition. In this paper, we prove that if the initial data is sufficiently close to a shock profile in $H^2$-norm, then the global solution of the Cauchy problem tends to the smooth manifold formed by the parametrized shock profiles as time goes to infinity. This is achieved using the method of $a$-contraction with shifts, which does not require the zero mass condition.