Stability of planar shock wave for the 3-dimensional compressible Navier-Stokes-Poisson equations

TL;DR

This paper proves the stability of planar viscous shock waves in a 3D compressible Navier-Stokes-Poisson system under small three-dimensional perturbations, extending previous results from one-dimensional to three-dimensional stability analysis.

Contribution

It demonstrates the stability of viscous shock waves in a 3D setting for the Navier-Stokes-Poisson system under small 3D perturbations, using mode decomposition and advanced estimates.

Findings

Zero mode and non-zero mode both tend to zero over time.

Non-zero mode exhibits exponential decay.

Stability holds under small 3D perturbations.

Abstract

This paper is concerned with the stability of planar viscous shock wave for the 3-dimensional compressible Navier-Stokes-Poisson (NSP) system in the domain $\Omega:=\mathbb{R}\times \mathbb{T}^2$ with $\mathbb{T}^2=(\mathbb{R}/\mathbb{Z})^2$. The stability problem of viscous shock under small 1-dimensional perturbations was solved in Duan-Liu-Zhang [7]. In this paper, we prove the viscous shock is still stable under small 3-d perturbations. Firstly, we decompose the perturbation into the zero mode and non-zero mode. Then we can show that both the perturbation and zero-mode time-asymptotically tend to zero by the anti-derivative technique and crucial estimates on the zero-mode. Moreover, we can further prove that the non-zero mode tends to zero with exponential decay rate. The key point is to estimate the non-zero mode of nonlinear terms involving electronic potential, see Lemma 6.1 below.