Spectral stability of shock profiles for the Navier-Stokes-Poisson system

TL;DR

This paper analyzes the spectral stability of small-amplitude shock profiles in the one-dimensional Navier-Stokes-Poisson system, revealing bounds on spectra and the simplicity of the zero eigenvalue, with implications for plasma shock stability.

Contribution

It develops an Evans-function framework extended into the essential spectrum to prove the simplicity of the zero eigenvalue for shock profiles in plasma models.

Findings

Bounded the essential spectrum and point spectrum.

Established the zero eigenvalue is simple.

Connected eigenvalue properties to shock transversality and hyperbolic stability.

Abstract

We investigate the spectral stability of small-amplitude shock profiles for the one-dimensional isothermal Navier-Stokes-Poisson system, which describes ion dynamics in a collision-dominated plasma. Specifically, we establish (i) bounds on the essential spectrum, (ii) bounds on the point spectrum, and (iii) simplicity of the zero eigenvalue for the linearized operator about the profile in $L^2$. The result in (i) shows that the zero eigenvalue arising from translation invariance is embedded in the essential spectrum. Consequently, the standard Evans function approach cannot be applied directly to prove (iii). To resolve this, we employ an Evans-function framework that extends into regions of the essential spectrum, thereby enabling us to compute the derivative of the Evans function at the origin. Our result establishes that this derivative admits a factorization into two factors: one associated with transversality of the connecting profile and the other with hyperbolic stability of the corresponding shock of the quasi-neutral Euler system. We further show that both factors are nonzero, which implies simplicity of the zero eigenvalue.