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

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.

💡 Research Summary

The paper studies the spectral stability of small‑amplitude shock profiles for the one‑dimensional isothermal Navier‑Stokes‑Poisson (NSP) system, which models ion dynamics in a collision‑dominated plasma. The NSP system consists of mass conservation, momentum balance, and a Poisson equation for the electric potential, with the electron density given by the Boltzmann relation (n_e=e^{\phi}). Shock profiles are traveling‑wave solutions that connect two constant far‑field states and satisfy the Rankine‑Hugoniot conditions; under the Lax entropy condition (v_+>v_-) and a smallness assumption on the amplitude (\delta_S), existence and uniqueness (up to translation) of such profiles are known from previous work.

The authors linearize the NSP equations about a given shock profile ((\bar v,\bar u,\bar\phi)). Because the Poisson equation is non‑local, the linearized electric potential can be expressed as (\phi = A^{-1}B,v) where (A) and (B) are differential operators. Substituting this relation yields a closed linear operator (L) acting on the perturbation vector (U=(v,u)^T). The operator is non‑local and of second order, but it is shown to be closed and densely defined on (L^2(\mathbb R)^2).

The spectral analysis proceeds in three main steps:

1. Essential Spectrum – By letting (x\to\pm\infty), the coefficients of (L) converge to constant matrices, defining limiting operators (L^\pm). The dispersion relations of (L^\pm) are computed, and standard arguments (e.g., Henry, Pego) give that the essential spectrum (\sigma_{\rm ess}(L)) lies entirely in the left half‑plane. However, because of translation invariance, the zero eigenvalue is embedded in (\sigma_{\rm ess}(L)), so there is no spectral gap.

2. Point Spectrum – To treat eigenvalues with non‑negative real part, the authors introduce integrated (anti‑derivative) variables, obtaining an auxiliary operator (\tilde L) whose point spectrum coincides with that of (L) away from the origin. Energy estimates of higher order (proved in an appendix) together with a contradiction argument show that no eigenvalues with (\Re\lambda\ge0) exist except possibly (\lambda=0). Thus (\sigma_{\rm pt}(L)\subset\mathbb C\setminus\Omega) where (\Omega={\Re\lambda\ge0}\setminus{0}).

3. Simplicity of the Zero Eigenvalue – The standard Evans‑function method cannot be applied directly because (\lambda=0) lies in the essential spectrum. The authors adopt the extended Evans‑function framework of Zumbrun–Howard and Mascia–Zumbrun. Using the conjugation lemma, they construct analytic bases of solutions decaying as (x\to\pm\infty) on a small disc (B(0,r)) that lies inside the region of consistent splitting. This yields an Evans function (D(\lambda)) analytic on (B(0,r)). A careful calculation shows that \