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

Wanyong Shim

TL;DR

This work analyzes the spectral stability of small-amplitude shock profiles for the one-dimensional isothermal Navier-Stokes-Poisson system, which models ion dynamics in a collision-dominated plasma. The authors formulate the linearized operator with a nonlocal Poisson coupling and decompose the spectrum into essential and point parts, proving that the essential spectrum does not intrude into the right half-plane except for the embedded zero and that no nonzero eigenvalues lie in the unstable half-plane. A key contribution is the use of an Evans-function framework extended into the essential spectrum, enabling a precise factorization of the derivative at the origin: $\mathcal{D}'(0)=\Gamma\Delta$, where $\Gamma$ encodes transversality of the shock connection and $\Delta$ is the Liu–Majda determinant for the associated hyperbolic system; both factors are shown nonzero. Consequently, the zero eigenvalue is simple, and the shock profile is spectrally stable in $L^2$ for sufficiently small shock strength. This spectral information complements nonlinear stability results and clarifies the impact of the Poisson coupling and the quasi-neutral Euler limit on 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.

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

TL;DR

This work analyzes the spectral stability of small-amplitude shock profiles for the one-dimensional isothermal Navier-Stokes-Poisson system, which models ion dynamics in a collision-dominated plasma. The authors formulate the linearized operator with a nonlocal Poisson coupling and decompose the spectrum into essential and point parts, proving that the essential spectrum does not intrude into the right half-plane except for the embedded zero and that no nonzero eigenvalues lie in the unstable half-plane. A key contribution is the use of an Evans-function framework extended into the essential spectrum, enabling a precise factorization of the derivative at the origin: , where encodes transversality of the shock connection and is the Liu–Majda determinant for the associated hyperbolic system; both factors are shown nonzero. Consequently, the zero eigenvalue is simple, and the shock profile is spectrally stable in for sufficiently small shock strength. This spectral information complements nonlinear stability results and clarifies the impact of the Poisson coupling and the quasi-neutral Euler limit on 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 . 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.
Paper Structure (15 sections, 19 theorems, 220 equations)

This paper contains 15 sections, 19 theorems, 220 equations.

Table of Contents

  1. Introduction
  2. Shock profiles for the Navier--Stokes--Poisson system
  3. Main result
  4. Main ideas of the proof
  5. Linearization about the profile
  6. Essential spectrum bounds
  7. Point spectrum bounds
  8. Simplicity of the zero eigenvalue
  9. Construction of the Evans function
  10. Computation of the Evans function
  11. The conjugation lemma
  12. Solvability of the linearized Poisson equation
  13. Domain and closedness of the linearized operator
  14. Higher-order energy estimates
  15. Transversality of the shock profiles

Key Result

Lemma 1.1

For given $(v_-,u_-)$ with $v_->0$, there exists a positive constant $\delta_0>0$ such that for any triplet $(v_+,u_+,s)$ satisfying RH, Lax, and the ODE system waveeq admits a unique (up to a shift) shock profile $(\bar{v},\bar{u},\bar{\phi})(x)$ satisfying ffcond and for some positive constants $\underline{C}$, $\overline{C}$. Moreover, the unique solution satisfying $\bar{v}(0)=\frac{v_-+v_+}

Theorems & Definitions (37)

  • Lemma 1.1: DLZ, Proposition 1.9
  • Theorem 1.2
  • Remark 1.3
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Proposition 4.1
  • Lemma 4.2
  • proof
  • Lemma 5.1
  • ...and 27 more