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Instability of shear flows with neutral embedded eigenvalues

Hui Li, Siqi Ren, Yuxi Wang, Guoqing Zhang

TL;DR

The work demonstrates that monotone two-dimensional shear flows in the neutral state—where the Rayleigh operator has embedded real eigenvalues—can exhibit instability despite the absence of spectral growth. By leveraging a representation formula for the stream function and a limiting-absorption framework, the authors show that non-normality of the Rayleigh operator drives transient and, in some cases, linear growth via embedded-eigenvalue structures. The results distinguish simple and multiple embedded eigenvalues, showing that simple cases yield growth through a large eigenmode projection, while multiple eigenvalues yield linear-in-time growth via an associated function; these instabilities persist under small viscosity. The findings illuminate the subtle and intrinsic instability mechanisms in neutral shear flows, with implications for inviscid damping and transition scenarios in fluid dynamics.

Abstract

We study the linear stability of a class of monotone shear flows. When the associated Rayleigh operator possesses a neutral embedded eigenvalue, we show that solutions of the linearized system may exhibit arbitrarily large growth in both the $L^\infty$ and $L^2$ norms. Moreover, when the embedded eigenvalue is multiple, we prove that the instability becomes stronger and explicitly construct solutions that grow linearly in time. This instability originates from the non-normality of the Rayleigh operator.

Instability of shear flows with neutral embedded eigenvalues

TL;DR

The work demonstrates that monotone two-dimensional shear flows in the neutral state—where the Rayleigh operator has embedded real eigenvalues—can exhibit instability despite the absence of spectral growth. By leveraging a representation formula for the stream function and a limiting-absorption framework, the authors show that non-normality of the Rayleigh operator drives transient and, in some cases, linear growth via embedded-eigenvalue structures. The results distinguish simple and multiple embedded eigenvalues, showing that simple cases yield growth through a large eigenmode projection, while multiple eigenvalues yield linear-in-time growth via an associated function; these instabilities persist under small viscosity. The findings illuminate the subtle and intrinsic instability mechanisms in neutral shear flows, with implications for inviscid damping and transition scenarios in fluid dynamics.

Abstract

We study the linear stability of a class of monotone shear flows. When the associated Rayleigh operator possesses a neutral embedded eigenvalue, we show that solutions of the linearized system may exhibit arbitrarily large growth in both the and norms. Moreover, when the embedded eigenvalue is multiple, we prove that the instability becomes stronger and explicitly construct solutions that grow linearly in time. This instability originates from the non-normality of the Rayleigh operator.
Paper Structure (25 sections, 18 theorems, 277 equations, 1 figure)

This paper contains 25 sections, 18 theorems, 277 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Main results
  4. Main ideas
  5. Toy model
  6. Case with simple eigenvalue
  7. Case with multiple eigenvalue
  8. Notations
  9. Organization
  10. Homogeneous Rayleigh equation
  11. Solving the homogeneous Rayleigh equation
  12. Regularity estimates for the homogeneous solutions
  13. Properties of $\mathcal{J}_1(c_r)$ and $\mathcal{J}_2(c_r)$
  14. Representation formula of the stream function
  15. The limiting absorption principle
  16. ...and 10 more sections

Key Result

Theorem 1.2

Let the background shear flow $b(y)$ satisfy Assumption assum. For any $M>0$, there exist $\omega_{\text{in}}$ and $T$ such that and the solution $\omega(t,y)$ to eq-linEuler-k with initial data $\omega_{\text{in}}$ satisfies

Figures (1)

  • Figure 1: integration contour

Theorems & Definitions (35)

  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 25 more