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Linear inviscid damping and vorticity depletion for shear flows

Dongyi Wei, Zhifei Zhang, Weiren Zhao

TL;DR

<3-5 sentence high-level summary>This work establishes linear inviscid damping for the 2-D Euler equations in a finite channel around a broad class of shear flows, under the key spectral assumption that the linearized operator has no embedding eigenvalues. It develops a resolvent-based framework built on the Rayleigh equation, including a limiting absorption principle and a precise odd/even decomposition for symmetric flows, to derive explicit decay rates for the velocity and to demonstrate vorticity depletion at stationary streamlines. A detailed analysis of key quantities and integral operators yields sharp decay and scattering results for symmetric base flows (e.g., Poiseuille), extending known monotone-flow results to non-monotone settings. The findings illuminate an enhanced damping mechanism tied to vorticity depletion and provide rigorous tools for understanding linear stability and potential extensions to nonlinear settings in regimes with stationary streamlines.

Abstract

In this paper, we prove the linear damping for the 2-D Euler equations around a class of shear flows under the assumption that the linearized operator has no embedding eigenvalues. For the symmetric flows, we obtain the explicit decay estimates of the velocity, which is the same as one for monotone shear flows. We confirm a new dynamical phenomena found by Bouchet and Morita: the depletion of the vorticity at the stationary streamlines, which could be viewed as a new mechanism leading to the damping for the base flows with stationary streamlines.

Linear inviscid damping and vorticity depletion for shear flows

TL;DR

<3-5 sentence high-level summary>This work establishes linear inviscid damping for the 2-D Euler equations in a finite channel around a broad class of shear flows, under the key spectral assumption that the linearized operator has no embedding eigenvalues. It develops a resolvent-based framework built on the Rayleigh equation, including a limiting absorption principle and a precise odd/even decomposition for symmetric flows, to derive explicit decay rates for the velocity and to demonstrate vorticity depletion at stationary streamlines. A detailed analysis of key quantities and integral operators yields sharp decay and scattering results for symmetric base flows (e.g., Poiseuille), extending known monotone-flow results to non-monotone settings. The findings illuminate an enhanced damping mechanism tied to vorticity depletion and provide rigorous tools for understanding linear stability and potential extensions to nonlinear settings in regimes with stationary streamlines.

Abstract

In this paper, we prove the linear damping for the 2-D Euler equations around a class of shear flows under the assumption that the linearized operator has no embedding eigenvalues. For the symmetric flows, we obtain the explicit decay estimates of the velocity, which is the same as one for monotone shear flows. We confirm a new dynamical phenomena found by Bouchet and Morita: the depletion of the vorticity at the stationary streamlines, which could be viewed as a new mechanism leading to the damping for the base flows with stationary streamlines.

Paper Structure

This paper contains 32 sections, 40 theorems, 554 equations.

Table of Contents

  1. Introduction
  2. New dynamic phenomena: vorticity depletion
  3. Sketch and key ideas of the proof
  4. The Homogeneous Rayleigh Equation for symmetric flow
  5. Basic properties of symmetric flow
  6. Rayleigh integral operator
  7. Existence of the solution
  8. Uniform estimates of the solution
  9. Spectral analysis of the linearized operator
  10. Spectrum and embedding eigenvalues
  11. A criterion on embedding eigenvalues
  12. The limiting absorption principle
  13. Linear damping and vorticity depletion
  14. The Inhomogeneous Rayleigh Equation for symmetric flow
  15. Representation formula of the solution
  16. ...and 17 more sections

Key Result

Theorem 1.1

Assume that $u(y)\in \mathcal{K}$ and the linearized operator $\mathcal{R}_\alpha$ defined by def:ray ope has no embedding eigenvalues. Assume that $\widehat{\omega}_0(\alpha,y)\in H^1_y(-1,1)$ and $P_{\mathcal{R}_\alpha}\widehat{\psi}_0(\alpha,y)=0$, where $\psi_0$ is the stream function and $P_{\m In particular, $\lim_{t\to+\infty}\|\widehat{V}(t,\alpha,\cdot)\|_{L^2_y}=0$.

Theorems & Definitions (80)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Lemma 4.1
  • proof
  • Definition 4.2
  • Definition 4.3
  • Definition 4.4
  • Definition 4.5
  • Lemma 4.6
  • ...and 70 more