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Long time instability of the Couette flow in low Gevrey spaces

Yu Deng, Nader Masmoudi

TL;DR

This paper establishes a nonlinear instability of the 2D Euler Couette flow for perturbations that lie below the Gevrey-2 threshold, showing that critical regularity is violated via a completely nonlinear energy cascade. The authors develop a coordinate change to a transformed system, perform a detailed linear analysis with a precise Orr-type toy model, and then implement a Taylor-expansion iterative scheme to construct an approximate nonlinear solution whose growth is sustained by a nonlinear cascade. Central to the approach are sharp Fourier localization techniques, recurrence relations capturing mode-to-mode transfer, and robust energy estimates that control higher-order nonlinear terms. The result sharpens the understanding of stability regimes by proving instability just below the known Gevrey-stable regime, and it highlights the delicate balance between linear transient growth and nonlinear energy transfer in shear flows.

Abstract

We prove the instability of the Couette flow if the disturbances is less smooth than the Gevrey space of class 2. This shows that this is the critical regularity for this problem since it was proved in [5] that stability and inviscid damping hold for disturbances which are smoother than the Gevrey space of class 2. A big novelty is that this critical space is due to an instability mechanism which is completely nonlinear and is due to some energy cascade.

Long time instability of the Couette flow in low Gevrey spaces

TL;DR

This paper establishes a nonlinear instability of the 2D Euler Couette flow for perturbations that lie below the Gevrey-2 threshold, showing that critical regularity is violated via a completely nonlinear energy cascade. The authors develop a coordinate change to a transformed system, perform a detailed linear analysis with a precise Orr-type toy model, and then implement a Taylor-expansion iterative scheme to construct an approximate nonlinear solution whose growth is sustained by a nonlinear cascade. Central to the approach are sharp Fourier localization techniques, recurrence relations capturing mode-to-mode transfer, and robust energy estimates that control higher-order nonlinear terms. The result sharpens the understanding of stability regimes by proving instability just below the known Gevrey-stable regime, and it highlights the delicate balance between linear transient growth and nonlinear energy transfer in shear flows.

Abstract

We prove the instability of the Couette flow if the disturbances is less smooth than the Gevrey space of class 2. This shows that this is the critical regularity for this problem since it was proved in [5] that stability and inviscid damping hold for disturbances which are smoother than the Gevrey space of class 2. A big novelty is that this critical space is due to an instability mechanism which is completely nonlinear and is due to some energy cascade.

Paper Structure

This paper contains 50 sections, 28 theorems, 470 equations.

Table of Contents

  1. Introduction
  2. Backgrounds, and the main theorem
  3. Inviscid damping and stability
  4. Orr growth mechanism and instability
  5. The main result
  6. Ideas of the proof
  7. The choice of data, and setup
  8. The linearized system around $\underline{f}$
  9. Linear analysis, and a more precise toy model
  10. Nonlinear analysis, and the Taylor expansion
  11. Further discussions
  12. Asymptotic instability
  13. Genericity
  14. Plan of the paper
  15. Choice of coordinates
  16. ...and 35 more sections

Key Result

Theorem 1.1

Let $\lambda>\lambda'>0$ and $1/2<s\leq 1$, and let $\varepsilon>0$ be sufficiently small. Suppose the initial data $\omega(0)$ satisfies $\|\omega(0)\|_{\mathcal{G}^{\lambda,s}}\leq\varepsilon$, and the assumptions where the Gevrey norm is defined by then there exists a unique global solution $\omega$ to (euler). Moreover, the solution satisfies that where $\Psi(y)=y+O(\varepsilon)$ is another

Theorems & Definitions (55)

  • Theorem 1.1: Bedrossian-Masmoudi BM15
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof : Proof of Theorem \ref{['main']} assuming Proposition \ref{['main1.5']}
  • Remark 2.3
  • Proposition 3.1
  • Proposition 4.1
  • ...and 45 more