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Asymptotic stability for the Couette flow in the 2D Euler equations

Jacob Bedrossian, Nader Masmoudi

TL;DR

The paper establishes nonlinear asymptotic stability of the Couette flow within the 2D Euler equations for small Gevrey-class perturbations, showing convergence to a nearby shear and controlled damping of the velocity, while enstrophy transfers to small scales. It introduces a solution-adaptive coordinate change and a time-dependent Gevrey norm informed by a toy model of Orr-type frequency cascades, and develops a para-differential energy framework plus precision elliptic estimates to manage quasi-linear interactions. The results demonstrate that velocity converges strongly in $L^2$ to a near-Couette profile and that the vorticity undergoes a weak, phase-mixed evolution with potential loss of enstrophy to high frequencies, providing a nonlinear analogue of Landau damping for fluids. The work highlights the role of regularity in nonlinear stability and lays groundwork for extensions to more general shear flows and boundary conditions, offering a rigorous connection between phase mixing and turbulent-like transport in 2D fluids.

Abstract

In this expository note we discuss our recent work [arXiv:1306.5028] on the nonlinear asymptotic stability of shear flows in the 2D Euler equations of ideal, incompressible flow. In that work it is proved that perturbations to the Couette flow which are small in a suitable regularity class converge strongly in $L^2$ to a shear flow which is close to the Couette flow. Enstrophy is mixed to small scales by an almost linear evolution and is generally lost in the weak limit as t -> +/- infinity. In this note we discuss the most important physical and mathematical aspects of the result and the key ideas of the proof.

Asymptotic stability for the Couette flow in the 2D Euler equations

TL;DR

The paper establishes nonlinear asymptotic stability of the Couette flow within the 2D Euler equations for small Gevrey-class perturbations, showing convergence to a nearby shear and controlled damping of the velocity, while enstrophy transfers to small scales. It introduces a solution-adaptive coordinate change and a time-dependent Gevrey norm informed by a toy model of Orr-type frequency cascades, and develops a para-differential energy framework plus precision elliptic estimates to manage quasi-linear interactions. The results demonstrate that velocity converges strongly in to a near-Couette profile and that the vorticity undergoes a weak, phase-mixed evolution with potential loss of enstrophy to high frequencies, providing a nonlinear analogue of Landau damping for fluids. The work highlights the role of regularity in nonlinear stability and lays groundwork for extensions to more general shear flows and boundary conditions, offering a rigorous connection between phase mixing and turbulent-like transport in 2D fluids.

Abstract

In this expository note we discuss our recent work [arXiv:1306.5028] on the nonlinear asymptotic stability of shear flows in the 2D Euler equations of ideal, incompressible flow. In that work it is proved that perturbations to the Couette flow which are small in a suitable regularity class converge strongly in to a shear flow which is close to the Couette flow. Enstrophy is mixed to small scales by an almost linear evolution and is generally lost in the weak limit as t -> +/- infinity. In this note we discuss the most important physical and mathematical aspects of the result and the key ideas of the proof.

Paper Structure

This paper contains 8 sections, 2 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Asymptotic Stability
  3. Physical and Mathematical aspects of the proof
  4. Coordinate transformation
  5. Weakly nonlinear effects and the toy model
  6. Main Energy Estimates
  7. Elliptic estimates
  8. Conclusion

Key Result

Theorem 1

For all $1/2 < s \leq 1$, $\lambda_0 > \lambda^\prime > 0$ there exists an $\epsilon_0 = \epsilon_0(\lambda_0,\lambda^\prime,s) \leq 1/2$ such that for all $\epsilon \leq \epsilon_0$ if $\omega_{in}$ satisfies $\int \omega_{in} dx dy= 0$, $\int \left\vert y\omega_{in}(x,y)\right\vert dx dy < \eps then there exists $f_\infty$ with $\int f_\infty dxdy = 0$ and $\|f_\infty\|_{\lambda^\prime} \less

Theorems & Definitions (3)

  • Theorem 1
  • Remark
  • Corollary 1