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Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

Jacob Bedrossian, Nader Masmoudi

TL;DR

<3-5 sentence high-level summary> This work establishes nonlinear inviscid damping for perturbations near planar Couette flow in the 2D Euler equations on $\mathbb{T}\times\mathbb{R}$, proving asymptotic stability for perturbations in a Gevrey class with $\tfrac12<s\le1$. The authors develop a solution-adaptive coordinate transform and a bespoke time-dependent Gevrey norm, built from a toy-model growth mechanism around Orr critical times, to control the quasi-linear evolution. A detailed bootstrap argument combines precise elliptic estimates, transport and reaction analyses, and careful coefficient-control of the evolving coordinate system to obtain uniform bounds and strong convergence of the velocity to a nearby shear. The results provide the first nonlinear, rigorous confirmation of Orr-type inviscid damping for 2D Euler and reveal enstrophy transfer to small scales, with implications for phase-m mixing and long-time dynamics in hydrodynamic stability.

Abstract

We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on $\Torus \times \Real$. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L^2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically driven to small scales by a linear evolution and weakly converges as $t \rightarrow \pm\infty$. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. This convergence was formally derived at the linear level by Kelvin in 1887 and it occurs at an algebraic rate first computed by Orr in 1907; our work appears to be the first rigorous confirmation of this behavior on the nonlinear level.

Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

TL;DR

<3-5 sentence high-level summary> This work establishes nonlinear inviscid damping for perturbations near planar Couette flow in the 2D Euler equations on , proving asymptotic stability for perturbations in a Gevrey class with . The authors develop a solution-adaptive coordinate transform and a bespoke time-dependent Gevrey norm, built from a toy-model growth mechanism around Orr critical times, to control the quasi-linear evolution. A detailed bootstrap argument combines precise elliptic estimates, transport and reaction analyses, and careful coefficient-control of the evolving coordinate system to obtain uniform bounds and strong convergence of the velocity to a nearby shear. The results provide the first nonlinear, rigorous confirmation of Orr-type inviscid damping for 2D Euler and reveal enstrophy transfer to small scales, with implications for phase-m mixing and long-time dynamics in hydrodynamic stability.

Abstract

We prove asymptotic stability of shear flows close to the planar Couette flow in the 2D inviscid Euler equations on . That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L^2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically driven to small scales by a linear evolution and weakly converges as . The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. This convergence was formally derived at the linear level by Kelvin in 1887 and it occurs at an algebraic rate first computed by Orr in 1907; our work appears to be the first rigorous confirmation of this behavior on the nonlinear level.

Paper Structure

This paper contains 45 sections, 26 theorems, 418 equations.

Table of Contents

  1. Introduction
  2. Statement and discussion
  3. Notation and conventions
  4. Proof of Theorem \ref{['thm:Main']}
  5. Coordinate transform
  6. Main energy estimate
  7. Conclusion of proof
  8. Growth mechanism and construction of $A$
  9. Construction of $w$
  10. Formal derivation of toy model
  11. Construction of $w$
  12. Total growth of $w_k(t,\eta)$
  13. Properties of $w$ and $J$
  14. Product lemma and other basic properties of $A$
  15. Elliptic Estimates
  16. ...and 30 more sections

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 $\left\lVert f_\infty\right\rVert_{\

Theorems & Definitions (54)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Corollary 1
  • Corollary 2
  • ...and 44 more