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On the stability threshold for the 3D Couette flow in Sobolev regularity

Jacob Bedrossian, Pierre Germain, Nader Masmoudi

TL;DR

The paper establishes a nonlinear stability threshold for the 3D Couette flow in Sobolev spaces at high Reynolds numbers, proving that disturbances with ||u_in||_{H^σ} ≤ δ Re^{-3/2} (for σ>9/2) yield global-in-time solutions that stay within O(Re^{-1/2}) of Couette and converge to x-independent streaks as t grows. The authors develop a finite-regularity framework using Fourier multipliers (including ghost energy) to balance lift-up and nonlinear growth against enhanced dissipation and inviscid damping, together with a careful bootstrap that tracks Q^1,Q^2,Q^3 and velocity components across zero/nonzero X-frequencies. Key contributions include a rigorous construction of M, m multipliers, sharp control of critical-time interactions, and a detailed treatment of nonlinear pressure/stretching terms that closes the energy estimates; numerics on Re-thresholds are consistent with the rigorous bound. This work advances finite-regularity subcritical instability analysis in fluid mechanics and lays groundwork for understanding transition dynamics near the stability threshold in high-Re flows. The results align with numerical estimates (≈ Re^{-31/20}) and suggest that finite regularity thresholds may be robust to noise compared to infinite-regularity results, with implications for subcritical transition theory.

Abstract

We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number $\textbf{Re}$. Our goal is to estimate how the stability threshold scales in $\textbf{Re}$: the largest the initial perturbation can be while still resulting in a solution that does not transition away from Couette flow. In this work we prove that initial data which satisfies $\| u_{in} \|_{H^σ} \leq δ\textbf{Re}^{-3/2}$ for any $σ> 9/2$ and some $δ= δ(σ) > 0$ depending only on $σ$, is global in time, remains within $O(\textbf{Re}^{-1/2})$ of the Couette flow in $L^2$ for all time, and converges to the class of "2.5 dimensional" streamwise-independent solutions referred to as streaks for times $t \gtrsim \textbf{Re}^{1/3}$. Numerical experiments performed by Reddy et. al. with "rough" initial data estimated a threshold of $\sim \textbf{Re}^{-31/20}$, which shows very close agreement with our estimate.

On the stability threshold for the 3D Couette flow in Sobolev regularity

TL;DR

The paper establishes a nonlinear stability threshold for the 3D Couette flow in Sobolev spaces at high Reynolds numbers, proving that disturbances with ||u_in||_{H^σ} ≤ δ Re^{-3/2} (for σ>9/2) yield global-in-time solutions that stay within O(Re^{-1/2}) of Couette and converge to x-independent streaks as t grows. The authors develop a finite-regularity framework using Fourier multipliers (including ghost energy) to balance lift-up and nonlinear growth against enhanced dissipation and inviscid damping, together with a careful bootstrap that tracks Q^1,Q^2,Q^3 and velocity components across zero/nonzero X-frequencies. Key contributions include a rigorous construction of M, m multipliers, sharp control of critical-time interactions, and a detailed treatment of nonlinear pressure/stretching terms that closes the energy estimates; numerics on Re-thresholds are consistent with the rigorous bound. This work advances finite-regularity subcritical instability analysis in fluid mechanics and lays groundwork for understanding transition dynamics near the stability threshold in high-Re flows. The results align with numerical estimates (≈ Re^{-31/20}) and suggest that finite regularity thresholds may be robust to noise compared to infinite-regularity results, with implications for subcritical transition theory.

Abstract

We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number . Our goal is to estimate how the stability threshold scales in : the largest the initial perturbation can be while still resulting in a solution that does not transition away from Couette flow. In this work we prove that initial data which satisfies for any and some depending only on , is global in time, remains within of the Couette flow in for all time, and converges to the class of "2.5 dimensional" streamwise-independent solutions referred to as streaks for times . Numerical experiments performed by Reddy et. al. with "rough" initial data estimated a threshold of , which shows very close agreement with our estimate.

Paper Structure

This paper contains 79 sections, 21 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. Presentation of the problem
  3. Background and previous work
  4. Streak solutions
  5. Linear effects
  6. The linearized problem
  7. Lift up
  8. Inviscid damping
  9. Enhanced dissipation
  10. Vorticity stretching and kinetic energy cascade
  11. Statement of results
  12. Brief discussion of the results and new ideas
  13. Preliminaries and outline of the proof
  14. Notations
  15. Miscellaneous
  16. ...and 64 more sections

Key Result

Proposition 1.1

Let $\nu \in [0,\infty)$, $u_{in} \in H^{5/2+}$ be divergence free and independent of $x$, that is, $u_{in}(x,y,z) = u_{in}(y,z)$, and denote by $u(t)$ the corresponding unique strong solution to NSC with initial data $u_{in}$. Then $u(t)$ is global in time and for all $T > 0$, $u(t) \in L^\infty( ( and $u^1$ solves the (linear) forced advection-diffusion equation

Theorems & Definitions (36)

  • Proposition 1.1: Streak solutions
  • Proposition 1.2: Linearized Navier-Stokes
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2: Local existence, continuation, and propagation of analyticity
  • Lemma 2.3
  • Proposition 2.1
  • Lemma 2.4
  • ...and 26 more