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Quantitative stability for the 2D Couette flow on the infinite channel with non-slip boundary condition

Qionglei Chen, Zhen Li, Changxing Miao

TL;DR

This work addresses the quantitative stability of the 2D Couette flow on the infinite channel $\mathbb{R}\times[-1,1]$ with no-slip boundary, establishing a nonlinear transition threshold $\gamma\le \tfrac{1}{2}$. The authors develop a comprehensive framework combining resolvent estimates for the linear Orr–Sommerfeld operator (under both Navier-slip and no-slip boundaries) with space-time estimates of the linearized Navier–Stokes equations, then leverage bilinear controls to handle the nonlinear terms. A key novelty is the division of the long-wave frequency range into low and intermediate regimes using a critical division at $|k|=10\nu$, justified by Wirtinger’s inequality and sharp Airy-function analysis, enabling uniform low-frequency control and a refined intermediate-frequency analysis. The linear estimates feed into a nonlinear stability argument that yields the transition threshold $\gamma\le \tfrac{1}{2}$, and the authors also prove enhanced dissipation for frequencies $|k|\ge \nu^{1-}$ via a detailed spectral-time evolution analysis. Overall, the paper extends quantitative stability results from periodic domains to the whole real line in the $x$-direction, capturing long-wave effects and boundary-layer phenomena essential for understanding laminar-turbulent transitions in shear flows.

Abstract

In this paper, we investigate the quantitative stability for the 2D Couette flow on the infinite channel $\mathbb{R}\times [-1,1]$ with non-slip boundary condition. Compared to the case $\mathbb{T}\times [-1,1]$, we establish the stability in the context of long wave associated with the frequency range $0\leq |k|<1$ by developing the resolvent estimate argument. The new ingredient is to discover the key division point at $10ν$ in the frequency interval $(0,1)$ by the sharp Sobolev constant in Wirtinger's inequality together with the refined estimates of the Airy function in the interval $(0,1)$, and then we establish the space-time estimates on the low-frequency $0\leq |k|\leq 10 ν$ and the intermediate-frequency $ 10 ν\leq |k|<1$, respectively. As an application of the space-time estimates, we obtain the nonlinear transition threshold to be $γ\leq\frac{1}{2}$.Meanwhile, we also show that when the frequencies $|k|\geq ν^{1-}$, the enhanced dissipation effect occurs for the linearized Navier-Stokes equations.

Quantitative stability for the 2D Couette flow on the infinite channel with non-slip boundary condition

TL;DR

This work addresses the quantitative stability of the 2D Couette flow on the infinite channel with no-slip boundary, establishing a nonlinear transition threshold . The authors develop a comprehensive framework combining resolvent estimates for the linear Orr–Sommerfeld operator (under both Navier-slip and no-slip boundaries) with space-time estimates of the linearized Navier–Stokes equations, then leverage bilinear controls to handle the nonlinear terms. A key novelty is the division of the long-wave frequency range into low and intermediate regimes using a critical division at , justified by Wirtinger’s inequality and sharp Airy-function analysis, enabling uniform low-frequency control and a refined intermediate-frequency analysis. The linear estimates feed into a nonlinear stability argument that yields the transition threshold , and the authors also prove enhanced dissipation for frequencies via a detailed spectral-time evolution analysis. Overall, the paper extends quantitative stability results from periodic domains to the whole real line in the -direction, capturing long-wave effects and boundary-layer phenomena essential for understanding laminar-turbulent transitions in shear flows.

Abstract

In this paper, we investigate the quantitative stability for the 2D Couette flow on the infinite channel with non-slip boundary condition. Compared to the case , we establish the stability in the context of long wave associated with the frequency range by developing the resolvent estimate argument. The new ingredient is to discover the key division point at in the frequency interval by the sharp Sobolev constant in Wirtinger's inequality together with the refined estimates of the Airy function in the interval , and then we establish the space-time estimates on the low-frequency and the intermediate-frequency , respectively. As an application of the space-time estimates, we obtain the nonlinear transition threshold to be .Meanwhile, we also show that when the frequencies , the enhanced dissipation effect occurs for the linearized Navier-Stokes equations.
Paper Structure (11 sections, 25 theorems, 262 equations)

This paper contains 11 sections, 25 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. The Resolvent Estimates of the Linear Operator
  3. The resolvent estimates with the Navier-slip boundary condition
  4. Resolvent estimates with non-slip boundary condition
  5. Space-Time Estimates of the Linearized Navier-Stokes Equations
  6. Space-time estimates of low frequencies $0\leqslant |k|<10\nu$
  7. Space-time estimates of intermediate frequencies $10\nu\leqslant |k|<1$
  8. Bilinear Estimates and Nonlinear Transition Threshold
  9. Enhanced Dissipation Estimates
  10. Some Basic Estimates
  11. Airy Function

Key Result

Theorem 1.1

Suppose that $u^{in}\in H^1_0\cap H^2(\Omega)$ with $\mathrm{div} u^{in}=0$ in pertu. Let $0<\nu_0\leqslant 1$ and $c$ be a suitably small constant. Then for $0<\nu\leqslant\nu_0$ and $\|u^{in}\|_{H^2}\leqslant c\nu^{\frac{1}{2}}$, the solution $u$ to pertu satisfies where

Theorems & Definitions (40)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 30 more