Transition threshold for the Navier-Stokes-Coriolis system at high Reynolds numbers

TL;DR

The paper analyzes the nonlinear transition from laminar to turbulent flow for the three-dimensional Navier–Stokes–Coriolis system near Couette flow at high Reynolds numbers. It develops an anisotropic Sobolev framework and a suite of dispersive Strichartz estimates to control zero and non-zero Fourier modes, enabling an improved stability threshold α>2/3 (and α≥5/6 for the alternate domain) and global-in-time stability. The authors introduce new good unknowns, moving-frame coordinates, and a specialized energy functional to harness rotation-induced dispersion and mixing, achieving inviscid damping and enhanced dissipation for non-zero modes and dispersive decay for zero modes. The results include detailed a priori estimates and long-time asymptotics for both zero and non-zero modes, applicable to two domain configurations, with precise decay rates and norm bounds. This advances the transition-threshold program for rotating fluids by leveraging rotational dispersion to lower the required perturbation size for stability.

Abstract

The transition mechanism from laminar flow to turbulent flow is a central problem in hydrodynamic stability theory. To shed light on this transition mechanism, Trefethen et al.({\it \small Science 1993}) proposed the transition threshold problem, aiming to quantify the magnitude of perturbations required to trigger instability and determine their scaling with the Reynolds number. In this paper, we investigate the transition threshold of Couette flow for the three-dimensional incompressible Navier-Stokes-Coriolis system in the high Reynolds number regime ($\mathrm{Re}\gg 1$). By exploiting the combined effects of rotation (dispersion) and mixing mechanisms, we derive an improved stability threshold scaling in $\mathrm{Re}$. Precisely, we show that if the initial perturbation satisfies $$\|v_{in}-(y, 0, 0)\|_{\tilde{H}(\mathbb T \times \mathbb D)}\leq ε_0 \,\mathrm{Re}^{-α},$$ with any $α>\frac 23$ and $\tilde{H}=H^6 \cap W^{3,1}$ for $\mathbb D=\mathbb{R}^2$, and with any $α\geq\frac 56$ and $\tilde{H}=H^6$ for $\mathbb D=\mathbb{R}\times\mathbb{T}$, the corresponding solution of the Navier-Stokes-Coriolis system exists globally in time and remains asymptotically close to the Couette flow. The main analytical challenge arises from the anisotropic nature of the estimates for the zero modes and from the interactions between zero and non-zero modes, which we address using an anisotropic Sobolev space directly tailored to the zero modes. Additionally, we introduce a new dispersive structure for the zero modes and derive suitable Strichartz-type estimates. These tools enable us to exploit both the nonlinear structure and the improved dispersive behavior of certain good components of the zero modes, which play a crucial role in achieving the improved stability threshold.

Theorems & Definitions (50)

• Theorem 1.1: Nonlinear stability on ${\mathbb T}\times{\mathbb R}^2$
• Theorem 1.2: Asymptotic behavior on ${\mathbb T}\times{\mathbb R}^2$
• Remark 1.3
• Theorem 1.4: Nonlinear stability on ${\mathbb T}\times{\mathbb R}\times{\mathbb T}$
• Theorem 1.5: Asymptotic behavior on ${\mathbb T}\times{\mathbb R}\times{\mathbb T}$
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof