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

TL;DR

This paper investigates the transition threshold from laminar to turbulent flow in the Navier-Stokes-Coriolis system at high Reynolds numbers, deriving improved stability criteria by analyzing anisotropic effects and dispersive structures.

Contribution

It introduces a new analytical framework combining anisotropic Sobolev spaces and dispersive estimates to improve the stability threshold scaling for Couette flow under rotation.

Findings

Derived stability threshold scaling with Re for Couette flow.

Established global existence and asymptotic stability under certain perturbations.

Developed new dispersive and anisotropic analytical tools for zero mode analysis.

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 $$\|v_{in}-(y, 0, 0)\|_{\tilde{H}(\mathbb T \times \mathbb D)}\leq \epsilon_0 \,\mathrm{Re}^{-\alpha},$$ with any $\alpha>\frac 23$ and $\tilde{H}=H^6 \cap W^{3,1}$ for $\mathbb D=\mathbb{R}^2$, and with any $\alpha \geq\frac 56$ 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.