Stability of the Couette flow for 3D Navier-Stokes equations with rotation

TL;DR

This paper investigates the stability of 3D rotating Couette flow at high Reynolds numbers, introducing new analytical tools to handle rotation effects and establishing a stability threshold based on initial perturbation size.

Contribution

The study develops novel unknowns to effectively analyze enhanced dissipation and inviscid damping in rotating flows, establishing a quantitative stability threshold for perturbations.

Findings

Established stability threshold: initial perturbation norm < δ Re^{-2}

Introduced new unknowns capturing dissipation and damping effects

Proved stability of Couette flow under high Reynolds number with rotation

Abstract

Rotation significantly influences the stability characteristics of both laminar and turbulent shear flows. This study examines the stability threshold of the three-dimensional Navier-Stokes equations with rotation, in the vicinity of the Couette flow at high Reynolds numbers ($\mathbf{Re}$) in the periodical domain $\mathbb{T} \times \mathbb{R} \times \mathbb{T}$, where the rotational strength is equivalent to the Couette flow. Compared to the classical Navier-Stokes equations, rotation term brings us more two primary difficulties: the linear coupling term involving in the equation of $u^2$ and the lift-up effect in two directions. To address these difficulties, we introduce two new good unknowns that effectively capture the phenomena of enhanced dissipation and inviscid damping to suppress the lift-up effect. Moreover, we establish the stability threshold for initial perturbation $\left\|u_{\mathrm{in}}\right\|_{H^{\sigma}} < \delta \mathbf{Re}^{-2}$ for any $\sigma > \frac{9}{2}$ and some $\delta=\delta(\sigma)>0$ depending only on $\sigma$.