On the Sobolev stability threshold for 3D Navier-Stokes equations with rotation near the Couette flow

TL;DR

This paper studies the stability of 3D rotating Navier-Stokes equations near Couette flow, showing that rotation enhances stability thresholds and prevents transition to turbulence at high Reynolds numbers.

Contribution

It demonstrates that Coriolis force improves the stability threshold for perturbations in 3D Navier-Stokes with rotation, extending previous results by quantifying the effect of rotation on stability.

Findings

Rotation induces a dispersion mechanism that cancels lift-up effects.

Stability threshold improves from Re^{-3/2} to Re^{-1} due to Coriolis force.

Solutions remain globally stable under small initial perturbations at high Reynolds numbers.

Abstract

Rotation is a crucial characteristic of fluid flow in the atmosphere and oceans, which is present in nearly all meteorological and geophysical models. The global existence of solutions to the 3D Navier-Stokes equations with large rotation has been established through the dispersion effect resulting from Coriolis force (i.e., rotation). In this paper, we investigate the dynamic stability of periodic, plane Couette flow in the three-dimensional Navier-Stokes equations with rotation at high Reynolds number $\mathbf{Re}$. Our aim is to determine the stability threshold index on $\mathbf{Re}$: the maximum range of perturbations within which the solution remains stable. Initially, we examine the linear stability effects of a linearized perturbed system. Comparing our results with those obtained by Bedrossian, Germain, and Masmoudi [Ann. Math. 185(2): 541--608 (2017)], we observe that mixing effects (which correspond to enhanced dissipation and inviscid damping) arise from Couette flow while Coriolis force acts as a restoring force inducing a dispersion mechanism for inertial waves that cancels out lift-up effects occurred at zero frequency velocity. This dispersion mechanism exhibits favorable algebraic decay properties distinct from those observed in classical 3D Navier-Stokes equations. Consequently, we demonstrate that if initial data satisfies $\left\|u_{\mathrm{in}}\right\|_{H^{\sigma}}<\delta \mathbf{Re}^{-1}$ for any $\sigma>\frac{9}{2}$ and some $\delta=\delta(\sigma)>0$ depending only on $\sigma$, then the solution to the 3D Navier-Stokes equations with rotation is global in time without transitioning away from Couette flow. In this sense, Coriolis force contributes as a factor enhancing fluid stability by improving its threshold from $\frac{3}{2}$ to 1.