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

Rotation is one of the most important features of fluid flow in the atmosphere and oceans, which appears in almost all meteorological and geophysical models. When the speed of rotation is sufficiently large, the global existence of strong solution to the 3D Navier-Stokes equations with rotation has been obtained by the dispersion effect coming from Coriolis force (i.e., rotation). In this paper, we study the dynamic stability of the periodic, plane Couette flow in the three-dimensional Navier-Stokes equations with rotation at high Reynolds number $\mathbf{Re}$. Our goal is to find the index of the stability threshold on $\mathbf{Re}$: the maximum range of perturbations in which the solution to the equations remains stable. We first study the linear stability effects of linearized perturbed system. Compared with the results of Bedrossian, Germain and Masmoudi [Ann. of Math. 185(2): 541--608 (2017)], mixing effects (which corresponds to enhanced dissipation and inviscid damping) arise from the Couette flow, Coriolis force acts as a restoring force which induces the dispersion mechanism of inertial waves and cancels the lift-up effect occurred in the zero frequency velocity. This dispersion mechanism bring good algebraic decay properties, which is different from the 3D classical Navier-Stokes equations. Therefore, we prove that the 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 $\sigma$, the resulting solution to the 3D Navier-Stokes equations with rotation is global in time and does not transition away from the Couette flow. In the sense, Coriolis force is a factor that contributes to the stability of the fluid, which improves the stability threshold from $\frac{3}{2}$ to $1$.