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 $\sigma$.
Categories: math.AP