On the stability of viscous three-dimensional rotating Couette flow

We study the stability of Couette flow in the 3d Navier-Stokes equations with rotation, as given by the Coriolis force. Hereby, the nature of linearized dynamics near Couette flow depends crucially on the force balance between background shearing and rotation, and includes lift-up or exponential instabilities, as well as a stable regime. In the latter, shearing resp. rotational inertial waves give rise to mixing and dispersive effects, which are relevant for distinct dynamical realms. Our main result quantifies these effects through enhanced dissipation and dispersive amplitude decay in both linear and nonlinear settings: in particular, we establish a nonlinear transition threshold which quantitatively improves over the setting without rotation (and increases further with rotation speed), showcasing its stabilizing effect.