Nonlinear stability of Ekman boundary layers in rotating stratified fluids

A stationary solution of the rotating Navier-Stokes equations with a boundary condition is called an Ekman boundary layer. This booklet constructs stationary solutions of the rotating Navier-Stokes-Boussinesq equations with stratification effects in the case when the rotating axis is not necessarily perpendicular to the horizon. We call such stationary solutions Ekman layers. This booklet shows the existence of a weak solution to an Ekman perturbed system, which satisfies the strong energy inequality. Moreover, we discuss the uniqueness of weak solutions and compute the decay rate of weak solutions with respect to time under some assumptions on the Ekman layers and the physical parameters. It is also shown that there exists a unique global-in-time strong solution of the perturbed system when the initial datum is sufficiently small. Comparing a weak solution satisfying the strong energy inequality with the strong solution implies that the weak solution is smooth with respect to time when time is sufficiently large.