Nonlinear boundary layers for rotating fluids

Abstract

We investigate the behaviour of rotating incompressible flows near a nonflat horizontal bottom. In the flat case, the velocity profile is given explicitly by a simple linear ODE. When bottom variations are taken into account, it is governed by a nonlinear PDE system, with far less obvious mathematical properties. We establish the well-posedness of this system and the asymptotic behaviour of the solution away from the boundary. In the course of the proof, we investigate in particular the action of pseudodifferential operators in nonlocalized Sobolev spaces. Our results extend an older paper of Gérard-Varet (J. Math. Pures Appl. $\left(9\right)$ 82:11 (2003), 1453–1498), restricted to periodic variations of the bottom, using the recent linear analysis of Dalibard and Prange (Anal. & PDE 7:6 (2014), 1253–1315).