Vol. 10, No. 1, 2017

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Nonlinear boundary layers for rotating fluids

Anne-Laure Dalibard and David Gérard-Varet

Vol. 10 (2017), No. 1, 1–42
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. (9) 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).

Keywords
fluid mechanics, geophysical fluids, Ekman layers, boundary layers
Mathematical Subject Classification 2010
Primary: 35Q30
Secondary: 35Q86
Milestones
Received: 3 November 2015
Revised: 12 May 2016
Accepted: 13 October 2016
Published: 30 January 2017
Authors
Anne-Laure Dalibard
Sorbonne Universités
UPMC Université Paris 06
CNRS, UMR 7598
Laboratoire Jacques-Louis Lions
4 place Jussieu
75005 Paris
France
David Gérard-Varet
Université Paris Diderot
Sorbonne Paris Cité
Institut de Mathématiques de Jussieu-Paris Rive Gauche
UMR 7586
F-75205 Paris
France