Vol. 7, No. 6, 2014

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Well-posedness of the Stokes–Coriolis system in the half-space over a rough surface

Anne-Laure Dalibard and Christophe Prange

Vol. 7 (2014), No. 6, 1253–1315
Abstract

This paper is devoted to the well-posedness of the stationary 3D Stokes–Coriolis system set in a half-space with rough bottom and Dirichlet data which does not decrease at space infinity. Our system is a linearized version of the Ekman boundary layer system. We look for a solution of infinite energy in a space of Sobolev regularity. Following an idea of Gérard-Varet and Masmoudi, the general strategy is to reduce the problem to a bumpy channel bounded in the vertical direction thanks to a transparent boundary condition involving a Dirichlet to Neumann operator. Our analysis emphasizes some strong singularities of the Stokes–Coriolis operator at low tangential frequencies. One of the main features of our work lies in the definition of a Dirichlet to Neumann operator for the Stokes–Coriolis system with data in the Kato space Huloc12.

Keywords
Stokes–Coriolis system, Ekman boundary layer, rough boundaries, Dirichlet to Neumann operator, Saint-Venant estimate, Kato spaces
Mathematical Subject Classification 2010
Primary: 35A22, 35C15, 35S99, 35A01
Secondary: 76U05, 35Q35, 35Q86
Milestones
Received: 23 April 2013
Revised: 28 January 2014
Accepted: 1 March 2014
Published: 18 October 2014
Authors
Anne-Laure Dalibard
Université Pierre et Marie Curie (Paris 6)
UMR 7598 Laboratoire Jacques-Louis Lions
F-75005 Paris
France
Centre national de la recherche scientifique
UMR 7598 Laboratoire Jacques-Louis Lions
F-75005 Paris
France
Christophe Prange
Department of Mathematics
The University of Chicago
5734 South University Avenue
Chicago, 60637
United States