The primitive equations on the large scale ocean under the small depth hypothesis

Changbing Hu; Roger Temam; Mohammed Ziane

doi:10.3934/dcds.2003.9.97

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2003, Volume 9, Issue 1: 97-131. Doi: 10.3934/dcds.2003.9.97

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The primitive equations on the large scale ocean under the small depth hypothesis

1.
The Institute for Scientific Computing & Applied Mathematics, Indiana University, Rawles Hall, Bloomington, IN 47405
2.
The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 E. 3rd St., Rawles Hall, Bloomington, IN 47405
3.
Department of Mathematics, University of Southern California, Los Angeles, CA 90089

Received: October 31, 2001

Revised: August 31, 2002

Published: December 2002

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Abstract

In this article we study the global existence of strong solutions of the Primitive Equations (PEs) for the large scale ocean under the small depth hypothesis. The small depth hypothesis implies that the domain $M_\varepsilon$ occupied by the ocean is a thin domain, its thickness parameter $\varepsilon$ is the aspect ratio between its vertical and horizontal scales. Using and generalizing the methods developed in [23], [24], we establish the global existence of strong solutions for initial data and volume and boundary 'forces', which belong to large sets in their respective phase spaces, provided $\varepsilon$ is sufficiently small. Our proof of the existence results for the PEs is based on precise estimates of the dependence of a number of classical constants on the thickness $\varepsilon$ of the domain. The extension of the results to the atmosphere or the coupled ocean and atmosphere or to other relevant boundary conditions will appear elsewhere.

Keywords:

Nonlinear partial differential equations,
primitive equations,
oceanography,
atmosphere sciences.

Mathematics Subject Classification: 35Q35, 35Q30, 76D03.

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