# American Institute of Mathematical Sciences

January  2003, 9(1): 97-131. doi: 10.3934/dcds.2003.9.97

## 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, United States 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, United States

Received  November 2001 Revised  September 2002 Published  November 2002

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.
Citation: Changbing Hu, Roger Temam, Mohammed Ziane. The primitive equations on the large scale ocean under the small depth hypothesis. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 97-131. doi: 10.3934/dcds.2003.9.97
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