Sobolev and Gevrey regularity results for the primitive equations in three space dimensions

Abstract

The aim of this article is to present a qualitative study of the Primitive Equations in a three-dimensional domain, with periodical boundary conditions. We start by recalling some already existing results regarding the existence locally in time of weak solutions and existence and uniqueness of strong solutions, and we prove the existence of very regular solutions, up to

-regularity. In the second part of the article we prove that the solution of the Primitive Equations belongs to a certain Gevrey class of functions.

Keywords:

• Primitive Equations
• Gevrey regularity
• 2000 Mathematics Subject Classifications:
• 35B65
• 35Q35
• 76D03

Acknowledgements

This work was supported in part by NSF Grant DMS 0305110, and by the Research Fund of Indiana University. The authors would like to thank Professor R. Temam for the help accorded in solving it and the Institute for Scientific Computing and Applied Mathematics at Indiana University for its hospitality during a part of this work.

Notes

Present address: Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, United Kingdom.

For the scalar products and the norms we use the same notations as in the real case.