Finite Dimensional Behaviors of the Primitive Equations Under Small Depth Assumption

Abstract

In this article, we study the asymptotic degrees of freedom for solutions to the primitive equation (PEs for brevity). More precisely, we will prove that the long-time behavior of solutions to PEs is determined by the set of either finite Fourier modes, line elements, or volume elements. Our results show that the long-time behavior of the PEs is determined by the baro-tropic flows that are independent of the vertical direction in ℝ³. This study builds upon the previous article by the author concerning the existence and uniqueness of strong solutions to the PEs in thin domains.

Keywords:

• Determining lines
• Determining modes
• Determining volume elements
• Long-time behavior
• Primitive equations
• Thin domain

AMS Subject Classification:

• 35Q35
• 76D03

ACKNOWLEDGMENTS

Part of this work had been done while the author was a visiting assistant professor in the Department of Mathematics at Texas A&M University from 2002 to 2005. He would like to acknowledge the warm and friendly hospitality he received.