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A squeezing property and its applications to a description of long-time behaviour in the three-dimensional viscous primitive equations

Published online by Cambridge University Press:  24 July 2014

Igor Chueshov
Igor Chueshov*
Affiliation:
Department of Mechanics and Mathematics, Karazin Kharkov National University, Kharkov 61022, Ukraine, chueshov@karazin.ua
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Abstract

We consider the three-dimensional viscous primitive equations with periodic boundary conditions. These equations arise in the study of ocean dynamics and generate a dynamical system in a Sobolev H1-type space. Our main result establishes the so-called squeezing property in the Ladyzhenskaya form for this system. As a consequence of this property we prove the finiteness of the fractal dimension of the corresponding global attractor, the existence of a finite number of determining modes and the ergodicity of a related random kick model. All these results provide new information concerning the long-time dynamics of oceanic motion.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 144 , Issue 4 , August 2014 , pp. 711 - 729
Copyright
Copyright © Royal Society of Edinburgh 2014 

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