A squeezing property and its applications to a description of long time behaviour in the 3D viscous primitive equations

We consider the 3D viscous primitive equations with periodic boundary conditions. These equations arise in the study of ocean dynamics and generate a dynamical system in a Sobolev H^1 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 (i) the finiteness of the fractal dimension of the corresponding global attractor, (ii) the existence of finite number of determining modes, and (iii) ergodicity of a related random kick model. All these results provide a new information concerning long time dynamics of oceanic motions.