Title: On the Question of Global Regularity for
       Three-dimensional Navier-Stokes
       Equations and Relevant Geophysical Models


Abstract. The basic problem faced in geophysical fluid dynamics is
that a mathematical description based only on fundamental physical
principles, the so-called the ``Primitive Equations'', is often
prohibitively expensive computationally, and hard to study
analytically. In this talk I will survey the main obstacles in
proving the global regularity for the three-dimensional
Navier-Stokes equations and their geophysical counterparts. Even
though the Primitive Equations look as if they are more difficult to
study analytically than the three-dimensional Navier-Stokes
equations I will show in this talk that they have a unique global
(in time) regular solution for all initial data.

Inspired by this work I will also provide a new global
regularity criterion for the three-dimensional Navier-Stokes
equations involving the pressure.

This is a joint work with Chongsheng Cao.