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Center for Nonlinear Studies |
The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, which are called the ``Primitive Equations'', is often prohibitively expensive computationally, and hard to study analytically. In this talk we will survey the main obstacles in proving the global regularity for the three-dimensional Navier--Stokes equations and their geophysical counterparts. However, taking advantage of certain geophysical balances and situations, such as geostrophic balance and the shallowness of the ocean and atmosphere, geophysicists derive more simplified and manageable models which are easier to study analytically. In particular, I will present the global well-posedness for the three-dimensional Benard convection problem in porous media, the global regularity for a three-dimensional viscous planetary geostrophic models, and three-dimensional primitive equations. Furthermore, these systems will be shown to have finite dimensional global attractors.