Nonlinear Analysis Seminar

March 19, 2004, 16:30-18:00   Faculty of Science and Technology, Keio University Bldg. 14 Room No. 413 (Discussion Room 43) Prof. Edriss S. Titi (Weizmann Institute, Israel) Mathematical Study of Certain Geophysical Models Abstract:      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, and the global regularity for a three dimensional viscous planetary geostrophic models. Furthermore, these systems will be shown to have finite dimensional global attractors. If time allows I will present the deriviation of certain two dimensional shallow water approximate models for the three dimensional Euler equations in a basin with slowly spatially varying topography, the so called ``Lake Equation" and ``Great Lake Equation", which should represent the behavior of the physical system on time and length scales of interest. These approximate models will be shown to be globally well-posed. I will also show that the Charney-Stommel model of the gulf-stream, which is a two dimensional damped driven shallow water model for ocean circulation, has a global attractor. Whether this attractor is finite or infinite dimensional is still an open question.

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