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Phone: (407) 823-6284;   Fax: (407) 823-6253;   MAP  207 2004-2005 MATHEMATICS COLLOQUIUM SERIES 02/08/05 Colloquium Dr. Edriss S. Titi University of California, Irvine and the Weitzmann Institute of Science Mathematical Analysis of Certain Analytic Sub-grid Scale Models of Turbulence Abstract:   In this talk he will discuss the mathematical difficulty in proving global regularity for the three-dimensional Navier-Stokes equations. Furthermore, he will show the global regularity for certain analytic three-dimensional sub-grid scale models of turbulence. This will include the Smagorinsky model, Navier--Stokes-alpha model, the Leray-alpha model and the Clark model. All these models are of nonlinear parabolic type and each has a finite dimensional global attractor. In some cases he will provide explicit bounds for the fractal and Hausdorff dimensions of these attractors, in terms of the relevant physical parameters. In addition, he will also prove the global regularity for the "shell model" of turbulence and show that it has a finite dimensional inertial manifold. Hence, this model can be reduced to a finite dimensional ordinary differential system. As a result, one can show that this model has a unique invariant measure associated with its dynamics when it is kicked randomly.