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Mathematical Study of Certain Geophysical Models: Global Regularity and Finite-time Blowup Results

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Mathematics for the Fluid Earth

The basic problem faced in geophysical uid 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 discuss 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 simpli ed 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. Even though the primitive equations look as if they are more dicult to study analytically than the three-dimensional Navier-Stokes equations I will show, on the one hand, that the viscous primitive equations have a unique global (in time) regular solution for all initial data. On the other hand, I will show that in the non-viscous (inviscid) case there is a one-parameter family of initial data for which the corresponding smooth solutions develop nite-time singularities (blowup).

This talk is part of the Isaac Newton Institute Seminar Series series.

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