Edriss S. Titi (Weizmann Institute and University of California)
Mathematical Study of Certain Geophysical Models: Global Regularity and Finite-time Blowup Results
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 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 simplified and manageable
models which are easier to study analytically. In particular, I will
present the global well-posedness for the three-dimensional B\'enard
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
difficult 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 finite-time singularities (blowup).
This is a joint work with Chongsheng Cao, Slim Ibrahim and Kenji Nakanishi.