### Mathematics Colloquium

** Edriss S. Titi
**

University of Cambridge,
Texas A&M University,
and
Weizmann Institute of Science

**Title:** Mathematical Analysis of Geophysical Models

**Date:** Friday, February 24, 2023

**Place and Time:** LOV 101, 3:05-3:55 pm

**Abstract.**

In this talk I will present some recent results concerning the global regularity of certain geophysical models. In particular, the three-dimensional Planetary Geostrophic and the Primitive Equations (PE) of oceanic and atmospheric dynamics with various anisotropic viscosity and turbulence mixing diffusion. However, in the non-viscous (inviscid) case it can be shown that, with or without rotation, the PE are linearly and nonlinearly ill-posed in the context of Sobolev spaces, and that there is a one-parameter family of initial data for which the corresponding smooth solutions of the primitive equations develop finite-time singularities (blowup). However, the PE will be shown to be well- posed in the space of real analytic functions, and I will discuss the effect of rotation on prolonging the life-span of analytic solutions.

Capitalizing on the above results, we can provide rigorous justification of the derivation of the viscous PE of planetary scale oceanic dynamics from the three-dimensional Navier-Stokes equations, for vanishing small values of the aspect ratio of the depth to horizontal width. Specifically, we can show that the Navier-Stokes equations, after being scaled appropriately by the small aspect ratio parameter of the physical domain, converge strongly to the primitive equations, globally and uniformly in time, and that the convergence rate is of the same order as the aspect ratio parameter.