Regularity results for some models in geophysical fluid dynamics

Resum

This thesis centers on the study of two di erent problems of partial di erential equations arising from geophysics and uid mechanics: the surface quasi-geostrophic equation and the so called, Incompressible Slice Model. The surface quasi-geostrophic equation is a two dimensional nonlo- cal partial di erential equation of geophysical importance, describing the evolution of a surface buoyancy in a rapidly rotating, strati ed potential vorticity uid. In the rst part of the talk, we will present some global regularity results for its dissipative analogue in the critical regime for the two dimensional sphere. After that, we will introduce the Incompressible Slice Model deal- ing with oceanic and atmospheric uid motions taking place in a ver- tical slice domain R2, with smooth boundary. The ISM can be understood as a toy model for the full 3D Euler-Boussinesq equa- tions. We will study the solution properties of the Incompressible Slice Model: characterizing a class of equilibrium solutions, establishing the local existence of solutions and providing a blow-up criterion.