The primitive equations describe hydrodynamical flows in thin layers of fluid (such as the atmosphere and the oceans). Due to the shallowness of the fluid layer the
the vertical motion is much smaller than the horizontal one and hence the former is modeled, in the primitive equations, by the hydrostatic balance. The primitive equations are considered to be a very good model
for large scale ocean circulations and for global atmospheric flows. As a result they are used in most global climate models. In this talk we will introduce a mathematical framework for studying various models of atmospheric and oceanic dynamics. In particular, the planetary geostrophic equations and the primitive equations. Furthermore, I will show the global well posedness of these equations.

This talk will give an introduction to optimal regularity as a tool to analyze (fully) nonlinear parabolic equations/systems. After a review of the major developments of the theory, the focus will shift to singular parabolic equations. It will be shown that optimal regularity results can be obtained for a large class of singular abstract Cauchy problems and, if time permits, applications of the theory will be presented.

We present a new adaptive numerical scheme for solving parabolic PDEs in
cartesian geometry. Applying a finite volume discretization with explicit
time integration, both of second order, we employ a fully adaptive
multiresolution scheme to represent the solution on locally refined nested
grids. The fluxes are evaluated on the adaptive grid. A dynamical adaption
strategy to advance the grid in time and to follow the time evolution of
the solution directly explaoits the multiresolution representation.
Applying this new method to several test probelms in one, two and three
space dimensions, like convection-diffucion, viscous Burgers and
reaction-diffusion equations, we show its second order accuracy and
demonstrate its computational efficiency.
This work is joint work with Olivier Roussel.