In many areas of applied sciences, engineering and technology there are three problems dealing with data and signals: (i) data compression; (ii) signal representations; and (iii) recovery of signals from partial or indirect information about the signals, often contaminated by noise. Major advances in these problems have been achieved in recent years where wavelets, multiresolution analysis, and kernel methods have played key roles. We consider problem (iii) and give an overview of specific contributions to inverse and ill-posed problems where reproducing kernel Hilbert spaces provide a natural setting.
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.
We study unique continuation properties of solutions of
linear and non-linear Schroedinger equations. In the nonlinear case we are interested in deducing uniqueness of the solution from information on the difference of two possible solutions at two different times.
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.