University of California-Irvine and Weizmann Institute of Science
Friday, October 27, 2006 - 4:00pm
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 survey the main obstacles in proving the global regularity for the three-dimensional Navier-Stokes equations and their geophysical counterparts. 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 in this talk that they have a unique global (in time) regular solution for all initial data.
We describe a method to improve both the accuracy and computational efficiency of a given finite difference scheme used to simulate a geophysical flow. The resulting modified scheme is at least as accurate as the original, has the same time step, and often uses the same spatial stencil. However, in certain parameter regimes it is higher order. As examples we apply the method to the shallow water equations, the Navier-Stokes equations, and to a sea breeze model.
I will talk about various lattice dynamical systems with long range interaction and related integro-differential evolution equations.
These arise in the modeling of phase transitions for a binary material, as models for the dispersal of organisms and from activity in families of neurons. Included here
are nonlocal analogs of the wave equation, Allen-Cahn and Cahn-Hilliard equations.
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.