The main goals of this project are concerned with the study of the regularity properties of the solutions of the nonlinear wave equations that arise in connection with the Einstein equations and more general quasilinear wave equations. In particular, we address the questions of local well posedness for the above equations. This project also investigates the global dispersive effects for the linear Schroedinger equation with variable coefficients.
The Einstein equations is one of the cornerstones of the theory of relativity. They play a fundamental role in the description of the structure of the universe. Since the equations can not be solved explicitly with exception of a few very special cases, we need to understand the qualitative properties of its solutions. We try to understand whether the solutions persist in time without exhibiting an abnormal behavior. To gain insight we do it for the more general class of equations. We also interested in understanding the connections between the classical and quantum behavior of particles. This leads us to the study of the behavior of solutions of the fundamental equation of the quantum mechanics, the Schroedinger equation.
