Regularity and Dispersive Properties of Evolution Equations

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The main goals of this project are concerned with the study of theregularity properties of the solutions of the nonlinear wave equationsthat arise in connection with the Einstein equations and more generalquasilinear wave equations. In particular, we address the questions oflocal well posedness for the above equations. This project alsoinvestigates the global dispersive effects for the linear Schroedingerequation with variable coefficients.The Einstein equations is one of the cornerstones of the theory ofrelativity. They play a fundamental role in the description of thestructure of the universe. Since the equations can not be solvedexplicitly with exception of a few very special cases, we need tounderstand the qualitative properties of its solutions. We try tounderstand whether the solutions persist in time without exhibitingan abnormal behavior. To gain insight we do it for the more generalclass of equations. We also interested in understanding the connectionsbetween the classical and quantum behavior of particles. This leadsus to the study of the behavior of solutions of the fundamentalequation of the quantum mechanics, the Schroedinger equation.