Regularity and Dispersive Properties of Evolution Equations

本项目的主要目标是研究与爱因斯坦方程和更一般的拟线性波动方程有关的非线性波动方程解的正则性。特别地，我们处理了上述方程的局部适定性问题。本项目还研究了变系数线性薛定谔方程的全局色散效应。爱因斯坦方程是相对论的基石之一。它们在描述宇宙结构方面起着重要作用。由于除了少数非常特殊的情况外，这些方程不能被明确地解出，我们需要了解其解的定性性质。我们试图了解这些溶液是否会持续存在而不会表现出异常行为。为了获得更深入的了解，我们对更一般的一类方程做这个。我们也对理解粒子的经典行为和量子行为之间的联系感兴趣。这导致了对量子力学基本方程——薛定谔方程——解的行为的研究。

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.