Regularity and Dispersive Properties of Evolution Equations

这个项目的主要目标是研究非线性波动方程解的正则性，这些非线性波动方程是与爱因斯坦方程和更一般的准线性波动方程有关的。特别地，我们解决了上述方程的局部适定性问题。本计画也研究变系数线性薛定谔方程的整体色散效应，爱因斯坦方程是相对论的基石之一。它们在描述宇宙结构中起着基础性的作用。由于方程除了极个别的特殊情况外是不可解的，因此我们需要了解其解的定性性质。我们试图理解这些解决方案是否能在不引起异常行为的情况下持续存在。为了获得更深入的了解，我们对更一般的方程类进行了研究。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.