Mathematical Sciences: Regularity Properties of Nonlinear Wave Equations

小行星9400258 一个中心主题，这一数学研究的理论，非线性偏微分方程的演变是规律性或故障的解决方案的重要例子，模拟物理世界。 这项工作不仅涉及什么是解决方案，而且还从物理的角度理解相应的物理理论的有效性的极限。 这项研究是针对在这方面的具体目标-发展相关的新技术，如闵可夫斯基时空的全局稳定性和爱因斯坦和其他经典场论的分析。 一个主要的目标是研究和降低，在可能的情况下，对初始数据集的正则性要求，以确保初始值问题是很好地提出。 简单场论的例子表明，寻找确保适定性的最优范数的问题与解的分解和正则性问题密切相关。 一个相关的方向，这项工作将追求关注的问题的非线性稳定性的闵可夫斯基度量相对于扰动的有限正则类的爱因斯坦真空方程。 该项目的主要研究领域，非线性偏微分方程，是目前正在调查的数学分析中最基本的工作的中心。 非线性方程的解已被证明具有相当大的规律性，而且方程本身表现出的结构比人们曾经认为的要多得多。 该项目继续采用该领域一些最有希望的调查途径。

9400258 Klainerman A central theme of this mathematical research in the theory of nonlinear partial differential equations of evolution is that of regularity or break-down of solutions to the important examples which model the physical world. The work concerns not only what is meant by a solution but also understanding from a physical point of view the very limits of validity of the corresponding physical theories. This research is directed at specific goals in this regard - the development of relevant new techniques such as global stability of the Minkowski space-time and the analysis of the Einstein and other classical field theories. A primary goal is to study and lower, where possible, the regularity requirements on the initial data sets which assure that the initial value problem is well posed. Examples from simpler field theories suggest that the question of finding the optimal norm assuring well-posedness is intimately connected to the issue of break-down and regularity of solutions. A related direction this work will pursue concerns the question of nonlinear stability of the Minkowski metric relative to perturbations in the limited regularity class of the Einstein vacuum equations. The primary area of this project's research, nonlinearpartial differential equations, is central to the mostfundamental work currently under investigation within mathematical analysis. Solutions to nonlinear equations have been shown to posses considerable regularity and the equations themselves exhibit far more structure than was once thought possible. This project continues some of the most promising avenues of investigation in the field.