Mathematical Sciences: Regularity Properties of Nonlinear Wave Equations

9400258 Klainerman 非线性演化偏微分方程理论数学研究的中心主题是模拟物理世界的重要例子的解的规律性或分解。 这项工作不仅涉及解决方案的含义，还涉及从物理角度理解相应物理理论的有效性极限。 这项研究针对这方面的具体目标——开发相关新技术，例如闵可夫斯基时空的全局稳定性以及爱因斯坦和其他经典场论的分析。 主要目标是研究并在可能的情况下降低对初始数据集的正则性要求，以确保初始值问题适定。 更简单的场论的例子表明，找到确保适定性的最优范数的问题与解的分解和规律性问题密切相关。 这项工作将追求的一个相关方向涉及闵可夫斯基度量相对于爱因斯坦真空方程的有限正则类中的扰动的非线性稳定性问题。 该项目研究的主要领域，非线性偏微分方程，是目前数学分析中正在研究的最基本工作的核心。 非线性方程的解已被证明具有相当大的规律性，并且方程本身表现出的结构比曾经认为可能的要多得多。 该项目延续了该领域一些最有前途的研究途径。

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.