Mathematical Sciences: Regularity Properties of Nonlinear Wave Equations

数学科学：非线性波动方程的正则性质

• 批准号: 9400258
• 负责人: Sergiu Klainerman
• 金额: $ 15万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9400258&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Regularity; Properties; Nonlinear

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.

9400258克莱纳曼这项关于演化的非线性偏微分方程论的数学研究的中心主题是对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，而且还涉及从物理角度理解相应物理理论的有效性的极限。这项研究是针对这方面的具体目标--发展相关的新技术，如Minkowski时空的整体稳定性，以及对爱因斯坦和其他经典场论的分析。一个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，以确保初值问题是良好的。更简单的场论的例子表明，找到确保适定性的最佳范数的问题与解的分解和正则性问题密切相关。这项工作将追求的一个相关方向是关于Minkowski度规相对于爱因斯坦真空方程的有限正则类扰动的非线性稳定性问题。这个项目研究的主要领域，非线性偏微分方程组，是目前数学分析中正在研究的最基本的工作的核心。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比以前认为可能的结构要多得多。该项目延续了该领域一些最有希望的调查途径。