Global Analysis of Non-Linear Wave Equations

这个项目关注的是涉及线性和非线性波动方程解的长时间行为的各种问题。这样的系统出现在数学物理的各种问题中，包括狭义和广义相对论的基本几何场论。该项目的基本关注点是了解此类系统在广泛背景下的渐近行为：从自发形成奇点等局部现象，到发生在无限时空尺度上的微妙色散特性。这些问题是偏微分方程理论中最核心和最困难的问题之一，这里提出的工作是最近的调查和主要研究者在几何波动方程中几个最重要问题上的进展的延续，包括能量临界波映射的爆破与全局正则性问题。线性和非线性波动方程作为模型出现在许多不同的物理领域，从电磁学、弹性力学的基本经典场论、线性化的引力辐射方程和完整的爱因斯坦场方程本身，到量子力学和量子场论的许多方程。主要研究者所要研究的数学问题都与描述这类物理系统的动力学性质的基本问题有关。对这些系统的数学理解的进步应该会导致更好的建模，预测和控制它们所支配的物理现象。PI计划不仅继续他最近的工作，而且还在这个令人兴奋和快节奏的研究领域培养几名新的研究生。

This project is concerned with a variety of problems involving the long-time behavior of solutions to linear and nonlinear wave equations. Such systems arise in a variety of problems in mathematical physics, including the underlying geometric field theories of special and general relativity. The basic concern of this project is to understand the asymptotic behavior of such systems in a wide range of contexts: from local phenomena such as the spontaneous formation of singularities, to delicate dispersive properties which take place over infinite space-time scales. Such questions are among the most central and difficult in the theory of partial differential equations, and the work proposed here is a continuation of recent investigations and advances of the principal investigator on several of the most important problems in geometric wave equations, including the blowup vs. global regularity question for energy critical wave-maps.Linear and nonlinear wave equations arise as models in many different areas of physics, from the underlying classical field theories of electro-magnetism, elasticity, the equations of linearized gravitational radiation, and the full Einstein field equations themselves, to many equations of quantum mechanics and quantum field theory. The mathematical issues to be studied by the principal investigator are all related to the basic question of describing the dynamical properties such physical systems. Progress on the mathematical understanding of these systems should lead to better modeling, prediction, and control of the physical phenomena they govern. The PI plans not only to continue his recent work, but also to train several new graduate students in this exciting and fast paced field of study.