Global Analysis of Non-Linear Wave Equations

这个项目涉及各种问题，涉及线性和非线性波动方程的解的长时间行为。这样的系统出现在数学物理中的各种问题中，包括狭义相对论和广义相对论的基本几何场理论。这个项目的基本关注是理解这类系统在广泛的背景下的渐近行为：从局部现象，如奇点的自发形成，到发生在无限时空尺度上的微妙的色散性质。这些问题是偏微分方程组理论中最核心和最困难的问题之一，这里提出的工作是主要研究人员最近对几何波动方程中几个最重要问题的研究和进展的继续，包括能量临界波图的爆破和全局正则性问题。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，从基本的电磁学、弹性力学、线性化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。首席研究员要研究的数学问题都与描述这类物理系统的动力学性质这一基本问题有关。在对这些系统的数学理解方面的进步应该导致对它们所支配的物理现象进行更好的建模、预测和控制。PI不仅计划继续他最近的工作，而且还计划在这个令人兴奋和快节奏的研究领域培训几名新的研究生。

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.