Analysis of Linear and Non-linear Wave Equations: Regularity, Asymptotics, and Blowup

分析线性和非线性波动方程：规律性，渐近性和blowup.摘要建议的研究雅各布K SterbenzThis项目是调查各种问题涉及的规律性和长期行为的解决方案的线性和非线性波动方程。其基本目标是了解这些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特性。最近在这一领域的研究的主题是几何波动方程的临界正则性结果，这种方程在临界尺寸的爆破现象，以及关键的线性和非线性模型的渐近稳定性。这是当前研究的一个非常活跃的领域。有效的策略和技术，产生洞察这些问题将有非平凡的重叠与许多其他领域的经典和现代数学，如光谱和散射理论，微分几何，真实的变量方法的谐波分析，以及建设和实施的数值模拟。这些系统出现的模型，许多不同的问题，在物理学，包括电磁学、非线性弹性力学、线性化引力辐射方程、量子力学和量子场论的基本经典场论。这里所要研究的数学问题都与描述线性和非线性物理系统的动力学性质的基本问题有关。 对这些问题的数学分析的进展应该会导致对这些物理现象的更好的理解，以及预测和控制。

Analysis of Linear and Nonlinear Wave Equations: regularity, asymptotics and blowup.Abstract of Proposed Research Jacob K SterbenzThis project is to investigate a variety of problems involving both the regularity and the long time behavior of solutions of linear and non-linear wave equations. The fundamental goal 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. The main themes of recent research in this area has been critical regularity results for geometric wave equations, blowup phenomena for such equations in critical dimensions, and the asymptotic stability of key linear and non-linear models. This is a very active field of current research. Effective strategies and techniques which yield insight into these problems will have non-trivial overlap with many other areas of classical and modern mathematics such as spectral and scattering theory, differential geometry, real variable methods of harmonic analysis, and the construction and implementation of numerical simulations.These systems arise as models of many different problems in physics, including the underlying classical field theories of electro-magnetism, nonlinear elasticity, the equations of linearized gravitational radiation, quantum mechanics and quantum field theory. The mathematical issues to be studied here all are related to the basic question of describing the dynamical properties of linear and nonlinear physical systems. Progress on the mathematical analysis of these problems should lead to a much better understanding, as well as prediction and perhaps control, of these physical phenomena.