Linear Partial Differential Equations on Singular Spaces

奇异空间上的线性偏微分方程

• 批准号: 1600023
• 负责人: Jared Wunsch
• 金额: $ 18万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600023&HistoricalAwards=false
• 关键词: Linear; Partial; Differential; Equations; Singular

The principal investigator will study the subtle correspondence between the movement of particles and the propagation of waves in a variety of geometric and physical situations. In the quantum world, this goes by the name of the "correspondence principle" and has long been understood in the simplest settings. However, many mysteries remain in the application of the correspondence principle to problems involving "bumpy" geometries or large scales. One of the principal investigator's projects addresses how waves (e.g., light waves or gravity waves) propagate on space-times of interest in Einstein's theory of general relativity, and in particular, how their decay is viewed by distant observers. This decay is influenced by the geometry of space-time on very large scales. Another part of the research will study how the familiar phenomenon of diffraction of waves influences the waves' rates of decay. This applies, for instance, to sound waves in the presence of sharp corners or cone points, and it is of practical importance in acoustics and in a variety of computational problems involving wave propagation.The project will study the asymptotic behavior of waves propagating on certain Lorentzian space-times such as arise in the theory of general relativity. A main goal is to understand the asymptotics of the Friedlander radiation field for a range of Lorentzian geometries, including asymptotically static space-times with asymptotically Euclidean ends (with long-range perturbations). The principal investigator will also study the decay of waves near their source in different geometric settings, especially in singular geometries. In particular, he will investigate the distribution of quantum resonances in a variety of geometries involving cone points and corners, as well as continue collaborations on more applied problems motivated by numerical analysis for high-frequency asymptotics of the Helmholtz equation. Finally, the principal investigator will study the closely related problem of the structure of the propagator for the Dirac-Coulomb equation, where the singularity of the potential diffracts the singularities of the solution, producing an outgoing spherical wave.

首席研究员将研究在各种几何和物理情况下粒子运动和波传播之间的微妙对应关系。在量子世界中，这被称为“对应原理”，长期以来一直在最简单的环境中被理解。然而，在将对应原理应用于涉及“凹凸不平”几何或大尺度的问题时，仍然存在许多谜团。首席研究员的一个项目是研究波（如光波或引力波）如何在爱因斯坦广义相对论中感兴趣的时空中传播，特别是远距离观测者如何观察它们的衰减。这种衰减在非常大的尺度上受到时空几何的影响。研究的另一部分将研究人们熟悉的波的衍射现象如何影响波的衰减速率。例如，这适用于存在尖角或锥点的声波，并且在声学和涉及波传播的各种计算问题中具有实际重要性。该项目将研究在某些洛伦兹时空中传播的波的渐近行为，例如在广义相对论中出现的波。主要目标是了解一系列洛伦兹几何的弗里德兰德辐射场的渐近性，包括具有渐近欧几里得端点的渐近静态时空（具有远程扰动）。首席研究员还将研究在不同几何环境下，特别是在奇异几何环境下，波在其源附近的衰减。特别是，他将研究涉及锥点和角的各种几何形状中的量子共振分布，并继续合作研究更多由亥姆霍兹方程高频渐近数值分析驱动的应用问题。最后，首席研究员将研究与狄拉克-库仑方程传播子结构密切相关的问题，其中势能的奇点绕射解的奇点，产生一个传出的球形波。