Linear Partial Differential Equations on Singular Spaces

奇异空间上的线性偏微分方程

• 批准号: 0401323
• 负责人: Jared Wunsch
• 金额: $ 7.2万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-15 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401323&HistoricalAwards=false
• 关键词: Linear; Partial; Differential; Equations; Singular

Proposal DMS-0401323Title: Linear partial differential equations on singular spacesP.I.: Jared Wunsch, Northwestern UniversityABSTRACTThe PI will study differential operators on manifolds with singular metricstructures. One project focuses on the wave equation on singular spaces;together with Melrose, the PI has shown that in general, a singularity ofa solution to the wave equation interacts with a cone point to produce a"diffracted" spherical wavefront emanating from the cone point. Thesingularity of the diffracted front may be weaker than that of theincident front if the latter is not too precisely focused on the conepoint. Melrose and the PI hope to generalize these results to morecomplex singular geometries, perhaps eventually to include a large classof stratified spaces. Another direction of research involves theSchroedinger equation on noncompact manifold endowed with "scattering"metrics (e.g. short-range perturbations of Euclidean spaces). Togetherwith Hassell and Tao, the PI is involved in a project to prove sharpStrichartz estimates for the Schroedinger equation in this geometricsetting. Hassell and the PI also hope to provide a precise description ofthe propagator for the Schroedinger equation, extending an earlier resultwhich yielded part of its Schwartz kernel.A central question in the mathematical theory of quantum mechanics is: what is the relationship between the classical dynamics of a particle andits corresponding quantum states? Insights into this problem have comenot only from the direct study of the quantum mechanical energy operatoror "Hamiltonian" itself, but also from studying a host of otherfundamental equations involving it, such as the heat equation, the waveequation, and, naturally, the time-dependent Schroedinger equation. Thefocus of the PI's research is thus the geometric analysis of several kindsof partial differential equations. One project investigates what happensto waves as they interact with (a certain generalization of) sharpcorners---the geometry of how wavefronts move can be quite subtle in thesecases owing to the effects of diffraction. Another project is to studythe structure of solutions of the Schroedinger equation on unboundedspaces; such solutions describe the time-evolution of a quantum particle. A related focus of research is to obtain certain estimates of solutions tothe Schroedinger equation in order to improve our understanding thenonlinear Schroedinger equation, which arises in nonlinear optics and thetheory of Bose-Einstein condensates, among other physical applications.

提案 DMS-0401323 标题：奇异空间上的线性偏微分方程 PI：Jared Wunsch，西北大学 摘要 PI 将研究具有奇异度量结构的流形上的微分算子。 一个项目重点关注奇异空间上的波动方程；PI 与 Melrose 一起表明，一般来说，波动方程解的奇异性与锥点相互作用，产生从锥点发出的“衍射”球面波前。 如果入射锋没有太精确地聚焦在锥点上，则衍射锋的奇异性可能弱于入射锋。 梅尔罗斯和 PI 希望将这些结果推广到更复杂的奇异几何形状，也许最终包括一大类分层空间。 另一个研究方向涉及具有“散射”度量（例如欧几里得空间的短程扰动）的非紧流形上的薛定谔方程。 PI 与 Hassell 和 Tao 一起参与了一个项目，以证明在此几何设置中对薛定谔方程的 SharpStrichartz 估计。 哈塞尔和 PI 还希望为薛定谔方程的传播子提供精确的描述，扩展早期的结果，产生部分施瓦茨核。量子力学数学理论的一个中心问题是：粒子的经典动力学与其相应的量子态之间的关系是什么？ 对这个问题的深入了解不仅来自对量子机械能算子或“哈密尔顿”本身的直接研究，还来自对涉及它的许多其他基本方程的研究，例如热方程、波动方程，当然还有与时间相关的薛定谔方程。 因此PI的研究重点是几种偏微分方程的几何分析。 一个项目研究了波与尖角（某种概括）相互作用时会发生什么——由于衍射效应，在这些情况下波前移动的几何形状可能非常微妙。 另一个项目是研究无界空间上薛定谔方程解的结构；这些解描述了量子粒子的时间演化。 研究的一个相关重点是获得薛定谔方程解的某些估计，以增进我们对非线性薛定谔方程的理解，该方程出现在非线性光学和玻色-爱因斯坦凝聚理论以及其他物理应用中。