Microlocal Analysis of Linear and Nonlinear Problems

线性和非线性问题的微局部分析

• 批准号: 1664683
• 负责人: Andras Vasy
• 金额: $ 20.4万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1664683&HistoricalAwards=false
• 关键词: Microlocal; Analysis; Linear; Nonlinear; Problems; Microlocal; Analysis; Linear; Nonlinear; Problems

This project develops and applies tools from the field of mathematics known as microlocal analysis. Roughly speaking, microlocal analysis devises methods to keep track of the position and frequency (or momentum) of waves (or, more generally, functions) simultaneously. The project's applications are to wave propagation and other related phenomena, as well as to inverse problems for determining a function from integrals along curves (the X-ray transform) and related problems for determining the structure of a material from boundary measurements. Although the project itself concerns their mathematical theory, the problems under investigation are closely connected to the physical world. Wave propagation is ubiquitous in nature, with electromagnetic waves, such as light, being one of the most prevalent examples. The theory of general relativity is another important physical example via (the recently detected) gravitational waves. Scattering theory for quantum particles (such as protons and electrons) is another subject governed by microlocal analysis, aspects of which enter into the description of both quantum waves at large distances and semiclassical phenomena (those in which Planck's constant can be regarded as small, often the case in chemistry). The inverse problems under study are also of broad significance. One application of the theory under development here is the determination of an unknown variable sound speed in an object via the measurement of travel times of waves, which for instance is relevant to imaging to interior of Earth using the travel times of earthquake waves.Parts of this project describe the long-time or far-field behavior of waves on curved space-times. Physically these arise in scattering theory and general relativity, including electromagnetic waves on a curved background. The microlocal approach to analysis on these spaces has made breakthroughs possible in work on linear and nonlinear problems on asymptotically (real) hyperbolic spaces as well as on Kerr-de Sitter space. The projects here aim to improve the understanding of Lorentzian scattering spaces, which include asymptotically Minkowski spaces (asymptotically flat spaces, which our universe approximates, at least locally, even if there is a small positive cosmological constant). Other projects concern the behavior of waves at edges -- specifically, the diffraction of the Rayleigh (surface) waves of elasticity. Yet another main area is inverse problems, building on recently-developed tools for spatially localized inversion of the geodesic X-ray transform, including the solution of fixed conformal class boundary rigidity: under suitable assumptions, one can determine a variable sound speed inside an object from the travel times of waves between points on the surface. The parts of the project in this area aim to extend the foregoing result to tensors, which describe anisotropic sound speeds in an appropriate sense; namely, the project will investigate boundary rigidity, for example, recovering a Riemannian metric from its boundary distance function.

该项目开发并应用了数学领域的工具，称为微局部分析。大致来说，微局部分析设计了方法，以跟踪波（或更一般而言，函数）的位置和频率（或动量）。该项目的应用是浪潮传播和其他相关现象，以及用于确定沿曲线积分（X射线变换）的功能的逆问题，以及确定从边界测量值的材料结构的相关问题。尽管该项目本身涉及他们的数学理论，但所研究的问题与物理世界密切相关。波传播本质上是普遍存在的，电磁波（例如光）是最普遍的例子之一。通过（最近检测到的）引力波是另一个重要的物理示例。量子颗粒（例如质子和电子）的散射理论是另一个受微世分析控制的受试者，其各个方面都涉及大距离和半经典现象的量子波的描述（planck常数可以被视为小的，通常是化学中的情况）。研究的反问题也具有广泛的意义。这里开发的理论的一种应用是通过测量波的行进时间来确定对象中未知的变量声速，例如，该波浪使用地震波的旅行时间与地球内部成像相关。从物理上讲，这些是在散射理论和一般相对论中出现的，包括弯曲背景上的电磁波。在这些空间上进行分析的微局部方法使得在渐近（真实）双曲线空间以及Kerr-DE安慰剂空间上的线性和非线性问题方面的突破成为可能。这里的项目旨在提高对洛伦兹散射空间的理解，其中包括渐近的minkowski空间（渐近平坦的空间，我们的宇宙在本地近似，即使存在一个小的积极宇宙学常数）。其他项目涉及边缘波的行为 - 特别是雷利（表面）弹性的衍射。另一个主要领域是逆问题，建立在最近开发的工具上，用于在空间定位的地球X射线变换的空间定位倒置，包括解决固定保形的类边界刚度的解决方案：在合适的假设下，可以确定对象内部从表面上的波浪中的波动时间来确定对象内部的可变声速。该区域的项目部分旨在将上述结果扩展到张量，以适当的意义描述各向异性音速；也就是说，该项目将研究边界刚度，例如，从其边界距离函数中恢复了Riemannian度量。