Harmonic Analysis and Geometric Partial Differential Equations

调和分析与几何偏微分方程

• 批准号: 0202139
• 负责人: Andrea Nahmod
• 金额: $ 10.2万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202139&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Geometric; Partial; Differential

Nahmod's research lies in the overlap of harmonic analysis, geometry and partial differential equations. It aims at studying the behavior of nonlinear waves arising in geometry, ferromagnetism and gauge field theories; and that of functions along vector fields whose integral curves lack sufficient curvature. In the first part the focus is in geometric partial differential equations. Of special interest are Schroedinger maps, Wave maps and other gauge field theories such as the Yang Mills equationsin Minkowski space-time. All of these equations model `wave like phenomena'. Their solutions arise as minimizers of the corresponding energy functionals. To conform with natural physical situations, it is of interest to study their existence, uniqueness under minimal regularity assumptions. These are difficult issues because the nonlinearities of these equations involve not just the solutions but also their derivatives. Nahmod will address these questions and plans to show that in the scale invariant set up solutions to the Cauchy initial value problem exist globally provided that the data is sufficiently small when measured relative to the critical regularity norm. She also plans stability issues; e.g. whether such a system remains close to its initial state as time evolves when the data has small energy. From a physical viewpoint the latter models whether such systems are close to equilibrium. The techniques exploit geometric aspects of these equations to extract crucial information -such as special structures in the nonlinearity- which is then used in the analysis. The method combines deep Fourier analysis with gauge theoretic geometric tools. The goal of the second part is the study of the Hilbert transform along vector fields and its associated maximal operator in two dimensions. Their treatment departs from the classical study of singular integrals for in the present situation, the singularity lives on some variety that is changing at each point. Nahmod will investigate how to develop time frequency techniques to study operators under no curvature assumptions. This is the case, for example, in studying differentiability properties of functions along vector fields. Partial differential equations are the mathematical models to the laws governing much of the phenomena in our physical world. The wave equation models the propagation of different kind of waves -such as light waves- in homogeneous media. Nonlinear models of conservative type arise in quantum mechanics while other variants appear for example in the study of vibrating systems and semiconductors. The nonlinear Schroedinger equation arises in various physical contexts in the description of nonlinear waves- such as propagation of a laser beam in a medium whose index of refraction is sensitive to the wave amplitude, water waves at the free surface of an ideal fluid as well as in plasma waves. Some of the interesting questions are those about local and global existence of solutions, uniqueness as well as long time behavior of global solutions. The role of mathematical analysis is to understand the behavior of the solutions to these equations, provide the tools to extract their quantitative and qualitative information and lay the foundations upon which methods to accurately approximate the solutions are developed. Fourier analysis and more generalized adapted frequency decompositions such as time-frequency analysis' consists in decomposing complex objects via `modulated waveforms' into basic building blocks which are localized and easy to understand, and then piecing them back together in a straightforward manner. It works very similarly to a musical score. The modulated waveforms have four attributes: amplitude (loudness), scale (duration), frequency (pitch) and position (instant it is played). The objects could be speech, radar signals, as well as oscillatory expressions arising in optics, AC ousting scattering, wave propagation and other phenomena of nonlocal nature.

Nahmod的研究在于谐波分析，几何和部分微分方程的重叠。它旨在研究在几何形状，铁磁和轨迹场理论中产生的非线性波的行为。以及沿矢量场的功能，其积分曲线缺乏足够的曲率。在第一部分中，重点是几何部分微分方程。特别感兴趣的是Schroedinger地图，波浪地图和其他规格场理论，例如Minkowski时空中的Yang Mills方程。所有这些方程都模型“像现象一样波浪”。它们的溶液作为相应能量功能的最小化器出现。为了符合自然的身体状况，研究它们的存在和在最少的规律性假设下的独特性是有意义的。这些是困难的问题，因为这些方程式的非线性不仅涉及解决方案，还涉及它们的衍生品。 Nahmod将解决这些问题，并计划表明，在规模不变的设置解决方案中，如果相对于关键规则规范进行测量，则在全球范围内存在针对Cauchy初始价值问题的解决方案。她还计划稳定问题；例如当数据具有较小的能量时，随着时间的发展，这种系统是否保持接近其初始状态。从物理角度来看，后者模型是否接近平衡。这些技术利用了这些方程式的几何方面来提取至关重要的信息，例如非线性中的特殊结构，然后在分析中使用。该方法将深度傅立叶分析与量规理论几何工具结合在一起。第二部分的目的是研究希尔伯特沿矢量场及其相关最大运算符的研究。他们的治疗与当前情况的奇异积分的古典研究不同，奇异性生活在每个点都在变化的各种。 Nahmod将研究如何在没有曲率假设下开发时间频率技术来研究操作员。例如，在研究沿矢量场函数的可不同性能时就是这种情况。部分微分方程是管理我们物理世界中许多现象的法律的数学模型。波方程模拟了均匀介质中不同类型波的传播 - 像光波一样的传播。保守类型的非线性模型在量子力学中出现，而其他变体则例如在振动系统和半导体的研究中出现。在非线性波的描述中，非线性Schroedinger方程在各种物理环境中出现，例如在介质中激光束传播，其折射索引对波振幅敏感，在理想流体的自由表面以及等离子波的自由表面上的水波。一些有趣的问题是关于解决方案的本地和全球存在，独特性以及全球解决方案的长时间行为的问题。数学分析的作用是了解解决这些方程的解决方案的行为，提供了提取其定量和定性信息的工具，并为确切近似于解决方案的方法奠定了基础。傅立叶分析和更广泛的适应频率分解，例如时频分析”，包括通过“调制波形”将复杂的对象分解为基本的构建块，这些构件是本地化且易于理解的，然后以简单的方式将它们拼凑在一起。它的工作原理与乐谱非常相似。调制波形具有四个属性：振幅（响度），比例（持续时间），频率（音高）和位置（即时播放）。这些物体可能是语音，雷达信号以及光学，驱散散射，波传播和其他非本地性质现象的振荡表达。