Aspects of Fluid Mechanics and Elasticity from the Point of View of Microlocal and Fourier Analysis

从微局部和傅里叶分析的角度看流体力学和弹性

• 批准号: 0708902
• 负责人: Anna Mazzucato
• 金额: $ 12.5万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0708902&HistoricalAwards=false
• 关键词: Aspects; Fluid; Mechanics; Elasticity; Point

Mathematical aspects of fluid mechanics and elasticity will be investigated using Fourier and microlocal analysis techniques. Despite recent developments, fundamental questions remain open in understanding fluid flow and elastic behavior in solids, in particular with respect to turbulence, elastic wave propagation, and singularity formation. A main goal is to obtain qualitative, but physically relevant, information from properties of solutions to the underlying differential equations. The complex phenomena observed in physical systems correspond to ill-posedness of the equations, in the form of instability, irregularity, and non-uniqueness of the corresponding solutions. Microlocal and Fourier analysis have proven effective tools for this investigation, as they encode the smoothness, size, and oscillations in a signal accurately and efficiently. Microlocal analysis provides crucial directional information in the presence of complex geometries, such as corners and cracks. Three main problems will be addressed. The first is dissipation of enstrophy, the mean square of vorticity, for incompressible 2D and quasi-geostrophic flows, and local decay of the energy spectrum for incompressible 3D flows using the Wigner transform. The second isanisotropic static elasticity on curved polyhedral domains with cracks. The third is identification of density and anisotropic elastic constants in the interior of a body from dynamic surface displacement-traction measurements. The proposed research consists of problems where the exchange between mathematics and other sciences has been fruitful. Fluid turbulence is a fundamental occurrence, which still lacks a complete understanding. It affects the way fluids transport and mix other substances with implications in global climate models, fish migration, and industrial design, for example. The mechanism by which vortices form and transfer energy at different length scales is central to turbulence and is one of the problems under study. Modeling of slow crack formation is important for structural stability in engineering. Mathematical analysis proposed in the second problem under study validates the results of computer simulations, which can be used to predict failure in elastic materials under mechanical stress. Identification of elastic response in materials from remote measurements gives rise to non-invasive, diagnostic imaging of the human body, and imaging of the earth's crust in seismology and oil exploration. The investigation proposed in the third problem aims at determining a priori when sufficient information in the data exists for image reconstruction.The overall goal of the proposal is to exploit mathematical results to advance understanding of physical phenomena with impact on real-life applications.

将使用傅里叶和微局域分析技术研究流体力学和弹性的数学方面。尽管最近取得了进展，但在理解固体中的流体流动和弹性行为方面，特别是在湍流、弹性波传播和奇点形成方面，基本问题仍然悬而未决。主要目标是从基础微分方程的解的属性中获取定性但物理相关的信息。在物理系统中观察到的复杂现象与方程的不适定性相对应，表现为相应解的不稳定性、不规则性和非唯一性。 微局域和傅里叶分析已被证明是这项研究的有效工具，因为它们准确有效地对信号的平滑度、大小和振荡进行编码。 微局部分析可在存在复杂几何形状（例如角和裂缝）的情况下提供关键的方向信息。将解决三个主要问题。 第一个是熵耗散，不可压缩 2D 和准地转流的涡度均方，以及使用维格纳变换的不可压缩 3D 流的能谱的局部衰减。 带有裂纹的弯曲多面体域上的第二个各向异性静弹性。 第三是通过动态表面位移-牵引测量来识别物体内部的密度和各向异性弹性常数。拟议的研究包括数学与其他科学之间的交流富有成果的问题。 流体湍流是一种基本现象，目前仍缺乏完整的认识。 它影响流体运输和混合其他物质的方式，对全球气候模型、鱼类迁徙和工业设计等产生影响。涡流在不同长度尺度上形成和传递能量的机制是湍流的核心，也是正在研究的问题之一。 缓慢裂纹形成的建模对于工程中的结构稳定性很重要。 研究中的第二个问题中提出的数学分析验证了计算机模拟的结果，可用于预测弹性材料在机械应力下的失效。 通过远程测量识别材料的弹性响应，可以实现人体的非侵入性诊断成像，以及地震学和石油勘探中的地壳成像。第三个问题中提出的研究旨在确定数据中何时存在足够的信息用于图像重建的先验。该提案的总体目标是利用数学结果来促进对物理现象的理解，从而影响现实生活中的应用。