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.

流体力学和弹性的数学方面将使用傅立叶和微局部分析技术研究。尽管最近发生了发展，但在理解固体中的流体流和弹性行为方面，尤其是关于湍流，弹性波传播和奇异性形成，基本问题仍然开放。一个主要目标是获取从解决方案的属性到基础微分方程的定性但与物理相关的信息。在物理系统中观察到的复杂现象对应于方程不稳定性，以不稳定性，不规则性和相应溶液的不唯一性的形式。 微局部和傅立叶分析已被证明是该研究的有效工具，因为它们可以准确有效地编码信号中的平滑度，大小和振荡。 微局部分析在存在复杂几何形状（例如角落和裂纹）的情况下提供了至关重要的方向信息。将解决三个主要问题。 第一个是腐蚀，涡流的均匀平方，用于不可压缩的2D和准地藻流，以及使用Wigner变换的不可压缩3D流的能量光谱的局部衰变。 带有裂缝的弯曲多面体结构域上的第二种质子静态弹性。 第三是通过动态表面位移量测量值鉴定人体内部中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的问题。 流体湍流是一种基本事件，仍然缺乏完全理解。 例如，它影响流体运输和混合其他物质的方式，例如在全球气候模型，鱼类迁移和工业设计中的影响。在不同长度尺度上形成和转移能量的机制对于湍流至关重要，是研究的问题之一。 慢速裂纹形成的建模对于工程的结构稳定性很重要。 在第二个问题中提出的数学分析验证了计算机模拟的结果，该结果可用于预测机械应力下的弹性材料的故障。 远程测量中材料中弹性反应的识别会导致人体的无创，诊断成像以及对地震学和石油探索中地壳的成像。第三个问题中提出的调查旨在确定数据中有足够的信息进行图像重建时的先验性。该提案的总体目标是利用数学结果，以提高对物理现象的理解，并对现实应用产生影响。