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

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进行研究。尽管最近的发展，基本问题仍然开放，在理解流体流动和固体中的弹性行为，特别是相对于湍流，弹性波传播，和奇点形成。一个主要的目标是获得定性的，但物理相关的，信息的解决方案的基本微分方程的属性。在物理系统中观察到的复杂现象对应于方程的不适定性，表现为相应解的不稳定性、不规则性和非唯一性。 微局部和傅立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编码了信号中的平滑度，大小和振荡。 微局部分析在存在复杂几何形状（如拐角和裂缝）的情况下提供关键的方向信息。将处理三个主要问题。 首先是耗散的涡度拟能，涡度的均方，不可压缩的2D和准地转流，和不可压缩的3D流的能量谱的局部衰减，使用Wigner变换。 第二类是含裂纹的弯曲多面体域的各向异性静弹性问题。 第三个是识别密度和各向异性弹性常数在内部的一个机构从动态表面位移牵引测量。拟议的研究包括数学和其他科学之间的交流已经富有成效的问题。 流体湍流是一种基本的现象，仍然缺乏完整的理解。 它会影响流体运输和混合其他物质的方式，例如对全球气候模型，鱼类迁徙和工业设计产生影响。涡旋的形成和能量在不同长度尺度上的传递机制是湍流的核心，也是正在研究的问题之一。 慢裂纹形成的数值模拟对工程结构的稳定性研究具有重要意义。 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弹性材料的失效。 从远程测量中识别材料中的弹性响应产生了对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。在第三个问题中提出的调查旨在确定一个先验时，在数据中存在足够的信息，图像reconstruction.The建议的总体目标是利用数学结果，以促进对物理现象的理解与现实生活中的应用的影响。

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.