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

将使用傅立叶和微局部分析技术研究流体力学和弹性的数学方面。尽管最近有了新的发展，但在理解固体中的流体流动和弹性行为，特别是关于湍流、弹性波传播和奇点形成方面，基本问题仍然是开放的。一个主要目标是从基本微分方程解的性质中获得定性的、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的病态，表现为相应解的不稳定性、不规则性和不唯一性。微局部分析和傅立叶分析已被证明是这项研究的有效工具，因为它们准确而有效地编码了信号中的平滑、大小和振荡。在存在复杂几何形状的情况下，微局部分析提供了关键的方向信息，例如拐角和裂缝。将解决三个主要问题。对于不可压缩的2D和准地转流，首先是涡度均方Enstrophy的耗散，对于不可压缩的3D流，使用Wigner变换是能量谱的局部衰减。第二种是含裂纹弯曲多面体区域上的各向异性静弹性。第三种是通过动态表面位移-牵引力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和其他科学之间的交流卓有成效的问题。流体湍流是一种基本现象，至今仍缺乏完整的认识。例如，它影响流体运输和混合其他物质的方式，并对全球气候模型、鱼类迁徙和工业设计产生影响。涡旋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。慢裂纹形成模型在工程上对结构稳定性具有重要意义。在研究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，该模拟可以用于预测弹性材料在机械应力下的破坏。通过远程测量识别材料中的弹性响应，可以对人体进行非侵入性的诊断成像，并在地震学和石油勘探中对地壳进行成像。在第三个问题中提出的调查旨在确定当数据中存在足够的信息用于图像重建的先验。该建议的总体目标是利用数学结果来促进对实际应用中的物理现象的理解。

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.