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里叶和微局部分析技术进行研究。尽管最近取得了一些进展，但在理解固体中的流体流动和弹性行为方面，特别是湍流、弹性波传播和奇点形成方面，基本问题仍然存在。主要目标是从微分方程的解的性质中获得定性的，但物理上相关的信息。在物理系统中观察到的复杂现象对应于方程的病态性，表现为相应解的不稳定性、不规则性和非唯一性。微局部分析和傅立叶分析已经被证明是这项研究的有效工具，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。微局部分析可以在拐角和裂缝等复杂几何形状的情况下提供关键的方向信息。将解决三个主要问题。首先是不可压缩二维流和准地转流的能量耗散（涡度均方），以及不可压缩三维流的能量谱的局部衰减（使用Wigner变换）。带裂纹的弯曲多面体域上的静态弹性。第三是通过动态地表位移-牵引测量确定物体内部的密度和各向异性弹性常数。所提出的研究包括数学和其他科学之间的交流卓有成效的问题。流体乱流是一种基本现象，但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，对全球气候模式、鱼类洄游和工业设计都有影响。涡旋在不同长度尺度上形成和传递能量的机制是湍流研究的核心，也是研究中的问题之一。慢裂纹形成的建模对工程结构的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，该结果可用于预测弹性材料在机械应力下的破坏。从远程测量中识别材料的弹性响应，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先验地确定当数据中存在足够的信息用于图像重建。该提案的总体目标是利用数学结果来推进对物理现象的理解，并对实际应用产生影响。

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.