Singular Problems in Continuum Mechanics

研究流体力学和弹性材料力学中具有奇异或近似奇异特性的几个问题。在项目的第一部分中，研究者的重点是理解和模拟复杂流体在不可压缩性要求下的流动行为，即保持流体所占的体积。许多流体几乎是不可压缩的，包括水，在某些条件下甚至包括空气。所研究的复杂行为是由于流体与刚性壁面之间的摩擦以及流体本身的搅拌。这两种现象都是流体力学中的主要现象，但仍然没有得到充分的严格理解。壁面摩擦会产生阻力，并在流动中产生涡流，从而导致湍流的发生。量化流体流动中搅拌和混合的影响对许多物理过程有直接影响，从挤压和成型，到有效燃烧，到污染物扩散。在项目的第二部分，研究者研究了时间周期波在弹性材料中的传播。通过记录材料在外加扰动下的弹性响应，波可以用于远程探测材料。这就是所谓的逆问题，因为材料的性质是从测量中推断出来的，而且对误差非常敏感。采用数学算法提高重建的可靠性。所研究的逆问题类型在地质勘探、地震预测和探地雷达中都有应用。该项目为研究生和本科生提供了培训机会。研究者使用分析和计算技术来研究不可压缩流体力学和可变形固体力学中的问题。本课题主要分为两个部分：1 .不可压缩流体力学：1 .粘性消失极限及边界层分析；最佳混合和不规则运输。2。弹性：弹性介质中时谐波的反边界问题。这些问题的特点是奇异性的存在，以偏微分方程（PDE）的奇异问题的形式出现，如Navier-Stokes方程的零粘度极限，或以偏微分方程中的奇异系数的形式出现，如在非lipschitz流下的混合问题，或偏微分方程的奇异域，如带角的域。该项目解决了流体力学中关于高雷诺数下不可压缩流体行为的一些基本开放性问题。它试图通过对特殊流动的严格分析来阐明边界层分离的机制和普朗特近似的有效性。本文还试图从输运方程和几何分析的角度来量化强无散度约束下流动的混合特性。在第二部分中，该项目解决了逆边界问题中重建算法的稳定性和性能，这对非侵入性成像技术产生了重大影响。奇点的出现以及分析和几何之间的相互作用是该项目反复出现的主题。各种各样的技术，在许多情况下以一种新颖的方式组合在一起，例如在混合问题中，被用来进行这项工作。该项目为研究生和本科生提供了培训机会。

The investigator studies several problems in fluid mechanics and mechanics of elastic materials that are characterized by a singular or nearly singular behavior. In the first part of the project, the investigator focuses on understanding and modeling the behavior of complex fluid flows under the requirement of incompressibility, that is, preserving the volume occupied by the fluid. Many fluids are approximately incompressible, including water and under some conditions even air. The complex behavior that is investigated is due to friction between the fluid and rigid walls, and to stirring of the fluid itself. Both are major phenomena in fluid mechanics and are still not fully understood rigorously. Wall friction creates drag and produces swirls in the flow that contribute to the onset of turbulence. Quantifying the effect of stirring and mixing in fluid flows has direct impact on many physical processes, from extrusion and molding, to efficient combustion, to pollutant dispersal. In the second part of the project, the investigator studies the propagation of time-periodic waves in elastic materials. Waves can be used to remotely probe materials by recording their elastic response under an applied disturbance. This is a so-called inverse problem, as material properties are inferred from measurements, and it is highly sensitive to errors. Mathematical algorithms are used to improve the reliability of the reconstruction. The type of inverse problem that is studied finds applications in geological exploration, earthquake prediction, and ground-penetrating radar. The project provides training opportunities for both graduate and undergraduate students.The investigator uses analytical and computational techniques to study problems in incompressible fluid mechanics and in mechanics of deformable solids. The project has two main parts: I. Incompressible Fluid Mechanics: I.a. Vanishing viscosity limit and boundary layer analysis; I.b. Optimal mixing and irregular transport. II. Elasticity: Inverse boundary problem for time-harmonic waves in elastic media. These problems are characterized by the presence of singularities, in the form of singular problems for partial differential equations (PDE), such as the zero-viscosity limit for the Navier-Stokes equations, or in the form of singular coefficients in the PDE, such as in mixing problems under non-Lipschitz flows, or singular domains for the PDE, such as domains with corners. The project addresses some fundamental open questions in fluid mechanics concerning the behavior of incompressible fluids at high Reynolds numbers. It seeks to shed light on the mechanism for boundary layer separation and the validity of Prandtl approximation through the rigorous analysis of special flows. It also seeks to quantify mixing properties of flows under the strong divergence-free constraint from the point of view of transport equations and geometric analysis. In the second part, the project addresses the stability and performance of reconstruction algorithms in inverse boundary problems, which have had a major impact on non-invasive imaging techniques. The appearance of singularities and the interplay between analysis and geometry are recurring themes of the project. A variety of techniques, in many cases combined in a novel way, such as in mixing problems, are used to carry out the work. The project provides training opportunities for both graduate and undergraduate students.