Applied Analysis of Partial Differential Equations and Related Inverse Problems in Mechanics

该项目涉及分析某些偏微分方程和相关的力学中出现的反问题，特别是可变形固体和不可压缩流体的连续力学。主要研究者的重点是数学可以影响理论理解和重要领域的实际实施的问题，如流体中的湍流，地震成像和统计力学。 该项目的目标是对所研究的物理系统的行为进行定性预测，同时开发具体而准确的近似模型。该项目由三个主要部分组成：（a）分析不可压缩流体流动：（a1）对称流动和相关边界层中的粘性极限消失;（a2）二维无粘性流体中的输运，与拟能耗散和弱解的唯一性有关。(b)弹性固体的分析：（b1）弹性静力学的混合边值/界面问题，更广泛地说，在多面体域中的椭圆算子，重点是广义有限元法;（b2）使用波包分析的弹性波反射和透射，以及在地震成像中的应用。（c）抛物型方程的绿色函数的计算：（c1）退化Fokker-Planck方程的封闭形式近似绿色函数及其在模型校正中的性能;（c2）半线性方程的扩展。正在调查的主题涉及尚未完全理解的现象，固有的多尺度，直接计算机模拟是具有挑战性的。在存在复杂性的情况下，特别需要精细的数学分析，例如非线性方程、奇异几何、不适定性和不稳定性，如逆问题的情况。主要研究者采用调和和微局部分析技术，结合微分几何思想，以解决这些挑战，并将项目的各个部分统一为一个有凝聚力的研究计划。该项目解决了弹性固体和不可压缩流体的数学分析中的几个开放问题。这些领域的进展对科学和工程的各个学科具有潜在影响。在项目的上述a）部分中，湍流在壁附近被放大，增强了流体的混合和传输，应用于从气候和污染模型到鱼类迁移模型的许多领域。在该项目的上述B）部分中，弹性成像已被用于地震学，以研究地球内部，并应用于地震预测，以及用于非侵入性医学成像，特别是弹性成像。界面问题，也是该项目的B）部分，模拟复合材料中的物理现象，如纤维增强聚合物和玻璃纤维，广泛应用于工业，从航空航天到健康。最后，福克-普朗克方程，在项目的上面部分c）中，出现在多粒子系统的统计力学中，更普遍地说，在概率中，应用于半导体，等离子体物理和或有债权的定价。主要研究者进行的研究结果通过参加专业会议和与其他学者以及美国和国外的从业人员合作进行传播，进一步增强了更广泛的影响。目前有两名研究生，其中一名是女性，正在研究该项目所涉及的问题。此外，首席研究员还指导了两名本科生，其中一名是女生，学习与该项目有关的研究经验。

This project concerns the analysis of certain partial differential equations and associated inverse problems arising in mechanics, in particular continuous mechanics of deformable solids and incompressible fluids. The focus of the principal investigator is on problems where mathematics can impact both a theoretical understanding and the practical implementation in important fields such as turbulence in fluids, seismic imaging, and statistical mechanics. The goal of the project is to make qualitative predictions on the behavior of the physical systems under study, and at the same time to develop concrete, yet accurate, approximate models. The project consists of three main parts: (a) Analysis of incompressible fluid flows: (a1) vanishing viscosity limits in flows with symmetry and the associated boundary layer; (a2) transport in two-dimensional inviscid fluids, in relation to enstrophy dissipation and uniqueness of weak solutions. (b) Analysis of elastic solids: (b1) mixed boundary value/interface problems for elastostatics, and more broadly for elliptic operators, in polyhedral domains, with emphasis on the generalized finite element method; (b2) reflection and transmission of elastic waves using wave packet analysis, and applications to seismic imaging. c) Computation of Green's functions for parabolic equations: (c1) closed-form approximate Green's function of degenerate Fokker-Planck equations, and their performance in model calibration; (c2) extension to semi-linear equations. The topics under investigation relate to phenomena not yet fully understood, inherently multiscale, where direct computer simulation is challenging. A refined mathematical analysis is particularly needed in the presence of complexities, in the form for example of nonlinear equations, singular geometries, illposedness and instability as in the case of inverse problems. The principal investigator employs techniques from harmonic and microlocal analysis, combined with differential geometric ideas, to address these challenges and unify the parts of the project into a cohesive research program.This project addresses several open issues in the mathematical analysis of elastic solids and incompressible fluids. Progress in these areas has potential impact on various disciplines in science and engineering. Turbulence, in part a) above of the project, is amplified near walls, enhancing mixing and transport in fluids with applications in many areas from climate and pollution models to models of fish migration. Elastic imaging, in part b) above of the project, has been used in seismology to study the earth's interior, with applications to earthquake prediction, and in non- invasive medical imaging, in particular elastography. Interface problems, also in part b) of the project, model physical phenomena in composite materials, such as fiber-reinforced polymers and fiberglass, with widespread applications to industry, from aerospace to health. Finally, Fokker-Planck equations, in part c) above of the project, arise in statistical mechanics of many-particle systems, and more generally in probability, with applications to semiconductors, plasma physics, and pricing of contingent claims. Results from the research carried out by the principal investigator are disseminated through participation at professional meetings and collaboration with other scholars, as well as practitioners, both in the US and abroad, further enhancing broader impact. Two current graduate students, one of which is female, are working on problems addressed in the project. In addition, the principal investigator has supervised two undergraduate students, one of which female, in research experiences related to the project.