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

本项目涉及力学中出现的某些偏微分方程和相关逆问题的分析，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在重要领域的实际应用的问题，如流体湍流、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。该项目包括三个主要部分：(a)不可压缩流体流动的分析；（a1）对称流动和相关边界层中消失的粘度极限；（a2）二维无粘流体中的输运，与弱解的熵耗散和唯一性有关。(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，是多面体域内椭圆算子的混合边值/界面问题，重点是广义有限元素法；（b2）利用波包分析弹性波的反射和透射，以及在地震成像中的应用。c)抛物型方程的格林函数的计算：（c1）退化的Fokker-Planck方程的近似格林函数的封闭形式及其在模型标定中的性能；（c2）半线性方程的扩展。正在研究的主题涉及尚未完全理解的现象，固有的多尺度，其中直接计算机模拟具有挑战性。在复杂的情况下，如非线性方程、奇异几何、逆问题中的病态和不稳定性等，特别需要精细的数学分析。首席研究员采用谐波和微局部分析技术，结合微分几何思想，来解决这些挑战，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都有潜在的影响。项目上文a)部分所述的湍流在墙壁附近被放大，加强了流体的混合和输送，在气候和污染模型以及鱼类洄游模型等许多领域都有应用。弹性成像（本项目以上b）部分)已用于地震学研究地球内部，并应用于地震预测，以及非侵入性医学成像，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，从航空航天到卫生。最后，在项目的c)部分中，福克-普朗克方程出现在多粒子系统的统计力学中，更普遍地出现在概率论中，应用于半导体、等离子体物理和或有索赔的定价。首席研究员的研究成果通过参加专业会议和与美国和国外的其他学者和从业者的合作来传播，进一步增强了更广泛的影响。两名在读研究生，其中一名是女性，正在研究这个项目中提到的问题。此外，首席研究员还指导了两名本科生（其中一名女性）进行与该项目相关的研究经验。

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.