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

力学中偏微分方程及相关反问题的应用分析

• 批准号: 1009713
• 负责人: Anna Mazzucato
• 金额: $ 19.11万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1009713&HistoricalAwards=false
• 关键词: Applied; Analysis; Partial; Differential; Equations

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.

该项目涉及对机理中某些部分微分方程和相关的逆问题的分析，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实施的问题，例如流体，地震成像和统计力学的湍流。 该项目的目的是对正在研究的物理系统的行为进行定性预测，同时开发具体但准确的近似模型。该项目由三个主要部分组成：（a）分析不可压缩的流体流量：（A1）在具有对称性和相关边界层的流中消失的粘度极限； （A2）在二维无粘性流体中转运，与弱体溶液的耗散性和独特性有关。 （b）弹性固体的分析：（b1）弹性固醇的混合边界值/界面问题，以及在多面体域中更广泛的椭圆算子，重点是广义有限元方法； （B2）使用波数据包分析以及对地震成像的应用反射和传播弹性波。 c）抛物线方程的Green功能的计算：（C1）封闭形式的近似绿色的绿色功能的fokker-Planck方程及其在模型校准中的性能； （C2）扩展到半线性方程。所研究的主题涉及尚未完全理解的现象，即直接计算机模拟具有挑战性。在存在复杂性的情况下，特别需要进行精致的数学分析，例如非线性方程，奇异的几何形状，不良性和不稳定性（如反向问题）。首席研究者采用了谐波和微环境分析的技术，再加上不同的几何思想，以应对这些挑战，并将项目的各个部分统一为凝聚力的研究计划。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。这些领域的进步对科学和工程学的各个学科有潜在的影响。在该项目上方的A部分A部分中，湍流在墙壁附近放大，从气候和污染模型到鱼类迁移模型，在许多地区的液体中加强了混合和运输。弹性成像在该项目的上方B）中已在地震学中用于研究地球内部，并在地震预测中应用，并在非侵入性医学成像中，特别是弹性学。界面问题，也是该项目的B部分中的，在复合材料中建模物理现象，例如纤维增强的聚合物和玻璃纤维，并广泛应用于行业，从航空航天到健康。最后，在该项目上方的c）中，福克 - 普兰克方程出现在许多粒子系统的统计力学上，更普遍地涉及概率，应用于半导体，等离子物理学和定价或有索赔的定价。首席研究人员进行的研究结果是通过参加专业会议和与其他学者以及在美国和国外的从业人员的专业会议和合作而传播的，从而进一步增强了更大的影响。两名现任研究生，其中一个是女性，正在研究该项目中解决的问题。此外，首席研究员还监督了两名本科生，其中一名是与该项目有关的研究经验。