Complex and Singular Behavior in Continuum Mechanics Models

这个项目的目标是研究流体和弹性材料的力学模型，这些模型的特征是在物理系统的响应中具有复杂和潜在的单一行为。这样的模型深深植根于应用程序中。该项目将特别研究地壳断层和岩石滑动的模型，空气动力学和流体动力学中围绕运动物体的微粘性流体的行为模型，以及飞机机翼周围或潜艇螺旋桨周围的模型，以及流体有效混合的模型，其应用范围从工业到环境过程。该项目的一个关键方面是使用严格的数学分析，并在可行的情况下结合模拟，以获得可用于预测方式的定量结果，并具有潜在的直接社会影响。例如，该项目的一部分内容是研究如何利用全球定位系统（GPS）的数据，通过一种数学算法来定位原本无法到达的埋藏断层，并估计断层沿线岩层的相对滑动，这是地震的一个预测指标。该项目为研究生和本科生，特别是妇女和代表性不足群体的成员提供培训机会。本项目旨在研究存在奇点的弹性和流体力学模型的各个方面。主要采用分析技术，但提出的问题的推动力来自应用，例如地球物理学中的位错模型，对流主导问题中混合的最佳边界，以及不可压缩流体的边界层分析。该项目由三个独立但又相互联系的部分组成：1 .不可压缩流体力学：1 .最佳混合扩散；奇异域边界层分析；2。弹性：地球物理学中位错模型的正反问题。该项目的统一方面是由于基础模型方程和不规则几何参数的不连续和不相容而导致的奇点的存在，以及对严格的定量估计的关注，例如混合的最佳边界和反问题的定量稳定性估计。该项目通过以一种新颖的方式结合已知技术，促进数学方法的发展，以挑战开放问题，例如在混合问题中，使用几何分析，偏微分方程和最优控制，并促进我们对重要物理过程的基本理解，例如湍流混合中的异常扩散和沿断层和微地震活动的地震间积聚建模，这可能会影响其他领域。特别是地球物理学和工程学。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The goal of this project is to study models in the mechanics of fluids and elastic materials characterized by a complex and potentially singular behavior in the response of the physical system. Such models are deeply rooted in applications. The project will investigate, in particular, models of faults and rock slippage in the Earth's crust, models for the behavior of slightly viscous fluids around moving bodies in aero- and hydro-dynamics, as well as around the wing of an airplane or around the propeller of a submarine, and models for effective mixing in fluids, which has applications ranging from industrial to environmental processes. A key aspect of the project is the use of rigorous mathematical analysis, combined with simulations whenever feasible, to obtain quantitative results that can be used in a predictive fashion, with potential direct societal impacts. For instance, one part of the project addresses how data from global positioning systems (GPS) can be used, through a mathematical algorithm, to locate buried faults which would be otherwise inaccessible, and to estimate the relative slip of the rock layers along the fault, a predictor of earthquakes. The project provides training opportunities for graduate and undergraduate students, particularly women and members of underrepresented groups. This project aims to study various aspects of models in elasticity and fluid mechanics in the presence of singularities. Analytical techniques will be employed primarily, but the impetus for the proposed problems comes from applications, such as models of dislocations in geophysics, optimal bounds for mixing in convection-dominated problems, and boundary layer analysis for incompressible fluids. The project consists of three separate, but connected, parts: I. Incompressible Fluid Mechanics: I.a. Optimal mixing with diffusion; I.b. Boundary layer analysis in singular domains; II. Elasticity: direct and inverse problems for models of dislocations in geophysics. The unifying aspects of the project are the presence of singularities due to discontinuities and incompatibilities in the parameters for the underlying model equations and to irregular geometries, and the focus on rigorous, quantitative estimates, such as optimal bounds in mixing and quantitative stability estimates in inverse problems. The project contributes to the development of mathematical approaches to challenging open problems by combining known techniques in a novel way, such as in mixing problems, where geometric analysis, partial differential equations, and optimal control are employed, and in advancing our basic understanding of important physical processes, such as anomalous diffusion in turbulent mixing and modeling of interseismic build-up along faults and microseismicity, which may impact other fields, in particular geophysics and engineering.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.