Partial Differential Equations for Incompressible Fluids and Elastic Solids

这个项目的重点是研究流体和弹性材料的行为，用所谓的偏微分方程进行数学建模。其目的是通过使用严格和定量的数学结果来加深我们对某些现象的理解。数值模拟也将在部分项目中使用。将被研究的现象是由具有潜在社会影响的现实生活问题的应用所驱动的，其特点是存在奇点以及不同长度和时间尺度之间的耦合。在项目的第一部分中，将考虑壁面和不可压缩流体之间的交换，研究注射和吸力对流体流动的影响，以及浸入流体中的多个物体的运动及其可能的碰撞的影响，如泥石流和沉积。在项目的第二部分，研究对象将是不可压缩流如何传输和变形定向物体，如磁性和导电流体，以及混合和扩散之间的相互作用，如在生物过程中。该项目的最后一部分涉及对地壳深处的断层进行建模，并利用全球定位系统（GPS）和卫星的数据进行监测，最终目标是预测地震事件的发生。该项目还为研究生和本科生，特别是代表性不足群体的成员提供培训机会。这个项目的重点是研究用偏微分方程建模的各种问题，这些问题涉及不可压缩流体和弹性固体的行为，使用解析和几何技术。所研究的问题是由基本的物理现象引起的，并带来具有挑战性的数学问题，例如存在碰撞的流固相互作用问题，以及弹性位错的非局部和非线性界面条件。该项目分为三个主要部分。第一部分研究无粘性和微粘性流体在边界注入和吸力作用下的行为，可用于控制管道和通道内的湍流流动和稳定粘性边界层。在允许滑移的情况下，刚体在粘性流体中的运动也将被研究，并应用于泥石流和沉积。第二部分涉及流体中矢量传输的混合和拉伸测量，例如磁流体力学，以及简并算子的增强耗散，应用于电、热流变流体和数学生物学。第三部分是地震断层及利用GPS和卫星数据进行断层监测。这些问题的进展可能会对其他领域产生影响，例如地球物理学和工程学。多尺度效应和奇点的存在给项目带来了凝聚力。虽然它的重点在于分析结果和技术，但一些提出的问题，如最佳混合和断层建模，有一个自然的计算对应物，将与合作者一起解决。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 focus of this project is to study the behavior of fluids and elastic materials that are modeled mathematically by so-called partial differential equations. The aim is to further our understanding of certain phenomena by utilizing rigorous and quantitative mathematical results. Numerical simulations will also be used in parts of the project. The phenomena that will be studied are motivated by applications to real-life problems with potential societal impacts and are characterized by the presence of singularities and the coupling between different length and time scales. In the first part of the project, the exchange between walls and incompressible fluids will be considered by studying the effect of injection and suction on fluid flow and the effect of the motion of multiple bodies immersed in the fluid and their possible collisions, as in debris flow and sedimentation. In the second part of the project, the object of investigation will be how an incompressible flow transports and deforms directional objects, as in magnetic and conducting fluids, and the interplay between mixing and diffusion, as in biological processes. The last part of the project concerns modeling of faults buried deep in the Earth's crust and their monitoring using data from Global Positioning Systems (GPS) and satellites, with the ultimate goal of predicting the onset of seismic events. The project provides also training opportunities for graduate and undergraduate students, particularly members of under-represented groups.The focus of this project is the study of various problems modeled by partial differential equations concerning the behavior of incompressible fluids and elastic solids, using analytic and geometric techniques. The problems under investigation are motivated by fundamental physical phenomena and bring about challenging mathematical questions, such as fluid-structure interaction problems in the presence of collisions, and non-local and non-linear interface conditions for elastic dislocations. The project is divided into three main parts. The first part concerns the behavior of inviscid and slightly viscous fluids with boundary injection and suction, which can be used to control turbulent flows in pipes and channels and stabilize the viscous boundary layer. The motion of rigid bodies in a viscous fluid when slippage is allowed will also be studied, with applications to debris flow and sedimentation. The second part concerns measures of mixing and stretching for transport of vectors by a flow, such as in magneto-hydrodynamics, and enhanced dissipation for degenerate operators, with applications in electro- and thermo-rheological fluids and mathematical biology. The third part concerns seismic faults and fault monitoring using GPS and satellite data. Progress on these problems is likely to have impact in other fields, such as geophysics and engineering. The presence of multi-scale effects and singularities gives cohesiveness to the project. While its focus lies on analytic results and techniques, several of the proposed problems, such as optimal mixing and modeling of faults, have a natural computational counterpart that will be addressed together with collaborators.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.