Partial Differential Equations for Incompressible Fluids and Elastic Solids

该项目的重点是研究通过所谓的偏微分方程数学建模的流体和弹性材料的行为。其目的是通过利用严格和定量的数学结果来加深我们对某些现象的理解。数值模拟也将用于该项目的一部分。将被研究的现象的动机是应用到现实生活中的问题与潜在的社会影响，其特点是存在的奇点和不同的长度和时间尺度之间的耦合。在该项目的第一部分，将通过研究注入和抽吸对流体流动的影响以及浸没在流体中的多个物体的运动及其可能的碰撞的影响，如泥石流和沉积作用，来考虑壁与不可压缩流体之间的交换。在该项目的第二部分，调查的对象将是不可压缩流如何传输和变形定向物体，如在磁性和导电流体，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深埋在地壳中的断层进行建模，并利用全球定位系统和卫星提供的数据进行监测，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提供培训机会，特别是代表性不足的群体成员。该项目的重点是研究各种问题，这些问题由关于不可压缩流体和弹性固体行为的偏微分方程建模，使用分析和几何技术。所调查的问题的动机是基本的物理现象，并带来了具有挑战性的数学问题，如存在碰撞的流体-结构相互作用问题，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。第一部分涉及无粘性和微粘性流体的行为与边界注入和抽吸，它可以用来控制管道和通道中的湍流和稳定的粘性边界层。还将研究允许滑动时刚体在粘性流体中的运动，并将其应用于泥石流和沉积。第二部分涉及的措施，混合和拉伸运输矢量的流动，如磁流体力学，并加强耗散退化运营商，在电和热流变流体和数学生物学的应用。第三部分涉及地震断层和利用GPS和卫星数据进行断层监测。在这些问题上的进展很可能会对其他领域产生影响，如物理学和工程学。多尺度效应和奇点的存在为项目提供了凝聚力。虽然它的重点在于分析结果和技术，几个建议的问题，如最佳混合和建模的故障，有一个自然的计算对应，将解决与合作者一起。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.