Partial Differential Equations for Incompressible Fluids and Elastic Solids

不可压缩流体和弹性固体的偏微分方程

基本信息

批准号：
2206453
负责人：
Anna Mazzucato
金额：
$ 37.44万
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：
2022
资助国家：
美国
起止时间：
2022-09-01 至 2025-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206453&HistoricalAwards=false
关键词：
Partial Differential Equations Incompressible Fluids

项目摘要

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.

该项目的重点是研究流体和弹性材料的行为，这些材料是通过所谓的部分微分方程对数学建模的。目的是通过利用严格和定量的数学结果来进一步了解某些现象。数值模拟也将用于项目的一部分。将研究的现象是由对潜在社会影响的现实生活问题的应用进行的，其特征是奇异性的存在以及不同的长度和时间尺度之间的耦合。在项目的第一部分中，将通过研究注射和抽吸对流体流动的影响以及浸入流体中的多个物体的影响及其可能的碰撞（如碎片流和沉积物中）来考虑墙壁与不可压缩的流体之间的交换。在项目的第二部分中，调查的对象将是不可压缩的流动如何转运和变形方向物体，例如在磁性和导电流体中，以及混合与扩散之间的相互作用，如生物过程中。该项目的最后一部分涉及对地壳深处埋藏的故障的建模，并使用来自全球定位系统（GPS）和卫星的数据进行监测，其最终目的是预测地震事件的发作。该项目还为研究生和本科生，尤其是代表性不足的群体的成员提供了培训机会。该项目的重点是研究通过分析和几何技术，该项目由部分微分方程建模的各种问题。所研究的问题是由基本的物理现象激发的，并提出了具有挑战性的数学问题，例如在存在碰撞的情况下流体结构相互作用问题，以及用于弹性脱位的非本地和非线性界面条件。该项目分为三个主要部分。第一部分涉及带有边界注入和吸力的无粘性和略微粘性的流体的行为，这些液体可用于控制管道和通道中的湍流，并稳定粘性边界层。当允许滑动时，还将研究粘性流体中刚体的运动，并应用于碎屑流和沉降。第二部分涉及通过流量（例如磁流动力学中的载体传输）进行混合和拉伸的量度，以及对退化运算符的耗散量的增强，以及在电流和热 - 流体流体和数学生物学中的应用。第三部分涉及使用GPS和卫星数据进行地震故障和故障监测。这些问题的进展可能会在其他领域（例如地球物理和工程学）上产生影响。多尺度效果和奇异性的存在使该项目具有凝聚力。尽管它的重点是分析结果和技术，但一些提出的问题（例如故障的最佳混合和建模）具有自然的计算对应方，该奖项将与合作者一起解决。该奖项反映了NSF的法定任务，并被认为是通过该基金会的知识分子功能和广泛的影响来评估CRITERIA的评估，并被认为是值得的。