Hydrodynamics of Collective Phenomena and Applications

集体现象的流体动力学及其应用

基本信息

批准号：
2107956
负责人：
Roman Shvydkoy
金额：
$ 33万
依托单位：
University of Illinois at Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2107956&HistoricalAwards=false
关键词：
Hydrodynamics Collective Phenomena Applications

项目摘要

Among the various collective phenomena observed in nature there are many that resemble the motion of a fluid. Such examples are abundant in biology, for example, a large flock of birds swinging constantly changing shape or a school of fish swirling in a milling pattern. Other examples include crowd dynamics and social networking, for example the formation of friend-clusters on social media, or dynamics of opinions among a large group of individuals. Examples also arise in technology, such as the coordinated fight of an escort of unmanned aerial vehicles or satellite navigation. All of these large systems are governed by models similar to those we use to study motion of a liquid, like water or gas. A set of new ideas on how to study this analogy recently developed into a new mathematical subject, called Hydrodynamics of Collective Behavior. This project will analyze hydrodynamic collective models both from the point of view of their mathematical properties and with a view towards their applications to self-organized dynamics and emergent phenomena.Central to the project will be analysis of the so-called Euler Alignment Systems (EAS for short). Particular focus will be placed on justification of a class of isentropic EASs via the hydrodynamic limit from a noisy kinetic Fokker-Plank model. In the framework of systems with singular communication introduced in PI’s earlier works, the project sets forth a program of research on understanding topological interactions prevalent in many biological systems. Such interactions enrich the models with a possibility of more diverse collective outcomes and, from theoretical perspective, endow them with a fractional parabolic structure. The regularity theory exploiting this parabolic structure will be developed in the context of unidirectional flocks. Applications of this research will be made to several fields of mathematics including the so-called opinion mean-field games and finding their Nash equilibria, alignment of interfacial profile in two-fluid porous media problem, and modeling turbulent energy cascade in 2D inviscid fluid flows.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.

在自然界中观察到的各种集体现象中，有许多类似于流体运动的例子，例如，一大群鸟不断地摇摆着改变形状，或者一群鱼以磨碎的方式旋转。例子包括人群动态和社交网络，例如社交媒体上朋友群的形成，或一大群人之间意见的动态，技术领域也有这样的例子，例如无人机护航或协调作战。所有这些大型系统都受到卫星导航的控制。通过类似于我们用来研究液体（如水或气体）运动的模型，一系列关于如何研究这种类比的新想法最近发展成为一个新的数学学科，称为集体行为的流体动力学，该项目将分析集体流体动力学。该项目的核心是对所谓的欧拉对准系统（简称 EAS）的分析。将被置于正当理由在 PI 早期作品中引入的具有奇异通信的系统框架中，该项目提出了一个了解许多生物系统中普遍存在的拓扑相互作用的研究计划。这种相互作用丰富了模型，使其具有更多样化的集体结果的可能性，并且从理论角度来看，赋予它们一种分数抛物线结构，利用这种抛物线结构的正则性理论将在单向的背景下发展。这项研究将应用于多个数学领域，包括所谓的意见平均场博弈和寻找纳什平衡、二流体多孔介质问题中的界面轮廓对齐以及二维无粘性中的湍流能量级联建模。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。