Hydrodynamics of Collective Phenomena and Applications

在自然界观察到的各种集体现象中，有许多类似于流体的运动。这样的例子在生物学中比比皆是，例如，一群不断改变形状的鸟在摇摆，一群鱼在旋转。其他例子还包括群体动力学和社交网络，例如社交媒体上朋友群的形成，或者一大群人之间的观点动态。这样的例子也出现在技术领域，例如无人驾驶飞行器护航或卫星导航的协同作战。所有这些大型系统都是由类似于我们用来研究液体运动的模型控制的，比如水或气体。关于如何研究这种类比的一系列新想法最近发展成为一门新的数学学科，称为集体行为流体动力学。本项目将从其数学性质的角度分析流体动力学集体模型，并将其应用于自组织动力学和紧急现象。该项目的核心将是分析所谓的欧拉校准系统（简称EAS）。特别的重点将放在一类等熵EASs的理由，通过流体动力极限从一个嘈杂的动力学Fokker-Plank模型。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，并从理论角度赋予它们分数抛物线结构。利用这种抛物线结构的正则性理论将在单向群的背景下发展。该研究将应用于几个数学领域，包括所谓的意见平均场博弈及其纳什平衡点的寻找，两流体多孔介质问题中界面剖面的对齐，以及二维无粘性流体流动中的湍流能量级联建模。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。

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.