Hydrodynamics of Collective Phenomena and Applications

在自然界中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，一大群鸟在不停地摆动，形状不断变化，或者一群鱼以碾磨的方式旋转。其他例子包括人群动态和社交网络，例如在社交媒体上形成朋友集群，或者在一大群人中动态地发表意见。在技术方面也出现了一些例子，例如无人驾驶飞行器护航或卫星导航的协同作战。所有这些大系统都由类似于我们用来研究液体运动的模型来管理，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，称为集体行为的流体力学。本项目将从流体动力学集体模型的数学性质以及它们在自组织动力学和涌现现象中的应用的角度进行分析。该项目的中心将是对所谓的欧拉排列系统(简称EAS)的分析。重点将放在通过噪声动力学福克-普朗克模型的水动力极限来证明一类等熵EASS的合理性。在Pi早期著作中介绍的具有单一通信的系统的框架内，该项目提出了一项关于理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，使其有可能产生更多样化的集体结果，并从理论上赋予了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羊群的背景下发展。这项研究将在几个数学领域得到应用，包括所谓的意见平均场博弈及其纳什均衡的寻找，双流体多孔介质问题中界面轮廓的对准，以及二维无粘流体流动中湍流能量级联的建模。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.