Mathematics of Collective Behavior: From Self-Organized Dynamics to Fluid Turbulence

集体行为理论是一个快速发展的数学领域，研究许多生物，社会和技术系统中出现的全球现象。这种系统的例子包括鸟群、鱼群、社交网络和人们之间的政治观点交换。虽然这些系统产生于看似不相关的背景，但它们有许多共同的自组织原则。该项目提出了一个研究这些原则的综合研究方案。更具体地说，它旨在了解集体系统中的“代理人”如何由其通信的自然法则驱动，仅使用本地交互就可以建立全球编队，例如羊群。该项目将系统地分类全球形态的类型，它们的结构和在外力存在下的稳定性，例如鸟群背景下的阵风或舆论动态背景下的媒体影响。在这个过程中，该项目将带来真实的应用和一个新的纯理论数学领域之间的联系，该领域研究扩散系统中的信息传播。该项目将涉及一名研究生，他们将参与理论方面以及拟议研究的数值和视觉实现。 该项目目标的技术实现涉及通过一个新的分数抛物方程系统对紧急动态进行建模。该系统的目的是通过仔细的长期分析其全球解决方案来预测“羊群”的最终状态行为。该模型的新功能包括一个自适应各向异性扩散内核作为全球行为的主要对齐机制。这一特点不仅是数学动机，但也观察到许多生物和社会的会众。在分析中使用的工具将被扩展到研究更一般的一类抛物和椭圆方程的正则性问题。该项目将解决经典定律（如能量守恒定律）的有效性问题，并将与著名的1949年Onsager猜想（研究此类定律成立的严格规律性条件）联系起来。该奖项反映了NSF的法定使命，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。

The theory of collective behavior is a rapidly developing area of mathematics that studies emergence of global phenomena in many biological, social and technological systems. Examples of such systems include flocks of birds, schools of fish, social networking, and exchange of political opinions among people. Although these systems arise from seemingly disconnected contexts, they share many common principles of self-organization. The project puts forward a comprehensive program of research to study those principles. More specifically, it aims to understand how "agents" in collective systems driven by their natural laws of communication can build global formations, such flocks, using only local interactions. This project will systematically classify types of global formations, their structure and stability under presence of an external force, such as gust of wind in the context of bird flocks or media influence in the context of opinion dynamics. On the way, the project will bring a connection between real applications and a new purely theoretical area of mathematics that studies spread of information in diffusive systems. The project will involve one graduate student, who will be involved in theoretical aspects as well as numerical and visual implementations of the proposed research. The technical implementation of the goals of the project involves modeling of emergent dynamics through a novel system of fractional parabolic equations. The system is designed to predict the end-state behavior of a "flock" through a careful long-time analysis of its global solutions. The new feature of proposed model involves inclusion of an adaptive anisotropic diffusion kernel as the main alignment mechanism of global behavior. This feature is not only mathematically motivated but is also observed in many biological and social congregations. The tools used in the analysis will be extended to study regularity problems in a more general class of parabolic and elliptic equations. The project will address validity of the classical laws such as energy conservation and will bring connection with the celebrated Onsager conjecture of 1949 that studies sharp regularity conditions under which such laws hold.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.