CAREER: Nonlocal partial differential equations in collective dynamics and fluid flow

职业：集体动力学和流体流动中的非局部偏微分方程

• 批准号: 2238219
• 负责人: Changhui Tan
• 金额: $ 40.42万
• 依托单位: University of South Carolina at Columbia
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2028-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2238219&HistoricalAwards=false
• 关键词: CAREER; Nonlocal; partial; differential; equations

The collective behaviors of large groups of similar animals, e.g., birds, insects, or fishes, are ubiquitous in nature. In recent times, the mathematical study of collective dynamics has become an active and fast-growing field of research. Many mathematical models of collective behavior rely on partial differential equations with nonlocal interactions to describe the resulting emergent behavior. It turns out that these models are intimately connected to other models traditionally used in fluid dynamics. The goal of this project is to study several models of nonlocal partial differential equations to model collective behavior or fluid flows, and to develop novel and robust analytical techniques to understand the collective behaviors driven by nonlocal structures. The training and professional development of graduate students and young researchers is an integral part of the project. This project studies three families of partial differential equations with shared nonlocal structures that can affect the solutions of the equations: existence, uniqueness, regularity, and long-time asymptotic behaviors. The first problem is on the compressible Euler system with nonlinear velocity alignment, which describes the remarkable flocking phenomenon in animal swarms. Global phenomena and asymptotic behaviors of the system will be investigated, with a focus on the nonlinearity in the velocity alignment. The second problem is on the pressureless Euler system, aiming at the long-standing question of the uniqueness of weak solutions. The plan is to approximate the system by the relatively well-studied Euler-alignment system in collective dynamics. The third problem is on the Euler-Monge-Ampère system which is closely related to the incompressible Euler equations in fluid dynamics. The embedded nonlocal geometric structure of the system will be explored, with interesting applications in optimal transport and mean-field games.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.

大量类似动物（例如鸟类，昆虫或鱼类）的集体行为本质上是无处不在的。最近，集体动力学的数学研究已成为一个积极且快速增长的研究领域。许多集体行为的数学模型都依赖于具有非本地相互作用的部分微分方程来描述由此产生的紧急行为。事实证明，这些模型与该项目目标传统上使用的其他模型密切相关，这是研究多种非本地部分偏微分方程的模型，以模拟集体行为或流体流，并开发出新颖和强大的分析技术，以了解由非局部结构驱动的集体行为。研究生和年轻研究人员的培训和专业发展是该项目不可或缺的一部分。该项目研究了三个具有共享非本地结构的部分微分方程家族，它们可以影响方程解决方案：存在，独特性，规律性和长期不对称行为。第一个问题是在具有非线性速度比对的竞争性Euler系统上，它描述了动物群中的显着植入现象。将研究系统的全球现象和不对称行为，重点是速度比对的非线性。第二个问题是无压力的欧拉系统，旨在探讨弱解决方案独特性的长期问题。该计划是通过集体动力学中的相对研究的Euler-Alignment系统近似系统。第三个问题是在Euler-Monge-ampère系统上，该系统与流体动力学中不可压缩的Euler方程密切相关。将探索系统的嵌入式非本地几何结构，并在最佳运输和平均场游戏中使用有趣的应用。该奖项反映了NSF的法定使命，并通过使用基金会的知识分子优点和更广泛的审查标准评估来诚实地获得支持。