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

鸟类、昆虫或鱼类等一大群相似动物的集体行为在自然界中无处不在。近年来，集体动力学的数学研究已成为一个活跃且快速发展的研究领域。许多集体行为的数学模型依赖于具有非局部相互作用的偏微分方程式来描述由此产生的紧急行为。事实证明，这些模型与传统上用于流体力学的其他模型密切相关。这个项目的目标是研究几种非局部偏微分方程模型来模拟集体行为或流体流动，并开发新的和健壮的分析技术来理解由非局部结构驱动的集体行为。研究生和青年研究人员的培训和专业发展是该项目不可分割的一部分。本课题研究了三类具有共享非局部结构的偏微分方程解的存在性、唯一性、正则性和长时间渐近行为。第一个问题是关于具有非线性速度排列的可压缩欧拉系统，它描述了动物群体中显著的群集现象。将研究系统的全局现象和渐近行为，重点是速度对准中的非线性。第二个问题是关于无压欧拉系统，针对的是长期存在的弱解的唯一性问题。我们的计划是用集体动力学中研究相对较好的欧拉排列系统来近似该系统。第三个问题是关于Euler-Monge-Ampère系统，它与流体力学中的不可压缩Euler方程密切相关。将探索该系统的嵌入式非局部几何结构，在最优传输和平均场游戏中具有有趣的应用。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。

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.