Nonlocal Transport Equations in Fluids, Swarming, and Traffic Flows

非局部模型与许多现实世界的现象有关，近几十年来一直是一个活跃和不断发展的研究领域。非局部相互作用的数学理论的发展在理解复杂结构方面起着重要的作用，在物理学、生物学和社会科学中有着丰富的应用。自然界中发现的非局部行为影响的一个例子是动物群体中的集体动力学，其中小规模的相互作用演变成有趣的全球现象。本项目为具有相似非局域性的模型开发新颖且稳健的分析技术。这些工具有助于促进对模型隐藏结构的理解，并最终在交通流等应用中产生影响，在这些应用中，它们可以用于研究如何将非本地通信集成到智能交通网络中，以提高效率并避免交通拥堵。研究生的培训和专业发展是该项目的一个组成部分。本课题研究了三类非局部输运方程。第一族包括描述动物群体群集现象的欧拉对准系统。目标是从施加径向对称开始，建立多维系统的全局适定性理论，并将该方法应用于其他模型，如欧拉-泊松方程等。第二部分包含描述重复微分下多项式根分布演化的非局部输运方程，目的是寻找该方程与微分过程之间的严格联系。最后是一系列非局部交通流模型，这些模型在过去十年中受到了广泛的关注，并分析了非局部相互作用的影响以及非局部现象如何有助于防止交通拥堵。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。

Nonlocal models are relevant to many real-world phenomena and have been an area of active and growing research in recent decades. The development of a mathematical theory of nonlocal interactions plays a significant role in the understanding of complex structures, with rich applications in physics, biology, and social sciences. One example of the effects of nonlocal behavior found in nature is the collective dynamics in animal swarms, where small-scale interactions emerge into intriguing global phenomena. This project develops novel and robust analytical techniques for models that share similar nonlocality. These tools help to advance the understanding of the hidden structures of the models, and ultimately have an impact in applications, such as in traffic flow, where they can be used to study how to integrate nonlocal communications into a smart traffic network to improve efficiency and avoid traffic congestions. The training and professional development of graduate students is an integral part of the project. The project studies three families of nonlocal transport equations. The first family includes the Euler-alignment system describing the flocking phenomenon for animal swarms. The goal is to establish a global well-posedness theory for the system in multi-dimensions, starting from imposing radial symmetry, and to apply the methodology to other models, such as the Euler-Poisson equations and more. The second includes a nonlocal transport equation which describes the evolution of the distribution of polynomial roots under repeated differentiation, the aim is to find a rigorous connection between this equation and the differentiation process. The last is a family of nonlocal traffic flow models, which have received extensive attention in the last decade, and are analyzed to understand the impact of the nonlocal interactions and how the nonlocal phenomenon can help to prevent traffic congestions.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.