Nonlocal Transport Equations in Fluids, Swarming, and Traffic Flows

流体、蜂群和交通流中的非局域传输方程

基本信息

批准号：
2108264
负责人：
Changhui Tan
金额：
$ 18.3万
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：
2021
资助国家：
美国
起止时间：
2021-08-01 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2108264&HistoricalAwards=false
关键词：
Nonlocal Transport Equations Fluids Swarming

项目摘要

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.

非局部模型与许多现实现象有关，并且在近几十年来一直是积极和不断增长的研究领域。非局部相互作用的数学理论的发展在对复杂结构的理解中起着重要作用，在物理，生物学和社会科学方面具有丰富的应用。在自然界中发现的非局部行为影响的一个例子是动物群中的集体动力学，小规模的相互作用出现在有趣的全球现象中。该项目为具有类似非局部性的模型开发了新颖和强大的分析技术。这些工具有助于促进对模型隐藏结构的理解，并最终对应用程序（例如在交通流中）产生影响，在该应用程序中，它们可以用于研究如何将非局部通信整合到智能交通网络中，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不可或缺的一部分。该项目研究了三个非本地运输方程的家族。第一家族包括描述动物群的羊群现象的欧拉对准系统。目的是在多维中为系统建立一个全球体系良好的理论，从施加径向对称性开始，并将方法应用于其他模型，例如Euler-Poisson方程等。第二个包括一个非局部传输方程，该方程描述了在重复分化下多项式根部分布的演变，其目的是在该方程和分化过程之间找到严格的联系。最后一个是一个非局部交通流量模型的家族，在过去十年中，它们受到了广泛的关注，并进行了分析，以了解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NSF的法定任务，并通过使用基础的智力效果和广泛的范围进行评估，并被视为值得通过评估来进行评估。