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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.

非局部模型与许多真实世界现象相关，近几十年来一直是一个活跃且不断增长的研究领域。非局域相互作用数学理论的发展在理解复杂结构方面发挥着重要作用，在物理学、生物学和社会科学中有着广泛的应用。自然界中发现的非局部行为的影响的一个例子是动物群体的集体动态，其中小规模的相互作用变成了有趣的全球现象。该项目为共享相似非局部性的模型开发了新颖和健壮的分析技术。这些工具有助于促进对模型隐藏结构的理解，并最终对应用程序产生影响，例如在交通流中，它们可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不可分割的一部分。该项目研究了三类非局部输运方程。第一个家族包括欧拉排列系统，该系统描述了动物群体的集群现象。目标是从施加径向对称性开始，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Euler-Poisson方程等。第二种方法包括一个描述多项式根的分布在重复微分下的演化的非局部迁移方程，目的是找到该方程与微分过程之间的严格联系。最后是一系列非本地交通流模型，这些模型在过去十年中受到了广泛的关注，并被分析以了解非本地相互作用的影响以及非本地现象如何有助于防止交通拥堵。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。