Regularity and Singularity Formation in Swarming and Related Fluid Models

集群及相关流体模型中的规律性和奇异性形成

基本信息

批准号：
1853001
负责人：
Changhui Tan
金额：
$ 11.82万
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：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1853001&HistoricalAwards=false
关键词：
Regularity Singularity Formation Swarming Related

项目摘要

Swarming is a commonly observed complex biological and sociological phenomenon. The internal interaction mechanism attracts a lot of attention in physics, engineering, biology, and social sciences. This project is devoted to developing a unified mathematical theory towards the understanding of the swarming dynamics, as well as other nonlocal models that share similar structures. These models are widely considered in fluid mechanics, meteorology, astrophysics, biology, and ecology. The study of the regularity and singularity formations of these equations will provide a firm theoretical foundation for these applications, and also help consolidate the validity of these models in describing the natural phenomena. The research will focus on understanding the nonlinear and nonlocal phenomena in swarming dynamics, and models having related structures in fluid mechanics. Three different but related models will be investigated. The first model is the Euler-Alignment system, which describes the flocking behavior in animal swarms. The goal is to develop a robust toolbox to analyze the nonlocal alignment operator and its balance with the drift nonlinearity. Similar behaviors are also observed in other fluid equations including porous medium flow, and surface quasi-geostrophic equations, which will be investigated using the same analytical techniques. The second model is the 2D inviscid Boussinesq equations. The global regularity is one of the outstanding problems in fluid dynamics. The idea is to construct solutions to capture the possible singularity formation, starting from some modified versions of the equations. The third model is the kinetic swarming system. The aim is to investigate the important relation between the kinetic equation and a variety of hydrodynamic limits. In particular, different alignment operators will be considered at the kinetic level. They are expected to lead to different macroscopic limits. All these three sub-projects will advance the mathematical understanding of nonlocal PDEs and related applications. They will also provide education and training to graduate and undergraduate students in this active field.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.

蜂群是一种通常观察到的复杂的生物学和社会学现象。内部互动机制在物理，工程，生物学和社会科学方面引起了很多关注。该项目致力于开发统一的数学理论，以理解蜂群动力学以及其他共享相似结构的非本地模型。这些模型在流体力学，气象，天体物理学，生物学和生态学中被广泛考虑。对这些方程式的规律性和奇异性形成的研究将为这些应用提供牢固的理论基础，并有助于巩固这些模型在描述自然现象中的有效性。这项研究将着重于了解蜂群动力学中的非线性和非局部现象，以及具有相关结构的模型。将研究三种不同但相关的模型。第一个模型是Euler-Alignment System，它描述了动物群中的植入行为。目的是开发一个强大的工具箱，以分析非本地对准操作员及其与漂移非线性的平衡。在其他流体方程（包括多孔培养基流量和表面准地藻方程）中也观察到了类似的行为，将使用相同的分析技术研究。第二个模型是2D Indiscid Boussinesq方程。全球规律性是流体动力学中出色的问题之一。这个想法是构建解决方案以捕获可能的奇异性形成，从一些方程式的一些修改版本开始。第三个模型是动力学群体。目的是研究动力学方程与多种流体动力学限制之间的重要关系。特别是，将在动力学水平上考虑不同的对齐操作员。预计它们会导致不同的宏观极限。所有这三个子项目都将提高对非局部PDE和相关应用的数学理解。他们还将为这个活跃领域的研究生和本科生提供教育和培训。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被视为值得通过评估来获得支持。