Regularity and Singularity Formation in Swarming and Related Fluid Models

蜂群是一种常见的复杂的生物学和社会学现象。内部相互作用机制在物理、工程、生物和社会科学领域受到广泛关注。该项目致力于发展一个统一的数学理论，以理解群体动力学，以及其他具有类似结构的非局部模型。这些模型被广泛应用于流体力学、气象学、天体物理学、生物学和生态学。研究这些方程的规则性和奇点形成将为这些应用提供坚实的理论基础，也有助于巩固这些模型在描述自然现象方面的有效性。重点研究群体动力学中的非线性和非局部现象，以及流体力学中具有相关结构的模型。我们将研究三种不同但相关的模型。第一个模型是欧拉对准系统，它描述了动物群体中的群集行为。目标是开发一个鲁棒工具箱来分析非局部对准算子及其与漂移非线性的平衡。在其他流体方程中也观察到类似的行为，包括多孔介质流动和表面准地转方程，这些方程将使用相同的分析技术进行研究。第二个模型是二维无粘Boussinesq方程。整体正则性问题是流体力学中的突出问题之一。这个想法是从一些修改过的方程开始，构造出捕捉可能的奇点形成的解。第三种模型是动态蜂群系统。目的是研究动力学方程与各种水动力极限之间的重要关系。特别是，不同的对准算子将考虑在动力学水平。预计它们会导致不同的宏观极限。所有这三个子项目将促进对非局部偏微分方程及其相关应用的数学理解。他们还将为这一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。

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.