Regularity and Singularity Formation in Swarming and Related Fluid Models

群集是一种常见的复杂的生物学和社会学现象。其内部相互作用机制在物理学、工程学、生物学和社会科学中引起了广泛的关注。该项目致力于发展一个统一的数学理论，以理解群集动力学，以及其他具有相似结构的非局部模型。这些模型在流体力学、气象学、天体物理学、生物学和生态学中被广泛考虑。研究这些方程的正则性和奇性形成，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描述自然现象的有效性。研究将集中在理解群集动力学中的非线性和非局部现象，以及具有流体力学相关结构的模型。将研究三种不同但相关的模型。第一个模型是欧拉-对齐系统，它描述了动物群体中的群集行为。我们的目标是开发一个强大的工具箱来分析非局部对准算子及其与漂移非线性的平衡。类似的行为也观察到在其他流体方程，包括多孔介质流，和表面准地转方程，这将使用相同的分析技术进行研究。第二种模型是二维无粘Boussinesq方程。整体正则性问题是流体力学中的一个突出问题。我们的想法是构造解决方案，以捕捉可能的奇点形成，从一些修改后的版本的方程。第三个模型是动力学群集系统。其目的是调查的动力学方程和各种水动力极限之间的重要关系。特别地，将在动力学水平上考虑不同的对准算子。预计它们将导致不同的宏观极限。所有这三个子项目将推进非局部偏微分方程的数学理解和相关的应用。该奖项反映了NSF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.