Small Scale and Singularity Formation in Fluids

该项目涉及流体力学和数学生物学的数学分析。流体无处不在，更好地理解流体运动在科学和工程中具有重要意义。该项目旨在促进对流体运动中的小规模地层的理解，这一过程出现在广泛的应用中，并与湍流的发展有关。该项目还将侧重于分析趋化性，细胞或其他生物制剂在响应外部化学刺激时的定向运动。这里的主要目标是了解和量化趋化性如何帮助促进从生殖到免疫系统功能的许多生物过程。许多趋化过程发生在流体中，扩散、流体流动和趋化性的基本效应之间的相互作用也将被研究。这些问题处于现代应用分析的前沿，需要开发适用于其他环境的新技术。该项目还将涉及培养博士后和研究生水平的初级研究人员。该项目的第一个方向是关注小尺度的形成和表面准地转（SQG）方程的补丁解的规则性损失。SQG方程出现在大气科学中，用于模拟温度锋等大规模天气现象。PI和合作者最近在斑块边界曲率的演化方程中发现了一个有趣的结构，并计划利用这一见解来获得关于大量流体中的病态和可能的奇点形成的新结果。第二个方向解决了二维Boussinesq系统和三维欧拉方程的小规模解决方案。该方向还寻求开发适合于深入了解三维欧拉和Navier-Stokes方程组解中可能形成的奇点的模型，这是由Tom Hou最近的数值模拟提出的。第三个方向关注耦合keller - segel -流体系统，探索流体平流抑制潜在奇异性的方法。这个项目的目的是在平流不是被动的和不处于扰动状态的情况下建立第一个严格的结果。最后一个方向是发展一类新的模型，解决趋化性对生物反应的影响。这项研究的目的是超越规律估计，严格推导出可能在应用中感兴趣的比例定律。将部署的各种技术包括新的比较原理，傅里叶和泛函分析方法，渐近分析技术，以及PDE正则性估计。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。

The project concerns the mathematical analysis of fluid mechanics and mathematical biology. Fluids are all around us, and better understanding of fluid motion is of importance in science and engineering. The project aims to advance understanding of small-scale formation in fluid motion, a process that appears in a wide range of applications and is related to development of turbulence. The project will also focus on analysis of chemotaxis, directed motion of cells or other biological agents in response to external chemical stimuli. Here the main goal is to understand and quantify how chemotaxis helps facilitate many biological processes from reproduction to immune system function. Many chemotactic processes take place in fluid, and interaction between the fundamental effects of diffusion, fluid flow and chemotaxis will also be studied. These problems are at the forefront of modern applied analysis and will require development of novel techniques that should be applicable in other settings. The project will also involve the training of junior researchers at a postdoctoral and graduate level.The first direction of the project is concerned with small scale formation and loss of regularity in patch solutions to the surface quasi-geostrophic (SQG) equation. The SQG equation appears in atmospheric science, where it is used to model large scale weather phenomena like temperature fronts. The PI and collaborators have recently discovered an intriguing structure in the evolution equation for curvature of the patch boundary, and plan to use this insight to obtain new results on ill-posedness and possible singularity formation in the bulk of the fluid. The second direction addresses small scale creation in solutions to the 2D Boussinesq system and 3D Euler equation. This direction also seeks to develop models suitable to gain insight into possible singularity formation in solutions of the 3D Euler and Navier-Stokes equations in the bulk, suggested by recent numerical simulations of Tom Hou. The third direction focuses on the coupled Keller-Segel-fluid system and explores the potential singularity suppression by fluid advection. The project aims to establish first rigorous results of this sort in the situation where advection is not passive and is not in a perturbative regime. The final direction is concerned with the development of a new class of models addressing the effect of chemotaxis on biological reactions. The research is intended to go beyond regularity estimates and rigorously derive scaling laws that may be of interest in applications. A variety of techniques that will be deployed include novel comparison principles, methods of Fourier and functional analysis, asymptotic analysis techniques, as well as PDE regularity estimates.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.