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。我们周围到处都是流体，更好地了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小尺度形成的理解，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响应外部化学刺激的定向运动。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。这些问题是现代应用分析的前沿，需要开发适用于其他环境的新技术。该项目还将涉及博士后和研究生水平的初级研究人员的培训。该项目的第一个方向是关注小尺度形成和地表准地转（SQG）方程斑块解的规律性损失。SQG方程出现在大气科学中，用于模拟大尺度天气现象，如温度锋。PI和合作者最近在补丁边界曲率的演化方程中发现了一个有趣的结构，并计划利用这一洞察力来获得关于大部分流体中不适定性和可能的奇点形成的新结果。第二个方向解决了2D Boussinesq系统和3D Euler方程的小规模创建。这个方向还寻求开发适合于深入了解3D欧拉和Navier-Stokes方程解中可能的奇点形成的模型，这是Tom Hou最近的数值模拟所建议的。第三个方向主要研究耦合的Keller-Segel-流体系统，探讨流体平流对潜在奇异性的抑制。该项目的目的是建立第一个严格的结果，这种情况下，平流是不是被动的，是不是在一个扰动制度。最后的方向是关注一类新的模型的发展，解决生物反应的趋化性的影响。这项研究的目的是超越规律性估计，并严格推导出可能在应用中感兴趣的标度律。将部署的各种技术包括新的比较原则，傅立叶和功能分析方法，渐近分析技术，以及PDE规律性estimations.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.