Small Scale and Singularity Formation in Fluids

流体中的小尺度和奇点形成

• 批准号: 2306726
• 负责人: Alexander Kiselev
• 金额: $ 45万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306726&HistoricalAwards=false
• 关键词: Small; Scale; Singularity; Formation; Fluids

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.

该项目涉及流体力学和数学生物学的数学分析。流体就在我们周围，更好地理解流体运动对于科学和工程非常重要。该项目旨在增进对流体运动中小规模形成的理解，这一过程出现在广泛的应用中，并且与湍流的发展有关。该项目还将重点分析趋化性、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繁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。该项目还将涉及博士后和研究生级别的初级研究人员的培训。该项目的第一个方向涉及表面准地转（SQG）方程的补丁解中的小尺度形成和规律性丧失。 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不适定性和可能的​​奇点形成的新结果。第二个方向涉及 2D Boussinesq 系统和 3D Euler 方程解的小规模创建。该方向还寻求开发适合深入了解 3D 欧拉和纳维-斯托克斯方程整体解中可能的奇点形成的模型，Tom Hou 最近的数值模拟表明。第三个方向侧重于耦合的凯勒-塞格尔流体系统，并探索流体平流潜在的奇点抑制。该项目旨在在平流不是被动且不处于扰动状态的情况下建立此类的第一个严格结果。最后一个方向涉及开发一类新型模型来解决趋化性对生物反应的影响。该研究旨在超越规律性估计，并严格推导应用中可能感兴趣的缩放定律。将采用的各种技术包括新颖的比较原理、傅里叶和泛函分析方法、渐近分析技术以及 PDE 正则估计。该奖项反映了 NSF 的法定使命，并通过使用基金会的知识进行评估，被认为值得支持。优点和更广泛的影响审查标准。