Small Scale and Singularity Formation in Fluids

流体中的小尺度和奇点形成

• 批准号: 2006372
• 负责人: Alexander Kiselev
• 金额: $ 43万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2006372&HistoricalAwards=false
• 关键词: Small; Scale; Singularity; Formation; Fluids

This award focuses on several directions in fluid mechanics. Fluids are all around us, and we can witness the complexity and subtleness of their properties in everyday life, in ubiquitous technology, and in dramatic phenomena such as tornados or hurricanes. There has been an enormous wealth of knowledge accumulated in the broad area of fluid mechanics, yet it is quite remarkable that many of the most fundamental and relevant in applications questions remain poorly understood. Of particular interest is the question whether solutions to equations describing fluid motion can spontaneously form singularities. Understanding singularities is important because they often correspond to dramatic, highly intense fluid motion, can indicate the range of applicability of the model, and are very difficult to resolve computationally. Moreover, there are indications that mechanisms responsible for singularity and small-scale formation are also involved in production of turbulence. The project aims to analyze singularity and small scale formation process for some of the key equations of fluid mechanics. Some equations with related structure inspired by problems originating in biology will be considered as well. The award will provide opportunities and research experiences for undergraduate and graduate students, and postdoctoral scholars.The award is concerned with analysis of several equations in fluid mechanics and related models motivated by biological phenomena. The main theme is the analysis of singularity and small scale formation in solutions. The surface quasi-geostrophic equation comes from atmospheric science, and models evolution of temperature near the Earth's surface. The incompressible porous media equation describes flow of fluid of variable density through porous media. One research direction will focus on better understanding of nonlinear mechanisms responsible for fast generation of small scales and potentially singularities in the solutions of these equations. Another direction aims to further analyze the Hou-Luo scenario for singularity formation in the three-dimensional Euler equation. The scenario was proposed several years ago based on extensive numerical simulations, and so far all simplified models and settings that have been analyzed rigorously suggest finite time blow up. Here the main goal will be to better understand the boundary layer where intense growth of vorticity is observed, and to design and explore models that are increasingly close to the actual equation. The research will also include analysis of equations modeling collective behavior and chemotaxis in biology. Here the focus is on analysis of regularity, long time dynamics, and boundary effects.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.

该奖项侧重于流体力学的几个方向。流体就在我们身边，我们可以在日常生活中，在无处不在的技术中，以及在龙卷风或飓风等戏剧性现象中目睹它们的复杂性和微妙性。在流体力学的广泛领域已经积累了大量的知识，但值得注意的是，许多最基本和最相关的应用问题仍然知之甚少。特别感兴趣的问题是描述流体运动的方程的解是否可以自发地形成奇点。理解奇点是很重要的，因为它们往往对应于戏剧性的，高强度的流体运动，可以指示模型的适用范围，并且很难通过计算解决。此外，有迹象表明，机制负责奇点和小规模的形成也参与生产的湍流。该项目旨在分析流体力学中一些关键方程的奇异性和小尺度形成过程。一些方程的相关结构的启发问题起源于生物学也将被考虑。该奖项将为本科生和研究生以及博士后学者提供机会和研究经验。该奖项涉及流体力学中几个方程的分析以及由生物现象激发的相关模型。主要的主题是分析奇异性和小规模形成的解决方案。地表准地转方程来源于大气科学，模拟近地表温度的演变。不可压缩多孔介质方程描述了变密度流体通过多孔介质的流动。一个研究方向将集中在更好地了解负责快速生成的小尺度和潜在的奇异性在这些方程的解决方案的非线性机制。另一个方向是进一步分析三维欧拉方程奇点形成的侯-罗方案。这种情况是几年前基于大量的数值模拟提出的，到目前为止，所有经过严格分析的简化模型和设置都表明有限时间爆炸。这里的主要目标将是更好地了解观察到涡量强烈增长的边界层，并设计和探索越来越接近实际方程的模型。该研究还将包括对生物学中集体行为和趋化性建模方程的分析。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。

项目成果

Random Search in Fluid Flow Aided by Chemotaxis

• DOI: 10.1007/s11538-022-01024-4
• 发表时间: 2021-07
• 期刊: Bulletin of Mathematical Biology
• 影响因子: 3.5
• 作者: Yishu Gong;Si-ming He;A. Kiselev

The Flow of Polynomial Roots Under Differentiation

微分下多项式根的流动

• DOI: 10.1007/s40818-022-00135-4
• 发表时间: 2022
• 期刊: Annals of PDE
• 影响因子: 2.8
• 作者: Kiselev, Alexander;Tan, Changhui
• 通讯作者: Tan, Changhui

Boundary layer models of the Hou-Luo scenario

后洛情景边界层模型

• DOI: 10.1016/j.jde.2021.07.007
• 发表时间: 2021
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: He, Siming;Kiselev, Alexander
• 通讯作者: Kiselev, Alexander

Global Regularity for a Nonlocal PDE Describing Evolution of Polynomial Roots Under Differentiation

描述微分下多项式根演化的非局部偏微分方程的全局正则性

• DOI: 10.1137/21m1422859
• 发表时间: 2022
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Kiselev, Alexander;Tan, Changhui
• 通讯作者: Tan, Changhui

Small Scale Formations in the Incompressible Porous Media Equation

• DOI: 10.1007/s00205-022-01830-z
• 发表时间: 2021-02
• 期刊: Archive for Rational Mechanics and Analysis
• 影响因子: 2.5
• 作者: A. Kiselev;Yao Yao-Yao