Small Scale and Singularity Formation in Fluids

该奖项侧重于流体力学的几个方向。流体无处不在，我们可以在日常生活、无处不在的技术以及龙卷风或飓风等戏剧性现象中看到它们特性的复杂性和微妙性。在流体力学的广阔领域中已经积累了大量的知识，然而，值得注意的是，许多最基本和相关的应用问题仍然知之甚少。特别令人感兴趣的问题是，描述流体运动的方程的解是否能自发地形成奇点。理解奇点是很重要的，因为它们通常对应于戏剧性的、高度激烈的流体运动，可以指示模型的适用范围，并且很难通过计算来解决。此外，有迹象表明，造成奇点和小尺度形成的机制也参与了湍流的产生。本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地层过程。此外，还将考虑一些由生物学问题所启发的具有相关结构的方程。该奖项将为本科生、研究生和博士后学者提供机会和研究经验。该奖项涉及对流体力学中的几个方程和由生物现象引起的相关模型的分析。主要的主题是分析解中的奇点和小尺度形成。地表准地转方程来源于大气科学，模拟了地表附近温度的演化。不可压缩多孔介质方程描述了变密度流体在多孔介质中的流动。一个研究方向将集中于更好地理解这些方程解中快速产生小尺度和潜在奇点的非线性机制。另一个方向是进一步分析三维欧拉方程中奇点形成的侯-罗情形。这种情景是几年前基于大量数值模拟提出的，迄今为止，所有经过严格分析的简化模型和设置都表明，爆炸是有限时间的。这里的主要目标将是更好地理解观察到强烈涡度增长的边界层，并设计和探索越来越接近实际方程的模型。这项研究还将包括分析生物学中集体行为和趋化性的方程。这里的重点是分析规律，长时间动力学和边界效应。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。

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.