Small Scale and Singularity Formation in Fluids

该奖项集中在流体力学的几个方向上。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见证它们特性的复杂性和微妙。在流体力学的广泛领域中积累了大量的知识，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。理解奇点很重要，因为它们通常对应于剧烈的、高度强烈的流体运动，可以指示模型的适用范围，并且很难通过计算解决。此外，有迹象表明，导致奇异性和小尺度形成的机制也参与了湍流的产生。该项目旨在分析流体力学的一些关键方程的奇异性和小尺度形成过程。还将考虑一些受生物学问题启发而具有相关结构的方程式。该奖项将为本科生和研究生以及博士后奖学金提供机会和研究经验。该奖项涉及分析流体力学中的几个方程和由生物现象驱动的相关模型。主要主题是分析解的奇异性和小尺度形成。地表准地转方程来自大气科学，它模拟了地球表面附近温度的演变。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在更好地理解导致小尺度快速产生的非线性机制以及这些方程的解的潜在奇异性。另一个方向是进一步分析三维欧拉方程中奇点形成的侯-罗情景。这一情景是几年前基于广泛的数值模拟提出的，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内会发生爆炸。这里的主要目标将是更好地了解观察到涡度强烈增长的边界层，并设计和探索越来越接近实际方程的模型。这项研究还将包括对模拟集体行为和生物学趋化性的方程的分析。这里的重点是对规律性、长期动态和边界效应的分析。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.