Workshop on Geometry and Analysis of Fluid Flows

该奖项支持参加2023年1月16日至20日在石溪大学举办的“流体流动的几何和分析研讨会”。流体力学的基本方程描述了许多自然现象，包括天气模拟中的空气运动，飞机的升力，工业应用中的流体混合，液体在管道中的流动，风和水产生的电力，以及血液在体内的流动。完整的方程太难显式求解，甚至很难用计算机求解，因此数学上的注意力集中在方程的性质上。这些性质包括给定初始条件的解是长期存在还是在有限时间内崩溃；解是稳定的并且可以用小误差预测，还是从根本上不稳定和不可预测；以及用于使方程更易于管理的简化近似的有效性。这些问题是困难的，也是长期存在的。例如，理想化的三维流体方程的长期存在的问题已经悬而未决了一个多世纪。分析和几何都被用来研究这样的问题，这次会议的主要目标是将在这两个领域具有专业知识的资深和初级研究人员聚集在一起，分享观点、技术和结果，并培训新一代数学家。研讨会将以不同数学背景的演讲者为特色，他们的专业知识横跨多个相关领域。研讨会将讨论的主题包括流体动力学中的自由边界问题，无限维群的几何，流体流动中的奇异极限，流体方程的适定性和正则性，以及数学和物理中的微分几何方法。一些讲座将集中讨论三维Euler方程和Navier-Stokes方程的整体存在性问题；粘性、可压缩性或表面张力趋于零时的极限行为；将流体流动描述为测地线的无限维几何；以及自由边界问题的适定性结果。组织者目前正在编写一本关于流体力学中几何和解析方法的教科书；这本书的草稿将在研讨会期间分发，以惠及participants.https://my.vanderbilt.edu/geoanalysisffThis，该奖项反映了美国国家科学基金会的法定使命，并已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 supports participation in the "Workshop on the Geometry and Analysis of Fluid Flows" hosted at Stony Brook University during the week of January 16-20, 2023. The fundamental equations of fluid mechanics describe many natural phenomena, including the motion of air in weather modeling, the lift of an airplane, mixing of fluids with applications in industry, flow of liquids through pipes, generation of electricity through wind and water, and the flow of blood through the body. The full equations are too difficult to solve explicitly or even by computer, hence mathematical attention focuses on properties of the equations. Such properties include whether solutions for given initial conditions exist for a long time or break down in finite time; whether solutions are stable and can be predicted with small errors, or fundamentally unstable and unpredictable; and the validity of simplifying approximations used to make the equations more manageable. These questions are difficult and longstanding. For instance, the question of long-time existence for the idealized three-dimensional fluid equations has remained open for well over a century. Both analysis and geometry have been used to study such questions, and the primary goal of this conference is to bring together senior and junior researchers with expertise in these two areas to share perspectives, techniques, and results, and to train the new generation of mathematicians.The workshop will feature a mathematically diverse group of speakers whose expertise spans multiple relevant areas. Topics to be discussed at the workshop include free boundary problems in fluid dynamics, the geometry of infinite-dimensional groups, singular limits in fluid flows, well-posedness and regularity of fluid equations, and differential geometric methods in mathematics and physics. Some of the talks will focus on issues of global existence for the three-dimensional Euler and Navier-Stokes equations; the limiting behavior as viscosity, compressibility, or surface tension approaches zero; the infinite-dimensional geometry describing fluid flows as geodesics; and well-posedness results for free boundary problems. The organizers are currently writing a textbook on geometric and analytic methods in fluid mechanics; a draft of this book will be distributed during the workshop for the benefit of participants.https://my.vanderbilt.edu/geoanalysisffThis 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.