Weak Solutions and Turbulence in Fluid Dynamics

流体动力学领域的一个主要目标是获得对湍流的详细和定量的理解，湍流是一种以速度的快速和不规则波动为特征的流体运动。湍流在无数的工程和工业应用中扮演着关键的角色，例如飞机的设计，必须了解湍流的产生，以实现节能和有效的技术。长期以来，人们一直相信湍流的内部运作可以用微分方程来理解，微分方程被用来模拟流体运动，包括欧拉方程。然而，这些方程的分析已被证明是如此困难，以至于湍流和流体动力学方程的解之间的许多关系仍然是推测和未解决的。最近，已经发展出一种新的方法来分析具有与湍流中所见类似的快速波动的流体方程的解。这种方法与几何数学学科中的问题密切相关。该研究项目旨在推动这种新方法发挥其最大潜力，以提高对湍流的理解并扩展数学分析的前沿。本项目为研究生提供研究训练机会。在技术层面上，PI将证明流体动力学方程（包括不可压缩欧拉方程）弱解结构的严格结果。研究结果将表明，在何种程度上，这种弱解可以显示出观察到的或预测到的湍流的性质。一个研究方向将涉及能量的反常耗散现象和昂萨格猜想的推广。PI还将研究欧拉方程和相关方程的一般弱解的动力结构的精细性质。本研究将在凸积分方法的基础上进行扩展，并根据需要引入新的几何或谐波分析工具。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。

A major goal in the field of fluid dynamics is to obtain a detailed and quantitative understanding of turbulence, which is a type of fluid motion characterized by rapid and irregular fluctuations in velocity. Turbulence plays a pivotal role in countless engineering and industrial applications such as the design of aircraft, where the production of turbulence must be understood to achieve energy efficient and effective technologies. It has long been believed that the inner workings of turbulence may be understood in terms of the differential equations that are being used to model fluid motion, including the Euler equations. However, the analysis of these equations has proved so difficult that much of the relationship between turbulence and solutions to the equations of fluid dynamics remains conjectural and unresolved. Recently, a new approach has been developed to analyze solutions to fluid equations that have rapid fluctuations comparable to what is seen in turbulence. This approach is intimately tied to questions in the mathematical discipline of geometry. This research project aims to push this new approach towards its fullest potential to improve the understanding of turbulence and to expand the frontiers of mathematical analysis. The project provides research training opportunities for graduate students.At a technical level, the PI will prove rigorous results on the structure of weak solutions to the equations of fluid dynamics, including the incompressible Euler equations. The results will address the extent to which such weak solutions can be shown to exhibit the properties observed or predicted of turbulent flows. One direction of research will relate to the phenomenon of anomalous dissipation of energy and generalizations of Onsager’s conjecture. The PI will also investigate fine properties of the dynamical structure of general weak solutions to the Euler and related equations. The research will implement and expand upon the method of convex integration, as well as introduce new tools of geometric or harmonic analysis as needed.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.