Intermittent Solutions of the Navier-Stokes Equations: From Onsager's Conjecture to Turbulence

本课题是研究流体运动的Navier-Stokes方程的一些基本开放性问题。这些方程被物理学家和工程师广泛应用于现实生活中，它们是在近两个世纪前引入的，用于描述粘性流体（如空气或水）的运动。然而，一些基本问题，如经典解的存在性和唯一性，仍然没有得到回答。因此，弱解的概念，其存在性是已知的，已被数学家广泛使用。PI将通过构造具有各种非物理性质的弱解来证明这种概念的局限性。另一方面，开发的技术将允许PI研究有望描述湍流的物理解。湍流通常被认为是经典物理学中最后一个未解决的问题，是自然界、工程和环境应用中普遍存在的一种基本的流体力学和空气动力学现象。这项研究将涉及研究生和博士后。Kolmogorov的紊流理论是建立在涡流在每个长度尺度上密集堆积的假设基础上的，这个假设在大量的实验和数值模拟中已经被证明是无效的。事实证明，这些涡流的排列有些松散，这种现象被称为间歇性。在这个项目中，PI将研究如何在数学上定义间歇性并通过实验测量；间歇性如何影响三维NSE弱解的正则性及其满足能量等式的能力；如何推导出间歇性的严格界限，并证明Navier-Stokes方程（NSE）解的尺度湍流定律。例如，PI将分析间歇性如何影响溶液异常释放或获得能量的能力。因此，将构造具有各种异常性质的间歇野解。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 project is to study some fundamental open questions concerning the Navier-Stokes equations of fluid motion. These equations, widely used by physicists and engineers for real-life applications, were introduced almost two centuries ago to describe the motion of viscous fluids, such as air or water. However, some essential questions, such as the existence and uniqueness of classical solutions, are still not answered. Therefore, a notion of weak solutions, whose existence is known, has been extensively used by mathematicians. The PI will demonstrate a limitation of this notion by constructing weak solutions with various unphysical properties. On the other hand, the developed technique will allow the PI to study physical solutions that are expected to describe turbulence. Turbulence, often referred to as the last unsolved problem in classical physics, is a fundamental and ubiquitous hydrodynamical and aerodynamical phenomenon occurring in nature, in engineering, and in environmental applications. This research will involve graduate students and postdocs.Kolmogorov's theory of turbulence is based on the assumption that eddies are densely packed at each length scale, which has been invalidated in numerous experiments and numerical simulations. It turns out that the eddies are packed somewhat loosely, a phenomenon called intermittency. In this project the PI will study how the intermittency can be defined mathematically and measured experimentally; how the intermittency affects regularity properties of weak solutions to the 3D NSE and their ability to satisfy the energy equality; how one can derive rigorous bounds on intermittency and justify scaling turbulence laws for solutions of the Navier-Stokes equations (NSE). For instance, the PI will analyze how the intermittency affects the ability of solutions to anomalously loose or gain energy. As a result, intermittent wild solutions with various anomalous properties will be constructed.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.