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

纳维-斯托克斯方程的间歇解：从昂萨格猜想到湍流

基本信息

批准号：
1909849
负责人：
Alexey Cheskidov
金额：
$ 16.5万
依托单位：
University of Illinois at Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1909849&HistoricalAwards=false
关键词：
Intermittent Solutions Navier Stokes Equations

项目摘要

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.

本课题主要研究流体运动的Navier-Stokes方程的一些基本问题。这些方程被物理学家和工程师广泛用于现实生活中，它们在近两个世纪前被引入来描述粘性流体的运动，如空气或水。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，弱解的概念被数学家广泛使用，它的存在是已知的。PI将通过构造具有各种非物理性质的弱解来证明这一概念的局限性。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。湍流通常被称为经典物理学中最后一个未解决的问题，它是自然界、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的湍流理论是基于在每个长度尺度上密集堆积涡流的假设，这一假设已经在大量的实验和数值模拟中被证明是无效的。结果发现，漩涡的排列有些松散，这种现象被称为间歇性。在这个项目中，PI将研究如何从数学上定义和通过实验测量间歇性；间歇性如何影响3D NSE弱解的正则性及其满足能量相等的能力；如何得出间歇性的严格界限并证明Navier-Stokes方程(NSE)解的湍流标度定律的合理性。例如，PI将分析间歇性如何影响溶液异常松散或获得能量的能力。因此，将构建具有各种异常属性的间歇性野生解决方案。该奖项反映了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。

项目成果

期刊论文数量（10）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Energy equality for the Navier–Stokes equations in weak-in-time Onsager spaces

DOI：
10.1088/1361-6544/ab60d3
发表时间：
2018-02
期刊：
Nonlinearity
影响因子：
1.7
作者：
A. Cheskidov;Xiaoyutao Luo
通讯作者：
A. Cheskidov;Xiaoyutao Luo

Discontinuity of weak solutions to the 3D NSE and MHD equations in critical and supercritical spaces

临界和超临界空间中 3D NSE 和 MHD 方程弱解的不连续性

DOI：
10.1016/j.jmaa.2019.123493
发表时间：
2020
期刊：
Journal of Mathematical Analysis and Applications
影响因子：
1.3
作者：
Cheskidov, Alexey;Dai, Mimi
通讯作者：
Dai, Mimi

The computation of wandering points on the global attractor by means of symmetry-breaking perturbations

通过对称破缺扰动计算全局吸引子上的漂移点

DOI：
发表时间：
2022
期刊：
Pure and applied functional analysis
影响因子：
0
作者：
Cheskidov, Alexey;Olson, Eric;Smith, Beau
通讯作者：
Smith, Beau

On the Determining Wavenumber for the Nonautonomous Subcritical SQG Equation

非自治次临界SQG方程波数的确定

DOI：
10.1007/s10884-019-09794-7
发表时间：
2019
期刊：
Journal of Dynamics and Differential Equations
影响因子：
1.3
作者：
Cheskidov, Alexey;Dai, Mimi
通讯作者：
Dai, Mimi

Susan Friedlander's Contributions in Mathematical Fluid Dynamics

苏珊·弗里德兰德 (Susan Friedlander) 在数学流体动力学方面的贡献

DOI：
10.1090/noti2237
发表时间：
2021
期刊：
Notices of the American Mathematical Society
影响因子：
0
作者：
Cheskidov, Alexey;Glatt-Holtz, Nathan;Pavlovic, Natasa;Shvydkoy, Roman;Vicol, Vlad
通讯作者：
Vicol, Vlad