A new approach to problems of global regularity for the 3D Navier-Stokes equations and other dissipative PDEs: the use of Kolmogorov's dissipation range and intermittency

解决 3D 纳维-斯托克斯方程和其他耗散偏微分方程全局正则性问题的新方法：使用柯尔莫哥洛夫耗散范围和间歇性

• 批准号: 1108864
• 负责人: Alexey Cheskidov
• 金额: $ 15.41万
• 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: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1108864&HistoricalAwards=false
• 关键词: new; approach; problems; global; regularity

The problems of blow-up for the Navier-Stokes and Euler equations have been extensively studied for decades using different techniques. This research is aimed at developing a new unified approach to the blow-up problem for the equations of incompressible fluid motion. Motivated by Kolmogorov's theory of turbulence, the project introduces a time-dependent dissipation wavenumber that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. This frequency separation will be used to obtain new regularity criteria, uniqueness conditions, local wellposedness results, conditions preventing type I blow-up. The developed technique will also be applied to related equations. The methods to be used will combine harmonic analysis tools and classical techniques for the Navier-Stokes and Euler equations.This research is devoted to several fundamental open questions concerning the equations of fluid motion. The equations were introduced almost two centuries ago but are still not well understood mathematically. Even though the equations are broadly used by physicists and engineers for real-life applications, and are widely believed to be an accurate representation of the physical phenomena involved, the existence and uniqueness of solutions is still not known. A mathematical proof of existence of solutions would justify the equations definitively. The proposed research is also expected to shed light on certain fundamental issues related to turbulence. Turbulence, often referred to as the last unsolved problem in classical physics, is a crucial phenomenon occurring in fluid flows around airplane bodies, vehicles, ships, and blades of turbines. A better mathematical understanding of turbulence will lead to improvements in the design of these objects.

Navier-Stokes方程和Euler方程的爆破问题已经被广泛研究了几十年，使用不同的技术。本研究的目的是发展一种新的统一的方法来解决不可压缩流体运动方程的爆破问题。受Kolmogorov湍流理论的启发，该项目引入了一种依赖于时间的耗散波数，该波数将欧拉动力学占主导地位的低模式与粘性力占主导地位的高模式分开。这种频率分离将被用来获得新的正则性准则，唯一性条件，局部适定性结果，防止I型爆破的条件。所开发的技术也将被应用到相关的方程。所使用的方法将结合联合收割机谐波分析工具和经典技术的Navier-Stokes和Euler equations.This研究致力于几个基本的开放问题有关的流体运动方程。这些方程是在两个世纪前提出的，但在数学上仍然没有得到很好的理解。尽管这些方程被物理学家和工程师广泛用于现实生活中的应用，并且被广泛认为是所涉及的物理现象的精确表示，但解的存在性和唯一性仍然未知。解的存在性的数学证明将明确地证明这些方程。拟议中的研究也有望阐明与湍流有关的某些基本问题。湍流，通常被称为经典物理学中最后一个未解决的问题，是发生在飞机机身，车辆，船舶和涡轮机叶片周围的流体流动中的关键现象。对湍流的更好的数学理解将导致这些物体设计的改进。