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

几十年来，人们用不同的方法广泛地研究了Navier-Stokes方程和Euler方程的爆破问题。本研究旨在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。受科尔莫戈洛夫湍流理论的启发，该项目引入了一个随时间变化的耗散波数，将以欧拉动力学为主的低模与粘性力占主导地位的高模分开。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型爆炸的条件。所开发的技术也将应用于相关方程。所使用的方法将结合调和分析工具和经典技术来求解Navier-Stokes方程和Euler方程。这项研究致力于关于流体运动方程的几个基本的开放问题。方程式是在近两个世纪前提出的，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实际应用，并被广泛认为是所涉及的物理现象的准确表示，但解的存在和唯一性仍然是未知的。解的存在性的数学证明将最终证明这些方程是正确的。这项拟议的研究预计还将阐明与湍流相关的某些基本问题。湍流通常被称为经典物理学中最后一个悬而未决的问题，它是在飞机机身、飞行器、船舶和涡轮机叶片周围的流体流动中发生的一种关键现象。对湍流的更好的数学理解将导致这些物体设计的改进。

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.