Onsager's conjecture and the energy of singular flows

为了理解流体运动的许多物理和数学方面，需要对能量传递进行逐比例的描述。这些包括从控制的Navier-Stokes方程组或惯性子尺度下的欧拉方程推导湍流定律；正则性问题和长时间渐近性问题。传统上用于处理湍流问题的传统分析方法仅限于与经验观测不一致的亚临界规则状态。然而，长期存在的Onsager猜想指出，尽管存在这些限制，欧拉方程允许特定的奇异解具有均匀各向同性湍流的共同标度特性。该项目的主要目标是奖励开发分析工具来研究这些解决方案。我们将通过二进尺度得到能量通量的新的频率局部估计；研究Besov空间范围内解的onsager临界光滑性；利用局部能量估计来研究各种间歇状态下的解决方案，包括经典的Kolmogorov和完全间歇状态。我们将重新讨论Navier-Stokes方程的Leray-Hopf解的一些基本正则性准则，以改进已知的能量相等的充分条件。湍流是流体要素在混合力作用下不断搅拌的复杂运动的复杂物理过程。汽车、飞机、轮船等后面的湍流尾迹是一种常见的日常现象。更好地理解这一现象有助于找到更有效的设计和节能解决方案。由于流体的湍流运动的复杂性，通常用统计方法来研究，这种方法涉及对大量实验数据或长时间内观察到的量进行平均。这项研究的目的是直接从控制方程中找到湍流运动的特定个体实现。该项目将有助于对1949年提出的所谓的昂萨格猜想（Onsager’s conjecture）进行新的展望，该猜想质疑经典流体运动方程描述湍流的能力。

A scale-by-scale description of the energy transfer is required for understanding many physical and mathematical aspects of fluid motion. These include derivation of turbulence laws from the governing Navier-Stokes system of equations, or Euler equations in the inertial subrange of scales; regularity problems and problems of long time asymptotic behavior. The conventional analytical methods traditionally used to approach the problems of turbulence are limited to subcritical regularity regimes not consistent with empirical observations. The long standing Onsager's conjecture states however that in spite of these limitations the Euler equations allow for particular singular solutions sharing common scaling properties of a homogeneous isotropic turbulence. The principal goal of the project is award developing analytical tools to study such solutions. We will obtain new frequency local estimates on the energy flux through dyadic scales; examine the Onsager-critical smoothness of solutions in the range of Besov spaces; use local energy estimates to study solutions in a variety of intermittency regimes including the classical Kolmogorov and fully intermittent regimes. We will revisit some of the fundamental regularity criteria for Leray-Hopf solutions of the Navier-Stokes equations to improve the known sufficient conditions for the energy equality.Turbulence is an intricate physical process of complex motion of fluid elements constantly stirred by a mixing force. Turbulent wakes behind cars, planes, ships, etc. are a common everyday phenomenon. A better understanding of this phenomenon leads to finding more effective designs and energy saving solutions. Due to its complexity a turbulent motion of fluid is usually studied by statistical methods involving averaging of observed quantities over a large number of experimental data or long periods of time.This research is directed to finding particular individual realizations of turbulent motion directly from the governing equations. The project will help to gain a new prospective on the so-called Onsager's conjecture, stated in 1949, which questions the ability of the classical equations of fluid motion to describe turbulence.