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级描述。这些问题包括从控制的Navier-Stokes方程组或尺度惯性子域中的欧拉方程推导出湍流定律；正则性问题和长时间渐近行为问题。传统上用来研究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性区域。然而，长期存在的Onsager猜想指出，尽管有这些限制，欧拉方程允许特定的奇异解共享均匀各向同性湍流的共同标度性质。该项目的主要目标是奖励开发分析工具来研究这种解决方案。我们将通过并矢尺度得到关于能量通量的新的频率局部估计；在Besov空间范围内检验解的Onsager-临界光滑性；使用局部能量估计来研究各种间歇状态下的解，包括经典的Kolmogorov和全间歇状态。我们将回顾Navier-Stokes方程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。浑浊是一个复杂的物理过程，流体元素在混合力的作用下不断搅拌。汽车、飞机、轮船等后面的颠簸尾流是日常生活中常见的现象。对这一现象的更好理解有助于找到更有效的设计和节能解决方案。由于流体湍流运动的复杂性，通常采用统计方法对大量实验数据或长时间内的观测量求平均。这项研究的目的是直接从控制方程中寻找湍流运动的具体实现。该项目将有助于对1949年提出的所谓Onsager猜想有一个新的看法，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.