Onsager's conjecture and the energy of singular flows

昂萨格猜想和奇异流能量

• 批准号: 0907812
• 负责人: Roman Shvydkoy
• 金额: $ 14.13万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0907812&HistoricalAwards=false
• 关键词: Onsager; conjecture; energy; singular; flows

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.

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尺度的描述。这些措施包括推导湍流法律从执政的Navier-Stokes方程组，或欧拉方程的惯性子范围内的尺度;规律性问题和问题的长期渐近行为。传统的分析方法，传统上用于接近湍流的问题是有限的亚临界规则性制度不符合经验的意见。长期存在的Onsager猜想指出，然而，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。我们将获得新的频率局部估计的能量通量通过并矢尺度;检查的Onsager临界光滑的解决方案的范围内的Besov空间;使用本地能量估计研究解决方案在各种不稳定性制度，包括经典的Kolmogorov和完全间歇性制度。我们将重新讨论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。湍流是流体元素在混合力的作用下不断搅动的复杂运动的复杂物理过程。在汽车、飞机、轮船等后面的湍流尾流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。由于流体湍流运动的复杂性，通常采用统计方法研究，包括对大量的实验数据或长时间的观测数据进行平均，这种研究的目的是直接从控制方程中找到湍流运动的具体实现。 该项目将有助于获得对1949年提出的所谓Onsager猜想的新展望，该猜想质疑经典流体运动方程描述湍流的能力。