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。其中包括从控制纳维-斯托克斯方程组或惯性尺度子范围内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。传统上用于解决湍流问题的传统分析方法仅限于与经验观察不一致的亚临界规律性范围。然而，长期存在的昂萨格猜想指出，尽管存在这些限制，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属性。该项目的主要目标是奖励开发分析工具来研究此类解决方案。我们将通过二进尺度获得能量通量的新频率局部估计；检查 Besov 空间范围内解的 Onsager 临界光滑度；使用局部能量估计来研究各种间歇状态下的解决方案，包括经典柯尔莫哥洛夫和完全间歇状态。我们将重新审视纳维-斯托克斯方程的勒雷-霍普夫解的一些基本正则性准则，以改进能量相等的已知充分条件。湍流是由混合力不断搅拌的流体元素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流是一种常见的日常现象。更好地理解这种现象可以找到更有效的设计和节能解决方案。由于其复杂性，流体的湍流运动通常通过统计方法来研究，涉及对大量实验数据或长时间段内观测到的量进行平均。这项研究旨在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 Onsager 猜想获得新的认识，该猜想对流体运动的经典方程描述湍流的能力提出了质疑。