Mechanisms for Energy Conservation in Onsager Supercritical Fluids

像水这样的流体湍流运动的复杂性带来了许多理论和技术挑战。从实际的角度来看，湍流定律在许多实际应用中都是至关重要的。他们支持飞机翼型的现代设计，或天气和气候预报模型的开发。湍流的特征之一称为反常能量耗散。当流体的运动如此混乱或粗糙，以至于经典的光滑动力学定律不再适用时，就会出现这种现象。在汽车轮毂支柱等许多常用的甩能机构中，都采用了异常耗能技术。虽然这很常见，但在某些情况下，即使在原本被认为是足以促进这种消散的湍流中，也不会发生能量耗散。人们已经观察到，在各种自然现象中，例如在飞机机翼后面发展起来的涡片，直到运动达到超临界状态时，能量才会耗散。该项目的目标是分离出在湍流或接近湍流的流体运动中负责能量保存或耗散的几种机制。一个主要的焦点是研究对称性在能量守恒中的作用。学生被包括在这个项目中。通过考虑能量守恒或耗散在欧拉方程所描述的流体流动中的作用，研究了消失粘性极限条件下欧拉方程及其粘性Navier-Stokes方程的弱解。这些方程已被证明从数值角度相当准确地描述了湍流，尽管从理论上讲它们提出了许多挑战。在Onsager之后，就正则性而言，当流动的平稳度降低到全导数的三分之一时，解达到湍流状态，也称为Onsager正则性。在这种正则性条件下，研究人员考察了四种主要的机制，作为能量守恒或耗散的候选者：基本控制方程的哈密顿结构，不可压缩条件，运动的基本标度对称性和输运性质，以及二维环境中的零粘性极限。最后一点将能量耗散与欧拉方程解的正则性联系起来。这个项目需要学生的积极参与。

The complexity of turbulent motion of fluids like water presents many theoretical as well as technological challenges. From the practical standpoint laws of turbulence are crucial in many real-life applications. They stand behind the modern design of a plane airfoil or development of weather and climate forecast models. One of the features of turbulence is called anomalous energy dissipation. This phenomenon arises when the motion of a fluid is so chaotic or rough that the the classical laws of smooth dynamics no longer apply. Anomalous energy dissipation is harnessed in many commonly used energy-dumping mechanisms, such as automobile wheel struts. Common though it is, in some cases energy dissipation does not occur even in what otherwise would be considered a flow turbulent enough to facilitate such dissipation. It has been observed that in various natural phenomena, such as vortex sheets that develop behind the wing of a plane, energy dissipation does not occur until motion reaches a supercritical state. The project goal is to isolate several mechanisms responsible for energy preservation or dissipation in fluid motion that is turbulent or nearly so. A main focus is on investigation of the role of symmetries in energy conservation. Students are included in the project. The investigator studies weak solutions to the Euler equation and its viscous Navier-Stokes counterpart in the vanishing viscosity limit regime, by considering the role of energy conservation or dissipation in the fluid flows described by these equations. The equations have been shown to describe turbulence rather accurately from a numerical point of view, although theoretically they present many challenges. Following Onsager, in terms of regularity a solution reaches its turbulent state when smoothness of the flow is reduced to a third of one full derivative, also known as Onsager regularity. In that regularity regime the investigator examines four main mechanisms as candidates responsible for energy conservation or dissipation: Hamiltonian structure of the underlying governing equation, incompressibility condition, basic scaling symmetries and transport nature of the motion, and the vanishing viscosity limit in the two-dimensional setting. The last point connects energy dissipation to regularity of solutions of the Euler equations. The project involves active participation of students.