Mechanisms for Energy Conservation in Onsager Supercritical Fluids

像水这样的流体的湍流运动的复杂性提出了许多理论和技术上的挑战。从实际的角度来看，湍流定律在许多实际应用中是至关重要的。它们支撑着飞机机翼的现代设计或天气和气候预报模型的发展。湍流的一个特征叫做反常能量耗散。当流体的运动非常混乱或粗糙，以致光滑动力学的经典定律不再适用时，就会出现这种现象。在许多常用的抛能机构中，如汽车轮杆，都利用了异常能量耗散。虽然它很常见，但在某些情况下，即使在被认为足够湍流以促进能量耗散的情况下，能量耗散也不会发生。已经观察到，在各种自然现象中，例如在飞机机翼后面形成的旋涡片，能量耗散直到运动达到超临界状态才发生。该项目的目标是在湍流或接近湍流的流体运动中分离出几个负责能量保存或耗散的机制。一个主要的焦点是研究对称在能量守恒中的作用。学生也包括在这个项目中。研究者通过考虑能量守恒或耗散在这些方程所描述的流体流动中的作用，研究了欧拉方程及其粘性Navier-Stokes对应的消失粘度极限域中的弱解。这些方程已被证明从数值角度相当准确地描述了湍流，尽管理论上它们提出了许多挑战。根据Onsager规则，当流动的平滑度降低到满导数的三分之一时，溶液达到湍流状态，也称为Onsager规则。在这个规则体系中，研究者考察了四种主要的机制，作为负责能量守恒或耗散的候选机制：潜在控制方程的哈密顿结构、不可压缩条件、基本尺度对称性和运动的输运性质，以及二维环境中消失的粘度极限。最后一点将能量耗散与欧拉方程解的规律性联系起来。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.