Mechanisms for Energy Conservation in Onsager Supercritical Fluids

水等流体湍流运动的复杂性提出了许多理论和技术挑战。 从实际的观点来看，湍流定律在许多实际应用中是至关重要的。 他们支持现代飞机机翼的设计或天气和气候预测模型的发展。 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，以至于光滑动力学的经典定律不再适用时，就会出现这种现象。 异常能量耗散在许多常用的能量倾销机制中被利用，例如汽车车轮支柱。 虽然这是常见的，但在某些情况下，即使在被认为是湍流足以促进这种耗散的情况下，也不会发生能量耗散。 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涡面，能量耗散直到运动达到超临界状态才发生。 该项目的目标是隔离几个机制，负责能量保存或耗散的流体运动是湍流或接近。 一个主要的重点是调查的作用，对称性的能量守恒。 学生被纳入该项目。 研究人员研究弱解欧拉方程及其粘性Navier-Stokes对应的消失粘度极限制度，通过考虑这些方程描述的流体流动中的能量守恒或耗散的作用。 从数值的角度来看，这些方程已经被证明可以相当准确地描述湍流，尽管理论上它们存在许多挑战。 在昂萨格之后，在正则性方面，当流动的平滑度降低到一个全导数的三分之一时，解达到其湍流状态，也称为昂萨格正则性。 在该规律性制度的调查员检查四个主要机制作为候选人负责能量守恒或耗散：基本的控制方程的哈密顿结构，不可压缩性条件，基本的缩放对称性和运输性质的运动，和消失的粘度限制在二维设置。 最后一点将能量耗散与欧拉方程解的正则性联系起来。 该项目涉及学生的积极参与。

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.