On The Limiting Behaviour of Regularizations of the Euler Equations With Vortex Sheet Initial Data

涡面模拟流体流动中的剪切层。 为了避免控制涡面动力学的欧拉方程中的有限时间奇异性，有必要对方程进行正则化。 该项目比较了目前重要的三种不同的正则化方法：涡滴方法，欧拉-α模型和物理粘度。 目前还不清楚当正则化参数接近零时，不同的正则化是否有不同的极限，本项目旨在通过数值研究和分析方法相结合来回答这个问题。 这个问题的答案是内在的数学兴趣和数值模拟的实际意义。本研究可能的结果包括：1）不同的正则化的欧拉方程给出不同的限制在一个固定的时间。 这将是感兴趣的本质上的数学原因和应用。 2)的限制是相同的，在最近的工作中观察到的涡滴方法的混沌特征存在于所有情况下，包括粘性的情况下。 这将使我们对真实的流体运动的物理学有重要的了解。 3)限制是相同的，并且不包括混沌特征。 这表明正则化可以引入人为的不规则性，并建议改进正则化方法。 这项工作旨在为这些可能的情况之一提供确凿的证据。 它将导致增加的欧拉方程和流体动力学与小粘度的正则化的理解。

Vortex sheets model shear layers in fluid flow. In order to avoid finite-time singularities in the Euler equations that govern the dynamics of a vortex sheet, it is necessary to regularize the equations. This project compares three different regularization methods of current importance: vortex blob methods, Euler-alpha models, and physical viscosity. At present it is unknown whether different regularizations have different limits as the regularization parameter approaches zero, and this project aims to answer this question through a combination of numerical studies and analytical methods. An answer to this question is of intrinsic mathematical interest and of practical importance for numerical simulations. Possible outcomes to this investigation include: 1) Different regularizations of the Euler equations give different limits at a fixed time. This would be of interest for both intrinsically mathematical reasons and for applications. 2) The limits are the same, and the chaotic features observed in recent work on the vortex blob method are present in all cases, including the viscous case. This would give significant insight into the physics of real fluid motion. 3) The limits are the same, and do not include the chaotic features. This would indicate that regularization can introduce artificial irregularities, and would suggest work for improved regularization methods. This work aims to present conclusive evidence towards one of these possible scenarios. It will result in increased understanding of regularizations of the Euler equations and of fluid dynamics with small viscosity.