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

涡旋片模拟流体流动中的剪切层。为了避免控制涡旋片动力学的欧拉方程中的有限时间奇点，有必要对方程进行正则化。本项目比较了当前重要的三种不同的正则化方法：涡旋blob方法、Euler-alpha模型和物理粘度。当正则化参数趋近于零时，不同的正则化是否有不同的极限，目前尚不清楚，本项目旨在通过数值研究与解析方法相结合的方法来回答这个问题。这个问题的答案具有内在的数学意义和数值模拟的实际重要性。本研究的可能结果包括：1)欧拉方程的不同正则化在固定时间给出不同的极限。这在本质上和应用上都是很有趣的。2)极限是相同的，在包括粘性情况在内的所有情况下都存在旋涡团法中观察到的混沌特征。这将对真实流体运动的物理学有重要的见解。3)极限相同，且不包含混沌特征。这将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。它将导致对欧拉方程的正则化和小粘度流体动力学的更多理解。

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.