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

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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.