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Description

We study singularity formation of 3-D vortex sheets using a new approach. First, we derive a leading order approximation to the boundary integral equation governing the 3-D vortex sheet. This leading order equation captures the most singular contribution of the integral equation. Moreover, after applying a transformation to the physical variables, we found that this leading order 3-D vortex sheet equation de-generates into a two-dimensional vortex sheet equation in the direction of the tangential velocity jump. This rather surprising result confirms that the tangential velocity jump is the physical driving force of the vortex sheet singularities. We also show that the singularity type of the three-dimensional problem is similar to that of the two-dimensional problem. Furthermore by using the abstract Cauchy-Kowalewski theorem, we prove the long time existence of 3-D vortex sheets for analytic initial data. The existence time can be arbitrarily close to the singularity time predicted by the leading order equation when the initial condition is near equilibrium. A generalized Moore's approximation to 3-D vortex sheets is introduced. Detailed numerical study will be provided to support the analytic results, and to reveal the generic form of the three-dimensional nature of the vortex sheet singularity.