## Recent progress in mathematical analysis of vortex sheets

*Sijue Wu*, University of Michigan, Ann Arbor

We consider the motion of the interface separating two domains
of the same fluid that moves with different velocities along the
tangential direction of the interface. The evolution of the interface
(the vortex sheet) is governed by the Birkhoff–Rott equations. We
investigate the specific nature of the vortex sheet motion, in
particular after the singularity formation; and consider the question
of the weakest possible assumptions such that the Birkhoff–Rott
equation makes sense. This leads us to introduce chord-arc curves to
this problem. We present three results. The first can be stated as the
following: Assume that the Birkhoff–Rott equation has a solution in a
weak sense and that the vortex strength is bounded away from 0 and
infinity. Moreover, assume that the solution gives rise to a vortex
sheet curve that is chord-arc. Then the curve is automatically
smooth, in fact analytic, for fixed time. The second and third results
demonstrate that the Birkhoff–Rott equation can be solved if and only
if ONLY half the initial data is given.

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