The classical vortex sheet is ill-posed: it exhibits the well-known Kelvin-Helmholtz instability. In the linearized equations of motion, surface tension removes the instability. It has been conjectured that surface tension also makes the full problem well-posed. We prove that this conjecture is correct using energy methods. In particular, for the initial value problem for vortex sheets with surface tension with sufficiently smooth data, it is proved that solutions exist locally in time, are unique, and depend continuously on the initial data. The analysis uses two important ideas from the numerical work of Hou, Lowengrub, and Shelley. First, the tangent angle and arclength of the vortex sheet are evolved instead of Cartesian variables. Second, instead of using a purely Lagrangian formulation, a special tangential velocity is used to simplify the evolution equations. The proof is valid in both the density-matched case and the case of fluids with different densities. A special case of the different-density result is the case of water waves with surface tension; this is the first proof of well-posedness which allows the wave to overturn.
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