Well-Posedness of Vortex Sheets with Surface Tension

Abstract

We study the initial value problem for two-dimensional, periodic vortex sheets with surface tension. We allow the upper and lower fluids to have different densities. Without surface tension, the 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 used rather than Cartesian variables. Second, instead of a purely Lagrangian formulation, a special tangential velocity is used in order to simplify the evolution equations. A special case of the result is well-posedness of water waves with surface tension; this is the first proof (with surface tension) which allows the wave to overturn.