We prove well-posedness for the initial value problem for a vortex sheet in 3D fluids, in the presence of surface tension. We first reformulate the problem by making a favorable choice of variables and parameterizations. We then perform energy estimates for the evolution equations. It is important to note that the Kelvin-Helmholtz instability is present for the vortex sheet in the absence of surface tension. Accordingly, we must construct the energy functional carefully with an eye toward the regularization of this instability. Well-posedness follows from the estimates.
"Well-posedness of 3D vortex sheets with surface tension." Commun. Math. Sci. 5 (2) 391 - 430, June 2007.