Regularization of the Kelvin-Helmholtz instability by surface tension

The Kelvin-Helmholtz instability is present in the motion of a vortex sheet without surface tension. This can be seen from the linearization of the equations of motion, and there have also been proofs of ill-posedness for the full nonlinear equations. In the presence of surface tension, the linearized equations no longer exhibit an instability, and it has been believed that the full equations should then be well-posed. In this paper, I sketch a proof that the vortex sheet with surface tension is well-posed in the case of both two- and three-dimensional fluids. The proof in the case of three-dimensional fluids is the joint work with Nader Masmoudi. The method is to first reformulate the problem using suitable variables and parametrizations, and then to perform energy estimates. The choice of variables and parametrizations in the two-dimensional case is the same as that of Hou et al. in a prior numerical work.