Computation of symmetric, time-periodic solutions of the vortex sheet with surface tension.

A numerical method is introduced for the computation of time-periodic vortex sheets with surface tension separating two immiscible, irrotational, two-dimensional ideal fluids of equal density. The approach is based on minimizing a nonlinear functional of the initial conditions and supposed period that is positive unless the solution is periodic, in which case it is zero. An adjoint-based optimal control technique is used to efficiently compute the gradient of this functional. Special care is required to handle singular integrals in the adjoint formulation. Starting with a solution of the linearized problem about the flat rest state, a family of smooth, symmetric breathers is found that, at quarter-period time intervals, alternately pass through a flat state of maximal kinetic energy, and a rest state in which all the energy is stored as potential energy in the interface. In some cases, the interface overturns before returning to the initial, flat configuration. It is found that the bifurcation diagram describing these solutions contains several disjoint curves separated by near-bifurcation events.

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.