Singularity formation in a model for the vortex sheet with surface tension

In a recent analytical study, the author has proved well-posedness of the vortex sheet with surface tension. This work included using a formulation of the problem introduced by Hou, Lowengrub, and Shelley for a numerical study of the same problem. The analytical study required identification of a term in the evolution equations which can be viewed as being responsible for the Kelvin-Helmholtz instability; this term is of lower order than the surface tension term. In the present work, the author introduces a simple model for the vortex sheet with surface tension which maintains the same dispersion relation and the same destabilizing force as in the vortex sheet with surface tension. For the model problem, it is found that finite-time singularities can form when the initial data is taken from a certain class. For the vortex sheet with surface tension, the only observed singularities thus far in numerical work have coincided with self-intersection of the fluid interface. There is no analogue of self-intersection in the model problem, and thus the singularities observed in the present work may well be related to a previously unobserved singularity for the full vortex sheet problem.