Singularities in complex interfaces

Abstract

We analyse an equation describing the motion of the material interface between two fluids in a pressure field. The interface can be expressed as the image of the unit circle under a certain time depending conformal map. This conformal transformation maps the exterior of the unit circle onto the region occupied by one of the fluids. The conformal map has singularities in the unit disc. As long as these singularities are close to the origin, the complicated non-local equation governing the evolution of the conformal map can be approximated by a somewhat simpler, local equation. We prove that there exist self-similar solutions of this equation, that they have singularities away from the origin, that these singularities hit in finite time the unit circle and that the self-similar blow up is stable to perturbations that respect the symmetry of the self-similar profile.