Local Well-Posedness for Fluid Interface Problems

In this paper, we prove the local well-posedness of the fluid interface problem with surface tension where the velocity fields are not assumed to be irrotational and the fluid domains are not assumed to be simply connected. Viewed as a Lagrangian system with the configuration space being an infinite dimensional manifold possessing many symmetries, this problem is reduced to the evolution of the interface, determined by its mean curvature, and the evolution of the rotational part of the velocity fields, determined by the symmetries. This framework also applies to several other fluid surface problems which are outlined in the paper.