On a Class of Geometry-Driven Free Boundary Problems

An abstract class of two-dimensional geometry-driven free boundary problems is considered in which the evolution of the Cauchy transform of the time-evolving domain satisfies a partial differential equation of conservation type. This idea generalizes an association between free boundary problems and the inverse gravitational problem. It is shown that this class of free boundary problems admits exact solutions expressible in terms of a finite set of time-evolving parameters. In addition, even when exact solutions are not available, a theorem is established which shows that solutions corresponding to certain initial conditions possess nontrivial conserved quantities. Two distinct problems of physical interest are shown to fall within this abstract class: the evolution of a fluid blob in a rotating Hele–Shaw cell and the evolution of highly viscous fluid under the effects of surface tension. Various aspects of these two problems are discussed.