On a class of geometry-driven free boundary problems (Q2783741)

The author considers a certain class of free boundary problems in the plane for which the Cauchy transform of the time-evolving domain satisfies a partial differential equation of conservation type. It is shown that such problems have exact solutions expressible in terms of a finite set of time-evolving parameters. There is given a theorem which shows that solutions corresponding to certain initial conditions possess nontrivial conserved quantities. It appears that the evolution of a fluid blob in a rotating Hele-Show cell as well as the evolution of highly viscous fluid under the effects of surface tension are distinct examples of the above class. Note that what the author calls the Cauchy transform bears also the names the Theodoresco transform and the Hilbert transform in the literature.