Dynamical Behaviour of a Soap Film Bridge Driven by an Electrostatic Force

We study a free boundary problem describing a tubular soap film bridge driven by an electrostatic force. To realize the electrostatic force, the soap film bridge is placed inside an outer metal cylinder and a voltage difference between this cylinder and the soap film is applied. Our model consists of a quasilinear parabolic equation for the evolution of the film coupled with an elliptic equation for the electrostatic potential in the unknown domain between outer cylinder and soap film bridge. For the rotationally symmetric case, we prove local well-posedness of this free boundary problem by recasting it as a single quasilinear parabolic equation with a non-local source term. Moreover, for large applied voltages, we show that solutions do not exist globally for a wide class of initial values.