On a Hele-Shaw Type Domain Evolution with Convected Surface Energy Density: The Third-Order Problem

We investigate a moving boundary problem with a gradient flow structure which generalizes Hele-Shaw flow driven solely by surface tension to the case of nonconstant surface tension coefficient taken along with the liquid particles at boundary. The resulting evolution problem is first order in time, contains a third-order nonlinear pseudodifferential operator and is degenerate parabolic. Well-posedness of this problem in Sobolev scales is proved by showing semiboundedness of the third order operator with respect to a variable symmetric bilinear form. Moreover, we show that any shape of the liquid domain near a ball is an equilibrium for some appropriately chosen distribution of the surface tension coefficient. Finally, a numerical example is given.