Seminar on partial differential equations

Weak solutions to a two-phase thin film equation with insoluble surfactant

We discuss a model describing the spreading of an insoluble surfactant on the upper surface of a viscous complete wetting two-phase thin film. Considering capillary effects as the only driving force, the system of evolution equations consists of two strongly coupled degenerated equations of fourth order describing the film heights of the fluids, which are additionally coupled to a second-order transport equation for the surfactant concentration. Owing to the degeneracy, it is in general not clear whether one can prove the existence of global solutions in a classical sense, which motivates the study of weak solutions. The proof of existence of nonnegative global weak solutions is based on a priori energy estimates and compactness arguments.