Surfactant Spreading on Thin Viscous Films: Nonnegative Solutions of A Coupled Degenerate System

We consider the Navier--Stokes system for an incompressible fluid coupled with a convection-diffusion equation for surfactant molecules on the free surface. The lubrication approximation leads to a coupled system of parabolic equations, consisting of a degenerate fourth-order equation for the film height and a second-order equation for the surfactant concentration. A proof based on energy estimates shows the existence of global weak solutions which in addition fulfill an integral inequality (entropy condition) which ensures positivity properties for the solution.