A thin film equation with a singular diffusion

In this paper, a thin film equation with a singular diffusion term ut+∇·(un∇Δu)+A∇·∇uuα=0 is studied with periodic boundary conditions in multidimensional space. It has a lot of applications in fluids theory such as draining of foams and the movement of contact lenses. In order to overcome the difficulty of singular diffusion term, the approximation problems need to be constructed and the entropy functional method is applied for the limit process. Moreover, a compactness argument shows the existence of nonnegative weak solutions.