A thin film approximation of the Muskat problem with gravity and   capillary forces
  Existence of nonnegative weak solutions is shown for a thin film approximation of the Muskat problem with gravity and capillary forces taken into account. The model describes the space-time evolution of the heights of the two fluid layers and is a fully coupled system of two fourth order degenerate parabolic equations. The existence proof relies on the fact that this system can be viewed as a gradient flow for the 2-Wasserstein distance in the space of probability measures with finite second moment. 
Philippe Laurencot (IMT), Bogdan-Vasile Matioc
math.AP