  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.
