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 LAURENÇOT. Bogdan-Vasile MATIOC. "A thin film approximation of the Muskat problem with gravity and capillary forces." J. Math. Soc. Japan 66 (4) 1043 - 1071, October, 2014. https://doi.org/10.2969/jmsj/06641043