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A countably infinite family of Liapunov functionals is constructed for thethin film Muskat problem, which is a second-order degenerate parabolic systemfeaturing cross-diffusion. More precisely, for each n $\ge$ 2 we construct anhomogeneous polynomial of degree n, which is convex on [0, $\infty$)^2 , withthe property that its integral is a Liapunov functional for the problem.Existence of global bounded non-negative weak solutions is then shown in onespace dimension.

Bounded weak solutions to the thin film Muskat problem via an infinite family of Liapunov functionals