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In this paper, we establish a novel approach to proving existence ofnon-negative weak solutions for degenerate parabolic equations of fourth order,like the Cahn-Hilliard and certain thin film equations. The consideredevolution equations are in the form of a gradient flow for a perturbedDirichlet energy with respect to a Wasserstein-like transport metric, and weaksolutions are obtained as curves of maximal slope. Our main assumption is thatthe mobility of the particles is a concave function of their spatial density. Aqualitative difference of our approach to previous ones is that essentialproperties of the solution - non-negativity, conservation of the total mass anddissipation of the energy - are automatically guaranteed by the constructionfrom minimizing movements in the energy landscape.

Cahn-Hilliard and Thin Film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics