Inserted: 25 jan 2012
Last Updated: 18 may 2012
Journal: Journal of differential equations
In this paper, we establish a novel approach to proving existence of non-negative weak solutions for degenerate parabolic equations of fourth order, like the Cahn-Hilliard and certain thin film equations. The considered evolution equations are in the form of a gradient flow for a perturbed Dirichlet energy with respect to a Wasserstein-like transport metric, weak solutions are obtained as curves of maximal slope. Our main assumption is that the mobility of the particles is a concave function of their spatial density. A qualitative difference of our approach to previous ones is that essential properties of the solution -- non-negativity, conservation of the total mass and dissipation of the energy -- are automatically guaranteed by the construction from minimizing movements in the energy landscape.
Keywords: Thin film equation, Metric gradient flow, Wasserstein metric, Fourth order parabolic equation, Cahn–Hilliard equation