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

TL;DR

This paper introduces a new method for proving the existence of non-negative weak solutions to degenerate fourth-order parabolic equations like Cahn-Hilliard and thin film equations, using a gradient flow framework in weighted Wasserstein metrics.

Contribution

It develops a novel approach based on minimizing movements in a Wasserstein-like metric to establish solutions with guaranteed non-negativity, mass conservation, and energy dissipation.

Findings

Existence of non-negative weak solutions for the equations.

Solutions preserve mass and dissipate energy.

Method applies to equations with concave mobility functions.

Abstract

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, and 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.