A two-phase two-fluxes degenerate Cahn-Hilliard model as constrained Wasserstein gradient flow

TL;DR

This paper introduces a non-local Cahn-Hilliard model with linear mobilities, proving the existence of weak solutions and demonstrating faster energy decay compared to classical models through numerical experiments.

Contribution

It provides a rigorous proof of weak solutions for a novel non-local Cahn-Hilliard model formulated as a constrained Wasserstein gradient flow.

Findings

Weak solutions exist for the non-local model.

Numerical experiments show faster energy decay.

The model differs from classical Cahn-Hilliard with linear mobilities.

Abstract

We study a non-local version of the Cahn-Hilliard dynamics for phase separation in a two-component incompressible and immiscible mixture with linear mobilities. In difference to the celebrated local model with nonlinear mobility, it is only assumed that the divergences of the two fluxes --- but not necessarily the fluxes themselves --- annihilate each other. Our main result is a rigorous proof of existence of weak solutions. The starting point is the formal representation of the dynamics as a constrained gradient flow in the Wasserstein metric. We then show that time-discrete approximations by means of the incremental minimizing movement scheme converge to a weak solution in the limit. Further, we compare the non-local model to the classical Cahn-Hilliard model in numerical experiments. Our results illustrate the significant speed-up in the decay of the free energy due to the higher degree of freedom for the velocity fields.