A Two-Phase Two-Fluxes Degenerate Cahn–Hilliard Model as Constrained Wasserstein Gradient Flow

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. Differently 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 the 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.