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

Clément Cancès (1, 2) , Daniel Matthes (3) , Flore Nabet (4)

1 RAPSODI - Reliable numerical approximations of dissipative systems
2 LPP - Laboratoire Paul Painlevé - UMR 8524
3 TUM - Technische Universität Munchen - Technical University Munich - Université Technique de Munich
4 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

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.

Mots clés: Multiphase flow; constrained Wasserstein gradient flow; Cahn-Hilliard type system

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Clément Cancès, Daniel Matthes, Flore Nabet. A two-phase two-fluxes degenerate Cahn-Hilliard model as constrained Wasserstein gradient flow. 2018. ⟨hal-01665338v2⟩