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

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.∇ · J tot = 0 where J tot = J 1 + J 2 1991 Mathematics Subject Classification. 35K65, 35K41, 49J40, 76T99.

Finite Volume approximation of a two-phase two fluxes degenerate Cahn–Hilliard model

“…The existence of a weak solution has been established in [8] by showing the convergence of a minimizing movement schemeà la Jordan et al [23]. Note that in [8], the case of a convex three-dimensional domain Ω is also addressed, but it relies on the fact that the 2 ( ) estimate on ∆ 1 yields a 2 ((0, ); 2 (Ω)) estimate on for which we don't have an equivalent at the discrete level. This is why we restrict our attention on the case Ω ⊂ R 2 (but not necessarily convex) in this paper.…”

“…Before entering the core of the paper, which is devoted to the convergence analysis of a finite volume scheme, let us briefly discuss the model under consideration, and in particular its difference with respect to the usual Cahn-Hilliard system. We refer to [8,29] for a more developed discussions on this purpose. The classical degenerate Cahn-Hilliard equation which is the closest one to our system (1.1)-(1.7) writes…”

A degenerate Cahn‐Hilliard model as constrained Wasserstein gradient flow

“…Our main result [3] is: Theorem 2.1 Given ρ 0 ∈ X withĒ( ρ 0 ) < ∞, then there exists at least one weak solution ( ρ t ) t≥0 to (1).…”

Energy stable numerical methods for porous media flow type problems

“…Such an estimate is the corner-stone in the study [89] where the convergence of an upstream mobility scheme was established for the Dupuit approximation of multiphase porous media flows. It was also used in [8], or in [90] where a degenerate Cahn-Hilliard system with a very similar mathematical structure to (1)-(4) (see [91,92]) was considered.…”