A degenerate Cahn‐Hilliard model as constrained Wasserstein gradient flow

Existence of solutions to a non‐local Cahn‐Hilliard model with degenerate mobility is considered. The PDE is written as a gradient flow with respect to the L2‐Wasserstein metric for two components that are coupled by an incompressibility constraint. Approximating solutions are constructed by means of an implicit discretization in time and variational methods.

Below, we show how (1) is derived from variational principles. And we indicate the proof for existence of weak solutions.
The first choice leads to the most classical Cahn-Hilliard model; for the second choice, E is the famous Flory-Huggins-deGennes energy [2]. Notice that in both cases, the integral involving the squared gradients is convex in ρ, whereas the integral involving ρη is not. To incorporate the volume constraint, let the functionalĒ : in Ω, andĒ( ρ) = +∞ otherwise.
Wasserstein metric. The Wasserstein metric W on probability densities on Ω can be defined in two ways, referred to as primal and dual formulation of the optimal transport problem: We shall need both formulations in the sequel. In the primal problem, one minimizes the kinetic energy over all velocity fields v : Ω → R n which are such that if all "mass particles" of ρ 1 are moved along v, then the resulting density is ρ 0 . In the dual problem, one optimizes over all Kantorovich potentials φ,φ : Ω → R that can be understood as optimal prices for transshipment. The optimizers are attained in both problems, and are related in the sense that v opt = ∇φ opt .
Formally, a curve ρ (·) : [0, ∞) → X is a solution to the gradient flow ofĒ w.r.t.Ŵ ifĝ ρt [∂ t ρ t , p] = − DĒ( ρ t )[ p] holds for any tangent vector p at ρ t : The differential on the right hand side is undefined unless p preserves the volume constraint, that is q = −p. In that case, we have for the solutions This produces the coupled system (1).
Time-discrete versions of the continuity equations for ρ and η in (1) are a consequence of the primal formulation (4) of the optimal transport problem: with v n τ = ∇φ n τ and w n τ = ∇ψ n τ being the optimal vector fields for the passage from ρ n−1 τ to ρ n τ , we conclude from (id +v n τ )#ρ n τ = ρ n−1 τ that Ω ρ n τ − ρ n−1 τ τ θ dx = ∇θ · ∇v n τ ρ n τ dx + O(τ ) for each θ ∈ C ∞ c (Ω), and likewise for η τ . To derive the Euler-Lagrange equation involving φ − ψ in (1), we use the dual formulation (5) of optimal transport, and comparē from above withĒ τ ( ρ ; ρ n−1 τ ) for some variation ρ of ρ n τ , and from below with the same expression as above, in which φ n τ , ψ n τ are replaced by the respective potentials φ , ψ for ρ . This provides a rigorous justification of the formal calculation leading to (6) above. Finally, to pass to the limit in the discrete system, we justify the following a priori estimate for the time-discrete approximation: