Convergence of a Robin boundary approximation for a Cahn–Hilliard system with dynamic boundary conditions

Abstract

We prove the existence of unique weak solutions to an extension of a Cahn-Hilliard model proposed recently by C Liu and H Wu (2019 Arch. Ration. Mech. Anal. 233 167-247), in which the new dynamic boundary condition is further generalised with an affine linear relation between the surface and bulk phase field variables. As a first approach to tackle more general and nonlinear relations, we investigate the existence of unique weak solutions to a regularisation by a Robin boundary condition. Included in our analysis is the case where there is no diffusion for the surface phase field, which causes new difficulties for the analysis of the Robin system. Furthermore, for the case of affine linear relations, we show the weak convergence of solutions as the regularisation parameter tends to zero, and derive an error estimate between the two models. This is supported by numerical experiments which also demonstrate some non-trivial dynamics for the extended Liu-Wu model that is not present in the original model.

Phase-field dynamics with transfer of materials: The Cahn–Hilliard equation with reaction rate dependent dynamic boundary conditions

“…The requirement ∈ ( ,0 ) −1 will be verified in Theorem 3.1. For a more detailed derivation of the gradient flow equation in similar situations see Section 3 of [26] and Section 3 of [32].…”

“…To prove the assertions, we construct approximate solutions via an implicit time discretisation of the gradient flow equation (2.4). This technique which goes back to [3] was first applied on a Cahn-Hilliard problem with dynamic boundary conditions in [26], and later also in [32]. In the subsequent proof, we will employ the same strategy.…”

“…Let us mention that a variant of the system (1.11) was proposed and investigated in [32], where equation (1.11b) is replaced by…”

“…The relation between and is described by the Robin type transmission condition n = ( ) − on Σ with > 0 and a function ∈ 2 (R) satisfying suitable growth conditions. In particular, it is rigorously established in [32] that, in the case ( ) = , solutions of this model converge to solutions of (1.11) in the limit → 0 in some suitable sense.…”

Convergence to equilibrium for a bulk–surface Allen–Cahn system coupled through a nonlinear Robin boundary condition

“…It will be interesting to perform corresponding analysis on the widely studied Cahn-Hilliard equation. While there are numerous contributions for the analysis of the Cahn-Hilliard equation with dynamic boundary conditions, amongst which we list the works [7, 9, 11, 13, 15, 19, 22-24, 35-37, 39, 44, 46], to the best of our knowledge, only recently the analysis of a corresponding Cahn-Hilliard model with affine linear relations and its Robin approximation has been performed in [32], but the analysis with a general transmission relation u| Γ = h(φ) between the bulk variable u and the surface variable φ has not yet received much attention. These will be the topics of our future study.…”