Convergence of a Robin boundary approximation for a Cahn–Hilliard system with dynamic boundary conditions. (3rd July 2020)

• Record Type: Journal Article
• Authors: Knopf, Patrik; Lam, Kei Fong
• Is Part Of: Nonlinearity. Volume 33:Number 8(2020)
• Page Start: 4191
• Page End: 4235
• Publication Date: 2020-07-03
• Subjects: Cahn–Hilliard equation -- dynamic boundary conditions -- penalisation via Robin boundary conditions -- gradient flow; 35A01 -- 35A02 -- 35A35 -- 35B40; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab8351
• Languages: English
• ISSNs: 0951-7715

Abstract

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.