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

## TL;DR

This paper proves the existence and uniqueness of solutions for a generalized Cahn--Hilliard system with dynamic boundary conditions, including a Robin boundary approximation, and analyzes convergence and error estimates with supporting numerical experiments.

## Contribution

It introduces a Robin boundary approximation for a Cahn--Hilliard system with dynamic boundary conditions and establishes convergence and error estimates for the approximation.

## Key findings

- Existence and uniqueness of weak solutions for the Robin boundary model.
- Weak convergence of solutions as the regularization parameter tends to zero.
- Numerical experiments reveal complex dynamics not seen in the original model.

## 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), in which the new dynamic boundary condition is further generalised with an affine linear relation between the surface and bulk order parameters. 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 order parameter, 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.