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.

中文翻译：

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具有动态边界条件的 Cahn-Hilliard 系统的 Robin 边界近似收敛

我们证明了 C.~Liu 和 H.~Wu (2019) 最近提出的 Cahn--Hilliard 模型扩展的唯一弱解的存在，其中新的动态边界条件进一步推广到表面和批量订单参数。作为处理更一般和非线性关系的第一种方法，我们研究了罗宾边界条件正则化的独特弱解的存在。我们的分析中包括了表面序参数没有扩散的情况，这给 Robin 系统的分析带来了新的困难。此外，对于仿射线性关系的情况，当正则化参数趋于零时，我们展示了解的弱收敛性，并导出了两个模型之间的误差估计。