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

¹ Fakultät für Mathematik, Universität Regensburg, 93053 Regensburg, Germany
² Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
³ Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
⁴ Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany

^* Corresponding author: patrik.knopf@ur.de

Received: 30 April 2020
Accepted: 21 December 2020

Abstract

The Cahn–Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for the Cahn–Hilliard equation have been proposed and investigated in recent times. Of particular interests are the model by Goldstein et al. [Phys. D 240 (2011) 754–766] and the model by Liu and Wu [Arch. Ration. Mech. Anal. 233 (2019) 167–247]. Both of these models satisfy similar physical properties but differ greatly in their mass conservation behaviour. In this paper we introduce a new model which interpolates between these previous models, and investigate analytical properties such as the existence of unique solutions and convergence to the previous models mentioned above in both the weak and the strong sense. For the strong convergences we also establish rates in terms of the interpolation parameter, which are supported by numerical simulations obtained from a fully discrete, unconditionally stable and convergent finite element scheme for the new interpolation model.