Weak Solutions of the Cahn--Hilliard System with Dynamic Boundary Conditions: A Gradient Flow Approach

The Cahn-Hilliard equation is the most common model to describe phase separation processes of a mixture of two components. For a better description of short-range interactions of the material with the solid wall, various dynamic boundary conditions have been considered in recent times. In this paper we study a Cahn-Hilliard model with a dynamic boundary condition that has recently been proposed by Liu and Wu [Arch. Ration. Mech. Anal., 233 (2019), pp. 167-247]. We prove the existence and uniqueness of a global weak solution to this model by interpreting the problem as a suitable gradient flow of the total free energy which contains volume as well as surface contributions. The formulation involves an inner product which couples bulk and surface quantities in an appropriate way. For our proof we use an implicit time discretization and we show that the obtained approximate solutions converge to a weak solution of the Cahn-Hilliard system. We point out that Liu and Wu used a slightly stronger notion of a weak solution in their corresponding well-posedness result. However, they needed very strong assumptions on the domain and its boundary which are not necessary for our approach