A Cahn-Hilliard-Gurtin model with dynamic boundary conditions
The authors study the following generalization of the Cahn-Hilliard system \[ \frac{\partial \rho}{\partial t} - a \cdot \nabla \frac{\partial \rho}{\partial t} = \text{div}(B\nabla \mu), \] \[ \mu-b \cdot \nabla \mu = \beta \frac{\partial \rho}{\partial t}-\alpha\Delta \rho + f(\rho) \;, \] where \(\rho\) is an order parameter and \(\mu\) is the chemical potential. This was proposed by [\textit{M. E. Gurtin}, Physica D 92, No. 3--4, 178--192 (1996; Zbl 0885.35121)], based on a balance law for internal microforces and a purely mechanical version of the second law. The authors actually consider a more general version, where \(a,\;B,\;\beta,\) and \(b\) are allowed to depend on the data of the problem \((\rho,\nabla\rho,\mu,\nabla\mu,\partial_t\rho)\). In order to take into account the interaction with the walls in confined systems, it is proposed that Gurtin's system should be combined with boundary conditions of the form \[ \frac{\partial \rho}{\partial t} + (a \cdot \nu)\frac{\partial \rho}{\partial t} = \delta \Delta_\Gamma \mu - \nu\cdot(B\nabla \mu), \] \[ \mu + (b \cdot \nu)\mu = \beta\frac{\partial \rho}{\partial t}- \eta\Delta_\Gamma \rho + \alpha\frac{\partial \rho}{\partial \nu}+g(\rho), \] where \(\Delta_\Gamma\) is the Laplace-Beltrami operator on the boundary \(\Gamma\), and \(\nu\) is the unit outward normal on \(\Gamma\). The main result of the paper yields the existence of a unique weak solution to the initial-boundary value problem, where the key assumption is coercivity. The proof of existence is based on a standard Galerkin method, while the uniqueness follows from energy type estimates for the difference of two solutions.
scientific article; zbMATH DE number 6144723