Goldstein, Gisèle Ruiz; Miranville, Alain A Cahn-Hilliard-Gurtin model with dynamic boundary conditions. (English) Zbl 1277.35202 Discrete Contin. Dyn. Syst., Ser. S 6, No. 2, 387-400 (2013). 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 [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. Reviewer: Dirk Blömker (Augsburg) Cited in 16 Documents MSC: 35K51 Initial-boundary value problems for second-order parabolic systems 35K55 Nonlinear parabolic equations 35D30 Weak solutions to PDEs 80A22 Stefan problems, phase changes, etc. 74N25 Transformations involving diffusion in solids 74A15 Thermodynamics in solid mechanics 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness Keywords:Galerkin method; energy estimates Citations:Zbl 0885.35121 PDFBibTeX XMLCite \textit{G. R. Goldstein} and \textit{A. Miranville}, Discrete Contin. Dyn. Syst., Ser. S 6, No. 2, 387--400 (2013; Zbl 1277.35202) Full Text: DOI