The Cahn-Hilliard equation with dynamic boundary conditions

Abstract

This paper is concerned with the following Cahn-Hilliard equation $ \psi _t = \Delta \mu, $ where $ \mu= -\Delta \psi -\psi +\psi ^3, $ subject on the boundary $\Gamma$ to the following dynamic boundary condition $ \sigma _s \Delta _{||} \psi - \partial _{\nu} \psi + h_s -g_s \psi = \frac{1}{\Gamma _s} \psi _t $ and $ \partial _{\nu} \mu =0, $ and the initial condition $ \psi |_{t=0}= \psi _0. $ This problem was recently proposed by physicists to describe spinodal decomposition of binary mixtures where the effective interaction between the wall (i.e., the boundary $\Gamma$) and two mixture components are short-ranged. The global existence and uniqueness of solutions to this initial boundary value problem with highest-order boundary conditions is proved.