On the Cahn–Hilliard equation with kinetic rate dependent dynamic boundary conditions and non-smooth potentials: Well-posedness and asymptotic limits

Abstract

We analyze a class of Cahn–Hilliard equations with kinetic rate dependent dynamic boundary conditions that describe possible short-range interactions between the binary mixture and the solid boundary. In the presence of surface diffusion on the boundary, the initial boundary value problem can be viewed as a transmission problem consisting of Cahn–Hilliard-type equations both in the bulk and on the boundary. We first establish the existence, uniqueness, and continuous dependence of global weak solutions. In the construction of weak solutions, an explicit convergence rate in terms of the parameter for the Yosida approximation is obtained. Under some additional assumptions, we further prove the existence and uniqueness of global strong solutions. Next, we study the asymptotic limit as the coefficient of the boundary diffusion goes to zero and show that the limit problem with a forward-backward dynamic boundary condition is well posed in a suitable weak formulation. At last, we investigate the asymptotic limits as the kinetic rate tends to zero and infinity, respectively. Our results are valid for a general class of bulk and boundary potentials with double-well structure, including the physically relevant logarithmic potential and the non-smooth double obstacle potential.