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

TL;DR

This paper studies a complex Cahn-Hilliard model with dynamic boundary conditions, proving well-posedness and analyzing asymptotic limits for various parameters, including non-smooth potentials relevant in physical applications.

Contribution

It establishes existence, uniqueness, and continuous dependence of solutions for a generalized Cahn-Hilliard equation with dynamic boundary conditions, including non-smooth potentials, and analyzes key asymptotic limits.

Findings

Proved global weak solution existence and uniqueness.

Derived explicit convergence rates for Yosida approximation.

Analyzed asymptotic limits as boundary diffusion and kinetic rate parameters tend to zero or infinity.

Abstract

We consider a class of Cahn-Hilliard equation 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 prove existence, uniqueness and continuous dependence of global weak solutions. In the construction of solutions, an explicit convergence rate in terms of the parameter for the Yosida approximation is established. Under some additional assumptions, we also obtain 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. Besides, we investigate the asymptotic limit 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.