The Cahn-Hilliard equation with forward-backward dynamic boundary condition via vanishing viscosity

TL;DR

This paper analyzes the asymptotic behavior of the Cahn-Hilliard equation with dynamic boundary conditions as surface diffusion vanishes, revealing a forward-backward boundary condition and establishing well-posedness and regularity results.

Contribution

It introduces a general framework for the limit of the Cahn-Hilliard system with dynamic boundary conditions, including nonlinear maximal monotone graphs, and proves well-posedness and regularity of the limit problem.

Findings

Limit problem has a forward-backward dynamic boundary condition.

Solution loses regularity in the limit, interpreted via subdifferential inclusion.

Error estimates are established for the vanishing diffusion parameter.

Abstract

An asymptotic analysis for a system with equation and dynamic boundary condition of Cahn-Hilliard type is carried out as the coefficient of the surface diffusion acting on the phase variable tends to 0, thus obtaining a forward-backward dynamic boundary condition at the limit. This is done in a very general setting, with nonlinear terms admitting maximal monotone graphs both in the bulk and on the boundary. The two graphs are related by a growth condition, with the boundary graph that dominates the other one. It turns out that in the limiting procedure the solution of the problem looses some regularity and the limit equation has to be properly interpreted in the sense of a subdifferential inclusion. However, the limit problem is still well-posed since a continuous dependence estimate can be proved. Moreover, in the case when the two graphs exhibit the same growth, it is shown that the solution enjoys more regularity and the boundary condition holds almost everywhere. An error estimate can also be shown, for a suitable order of the diffusion parameter.