Asymptotic analysis of the Allen-Cahn equation with dynamic boundary conditions of Cahn-Hilliard type

TL;DR

This paper investigates the asymptotic behavior of a coupled bulk-surface PDE system where the Allen-Cahn equation in the bulk interacts with a Cahn-Hilliard type boundary condition, analyzing well-posedness and error estimates for various limits.

Contribution

It introduces and analyzes a novel model coupling Allen-Cahn bulk dynamics with Cahn-Hilliard boundary conditions, including asymptotic analysis and error estimates.

Findings

Established well-posedness of the coupled system.

Derived error estimates for asymptotic limits.

Analyzed three types of limiting systems.

Abstract

Problems for partial differential equations coupled with dynamic boundary conditions can be viewed as a type of transmission problem between the bulk and its boundary. For the heat equation and the Allen-Cahn equation, various forms of such problems with dynamic boundary conditions are studied in this paper. In the case of the Cahn-Hilliard equation in the bulk, several models have been proposed in which the boundary equations and conditions differ. Recently, the vanishing surface diffusion limit has been investigated in more than one of these models. In such settings, the resulting dynamic boundary equation typically takes the form of a forward-backward parabolic equation. In this paper, we focus on a different model, in which the Allen-Cahn equation governs the bulk dynamics, while the boundary condition is of Cahn-Hilliard type. We analyze the asymptotic behavior of the system, including the well-posedness of the limiting problems and corresponding error estimates for the differences between solutions. These aspects are discussed for three types of limiting systems.