Cahn-Hilliard equation on the boundary with bulk condition of Allen-Cahn type

TL;DR

This paper investigates the well-posedness of a coupled bulk-surface PDE system involving the Cahn-Hilliard equation on the boundary with Allen-Cahn type conditions, establishing existence and continuous dependence of solutions.

Contribution

It extends the analysis of Cahn-Hilliard equations by incorporating dynamic boundary conditions of Allen-Cahn type and proves well-posedness for this coupled system.

Findings

Proved existence of solutions for the coupled system.

Established continuous dependence on initial data.

Extended previous results to include boundary dynamics of Allen-Cahn type.

Abstract

The well-posedness for a system of partial differential equations and dynamic boundary conditions is discussed. This system is a sort of transmission problem between the dynamics in the bulk $\Omega $ and on the boundary $\Gamma$. The Poisson equation for the chemical potential, the Allen-Cahn equation for the order parameter in the bulk $\Omega$ are considered as auxiliary conditions for solving the Cahn-Hilliard equation on the boundary $\Gamma$. Recently the well-posedness for the equation and dynamic boundary condition, both of Cahn-Hilliard type, was discussed. Based on this result, the existence of the solution and its continuous dependence on the data are proved.