Convergence of Cahn-Hilliard systems to the Stefan problem with dynamic boundary conditions

TL;DR

This paper demonstrates how the Stefan problem with dynamic boundary conditions can be approximated and understood through the convergence of Cahn-Hilliard systems, providing insights into the mushy region and potential numerical methods.

Contribution

It establishes the convergence of Cahn-Hilliard systems to the Stefan problem with dynamic boundary conditions, linking fourth-order systems to phase change modeling.

Findings

Characterization of the mushy region via asymptotic limits

Convergence proof of Cahn-Hilliard to Stefan problem

Potential for numerical applications of the Cahn-Hilliard approach

Abstract

This paper examines the well-posedness of the Stefan problem with a dynamic boundary condition. To show the existence of the weak solution, the original problem is approximated by a limit of an equation and dynamic boundary condition of Cahn-Hilliard type. By using this Cahn-Hilliard approach, it becomes clear that the state of the mushy region of the Stefan problem is characterized by an asymptotic limit of the fourth-order system, which has a double-well structure. This fact also raises the possibility of the numerical application of the Cahn-Hilliard system to the degenerate parabolic equation, of which the Stefan problem is one.