An Energetic Variational Approach for the Cahn--Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis

TL;DR

This paper introduces a new energetic variational model for the Cahn--Hilliard equation with dynamic boundary conditions, ensuring physical constraints and providing rigorous mathematical analysis of solutions and stability.

Contribution

It proposes a novel class of dynamic boundary conditions derived via an energetic variational approach, ensuring physical constraints and mathematical well-posedness.

Findings

Existence and uniqueness of global solutions established.

Asymptotic behavior and stability of energy minimizers characterized.

Model naturally satisfies mass conservation, energy dissipation, and force balance.

Abstract

The Cahn--Hilliard equation is a fundamental model that describes phase separation processes of binary mixtures. In recent years, several types of dynamic boundary conditions have been proposed in order to account for possible short-range interactions of the material with the solid wall. Our first aim in this paper is to propose a new class of dynamic boundary conditions for the Cahn--Hilliard equation in a rather general setting. The derivation is based on an energetic variational approach that combines the least action principle and Onsager's principle of maximum energy dissipation. One feature of our model is that it naturally fulfills three important physical constraints such as conservation of mass, dissipation of energy and force balance relations. Next, we provide a comprehensive analysis of the resulting system of partial differential equations. Under suitable assumptions, we prove the existence and uniqueness of global weak/strong solutions to the initial boundary value problem with or without surface diffusion. Furthermore, we establish the uniqueness of asymptotic limit as $t\to+\infty$ and characterize the stability of local energy minimizers for the system.