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

Chun Liu, Hao Wu

Research output: Contribution to journalArticle

4 Citations (Scopus)

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: 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 the asymptotic limit as t→ + ∞ and characterize the stability of local energy minimizers for the system.

Original languageEnglish (US)
Pages (from-to)167-247
Number of pages81
JournalArchive for Rational Mechanics and Analysis
Volume233
Issue number1
DOIs
StatePublished - Jul 1 2019

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Dynamic Boundary Conditions
Cahn-Hilliard Equation
Variational Approach
Mathematical Analysis
Boundary conditions
Surface Diffusion
Surface diffusion
Asymptotic Limit
Binary Mixtures
Phase Separation
Energy Dissipation
Systems of Partial Differential Equations
Strong Solution
Binary mixtures
Energy
Minimizer
Phase separation
Initial-boundary-value Problem
Boundary value problems
Partial differential equations

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematics (miscellaneous)
  • Mechanical Engineering

Cite this

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