Archive for Rational Mechanics and Analysis

, Volume 233, Issue 1, pp 167–247 | Cite as

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

  • Chun Liu
  • Hao WuEmail author


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 \({t\to+\infty}\) and characterize the stability of local energy minimizers for the system.


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The authorswould like to thank the anonymous referees for their careful reading of an initial version of this paper and for several helpful comments that allowed us to improve the presentation. The authors also want to thank Professors P. Colli, T. Fukao, C. Gal, H. Garcke, T.-Z. Qian and U. Stefanelli for helpful discussions.

Compliance with Ethical Standards


C. Liu is partially supported byNSF grantsDMS-1714401, DMS-1412005. H. Wu is partially supported by NNSFC grant No. 11631011 and the Shanghai Center for Mathematical Sciences at Fudan University.

Conflict of Interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA
  2. 2.School of Mathematical SciencesFudan UniversityShanghaiChina
  3. 3.Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University)Ministry of EducationShanghaiChina
  4. 4.Shanghai Key Laboratory for Contemporary Applied MathematicsFudan UniversityShanghaiChina

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