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

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

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, USA

    Chun Liu

  2. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Hao Wu

  3. Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, Shanghai, 200433, China

    Hao Wu

  4. Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China

    Hao Wu

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  1. Chun Liu
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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.

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Liu, C., Wu, H. An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis. Arch Rational Mech Anal 233, 167–247 (2019). https://doi.org/10.1007/s00205-019-01356-x

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