The Cahn–Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for the Cahn–Hilliard equation have been proposed and investigated in recent times. Of particular interests are the model by Goldstein et al. [Phys. D 240 (2011) 754–766] and the model by Liu and Wu [Arch. Ration. Mech. Anal. 233 (2019) 167–247]. Both of these models satisfy similar physical properties but differ greatly in their mass conservation behaviour. In this paper we introduce a new model which interpolates between these previous models, and investigate analytical properties such as the existence of unique solutions and convergence to the previous models mentioned above in both the weak and the strong sense. For the strong convergences we also establish rates in terms of the interpolation parameter, which are supported by numerical simulations obtained from a fully discrete, unconditionally stable and convergent finite element scheme for the new interpolation model.
Keywords: Cahn–Hilliard equation, dynamic boundary conditions, relaxation by Robin boundary conditions, gradient flow, finite element analysis
@article{M2AN_2021__55_1_229_0,
author = {Knopf, Patrik and Lam, Kei Fong and Liu, Chun and Metzger, Stefan},
title = {Phase-field dynamics with transfer of materials: {The} {Cahn{\textendash}Hilliard} equation with reaction rate dependent dynamic boundary conditions},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {229--282},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {1},
doi = {10.1051/m2an/2020090},
mrnumber = {4216838},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2020090/}
}
TY - JOUR AU - Knopf, Patrik AU - Lam, Kei Fong AU - Liu, Chun AU - Metzger, Stefan TI - Phase-field dynamics with transfer of materials: The Cahn–Hilliard equation with reaction rate dependent dynamic boundary conditions JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 229 EP - 282 VL - 55 IS - 1 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2020090/ DO - 10.1051/m2an/2020090 LA - en ID - M2AN_2021__55_1_229_0 ER -
%0 Journal Article %A Knopf, Patrik %A Lam, Kei Fong %A Liu, Chun %A Metzger, Stefan %T Phase-field dynamics with transfer of materials: The Cahn–Hilliard equation with reaction rate dependent dynamic boundary conditions %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 229-282 %V 55 %N 1 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2020090/ %R 10.1051/m2an/2020090 %G en %F M2AN_2021__55_1_229_0
Knopf, Patrik; Lam, Kei Fong; Liu, Chun; Metzger, Stefan. Phase-field dynamics with transfer of materials: The Cahn–Hilliard equation with reaction rate dependent dynamic boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 1, pp. 229-282. doi: 10.1051/m2an/2020090
[1] and , Convergence to equilibrium for the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. 67 (2007) 3176–3193. | MR | Zbl | DOI
[2] , Linear Functional Analysis – An Application-Oriented Introduction. Springer, London (2016). | MR | Zbl | DOI
[3] , and , Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser Basel (2008). | MR | Zbl
[4] and , The dynamics of nucleation for the Cahn-Hilliard equation. SIAM J. Appl. Math. 53 (1993) 990–1008. | MR | Zbl | DOI
[5] and , The Mathematical Theory of Finite Element Methods. Springer (2002). | MR | Zbl | DOI
[6] , and , On traces for in Lipschitz domains. J. Math. Anal. Appl. 276 (2002) 845–867. | MR | Zbl | DOI
[7] and , Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 2 (1958) 205–245.
[8] , and , On modeling and simulation of electrokinetic phenomena in two-phase flow with general mass densities. SIAM J. Appl. Math. 72 (2012) 1899–1925. | MR | Zbl | DOI
[9] , and , The Cahn-Hilliard equation with logarithmic potentials. Milan J. Math. 79 (2011) 561–596. | MR | Zbl | DOI
[10] and , Cahn-Hilliard equation with dynamic boundary conditions and mass constraint on the boundary. J. Math. Anal. Appl. 429 (2015) 1190–1213. | MR | DOI
[11] and , Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials. Nonlinear Anal. 127 (2015) 413–433. | MR | DOI
[12] , and , On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential. J. Math. Anal. Appl. 419 (2014) 972–994. | MR | Zbl | DOI
[13] , , and , A class of quasi-linear Allen-Cahn type equations with dynamic boundary conditions. Nonlinear Anal. 158 (2017) 32–59. | MR | DOI
[14] , The Cahn–Hilliard equation as a gradient flow. Master’s thesis, Simon Fraser University (2005).
[15] , Finite elements for the Beltrami operator on arbitrary surfaces. in: Lecture Notes in Mathematics, Springer, Berlin Heidelberg (1988) 142–155. | MR | Zbl | DOI
[16] and , Finite element methods for surface PDEs. Acta Numer. 22 (2013) 289–396. | MR | Zbl | DOI
[17] and , On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27 (1996) 404–424. | MR | Zbl | DOI
[18] and , Finite element analysis for a coupled bulk-surface partial differential equation. IMA J. Numer. Anal. 33 (2012) 377–402. | MR | Zbl | DOI
[19] and , On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96 (1986) 339–357. | MR | Zbl | DOI
[20] , and , Novel surface modes in spinodal decomposition. Phys. Rev. Lett. 79 (1997) 893–896. | DOI
[21] , , , , , and , Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall. J. Chem. Phys. 108 (1998) 3028–3037. | DOI
[22] and , Separation property and convergence to equilibrium for the equation and dynamic boundary condition of Cahn-Hilliard type with singular potential. To appear. In: Asymptot. Anal. DOI: (2020). | DOI | MR
[23] , A Cahn-Hilliard model in bounded domains with permeable walls. Math. Methods App. Sci. 29 (2006) 2009–2036. | MR | Zbl | DOI
[24] and , Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation. Discrete Contin. Dyn. Syst. 22 (2008) 1041–1063. | MR | Zbl | DOI
[25] , On Cahn-Hilliard systems with elasticity. Proc. Roy. Soc. Edinburgh 133A (2003) 307–331. | MR | Zbl | DOI
[26] and , Weak solutions of the Cahn-Hilliard equation with dynamic boundary conditions: A gradient flow approach. SIAM J. Math. Anal. 52 (2020) 340–369. | MR | DOI
[27] , and , A Cahn-Hilliard model in a domain with non permeable walls. Phys. D 240 (2011) 754–766. | MR | Zbl | DOI
[28] and , Two-phase flow with mass density contrast: stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model. J. Comput. Phys. 257 (2014) 708–725. | MR | DOI
[29] , and , On fully decoupled, convergent schemes for diffuse interface models for two-phase flow with general mass densities. Commun. Comput. Phys. 19 (2016) 1473–1502. | MR | DOI
[30] and , Error estimates for the Cahn-Hilliard equation with dynamic boundary conditions. Preprint [math.NA] (2020). | arXiv | MR
[31] , , , , , and , Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions. Comp. Phys. Comm. 133 (2001) 139–157. | MR | Zbl | DOI
[32] and , Convergence of a Robin boundary approximation for a Cahn-Hilliard system with dynamic boundary conditions. Nonlinearity 33 (2020) 4191–4235. | MR | DOI
[33] and , On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: Well-posedness, regularity theory and eigenvalue problems. Preprint [math.AP] (2020). | arXiv | MR
[34] , Passing from bulk to bulk-surface evolution in the Allen-Cahn equation. Nonlinear Differ. Equ. Appl. 20 (2013) 919–942. | MR | Zbl | DOI
[35] and , An energetic variational approach for the Cahn-Hilliard equation with dynamic boundary condition: model derivation and mathematical analysis. Arch. Ration. Mech. Anal. 233 (2019) 167–247. | MR | DOI
[36] , On convergent schemes for two-phase flow of dilute polymeric solutions. ESAIM:M2AN 52 (2018) 2357–2408. | MR | Numdam | DOI
[37] , On stable, dissipation reducing splitting schemes for two-phase flow of electrolyte solutions. Numer. Algorithms 80 (2018) 1361–1390. | MR | DOI
[38] , An efficient and convergent finite element scheme for Cahn-Hilliard equations with dynamic boundary conditions. Preprint [math.NA], accepted for publication in SIAM Journal on Numerical Analysis (2020). | arXiv | MR
[39] , and , Higher-order Cahn-Hilliard equations with dynamic boundary conditions. J. Math. Anal. Appl. 449 (2017) 1321–1339. | MR | DOI
[40] , The Cahn-Hilliard Equation: Recent Advances and Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA (2019). | MR | DOI
[41] and , The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. Discrete Contin. Dyn. Syst. 28 (2010) 275–310. | MR | Zbl | DOI
[42] , Time periodic solutions of Cahn-Hilliard systems with dynamic boundary conditions. AIMS Math. 3 (2018) 263–287. | DOI
[43] , Front migration in the nonlinear Cahn-Hilliard equation. Proc. R. Soc. Lond. A 422 (1989) 261–278. | MR | Zbl | DOI
[44] and , The Cahn-Hilliard equation with dynamic boundary conditions. Adv. Differ. Equ. 8 (2003) 83–110. | MR | Zbl
[45] and , Convergence of solutions to Cahn-Hilliard equation. Commun. Part. Differ. Equ. 24 (1999) 1055–1077. | MR | Zbl | DOI
[46] , Compacts sets in the space . Ann. Mat. Pura App. 146 (1986) 65–96. | MR | Zbl | DOI
[47] , Partial differential equations I. . Basic theory, 2nd edition. In: Vol. 115 of Applied Mathematical Sciences, Springer, New York (2011). | MR | Zbl | DOI
[48] and , Simulations of contact-line motion: slip and the dynamic contact angle. Phys. Rev. Lett. 63 (1989) 766–769. | DOI
[49] , Convergence to equilibrium for a Cahn-Hilliard model with the Wentzell boundary condition. Asymptot. Anal. 54 (2007) 71–92. | MR | Zbl
[50] and , Convergence to equilibrium for the Cahn-Hilliard equation with dynamic boundary conditions. J. Differ. Equ. 204 (2004) 511–531. | MR | Zbl | DOI
[51] , Asymptotic behavior of solution to the Cahn-Hilliard equation. Appl. Anal. 23 (1986) 165–184. | MR | Zbl | DOI
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Note to the reader: In the title you should read "Hilliard" instead of "Hillard" due to a typo, corrected online in August 2021.
