Abstract
The initial boundary value problem for a Cahn–Hilliard system subject to a dynamic boundary condition of Allen–Cahn type is treated. The vanishing of the surface diffusion on the dynamic boundary condition is the point of emphasis. By the asymptotic analysis as the diffusion coefficient tends to 0, one can expect that the solutions of the surface diffusion problem converge to the solution of the problem without the surface diffusion. This is actually the case, but the solution of the limiting problem naturally looses some regularity. Indeed, the system we investigate is rather complicate due to the presence of nonlinear terms including general maximal monotone graphs both in the bulk and on the boundary. The two graphs are related each to the other by a growth condition, with the boundary graph that dominates the other one. In general, at the asymptotic limit a weaker form of the boundary condition is obtained, but in the case when the two graphs exhibit the same growth the boundary condition still holds almost everywhere.
Similar content being viewed by others
References
Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, London (2010)
Brézis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Especes de Hilbert. Elsevier, Amsterdam (1973)
Brezzi, F., Gilardi, G.: Chapters 1–3 in Finite Element Handbook. In: Kardestuncer, H., Norrie, D.H. (eds.) McGraw–Hill Book Co., New York (1987)
Cahn, J.W.: On spinodal decomposition. Acta Metall. 9, 795–801 (1961)
Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 2, 258–267 (1958)
Calatroni, L., Colli, P.: Global solution to the Allen–Cahn equation with singular potentials and dynamic boundary conditions. Nonlinear Anal. 79, 12–27 (2013)
Cavaterra, C., Gal, C.G., Grasselli, M.: Cahn–Hilliard equations with memory and dynamic boundary conditions. Asymptot. Anal. 71, 123–162 (2011)
Colli, P., Fukao, T.: Cahn–Hilliard equation with dynamic boundary conditions and mass constraint on the boundary. J. Math. Anal. Appl. 429, 1190–1213 (2015)
Colli, P., Fukao, T.: Equation and dynamic boundary condition of Cahn–Hilliard type with singular potentials. Nonlinear Anal. 127, 413–433 (2015)
Colli, P., Fukao, T., Wu, H.: On a transmission problem for equation and dynamic boundary condition of Cahn–Hilliard type with nonsmooth potentials. to appear in Math. Nachr. https://doi.org/10.1002/mana.201900361
Colli, P., Gilardi, G., Nakayashiki, R., Shirakawa, K.: A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions. Nonlinear Anal. 158, 32–59 (2017)
Colli, P., Gilardi, G., Sprekels, J.: On the Cahn–Hilliard equation with dynamic boundary conditions and a dominating boundary potential. J. Math. Anal. Appl. 419, 972–994 (2014)
Colli, P., Gilardi, G., Sprekels, J.: Global existence for a nonstandard viscous Cahn–Hilliard system with dynamic boundary condition. SIAM J. Math. Anal. 49, 1732–1760 (2017)
Colli, P., Gilardi, G., Sprekels, J.: On a Cahn–Hilliard system with convection and dynamic boundary conditions. Ann. Mat. Pura Appl. 197(4), 1445–1475 (2018)
Colli, P., Visintin, A.: On a class of doubly nonlinear evolution equations. Commun. Par. Differ. Equ. 15, 737–756 (1990)
Gal, C.G.: Global well-posedness for the non-isothermal Cahn–Hilliard equation with dynamic boundary conditions. Adv. Differ. Equ. 12, 1241–1274 (2007)
Gal, C.G., Miranville, A.: Robust exponential attractors and convergence to equilibria for non-isothermal Cahn–Hilliard equations with dynamic boundary conditions. Discrete Contin. Dyn. Syst. Ser. S 2, 113–147 (2009)
Gilardi, G., Miranville, A., Schimperna, G.: On the Cahn–Hilliard equation with irregular potentials and dynamic boundary conditions. Commun. Pure Appl. Anal. 8, 881–912 (2009)
Grigor’yan, A.: Heat Kernel and Analysis on Manifolds. American Mathematical Society, International Press, Boston (2009)
Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod Gauthier-Villars, Paris (1969)
Lions, J.-L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications I. Springer, New York (1972)
Liu, C., Wu, H.: An energetic variational approach for the Cahn–Hilliard equation with dynamic boundary conditions: model derivation and mathematical analysis. Arch. Ration. Mech. Anal. 233, 167–247 (2019)
Miranville, A.: The Cahn–Hilliard equation and some of its variants. AIMS Math. 2, 479–544 (2017)
Miranville, A., Zelik, S.: Exponential attractors for the Cahn–Hilliard equation with dynamical boundary conditions. Math. Meth. Appl. Sci. 28, 709–735 (2005)
Novick-Cohen, A.: On the viscous Cahn–Hilliard equation. In: Material instabilities in continuum mechanics (Edinburgh, 1985–1986), Oxford Sci. Publ., Oxford Univ. Press, New York, 329–342 (1988)
Novick-Cohen, A., Pego, R.L.: Stable patterns in a viscous diffusion equation. Trans. Am. Math. Soc. 324, 331–351 (1991)
Racke, R., Zheng, S.: The Cahn–Hilliard equation with dynamic boundary conditions. Adv. Differ. Equ. 8, 83–110 (2003)
Scarpa, L.: Existence and uniqueness of solutions to singular Cahn–Hilliard equations with nonlinear viscosity terms and dynamic boundary conditions. J. Math. Anal. Appl. 469, 730–764 (2019)
Simon, J.: Compact sets in the spaces \(L^p(0,T;B)\). Ann. Mat. Pura. Appl. 146(4), 65–96 (1987)
Wu, H., Zheng, S.: Convergence to equilibrium for the Cahn–Hilliard equation with dynamic boundary conditions. J. Differ. Equ. 204, 511–531 (2004)
Acknowledgements
The authors warmly thank Professor Ken Shirakawa for his valuable advice about Lemmas A.1 and A.2. P. Colli points out his affiliation as Research Associate to the IMATI – C.N.R. Pavia, Italy, and acknowledges support from the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022) – Dept. of Mathematics “F. Casorati”, University of Pavia, and from the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). T. Fukao acknowledges the support from the JSPS KAKENHI Grant-in-Aid for Scientific Research(C), Japan, Grant Number 17K05321 and from the Grant Program of The Sumitomo Foundation, Grant Number 190367. Last but not least, the authors are very grateful to the reviewer for the careful reading of manuscript.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix A
Appendix A
We use the same setting as in the previous sections.
Lemma A.1
Assume (A2). Then (3.3) holds, that is,
for all \(\varepsilon \in (0,1]\) with the same constants \(\varrho \ge 1 \) and \(c_0 >0 \).
Proof
Thanks to [6, Lemma 4.4], it is known that
where \(\beta _{\Gamma ,\varepsilon \varrho }\) denotes the Yosida approximation of \(\beta _{\Gamma }\) with parameter \(\varepsilon \varrho \), i.e.,
Then, recalling that \(\varrho \ge 1\), we may invoke the fundamental property [2, Proposition 2.6, p. 28] of Yosida approximations, which implies that
because \(\varepsilon \le \varepsilon \varrho \). Thus we get the conclusion. \(\square \)
Lemma A.2
Assume (A2)\(^\prime \). Then (5.2) holds, that is,
for all \(\varepsilon \in (0,1]\) with the same constants \(\varrho \ge 1 \) and \(c_0 >0\).
Proof
In view of Lemma A.1, is enough to prove that
which is the same as
But this follows immediately from Lemma A.1 again. \(\square \)
Remark A.3
Comparing to previous works (see, e.g., [8, 9, 12]) in which the same kind of property (2.7) was assumed for the two maximal monotone graphs, the parameter of the Yosida regularizations is here the same for both graphs (see also [11, Section 2]) . Instead, in the approach devised in [6, Lemma 4.4] exactly the approximation \(\beta _{\Gamma , \varepsilon \varrho }\) defined by (5.10) was introduced and used for \(\beta _{\Gamma }\).
Rights and permissions
About this article
Cite this article
Colli, P., Fukao, T. Vanishing diffusion in a dynamic boundary condition for the Cahn–Hilliard equation. Nonlinear Differ. Equ. Appl. 27, 53 (2020). https://doi.org/10.1007/s00030-020-00654-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-020-00654-8
Keywords
- Cahn–Hilliard system
- Dynamic boundary condition
- Non-smooth potentials
- Convergence
- Well-posedness
- Regularity