Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T11:15:23.885Z Has data issue: false hasContentIssue false
  • English
  • Français

Cahn–Hilliard equations on an evolving surface

Part of: Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  16 June 2021

D. CAETANO Open the ORCID record for D. CAETANO [Opens in a new window]  and
C. M. ELLIOTT
D. CAETANO
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK emails: [email protected]; [email protected]
C. M. ELLIOTT
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK emails: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe a functional framework suitable to the analysis of the Cahn–Hilliard equation on an evolving surface whose evolution is assumed to be given a priori. The model is derived from balance laws for an order parameter with an associated Cahn–Hilliard energy functional and we establish well-posedness for general regular potentials, satisfying some prescribed growth conditions, and for two singular non-linearities – the thermodynamically relevant logarithmic potential and a double-obstacle potential. We identify, for the singular potentials, necessary conditions on the initial data and the evolution of the surfaces for global-in-time existence of solutions, which arise from the fact that integrals of solutions are preserved over time, and prove well-posedness for initial data on a suitable set of admissible initial conditions. We then briefly describe an alternative derivation leading to a model that instead preserves a weighted integral of the solution and explain how our arguments can be adapted in order to obtain global-in-time existence without restrictions on the initial conditions. Some illustrative examples and further research directions are given in the final sections.

Keywords

Cahn–Hilliardevolving surfaces

MSC classification

Primary: 35A01: Existence problems
Secondary: 35Q92: PDEs in connection with biology and other natural sciences
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 5: 30th Anniversary Special Issue , October 2021 , pp. 937 - 1000
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alphonse, A., Elliott, C. M. & Stinner, B. (2015) An abstract framework for parabolic PDEs on evolving spaces. Portugaliae Mathematica 71(1), 1–46.CrossRefGoogle Scholar
Alphonse, A., Elliott, C. M. & Stinner, B. (2015) On some linear parabolic PDEs on moving hypersurfaces. Interfaces Free Boundaries 17, 157–187.CrossRefGoogle Scholar
Alphonse, A., Elliott, C. M. & Terra, J. (2017) A coupled ligand-receptor bulk-surface system on a moving domain: well posedness, regularity, and convergence to equilibrium. SIAM J. Math. Anal. 50, 1544–1592.CrossRefGoogle Scholar
Baiocchi, C. (1965) Regolarità e unicità della soluzione di una equazione differenziale astratta. Rendiconti del Seminario Matematico delle Università di Padova 35(2), 380–417.Google Scholar
Barreira, R., Elliott, C. M. & Madzvamuse, A. (2011) The surface finite element method for pattern formation on evolving biological surfaces. J. Math. Biol. 63, 1095–1119.CrossRefGoogle ScholarPubMed
Blowey, J. F. & Elliott, C. M. (1991) The Cahn–Hilliard gradient theory for phase separation with non–smooth free energy Part I: Mathematical Analysis. Eur. J. Appl. Math. 2, 233–280.CrossRefGoogle Scholar
Blowey, J. F. & Elliott, C. M. (1992) The Cahn–Hilliard gradient theory for phase separation with non–smooth free energy Part II: Numerical Analysis. Eur. J. Appl. Math. 3, 147–179.CrossRefGoogle Scholar
Cahn, J. W. (1961) On spinodal decomposition. Acta Metall. Mater. 9, 795–801.CrossRefGoogle Scholar
Cahn, J. W., Elliott, C. M. & Novick-Cohen, A. (1996) The Cahn–Hilliard equation with a concentration dependent mobility: motion by minus the laplacian of the mean curvature. Eur. J. Appl. Math. 7, 287–301.CrossRefGoogle Scholar
Cahn, J. W. & Hilliard, J. E. (1958) Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 28, 258–267.CrossRefGoogle Scholar
Cherfile, L., Miranville, A. & Zelik, S. (2011) The Cahn-Hilliard equation with logarithmic potentials. Milan J. Math. 79(2), 561–596.CrossRefGoogle Scholar
Dai, S. & Du, Q. (2014) Coarsening mechanism for systems governed by the Cahn-Hilliard equation with degenerate diffusion mobility. Multiscal Model. Simul. 12(4), 1870–1889.CrossRefGoogle Scholar
Dai, S. & Du, Q. (2016) Weak solutions for the Cahn-Hilliard equation with degenerate mobility. Arch. Rational Mech. Anal. 219, 1161–1184.CrossRefGoogle Scholar
Debussche, A. & Dettori, L. (1995) On the Cahn-Hilliard equation with a logarithmic free energy. Nonlinear Anal. Theor. 24(10), 1491–1514.CrossRefGoogle Scholar
Deckelnick, K., Dziuk, G. & Elliott, C. M. (2005) Computation of geometric partial differential equations and mean curvature flow. Acta Numerica 14, 139–232.CrossRefGoogle Scholar
Dziuk, G. & Elliott, C. M. (2007) Finite elements on evolving surfaces. IMA J. Numer. Anal. 25, 385–407.Google Scholar
Dziuk, G. & Elliott, C. M. (2013) Finite element methods for surface partial differential equations. Acta Numerica 22, 289–396.CrossRefGoogle Scholar
Eilks, C. & Elliott, C. M. (2008) Numerical simulation of dealloying by surface dissolution via the evolving surface finite element method. J. Comput. Phys. 227(23), 9727–9741.CrossRefGoogle Scholar
Elliott, C. M. (1989) The Cahn-Hilliard Model for the Kinetics of Phase Separation , International Series of Numerical Mathematics, Vol. 88, Birkhäuser Verlag, Basel, Germany, pp. 35–73.Google Scholar
Elliott, C. M. & French, D. A. (1989) A nonconforming finite element method for the two-dimensional Cahn-Hilliard equation. SIAM J. Numer. Anal. 24(4), 884–903.CrossRefGoogle Scholar
Elliott, C. M., French, D. A. & Milner, F. A. (1989) A second order splitting method for the Cahn-Hilliard equation. Numerische Mathematik 54(5), 575–590.CrossRefGoogle Scholar
Elliott, C. M. & Garcke, H. (1996) On the Cahn-Hilliard equation with degenerate mobility. SIAM J. Math. Anal. 27(2), 404–423.CrossRefGoogle Scholar
Elliott, C. M. & Larsson, S. (1992) Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation. Math. Comput. 58(198), 603–630.CrossRefGoogle Scholar
Elliott, C. M. & Luckhaus, S. (1991) A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy. IMA Preprint Series, 887.Google Scholar
Elliott, C. M. & Ranner, T. (2015) Evolving surface finite element method for the Cahn–Hilliard equation. Numerische Mathematik 129(3), 483–534.CrossRefGoogle Scholar
Elliott, C. M. & Ranner, T. (2020) A unified theory for continuous-in-time evolving finite element space approximations to partial differential equations in evolving domains. IMA J. Numer. Anal., 1–150. doi: 10.1093/imanum/draa062.CrossRefGoogle Scholar
Elliott, C. M., Stinner, B. & Venkataraman, C. (2012) Modelling cell motility and chemotaxis with evolving surface finite elements. J. R. Soc. Interface 9(76), 3027–3044.CrossRefGoogle ScholarPubMed
Elliott, C. M. & Zheng, S. (1986) On the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 96(4), 339–357.CrossRefGoogle Scholar
Erlebacher, J., Aziz, M., Karma, A., Dimitrov, N. & Sieradzki, K. (2001) Evolution of nanoporosity in dealloying. Nature 410, 450–453.CrossRefGoogle ScholarPubMed
Garcke, H. (2013) Curvature driven interface evolution. Jahresbericht der Deutschen Mathematiker-Vereinigung 115(2), 63–100.CrossRefGoogle Scholar
Garcke, H. & Knopf, P. (2020) Weak solutions of the cahn-hilliard system with dynamic boundary conditions: a gradient flow approach. SIAM J. Math. Anal. 52(1), 340–369.CrossRefGoogle Scholar
Garcke, H., Lam, K. F. & Stinner, B. (2014) Diffuse interface modelling of soluble surfactants in two-phase flow. Commun. Math. Sci. 12(8), 1475–1522.CrossRefGoogle Scholar
Gilbarg, D. & Trudinger, N. S. Elliptic Partial Differential Equations of Second Order , Grundlehren der mathematischen Wissenschaften, Springer Verlag, 1998.Google Scholar
Heida, M. (2015) Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation. Appl. Math. 60(1), 51–90.CrossRefGoogle Scholar
Kinderlehrer, D. & Stampacchia, G. (1980) An Introduction to Variational Inequalities and their Applications, Academic Press, New York - London.Google Scholar
Lan, D., Son, D. T., Tang, B. Q. & Thuy, L. T. (2021) Quasilinear parabolic equations with first order terms and L1-data in moving domains. Nonlinear Anal. 206, 112233.CrossRefGoogle Scholar
Leoni, G. (2009) A First Course in Sobolev Spaces, American Mathematical Society, Providence, RI, USA.Google Scholar
Lions, J.-L. (1957) Sur les problèmes mixtes pour certains systèmes paraboliques dans les ouverts non cylindriques. Annales de l’institut Fourier 7, 143–182.CrossRefGoogle Scholar
Miranville, A. (2019) The Cahn-Hilliard Equation: Recent Advances and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.CrossRefGoogle Scholar
Naegele, P. (2015) Monotone Operator Theory for Unsteady Problems on Non-Cylindrical Domains. PhD thesis, University of Freiburg.Google Scholar
Novick Cohen, A. (2008) The Cahn-Hilliard equation. In: C. Dafermos and E. Feireisel (editors), Handbook of Differential Equations, Evolutionary Equations, Vol. 4, Elsevier.CrossRefGoogle Scholar
Novick-Cohen, A. & Segel, L. A. (1984) Nonlinear aspects of the Cahn-Hilliard equation. Physica D Nonlinear Phenomena 10(3), 277–298.CrossRefGoogle Scholar
O’Connor, D. & Stinner, B. (2016) The Cahn-Hilliard equation on an evolving surface. arXiv e-prints, arXiv:1607.05627.Google Scholar
Olshanskii, M., Xu, X. & Yushutin, V. (2021) A finite element method for Allen-Cahn equation on deforming surface. Comput. Math. Appl. 90, 148–158.CrossRefGoogle Scholar
Pego, R. L. (1986) Front migration in the nonlinear Cahn-Hilliard equation. Proc. R. Soc. London. Ser. A Math. Phys. Sci. 422, 261–278.Google Scholar
Prato, G. D. & Debussche, A. (1996) Stochastic Cahn-Hilliard equation. Nonlinear Anal. Theory Methods Appl. 26(2), 241–263.CrossRefGoogle Scholar
Rakotoson, J. M. & Temam, R. (2001) An optimal compactness theorem and application to elliptic-parabolic systems. Appl. Math. Lett. 14, 303–306.CrossRefGoogle Scholar
Robinson, J. C. (2001) Infinite Dimensional Dynamical Systems , Cambridge Texts in Applied Mathematics, Cambridge.Google Scholar
Venkataraman, C., Sekimura, T., Gaffney, E. A., Maini, P. K. & Madzvamuse, A. (2011) Modeling parr-mark pattern formation during the early development of amago trout. Phys. Rev. E 84, 041923.CrossRefGoogle ScholarPubMed
Vierling, M. (2014) Parabolic optimal control problems on evolving surfaces subject to point-wise box constraints on the control - theory and numerical realization. Interfaces Free Boundaries 16, 137–173.CrossRefGoogle Scholar
Yushutin, V., Quaini, A. & Olshanskii, M. (2020) Numerical modeling of phase separation on dynamic surfaces. J. Comput. Phys. 407, 109126.CrossRefGoogle Scholar
Zeidler, E. (1990) Nonlinear Functional Analysis and Its Applications. II/B: Nonlinear Monotone Operators, Springer-Verlag, New York.Google Scholar
Zimmermann, C., Toshniwal, D., Landis, C. M., Hughes, T. J. R., Mandadapu, K. K. & Sauer, R. A. (2019) An isogeometric finite element formulation for phase transitions on deforming surfaces. Comput. Methods Appl. Mech. Eng. 351, 441–477.CrossRefGoogle Scholar