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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]
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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
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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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