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Keywords:
viscous Cahn-Hilliard equation; viscous Allen-Cahn equation; Willmore regularization; well-posedness of models; global attractor; robust exponential attractors; anisotropy; simulations
Summary:
We consider the viscous Allen-Cahn and Cahn-Hilliard models with an additional term called the nonlinear Willmore regularization. First, we are interested in the well-posedness of these two models. Furthermore, we prove that both models possess a global attractor. In addition, as far as the viscous Allen-Cahn equation is concerned, we construct a robust family of exponential attractors, i.e.\ attractors which are continuous with respect to the perturbation parameter. Finally, we give some numerical simulations which show the effects of the viscosity term on the anisotropic and isotropic Cahn-Hilliard equation.
References:
[1] Allen, S. M., Cahn, J. W.: A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (1979), 1085-1095. DOI 10.1016/0001-6160(79)90196-2
[2] Babin, A. V., Vishik, M. I.: Attractors of Evolution Equations. Studies in Mathematics and Its Applications 25 North-Holland, Amsterdam (1992). MR 1156492 | Zbl 0778.58002
[3] Bai, F., Elliott, C. M., Gardiner, A., Spence, A., Stuart, A. M.: The viscous Cahn-Hilliard equation. I: Computations. Nonlinearity 8 131-160 (1995). MR 1328591 | Zbl 0818.35045
[4] Barrett, J. W., Blowey, J. F.: Finite element approximation of the Cahn-Hilliard equation with concentration dependent mobility. Math. Comput. (1999), 68 487-517. DOI 10.1090/S0025-5718-99-01015-7 | MR 1609678 | Zbl 1126.65321
[5] Cahn, J. W.: On spinodal decomposition. Acta Metall. 9 (1961), 795-801. DOI 10.1016/0001-6160(61)90182-1
[6] Cahn, J. W., Hilliard, J. E.: Free energy of a non-uniform system I: Interfacial free energy. J. Chem. Phys. 28 (1958), 258-267. DOI 10.1063/1.1744102
[7] Chen, F., Shen, J.: Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems. Commun. Comput. Phys. 13 (2013), 1189-1208. DOI 10.4208/cicp.101111.110512a | MR 2988885
[8] Cherfils, L., Gatti, S.: Robust family of exponential attractors for isotropic crystal models. Math. Methods Appl. Sci. 39 (2016), 1705-1729. DOI 10.1002/mma.3597 | MR 3499040 | Zbl 1339.35053
[9] Cherfils, L., Miranville, A., Zelik, S.: The Cahn-Hilliard equation with logarithmic potentials. Milan J. Math. 79 (2011), 561-596. DOI 10.1007/s00032-011-0165-4 | MR 2862028 | Zbl 1250.35129
[10] Eden, A., Foias, C., Nicolaenko, B., Temam, R.: Exponential Attractors for Dissipative Evolution Equations. Research in Applied Mathematics 37 Masson, Paris; Wiley, Chichester (1994). MR 1335230 | Zbl 0842.58056
[11] Efendiev, M., Miranville, A., Zelik, S.: Exponential attractors for a singularly perturbed Cahn-Hilliard system. Math. Nachr. 272 (2004), 11-31. DOI 10.1002/mana.200310186 | MR 2079758 | Zbl 1046.37047
[12] Efendiev, M., Zelik, S., Miranville, A.: Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems. Proc. R. Soc. Edinb., Sect. A, Math. 13 (2005), 703-730. DOI 10.1017/S030821050000408X | MR 2173336 | Zbl 1088.37005
[13] Elliott, C. M.: The Cahn-Hilliard model for the kinetics of phase separation. Mathematical Models for Phase Change Problems, Obidos 1988 Int. Ser. Numer. Math. 88 Birkkhäuser, Basel 35-73 (1989). MR 1038064 | Zbl 0692.73003
[14] Elliott, C. M., Stuart, A. M.: Viscous Cahn-Hilliard equation. II: Analysis. J. Differ. Equations 128 387-414 (1996). MR 1398327 | Zbl 0855.35067
[15] Fabrie, P., Galusinski, C., Miranville, A., Zelik, S.: Uniform exponential attractors for a singularly perturbed damped wave equation. Discrete Contin. Dyn. Syst. 10 (2004), 211-238. DOI 10.3934/dcds.2004.10.211 | MR 2026192 | Zbl 1060.35011
[16] Fife, P. C.: Models for phase separation and their mathematics. Electron. J. Differ. Equ. (electronic only) 2000 (2000), Paper No. 48, 26 pages. MR 1772733 | Zbl 0957.35062
[17] Gatti, S., Grasselli, M., Miranville, A., Pata, V.: A construction of a robust family of exponential attractors. Proc. Am. Math. Soc. 134 (2006), 117-127. DOI 10.1090/S0002-9939-05-08340-1 | MR 2170551 | Zbl 1078.37047
[18] Grinfeld, M., Novick-Cohen, A.: The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor. Trans. Am. Math. Soc. 351 2375-2406 (1999). DOI 10.1090/S0002-9947-99-02445-9 | MR 1650085 | Zbl 0927.35045
[19] Gurtin, M. E.: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92 (1996), 178-192. DOI 10.1016/0167-2789(95)00173-5 | MR 1387065 | Zbl 0885.35121
[20] Hecht, F.: New development in FreeFem++. J. Numer. Math. 20 (2012), 251-256. DOI 10.1515/jnum-2012-0013 | MR 3043640 | Zbl 1266.68090
[21] Makki, A., Miranville, A.: Well-posedness for one-dimensional anisotropic Cahn-Hilliard and Allen-Cahn systems. Electron. J. Differ. Equ. (electronic only) 2015 (2015), Paper No. 04, 15 pages. MR 3335735 | Zbl 1334.35106
[22] Makki, A., Miranville, A.: Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete Contin. Dyn. Syst. Ser. S (2016), 9 759-775. MR 3503639 | Zbl 1346.35029
[23] Miranville, A.: Asymptotic behavior of a sixth-order Cahn-Hilliard system. Cent. Eur. J. Math. 12 (2014), 141-154. MR 3121828 | Zbl 1286.35047
[24] Miranville, A., Pata, V., Zelik, S.: Exponential attractors for singularly perturbed damped wave equations: a simple construction. Asymptot. Anal. 53 (2007), 1-12. MR 2343457 | Zbl 1139.35030
[25] Miranville, A., Quintanilla, R.: A generalization of the Allen-Cahn equation. IMA J. Appl. Math. 80 410-430 (2015). DOI 10.1093/imamat/hxt044 | MR 3335166 | Zbl 1320.35346
[26] Miranville, A., Zelik, S.: Robust exponential attractors for Cahn-Hilliard type equations with singular potentials. Math. Methods Appl. Sci. 27 (2004), 545-582. DOI 10.1002/mma.464 | MR 2041814 | Zbl 1050.35113
[27] Miranville, A., Zelik, S.: Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions. Math. Methods Appl. Sci. 28 (2005), 709-735. DOI 10.1002/mma.590 | MR 2125817 | Zbl 1068.35020
[28] Miranville, A., Zelik, S.: Attractors for dissipative partial differential equations in bounded and unbounded domains. Handbook of Differential Equations: Evolutionary Equations, Vol. IV. C. M. Dafermos et al. Handbook of Differential Equations Elsevier, Amsterdam 103-200 (2008). DOI 10.1016/S1874-5717(08)00003-0 | MR 2508165 | Zbl 1221.37158
[29] Novick-Cohen, A.: On the viscous Cahn-Hilliard equation. Material Instabilities in Continuum Mechanics (Edinburgh, 1985-1986) Oxford Sci. Publ. Oxford Univ. Press, New York (1988), 329-342. MR 0970531 | Zbl 0632.76119
[30] Novick-Cohen, A.: The Cahn-Hilliard equation: Mathematical and modeling perspectives. Adv. Math. Sci. Appl. (1998), 8 965-985. MR 1657208 | Zbl 0917.35044
[31] Novick-Cohen, A.: The Cahn-Hilliard equation. Handbook of Differential Equations: Evolutionary Equations. Vol. 4 C. M. Dafermos et al. Handbook of Differential Equations Elsevier/North-Holland, Amsterdam 201-228 (2008). MR 2508166 | Zbl 1185.35001
[32] Saoud, B.: Attracteurs Pour des Systèmes Dissipatifs non Autonomes. PhD thesis, Université de Poitiers French (2011).
[33] Taylor, J. E., Cahn, J. W.: Diffuse interfaces with sharp corners and facets: Phase field models with strongly anisotropic surfaces. Physica D 112 (1998), 381-411. DOI 10.1016/S0167-2789(97)00177-2 | MR 1607466 | Zbl 0930.35201
[34] Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences 68 Springer, New York (1997). DOI 10.1007/978-1-4612-0645-3 | MR 1441312 | Zbl 0871.35001
[35] Torabi, S., Lowengrub, J., Voigt, A., Wise, S.: A new phase-field model for strongly anisotropic systems. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 465 (2009), 1337-1359. DOI 10.1098/rspa.2008.0385 | MR 2500806 | Zbl 1186.80014
[36] Wise, S., Kim, J., Lowengrub, J.: Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method. J. Comput. Phys. 226 (2007), 414-446. DOI 10.1016/j.jcp.2007.04.020 | MR 2356365 | Zbl 1310.82044
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