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Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions

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  • Our aim in this paper is to prove the existence and uniqueness of solutions to Cahn-Hilliard and Allen-Cahn type equations based on a modification of the Ginzburg-Landau free energy proposed in [12] (see also [16]) which takes into account strong anisotropy effects. In particular, the free energy contains a regularization term, called Willmore regularization.
    Mathematics Subject Classification: Primary: 35B45; Secondary: 35K55.

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  • [1]

    S. M. Allen and J. W. Cahn, A microscopic 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]

    J. Berry, K. R. Elder and M. Grant, Simulation of an atomistic dynamic field theory for monatomic liquids: Freezing and glass formation, Phys. Rev. E, 77 (2008), 061506.doi: 10.1103/PhysRevE.77.061506.

    [3]

    J. Berry, M. Grant and K. R. Elder, Diffusive atomistic dynamics of edge dislocations in two dimensions, Phys. Rev. E, 73 (2006), 031609.doi: 10.1103/PhysRevE.73.031609.

    [4]

    J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.

    [5]

    G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Ration. Mech. Anal., 92 (1986), 205-245.doi: 10.1007/BF00254827.

    [6]

    F. Chen and J. Shen, Efficient energy stable schemes with spectral discretization in space for anisotropic Cahn-Hilliard systems, Commun. Comput. Phys., 13 (2013), 1189-1208.

    [7]

    P. G. de Gennes, Dynamics of fluctuations and spinodal decomposition in polymer blends, J. Chem. Phys., 72 (1980), 4756-4763.doi: 10.1063/1.439809.

    [8]

    P. Galenko, D. Danilov and V. Lebedev, Phase-field-crystal and Swift-Hohenberg equations with fast dynamics, Phys. Rev. E, 79 (2009), 051110, 11pp.doi: 10.1103/PhysRevE.79.051110.

    [9]

    G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E, 47 (1993), 4289-4300.doi: 10.1103/PhysRevE.47.4289.

    [10]

    G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E, 47 (1993), 4301-4312.doi: 10.1103/PhysRevE.47.4301.

    [11]

    Z. Hu, S. M. Wise, C. Wang and J. S. Lowengrub, Stable finite difference, nonlinear multigrid simulation of the phase field crystal equation, J. Comput. Phys., 228 (2009), 5323-5339.doi: 10.1016/j.jcp.2009.04.020.

    [12]

    R. Kobayashi, Modeling and numerical simulations of dendritic crystal growth, Phys. D, 63 (1993), 410-423.doi: 10.1016/0167-2789(93)90120-P.

    [13]

    M. Korzec, P. Nayar and P. Rybka, Global weak solutions to a sixth order Cahn-Hilliard type equation, SIAM J. Math. Anal., 44 (2012), 3369-3387.doi: 10.1137/100817590.

    [14]

    M. Korzec and P. Rybka, On a higher order convective Cahn-Hilliard type equation, SIAM J. Appl. Math., 72 (2012), 1343-1360.doi: 10.1137/110834123.

    [15]

    A. Miranville, Existence of solutions for a one-dimensional Allen-Cahn equation, J. Appl. Anal. Comput., 3 (2013), 265-277.

    [16]

    A. Makki and A. Miranville, Well-posedness for one-dimensional anisotropic Cahn-Hilliard and Allen-Cahn systems, Electron. J. Differential Equaytions, (2015), 1-15.

    [17]

    T. V. Savina, A. A. Golovin, S. H. Davis, A. A. Nepomnyashchy and P. W. Voorhees, Faceting of a growing crystal surface by surface diffusion, Phys. Rev. E, 67 (2003), 021606.doi: 10.1103/PhysRevE.67.021606.

    [18]

    S. Torabi, J. Lowengrub, A. Voigt and S. Wise, A new phase-field model for strongly anisotropic systems, Proc. R. Soc. A, 465 (2009), 1337-1359.doi: 10.1098/rspa.2008.0385.

    [19]

    J. E. Taylor and J. W. Cahn, Diffuse interfaces with sharp corners and facets: Phase-field models with strongly anisotropic surfaces, Phys. D, 112 (1998), 381-411.doi: 10.1016/S0167-2789(97)00177-2.

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