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June  2016, 9(3): 759-775. doi: 10.3934/dcdss.2016027

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

Ahmad Makki 1, and  Alain Miranville 2,

1. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France
2. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348 - SP2MI, Boulevard Marie et Pierre Curie - Téléport 2, 86962 Chasseneuil Futuroscope Cedex

Received  March 2015 Revised  April 2015 Published  April 2016

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.
Keywords: well-posedness., Willmore regularization, anisotropy, Allen-Cahn equation, Cahn-Hilliard equation.
Mathematics Subject Classification: Primary: 35B45; Secondary: 35K5.
Citation: Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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