doi: 10.3934/dcdsb.2018308

On the Cahn-Hilliard/Allen-Cahn equations with singular potentials

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. 

Xiamen University, School of Mathematical Sciences, Xiamen, Fujian, China

3. 

Université Libanaise, Laboratoire de Mathématiques - EDST, Faculté des Sciences, Hadath, Liban

Received  February 2018 Revised  June 2018 Published  November 2018

The purpose of this work is to prove the existence and uniqueness of the solution for a Cahn-Hilliard/Allen-Cahn system with singular potentials (and, in particular, the thermodynamically relevant logarithmic potentials). We also prove the existence of the global attractor. Finally, we show further regularity results and we prove a strict separation property (from the pure states) in one space dimension.

Citation: Alain Miranville, Wafa Saoud, Raafat Talhouk. On the Cahn-Hilliard/Allen-Cahn equations with singular potentials. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018308
References:
[1]

A. Adams and J. Fournier, Sobolev Spaces, 2nd edition, Academic Press, 2003.

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, 1st edition, Elsevier, Amsterdam, 1992.

[3]

D. BrochetD. Hilhorst and A. Novick-Cohen, Finite-Dimensional exponential attractor for a model for order-disorder and phase separation, Appl. Math. Lett., 7 (1994), 83-87. doi: 10.1016/0893-9659(94)90118-X.

[4]

J. W. Cahn and A. Novick-Cohen, Evolution equations for phase separation and ordering in binary alloys, Statistical Phys., 76 (1994), 877-909.

[5]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I, Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.

[6]

L. CherfilsA. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic nonlinear terms, Milan J. Math., 79 (2011), 561-596. doi: 10.1007/s00032-011-0165-4.

[7]

R. Dal PassoL. Giacomelli and A. Novick-Cohen, Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility, Interfaces Free Boundaries, 1 (1999), 199-226. doi: 10.4171/IFB/9.

[8]

A. GiorginiM. Grasselli and A. Miranville, The Cahn-Hilliard-Oono equation with singular potential, Math. Models Methods Appl. Sci., 27 (2017), 2485-2510. doi: 10.1142/S0218202517500506.

[9]

P. C. MillettS. RokkamA. El-AzabM. Tonks and D. Wolf, Void nucleation and growth in irradiated polycrystalline metals: A phase-field model, Modelling Simul. Mater. Sci. Eng., 17 (2009), 0064-003. doi: 10.1088/0965-0393/17/6/064003.

[10]

A. Miranville, Finite dimensional global attractor for a class of doubly nonlinear parabolic equations, Cent. Eur. J. Math., 4 (2006), 163-182. doi: 10.1007/s11533-005-0010-5.

[11]

A. Miranville, The Cahn-Hilliard equation and some of its variants, AIMS Math., 2 (2017), 479-544. doi: 10.3934/Math.2017.2.479.

[12]

A. MiranvilleW. Saoud and R. Talhouk, Asymptotic behavior of a model for order-disorder and phase separation, Asympt. Anal., 103 (2017), 57-76. doi: 10.3233/ASY-171419.

[13]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582. doi: 10.1002/mma.464.

[14]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbouded domains, in Handbook of Differential Equations, Evolutionary Partial Differential Equations (eds. C.M. Dafermos and M. Pokorny), Elsevier, Amsterdam, (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0.

[15]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Cont. Dyn. Sys., 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275.

[16]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.

[17]

A. Novick-Cohen, Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system, Phys. D, 137 (2000), 1-24. doi: 10.1016/S0167-2789(99)00162-1.

[18]

A. Novick-Cohen and L. Peres Hari, Geometric motion for a degenerate Allen-Cahn/Cahn-Hilliard system: The partial wetting case, Phys. D, 209 (2005), 205-235. doi: 10.1016/j.physd.2005.06.028.

[19]

S. RokkamA. El-AzabP. Millett and D. Wolf, Phase field modeling of void nucleation and growth in irradiated metals, Modelling Simul. Mater. Sci. Eng., 17 (2009), 0064-002.

[20]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, American Mathematical Society, 2001. doi: 10.1090/chel/343.

[21]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, Volume 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[22]

M. R. TonksD. GastonP. C. MillettD. Andrs and P. Talbot, An object-oriented finite element framework for multiphysics phase field simulations, Comput. Mater. Sci., 51 (2012), 20-29. doi: 10.1016/j.commatsci.2011.07.028.

[23]

L. Wang, J. Lee, M. Anitescu, A. E. Azab, L. C. Mcinnes, T. Munson and B. Smith, A differential variational inequality approach for the simulation of heterogeneous materials, in Proc. SciDAC, 2011.

[24]

Y. XiaY. Xu and C. W. Shu, Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system, Commun. Comput. Phys., 5 (2009), 821-835.

[25]

C. YangX. C. CaiD. E. Keyes and M. Pernice, NKS method for the implicit solution of a coupled allen-cahn/cahn-hilliard system, Domain Decomposition Methods in Science and Engineering, 21 (2014), 819-827. doi: 10.1007/978-3-319-05789-7_79.

show all references

References:
[1]

A. Adams and J. Fournier, Sobolev Spaces, 2nd edition, Academic Press, 2003.

[2]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, 1st edition, Elsevier, Amsterdam, 1992.

[3]

D. BrochetD. Hilhorst and A. Novick-Cohen, Finite-Dimensional exponential attractor for a model for order-disorder and phase separation, Appl. Math. Lett., 7 (1994), 83-87. doi: 10.1016/0893-9659(94)90118-X.

[4]

J. W. Cahn and A. Novick-Cohen, Evolution equations for phase separation and ordering in binary alloys, Statistical Phys., 76 (1994), 877-909.

[5]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I, Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. doi: 10.1063/1.1744102.

[6]

L. CherfilsA. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic nonlinear terms, Milan J. Math., 79 (2011), 561-596. doi: 10.1007/s00032-011-0165-4.

[7]

R. Dal PassoL. Giacomelli and A. Novick-Cohen, Existence for an Allen-Cahn/Cahn-Hilliard system with degenerate mobility, Interfaces Free Boundaries, 1 (1999), 199-226. doi: 10.4171/IFB/9.

[8]

A. GiorginiM. Grasselli and A. Miranville, The Cahn-Hilliard-Oono equation with singular potential, Math. Models Methods Appl. Sci., 27 (2017), 2485-2510. doi: 10.1142/S0218202517500506.

[9]

P. C. MillettS. RokkamA. El-AzabM. Tonks and D. Wolf, Void nucleation and growth in irradiated polycrystalline metals: A phase-field model, Modelling Simul. Mater. Sci. Eng., 17 (2009), 0064-003. doi: 10.1088/0965-0393/17/6/064003.

[10]

A. Miranville, Finite dimensional global attractor for a class of doubly nonlinear parabolic equations, Cent. Eur. J. Math., 4 (2006), 163-182. doi: 10.1007/s11533-005-0010-5.

[11]

A. Miranville, The Cahn-Hilliard equation and some of its variants, AIMS Math., 2 (2017), 479-544. doi: 10.3934/Math.2017.2.479.

[12]

A. MiranvilleW. Saoud and R. Talhouk, Asymptotic behavior of a model for order-disorder and phase separation, Asympt. Anal., 103 (2017), 57-76. doi: 10.3233/ASY-171419.

[13]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582. doi: 10.1002/mma.464.

[14]

A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbouded domains, in Handbook of Differential Equations, Evolutionary Partial Differential Equations (eds. C.M. Dafermos and M. Pokorny), Elsevier, Amsterdam, (2008), 103–200. doi: 10.1016/S1874-5717(08)00003-0.

[15]

A. Miranville and S. Zelik, The Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, Discrete Cont. Dyn. Sys., 28 (2010), 275-310. doi: 10.3934/dcds.2010.28.275.

[16]

T. NagaiT. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.

[17]

A. Novick-Cohen, Triple-junction motion for an Allen-Cahn/Cahn-Hilliard system, Phys. D, 137 (2000), 1-24. doi: 10.1016/S0167-2789(99)00162-1.

[18]

A. Novick-Cohen and L. Peres Hari, Geometric motion for a degenerate Allen-Cahn/Cahn-Hilliard system: The partial wetting case, Phys. D, 209 (2005), 205-235. doi: 10.1016/j.physd.2005.06.028.

[19]

S. RokkamA. El-AzabP. Millett and D. Wolf, Phase field modeling of void nucleation and growth in irradiated metals, Modelling Simul. Mater. Sci. Eng., 17 (2009), 0064-002.

[20]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, AMS Chelsea Publishing, American Mathematical Society, 2001. doi: 10.1090/chel/343.

[21]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, Volume 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.

[22]

M. R. TonksD. GastonP. C. MillettD. Andrs and P. Talbot, An object-oriented finite element framework for multiphysics phase field simulations, Comput. Mater. Sci., 51 (2012), 20-29. doi: 10.1016/j.commatsci.2011.07.028.

[23]

L. Wang, J. Lee, M. Anitescu, A. E. Azab, L. C. Mcinnes, T. Munson and B. Smith, A differential variational inequality approach for the simulation of heterogeneous materials, in Proc. SciDAC, 2011.

[24]

Y. XiaY. Xu and C. W. Shu, Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system, Commun. Comput. Phys., 5 (2009), 821-835.

[25]

C. YangX. C. CaiD. E. Keyes and M. Pernice, NKS method for the implicit solution of a coupled allen-cahn/cahn-hilliard system, Domain Decomposition Methods in Science and Engineering, 21 (2014), 819-827. doi: 10.1007/978-3-319-05789-7_79.

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