February  2014, 7(1): 127-137. doi: 10.3934/dcdss.2014.7.127

On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation

1. 

Institute of Applied and Computational Mathematics, FO.R.T.H., and Department of Applied Mathematics, University of Crete, P.O. Box 2208, Heraklion, Crete 71409, Greece

2. 

Archimedes Center for Modeling, Analysis and Computation (ACMAC), Department of Applied Mathematics, University of Crete, P.O. Box 2208, Heraklion, Crete 71409, Greece

Received  February 2012 Revised  June 2012 Published  July 2013

In this manuscript, we consider a Cahn-Hilliard/Allen-Cahn equation is introduced in [17]. We give an existence of the solution, slightly improved from [18]. We also present a stochastic version of this equation in [3].
Citation: Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127
References:
[1]

Nicholas D. Alikakos, Peter W. Bates and Xinfu Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model,, Arch. Rational Mech. Anal., 128 (1994), 165.  doi: 10.1007/BF00375025.  Google Scholar

[2]

Dimitra Antonopoulou and Georgia Karali, Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains,, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 31.  doi: 10.3934/dcdsb.2011.16.31.  Google Scholar

[3]

Dimitra Antonopoolou, Georgia Karali, Anne Millet and Yuko Nagase, Existence of solution and of its density for a Stochastic Cahn-Hilliard/Allen-Cahn equation,, preprint., ().   Google Scholar

[4]

Kenneth A. Brakke, "The Motion of a Surface by its Mean Curvature,", Mathematical Notes, 20 (1978).   Google Scholar

[5]

Caroline Cardon-Weber, Cahn-Hilliard stochastic equation: Existence of the solution and of its density,, Bernoulli, 7 (2001), 777.  doi: 10.2307/3318542.  Google Scholar

[6]

Yun Gang Chen, Yoshikazu Giga and Shun'ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations,, J. Differential Geom., 33 (1991), 749.   Google Scholar

[7]

Giuseppe Da Prato and Arnaud Debussche, Stochastic Cahn-Hilliard equation,, Nonlinear Anal., 26 (1996), 241.  doi: 10.1016/0362-546X(94)00277-O.  Google Scholar

[8]

Weinan E, Weiqing Ren and Eric Vanden-Eijnden, Minimum action method for the study of rare events,, Comm. Pure Appl. Math., 57 (2004), 637.  doi: 10.1002/cpa.20005.  Google Scholar

[9]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature I,, J. Differential Geom., 33 (1991), 635.   Google Scholar

[10]

William G. Faris and Giovanni Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise,, J. Phys. A: Math. Gen., 15 (1982), 3025.  doi: 10.1088/0305-4470/15/10/011.  Google Scholar

[11]

Jin Feng and Markos A. Katsoulakis, A comparison principle for Hamilton-Jacobi equations related to controlled gradient flows in infinite dimensions,, Arch. Ration. Mech. Anal., 192 (2009), 275.  doi: 10.1007/s00205-008-0133-5.  Google Scholar

[12]

Paul C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces,", CBMS-NSF Regional Conference Series in Applied Mathematics, 53 (1988).  doi: 10.1137/1.9781611970180.  Google Scholar

[13]

M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems," (English summary), Second edition, (1998).  doi: 10.1007/978-1-4612-0611-8.  Google Scholar

[14]

, Yannis Goumas and Takashi Suzuki,, work in progress., ().   Google Scholar

[15]

M. Hildebrand and A. S. Mikhailov, Mesoscopic modeling in the kinetic theory of adsorbates,, J. Phys. Chem., 100 (1996), 19089.  doi: 10.1021/jp961668w.  Google Scholar

[16]

Tom Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature,, J. Differential Geom., 38 (1993), 417.   Google Scholar

[17]

Georgia Karali and Markos A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution,, J. Differential Equations, 235 (2007), 418.  doi: 10.1016/j.jde.2006.12.021.  Google Scholar

[18]

Georgia Karali and Tonia Ricciardi, On the convergence of a fourth order evolution equation to the Allen-Cahn equation,, Nonlinear Anal., 72 (2010), 4271.  doi: 10.1016/j.na.2010.02.003.  Google Scholar

[19]

Markos A. Katsoulakis and Dionisios G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution,, Phys. Rev. Lett., 84 (2000), 1511.  doi: 10.1103/PhysRevLett.84.1511.  Google Scholar

[20]

Robert V. Kohn, Felix Otto, Maria G. Reznikoff and Eric Vanden-Eijinden, Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation,, Comm. Pure Appl. Math., 60 (2007), 393.  doi: 10.1002/cpa.20144.  Google Scholar

[21]

Robert V. Kohn, Maria G. Reznikoff and Yoshihiro Tonegawa, Sharp-interface limit of the Allen-Cahn action functional in one space dimension,, Calc. Var. Partial Differential Equations, 25 (2006), 503.  doi: 10.1007/s00526-005-0370-5.  Google Scholar

[22]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", (French) Dunod; Gauthier-Villars, (1969).   Google Scholar

[23]

L. Mugnai and Röger, The Allen-Cahn action functional in higher dimensions,, Interfaces Free Bound., 10 (2008), 45.  doi: 10.4171/IFB/179.  Google Scholar

[24]

Yuko Nagase, Action minimization for an Allen-Cahn equation with an unequal double-well potential,, Manuscripta Mathematica, 137 (2012), 81.  doi: 10.1007/s00229-011-0458-5.  Google Scholar

[25]

R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation,, Proc. Roy. Soc. London Ser. A, 422 (1989), 261.  doi: 10.1098/rspa.1989.0027.  Google Scholar

[26]

Jacob Rubinstein, Peter Sternberg and Joseph B. Keller, Fast reaction, slow diffusion, and curve shortening,, SIAM J. Appl. Math., 49 (1989), 116.  doi: 10.1137/0149007.  Google Scholar

[27]

Roger Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Second edition, 68 (1997).   Google Scholar

[28]

Eric Vanden-Eijnden and Maria G.Westdickenberg, Rare events in stochastic partial differential equations on large spatial domains,, J. Stat. Phys., 131 (2008), 1023.  doi: 10.1007/s10955-008-9537-8.  Google Scholar

[29]

John B. Walsh, An introduction to stochastic partial differential equations,, in, 1180 (1986), 265.  doi: 10.1007/BFb0074920.  Google Scholar

[30]

Maria G. Westdickenberg, Rare events, action minimization, and sharp interface limits,, in, 44 (2008), 217.   Google Scholar

[31]

Maria. G. Westdickenberg and Yoshihiro Tonegawa, Higher multiplicity in the one-dimensional Allen-Cahn action functional,, Indiana Univ. Math. J., 56 (2007), 2935.  doi: 10.1512/iumj.2007.56.3182.  Google Scholar

show all references

References:
[1]

Nicholas D. Alikakos, Peter W. Bates and Xinfu Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model,, Arch. Rational Mech. Anal., 128 (1994), 165.  doi: 10.1007/BF00375025.  Google Scholar

[2]

Dimitra Antonopoulou and Georgia Karali, Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains,, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 31.  doi: 10.3934/dcdsb.2011.16.31.  Google Scholar

[3]

Dimitra Antonopoolou, Georgia Karali, Anne Millet and Yuko Nagase, Existence of solution and of its density for a Stochastic Cahn-Hilliard/Allen-Cahn equation,, preprint., ().   Google Scholar

[4]

Kenneth A. Brakke, "The Motion of a Surface by its Mean Curvature,", Mathematical Notes, 20 (1978).   Google Scholar

[5]

Caroline Cardon-Weber, Cahn-Hilliard stochastic equation: Existence of the solution and of its density,, Bernoulli, 7 (2001), 777.  doi: 10.2307/3318542.  Google Scholar

[6]

Yun Gang Chen, Yoshikazu Giga and Shun'ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations,, J. Differential Geom., 33 (1991), 749.   Google Scholar

[7]

Giuseppe Da Prato and Arnaud Debussche, Stochastic Cahn-Hilliard equation,, Nonlinear Anal., 26 (1996), 241.  doi: 10.1016/0362-546X(94)00277-O.  Google Scholar

[8]

Weinan E, Weiqing Ren and Eric Vanden-Eijnden, Minimum action method for the study of rare events,, Comm. Pure Appl. Math., 57 (2004), 637.  doi: 10.1002/cpa.20005.  Google Scholar

[9]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature I,, J. Differential Geom., 33 (1991), 635.   Google Scholar

[10]

William G. Faris and Giovanni Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise,, J. Phys. A: Math. Gen., 15 (1982), 3025.  doi: 10.1088/0305-4470/15/10/011.  Google Scholar

[11]

Jin Feng and Markos A. Katsoulakis, A comparison principle for Hamilton-Jacobi equations related to controlled gradient flows in infinite dimensions,, Arch. Ration. Mech. Anal., 192 (2009), 275.  doi: 10.1007/s00205-008-0133-5.  Google Scholar

[12]

Paul C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces,", CBMS-NSF Regional Conference Series in Applied Mathematics, 53 (1988).  doi: 10.1137/1.9781611970180.  Google Scholar

[13]

M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems," (English summary), Second edition, (1998).  doi: 10.1007/978-1-4612-0611-8.  Google Scholar

[14]

, Yannis Goumas and Takashi Suzuki,, work in progress., ().   Google Scholar

[15]

M. Hildebrand and A. S. Mikhailov, Mesoscopic modeling in the kinetic theory of adsorbates,, J. Phys. Chem., 100 (1996), 19089.  doi: 10.1021/jp961668w.  Google Scholar

[16]

Tom Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature,, J. Differential Geom., 38 (1993), 417.   Google Scholar

[17]

Georgia Karali and Markos A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution,, J. Differential Equations, 235 (2007), 418.  doi: 10.1016/j.jde.2006.12.021.  Google Scholar

[18]

Georgia Karali and Tonia Ricciardi, On the convergence of a fourth order evolution equation to the Allen-Cahn equation,, Nonlinear Anal., 72 (2010), 4271.  doi: 10.1016/j.na.2010.02.003.  Google Scholar

[19]

Markos A. Katsoulakis and Dionisios G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution,, Phys. Rev. Lett., 84 (2000), 1511.  doi: 10.1103/PhysRevLett.84.1511.  Google Scholar

[20]

Robert V. Kohn, Felix Otto, Maria G. Reznikoff and Eric Vanden-Eijinden, Action minimization and sharp-interface limits for the stochastic Allen-Cahn equation,, Comm. Pure Appl. Math., 60 (2007), 393.  doi: 10.1002/cpa.20144.  Google Scholar

[21]

Robert V. Kohn, Maria G. Reznikoff and Yoshihiro Tonegawa, Sharp-interface limit of the Allen-Cahn action functional in one space dimension,, Calc. Var. Partial Differential Equations, 25 (2006), 503.  doi: 10.1007/s00526-005-0370-5.  Google Scholar

[22]

J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,", (French) Dunod; Gauthier-Villars, (1969).   Google Scholar

[23]

L. Mugnai and Röger, The Allen-Cahn action functional in higher dimensions,, Interfaces Free Bound., 10 (2008), 45.  doi: 10.4171/IFB/179.  Google Scholar

[24]

Yuko Nagase, Action minimization for an Allen-Cahn equation with an unequal double-well potential,, Manuscripta Mathematica, 137 (2012), 81.  doi: 10.1007/s00229-011-0458-5.  Google Scholar

[25]

R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation,, Proc. Roy. Soc. London Ser. A, 422 (1989), 261.  doi: 10.1098/rspa.1989.0027.  Google Scholar

[26]

Jacob Rubinstein, Peter Sternberg and Joseph B. Keller, Fast reaction, slow diffusion, and curve shortening,, SIAM J. Appl. Math., 49 (1989), 116.  doi: 10.1137/0149007.  Google Scholar

[27]

Roger Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics,", Second edition, 68 (1997).   Google Scholar

[28]

Eric Vanden-Eijnden and Maria G.Westdickenberg, Rare events in stochastic partial differential equations on large spatial domains,, J. Stat. Phys., 131 (2008), 1023.  doi: 10.1007/s10955-008-9537-8.  Google Scholar

[29]

John B. Walsh, An introduction to stochastic partial differential equations,, in, 1180 (1986), 265.  doi: 10.1007/BFb0074920.  Google Scholar

[30]

Maria G. Westdickenberg, Rare events, action minimization, and sharp interface limits,, in, 44 (2008), 217.   Google Scholar

[31]

Maria. G. Westdickenberg and Yoshihiro Tonegawa, Higher multiplicity in the one-dimensional Allen-Cahn action functional,, Indiana Univ. Math. J., 56 (2007), 2935.  doi: 10.1512/iumj.2007.56.3182.  Google Scholar

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