# American Institute of Mathematical Sciences

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 and 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), 165-205. doi: 10.1007/BF00375025. [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), 31-55. doi: 10.3934/dcdsb.2011.16.31. [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., (). [4] Kenneth A. Brakke, "The Motion of a Surface by its Mean Curvature," Mathematical Notes, 20, Princeton University Press, Princeton, N.J., 1978. [5] Caroline Cardon-Weber, Cahn-Hilliard stochastic equation: Existence of the solution and of its density, Bernoulli, 7 (2001), 777-816. doi: 10.2307/3318542. [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), 749-786. [7] Giuseppe Da Prato and Arnaud Debussche, Stochastic Cahn-Hilliard equation, Nonlinear Anal., 26 (1996), 241-263. doi: 10.1016/0362-546X(94)00277-O. [8] Weinan E, Weiqing Ren and Eric Vanden-Eijnden, Minimum action method for the study of rare events, Comm. Pure Appl. Math., 57 (2004), 637-656. doi: 10.1002/cpa.20005. [9] L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom., 33 (1991), 635-681. [10] William G. Faris and Giovanni Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise, J. Phys. A: Math. Gen., 15 (1982), 3025-3055. doi: 10.1088/0305-4470/15/10/011. [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), 275-310. doi: 10.1007/s00205-008-0133-5. [12] Paul C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. doi: 10.1137/1.9781611970180. [13] M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems," (English summary) Second edition, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0611-8. [14] , Yannis Goumas and Takashi Suzuki,, work in progress., (). [15] M. Hildebrand and A. S. Mikhailov, Mesoscopic modeling in the kinetic theory of adsorbates, J. Phys. Chem., 100 (1996), 19089-19101. doi: 10.1021/jp961668w. [16] Tom Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom., 38 (1993), 417-461. [17] Georgia Karali and Markos A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution, J. Differential Equations, 235 (2007), 418-438. doi: 10.1016/j.jde.2006.12.021. [18] Georgia Karali and Tonia Ricciardi, On the convergence of a fourth order evolution equation to the Allen-Cahn equation, Nonlinear Anal., 72 (2010), 4271-4281. doi: 10.1016/j.na.2010.02.003. [19] Markos A. Katsoulakis and Dionisios G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution, Phys. Rev. Lett., 84 (2000), 1511-1514. doi: 10.1103/PhysRevLett.84.1511. [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), 393-438. doi: 10.1002/cpa.20144. [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), 503-534. doi: 10.1007/s00526-005-0370-5. [22] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French) Dunod; Gauthier-Villars, Paris, 1969. [23] L. Mugnai and Röger, The Allen-Cahn action functional in higher dimensions, Interfaces Free Bound., 10 (2008), 45-78. doi: 10.4171/IFB/179. [24] Yuko Nagase, Action minimization for an Allen-Cahn equation with an unequal double-well potential, Manuscripta Mathematica, 137 (2012), 81-106. doi: 10.1007/s00229-011-0458-5. [25] R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278. doi: 10.1098/rspa.1989.0027. [26] Jacob Rubinstein, Peter Sternberg and Joseph B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133. doi: 10.1137/0149007. [27] Roger Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. [28] Eric Vanden-Eijnden and Maria G.Westdickenberg, Rare events in stochastic partial differential equations on large spatial domains, J. Stat. Phys., 131 (2008), 1023-1038. doi: 10.1007/s10955-008-9537-8. [29] John B. Walsh, An introduction to stochastic partial differential equations, in "École d'été de probabilités de Saint-Flour, XIV-1984," Lecture Notes in Math., 1180, Springer, Berlin, (1986), 265-439. doi: 10.1007/BFb0074920. [30] Maria G. Westdickenberg, Rare events, action minimization, and sharp interface limits, in "Singularities in PDE and the Calculus of Variations," CRM Proc. Lecture Notes, 44, Amer. Math. Soc., Providence, RI, (2008), 217-231. [31] Maria. G. Westdickenberg and Yoshihiro Tonegawa, Higher multiplicity in the one-dimensional Allen-Cahn action functional, Indiana Univ. Math. J., 56 (2007), 2935-2989. doi: 10.1512/iumj.2007.56.3182.

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##### 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), 165-205. doi: 10.1007/BF00375025. [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), 31-55. doi: 10.3934/dcdsb.2011.16.31. [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., (). [4] Kenneth A. Brakke, "The Motion of a Surface by its Mean Curvature," Mathematical Notes, 20, Princeton University Press, Princeton, N.J., 1978. [5] Caroline Cardon-Weber, Cahn-Hilliard stochastic equation: Existence of the solution and of its density, Bernoulli, 7 (2001), 777-816. doi: 10.2307/3318542. [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), 749-786. [7] Giuseppe Da Prato and Arnaud Debussche, Stochastic Cahn-Hilliard equation, Nonlinear Anal., 26 (1996), 241-263. doi: 10.1016/0362-546X(94)00277-O. [8] Weinan E, Weiqing Ren and Eric Vanden-Eijnden, Minimum action method for the study of rare events, Comm. Pure Appl. Math., 57 (2004), 637-656. doi: 10.1002/cpa.20005. [9] L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom., 33 (1991), 635-681. [10] William G. Faris and Giovanni Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise, J. Phys. A: Math. Gen., 15 (1982), 3025-3055. doi: 10.1088/0305-4470/15/10/011. [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), 275-310. doi: 10.1007/s00205-008-0133-5. [12] Paul C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. doi: 10.1137/1.9781611970180. [13] M. I. Freidlin and A. D. Wentzell, "Random Perturbations of Dynamical Systems," (English summary) Second edition, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0611-8. [14] , Yannis Goumas and Takashi Suzuki,, work in progress., (). [15] M. Hildebrand and A. S. Mikhailov, Mesoscopic modeling in the kinetic theory of adsorbates, J. Phys. Chem., 100 (1996), 19089-19101. doi: 10.1021/jp961668w. [16] Tom Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differential Geom., 38 (1993), 417-461. [17] Georgia Karali and Markos A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution, J. Differential Equations, 235 (2007), 418-438. doi: 10.1016/j.jde.2006.12.021. [18] Georgia Karali and Tonia Ricciardi, On the convergence of a fourth order evolution equation to the Allen-Cahn equation, Nonlinear Anal., 72 (2010), 4271-4281. doi: 10.1016/j.na.2010.02.003. [19] Markos A. Katsoulakis and Dionisios G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution, Phys. Rev. Lett., 84 (2000), 1511-1514. doi: 10.1103/PhysRevLett.84.1511. [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), 393-438. doi: 10.1002/cpa.20144. [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), 503-534. doi: 10.1007/s00526-005-0370-5. [22] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires," (French) Dunod; Gauthier-Villars, Paris, 1969. [23] L. Mugnai and Röger, The Allen-Cahn action functional in higher dimensions, Interfaces Free Bound., 10 (2008), 45-78. doi: 10.4171/IFB/179. [24] Yuko Nagase, Action minimization for an Allen-Cahn equation with an unequal double-well potential, Manuscripta Mathematica, 137 (2012), 81-106. doi: 10.1007/s00229-011-0458-5. [25] R. L. Pego, Front migration in the nonlinear Cahn-Hilliard equation, Proc. Roy. Soc. London Ser. A, 422 (1989), 261-278. doi: 10.1098/rspa.1989.0027. [26] Jacob Rubinstein, Peter Sternberg and Joseph B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133. doi: 10.1137/0149007. [27] Roger Temam, "Infinite-Dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. [28] Eric Vanden-Eijnden and Maria G.Westdickenberg, Rare events in stochastic partial differential equations on large spatial domains, J. Stat. Phys., 131 (2008), 1023-1038. doi: 10.1007/s10955-008-9537-8. [29] John B. Walsh, An introduction to stochastic partial differential equations, in "École d'été de probabilités de Saint-Flour, XIV-1984," Lecture Notes in Math., 1180, Springer, Berlin, (1986), 265-439. doi: 10.1007/BFb0074920. [30] Maria G. Westdickenberg, Rare events, action minimization, and sharp interface limits, in "Singularities in PDE and the Calculus of Variations," CRM Proc. Lecture Notes, 44, Amer. Math. Soc., Providence, RI, (2008), 217-231. [31] Maria. G. Westdickenberg and Yoshihiro Tonegawa, Higher multiplicity in the one-dimensional Allen-Cahn action functional, Indiana Univ. Math. J., 56 (2007), 2935-2989. doi: 10.1512/iumj.2007.56.3182.
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