Asymptotics for a generalized Cahn-Hilliard equation
with forcing terms

Dimitra Antonopoulou; Georgia Karali; Georgios T. Kossioris

doi:10.3934/dcds.2011.30.1037

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November 2011, 30(4): 1037-1054. doi: 10.3934/dcds.2011.30.1037

Asymptotics for a generalized Cahn-Hilliard equation with forcing terms

Dimitra Antonopoulou ^1, , Georgia Karali ^1, and Georgios T. Kossioris ^2,

1.	Department of Applied Mathematics, University of Crete, 714 09 Heraklion, Greece, Greece
2.	Department of Mathematics, University of Crete, GR-714 09 Heraklion, Crete, Greece

Received February 2010 Revised July 2010 Published May 2011

Abstract
Related Papers

Motivated by the physical theory of Critical Dynamics the Cahn-Hilliard equation on a bounded space domain is considered and forcing terms of general type are introduced. For such a rescaled equation the limiting inter-face problem is studied and the following are derived: (i) asymptotic results indicating that the forcing terms may slow down the equilibrium locally or globally, (ii) the sharp interface limit problem in the multidimensional case demonstrating a local influence in phase transitions of terms that stem from the chemical potential, while free energy independent terms act on the rest of the domain, (iii) a limiting non-homogeneous linear diffusion equation for the one-dimensional problem in the case of deterministic forcing term that follows the white noise scaling.

Keywords: Generalized Cahn-Hilliard equation, Hele-Shaw problem., asymptotics.

Mathematics Subject Classification: Primary: 35K55, 35K5.

Citation: Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037

References:

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[2]	N. D. Alikakos, G. Fusco and G. Karali, Ostwald ripening in two dimensions- The rigorous derivation of the equations from Mullins-Sekerka dynamics,, Journ. of Differential Equations, 205 (2004), 1. Google Scholar
[3]	T. Antal, M. Droz, J. Magnin and Z. Rácz, Formation of liesengang patterns: A spinodal decomposition scenarion,, Phys. Rev. Lett., 83 (1999), 2880. doi: 10.1103/PhysRevLett.83.2880. Google Scholar
[4]	G. Belletini, M. S. Gelli, S. Luckhaus and M. Novaga, Deterministic equivalent for the Allen Cahn energy of a scaling law in the Ising model,, Calc. Var., 26 (2006), 429. doi: 10.1007/s00526-006-0012-6. Google Scholar
[5]	P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard Equation, Part I,, Journ. of Differential Equations, 111 (1994), 421. Google Scholar
[6]	P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard Equation, Part II,, Journ. of Differential Equations, 117 (1995), 165. Google Scholar
[7]	D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hilliard models,, in, (2005), 1. Google Scholar
[8]	J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102. Google Scholar
[9]	J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system II. Thermodynamic basis,, J. Chem. Phys., 30 (1959), 1121. doi: 10.1063/1.1730145. Google Scholar
[10]	G. Cagninalp and P. C. Fife, Dynamics of layered interfaces arising from phase boundaries,, SIAM J. Appl. Math., 48 (1988), 506. doi: 10.1137/0148029. Google Scholar
[11]	H. Cook, Brownian motion in spinodal decomposition,, Acta Metallurgica, 18 (1970), 297. doi: 10.1016/0001-6160(70)90144-6. Google Scholar
[12]	G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar
[13]	G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation,, Nonlinear Anal., 26 (1996), 241. doi: 10.1016/0362-546X(94)00277-O. Google Scholar
[14]	A. Debussche and L. Zambotti, Conservative stochastic Cahn-Hilliard equation with reflection,, Ann. Probab., 35 (2007), 1706. doi: 10.1214/009117906000000773. Google Scholar
[15]	Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises,, SIAM J. Num. Anal., 40 (2002), 1421. doi: 10.1137/S0036142901387956. Google Scholar
[16]	C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation,, Arch. Rat. Mech. Anal., 96 (1986), 339. doi: 10.1007/BF00251803. Google Scholar
[17]	N. Elezovic and A. Mikelic, On the stochastic Cahn-Hilliard equation,, Nonlinear Anal., 16 (1991), 1169. doi: 10.1016/0362-546X(91)90204-E. Google Scholar
[18]	X. Chen, X. Hong and F. Yi, Existence, uniqueness and regularity of solutions of Mullins-Sekerka problem,, Comm. Partial Differential Equations, 21 (1996), 1705. Google Scholar
[19]	P. C. Fife, Models for phase separation and their mathematics,, El. Journ. Diff. E., 48 (2000), 1. Google Scholar
[20]	T. Funaki, The scaling limit for a Stochastic PDE and the separation of phases,, Probab. Theory Relat. Fields., 102 (1995), 221. doi: 10.1007/BF01213390. Google Scholar
[21]	T. Funaki, Singular limit for stochastic reaction-diffusion equation and generation of random interfaces,, Acta Math. Sinica, 15 (1999), 407. doi: 10.1007/BF02650735. Google Scholar
[22]	T. Funaki, Singular limit for reaction-diffusion equation with self-similar Gaussian noise,, in Proceedings of Taniguchi symposium, (2000), 132. Google Scholar
[23]	M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178. doi: 10.1016/0167-2789(95)00173-5. Google Scholar
[24]	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
[25]	P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena,, J. Rev. Mod. Phys., 49 (1977), 435. doi: 10.1103/RevModPhys.49.435. Google Scholar
[26]	G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces,", Institute of Mathematical Statistics, 26 (1995). Google Scholar
[27]	G. Karali, Phase boundaries motion preserving the volume of each connected component,, Asymptotic Analysis, 49 (2006), 17. Google Scholar
[28]	G. Karali and M. A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution,, J. Differential Equations, 235 (2007), 418. Google Scholar
[29]	M. A. Katsoulakis and D. G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution,, Phys. Rev. Letters, 84 (2000), 1511. doi: 10.1103/PhysRevLett.84.1511. Google Scholar
[30]	K. Kitahara, Y. Oono and D. Jasnow, Phase separation dynamics and external force field,, Mod. Phys. Letters B, 2 (1988), 765. doi: 10.1142/S0217984988000461. Google Scholar
[31]	M. Katsoulakis, G. Kossioris and O. Lakkis, Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem,, Interfaces Free Bound, 9 (2007), 1. doi: 10.4171/IFB/154. Google Scholar
[32]	G. Kossioris and G. Zouraris, Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise,, ESAIM: Mathematical Modelling and Numerical Analysis, (). Google Scholar
[33]	L. D. Landau and E. M. Lifshitz, "Statistical Physics Part 1,", Course of Theoretical Physics, 5 (1994). Google Scholar
[34]	J. S. Langer, Theory of spinodal decomposition in alloys,, Ann. of Phys., 65 (1971), 53. doi: 10.1016/0003-4916(71)90162-X. Google Scholar
[35]	Di Liu, Convergence of the spectral method for stochastic Ginzburg-Landau equation driven by space-time white noise,, Comm. Math. Sci., 1 (2003), 361. Google Scholar
[36]	B. Øksendal, "Stochastic Differential Equations,", Springer, (2003). Google Scholar
[37]	R. L. Pego, Front migration in the non-linear Cahn-Hilliard equation,, Proc. R. Soc. Lond. A, 422 (1989), 261. doi: 10.1098/rspa.1989.0027. Google Scholar
[38]	J. Printems, On the discretization in time of parabolic stochastic partial differential equations,, Mathematical Modelling and Numerical Analysis, 35 (2001), 1055. doi: 10.1051/m2an:2001148. Google Scholar
[39]	T. M. Rogers, K. R. Elder and R. C. Desai, Numerical study of the late stages of spinodal decomposition,, Phys. Rev. B, 37 (1988), 9638. doi: 10.1103/PhysRevB.37.9638. Google Scholar
[40]	Y. Yan, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise,, BIT Numerical Mathematics, 44 (2004), 829. doi: 10.1007/s10543-004-3755-5. Google Scholar
[41]	Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations,, SIAM J. Numer. Anal., 43 (2005), 1363. doi: 10.1137/040605278. Google Scholar
[42]	J. B. Walsh, An introduction to stochastic partial differential equations,, Lecture Notes in Math., 984 (1986), 265. doi: 10.1007/BFb0074920. Google Scholar
[43]	Quan-Fang Wang and Shin-ichi Nakagiri, Weak solutions of Cahn-Hilliard equations having forcing terms and optimal control problems,, Mathematical Models in Functional Equations, 1128 (2000), 172. Google Scholar

show all references

References:

[1]	N. D. Alikakos, P. W. Bates and X. Chen, Convergence of the Cahn-Hilliard equation to the Hele-Shaw model,, Arch. Rat. Mech. Anal., 128 (1994), 165. doi: 10.1007/BF00375025. Google Scholar
[2]	N. D. Alikakos, G. Fusco and G. Karali, Ostwald ripening in two dimensions- The rigorous derivation of the equations from Mullins-Sekerka dynamics,, Journ. of Differential Equations, 205 (2004), 1. Google Scholar
[3]	T. Antal, M. Droz, J. Magnin and Z. Rácz, Formation of liesengang patterns: A spinodal decomposition scenarion,, Phys. Rev. Lett., 83 (1999), 2880. doi: 10.1103/PhysRevLett.83.2880. Google Scholar
[4]	G. Belletini, M. S. Gelli, S. Luckhaus and M. Novaga, Deterministic equivalent for the Allen Cahn energy of a scaling law in the Ising model,, Calc. Var., 26 (2006), 429. doi: 10.1007/s00526-006-0012-6. Google Scholar
[5]	P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard Equation, Part I,, Journ. of Differential Equations, 111 (1994), 421. Google Scholar
[6]	P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard Equation, Part II,, Journ. of Differential Equations, 117 (1995), 165. Google Scholar
[7]	D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hilliard models,, in, (2005), 1. Google Scholar
[8]	J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102. Google Scholar
[9]	J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system II. Thermodynamic basis,, J. Chem. Phys., 30 (1959), 1121. doi: 10.1063/1.1730145. Google Scholar
[10]	G. Cagninalp and P. C. Fife, Dynamics of layered interfaces arising from phase boundaries,, SIAM J. Appl. Math., 48 (1988), 506. doi: 10.1137/0148029. Google Scholar
[11]	H. Cook, Brownian motion in spinodal decomposition,, Acta Metallurgica, 18 (1970), 297. doi: 10.1016/0001-6160(70)90144-6. Google Scholar
[12]	G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions,", Cambridge University Press, (1992). doi: 10.1017/CBO9780511666223. Google Scholar
[13]	G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation,, Nonlinear Anal., 26 (1996), 241. doi: 10.1016/0362-546X(94)00277-O. Google Scholar
[14]	A. Debussche and L. Zambotti, Conservative stochastic Cahn-Hilliard equation with reflection,, Ann. Probab., 35 (2007), 1706. doi: 10.1214/009117906000000773. Google Scholar
[15]	Q. Du and T. Zhang, Numerical approximation of some linear stochastic partial differential equations driven by special additive noises,, SIAM J. Num. Anal., 40 (2002), 1421. doi: 10.1137/S0036142901387956. Google Scholar
[16]	C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation,, Arch. Rat. Mech. Anal., 96 (1986), 339. doi: 10.1007/BF00251803. Google Scholar
[17]	N. Elezovic and A. Mikelic, On the stochastic Cahn-Hilliard equation,, Nonlinear Anal., 16 (1991), 1169. doi: 10.1016/0362-546X(91)90204-E. Google Scholar
[18]	X. Chen, X. Hong and F. Yi, Existence, uniqueness and regularity of solutions of Mullins-Sekerka problem,, Comm. Partial Differential Equations, 21 (1996), 1705. Google Scholar
[19]	P. C. Fife, Models for phase separation and their mathematics,, El. Journ. Diff. E., 48 (2000), 1. Google Scholar
[20]	T. Funaki, The scaling limit for a Stochastic PDE and the separation of phases,, Probab. Theory Relat. Fields., 102 (1995), 221. doi: 10.1007/BF01213390. Google Scholar
[21]	T. Funaki, Singular limit for stochastic reaction-diffusion equation and generation of random interfaces,, Acta Math. Sinica, 15 (1999), 407. doi: 10.1007/BF02650735. Google Scholar
[22]	T. Funaki, Singular limit for reaction-diffusion equation with self-similar Gaussian noise,, in Proceedings of Taniguchi symposium, (2000), 132. Google Scholar
[23]	M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance,, Physica D, 92 (1996), 178. doi: 10.1016/0167-2789(95)00173-5. Google Scholar
[24]	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
[25]	P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena,, J. Rev. Mod. Phys., 49 (1977), 435. doi: 10.1103/RevModPhys.49.435. Google Scholar
[26]	G. Kallianpur and J. Xiong, "Stochastic Differential Equations in Infinite Dimensional Spaces,", Institute of Mathematical Statistics, 26 (1995). Google Scholar
[27]	G. Karali, Phase boundaries motion preserving the volume of each connected component,, Asymptotic Analysis, 49 (2006), 17. Google Scholar
[28]	G. Karali and M. A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution,, J. Differential Equations, 235 (2007), 418. Google Scholar
[29]	M. A. Katsoulakis and D. G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution,, Phys. Rev. Letters, 84 (2000), 1511. doi: 10.1103/PhysRevLett.84.1511. Google Scholar
[30]	K. Kitahara, Y. Oono and D. Jasnow, Phase separation dynamics and external force field,, Mod. Phys. Letters B, 2 (1988), 765. doi: 10.1142/S0217984988000461. Google Scholar
[31]	M. Katsoulakis, G. Kossioris and O. Lakkis, Noise regularization and computations for the 1-dimensional stochastic Allen-Cahn problem,, Interfaces Free Bound, 9 (2007), 1. doi: 10.4171/IFB/154. Google Scholar
[32]	G. Kossioris and G. Zouraris, Fully-discrete finite element approximations for a fourth-order linear stochastic parabolic equation with additive space-time white noise,, ESAIM: Mathematical Modelling and Numerical Analysis, (). Google Scholar
[33]	L. D. Landau and E. M. Lifshitz, "Statistical Physics Part 1,", Course of Theoretical Physics, 5 (1994). Google Scholar
[34]	J. S. Langer, Theory of spinodal decomposition in alloys,, Ann. of Phys., 65 (1971), 53. doi: 10.1016/0003-4916(71)90162-X. Google Scholar
[35]	Di Liu, Convergence of the spectral method for stochastic Ginzburg-Landau equation driven by space-time white noise,, Comm. Math. Sci., 1 (2003), 361. Google Scholar
[36]	B. Øksendal, "Stochastic Differential Equations,", Springer, (2003). Google Scholar
[37]	R. L. Pego, Front migration in the non-linear Cahn-Hilliard equation,, Proc. R. Soc. Lond. A, 422 (1989), 261. doi: 10.1098/rspa.1989.0027. Google Scholar
[38]	J. Printems, On the discretization in time of parabolic stochastic partial differential equations,, Mathematical Modelling and Numerical Analysis, 35 (2001), 1055. doi: 10.1051/m2an:2001148. Google Scholar
[39]	T. M. Rogers, K. R. Elder and R. C. Desai, Numerical study of the late stages of spinodal decomposition,, Phys. Rev. B, 37 (1988), 9638. doi: 10.1103/PhysRevB.37.9638. Google Scholar
[40]	Y. Yan, Semidiscrete Galerkin approximation for a linear stochastic parabolic partial differential equation driven by an additive noise,, BIT Numerical Mathematics, 44 (2004), 829. doi: 10.1007/s10543-004-3755-5. Google Scholar
[41]	Y. Yan, Galerkin finite element methods for stochastic parabolic partial differential equations,, SIAM J. Numer. Anal., 43 (2005), 1363. doi: 10.1137/040605278. Google Scholar
[42]	J. B. Walsh, An introduction to stochastic partial differential equations,, Lecture Notes in Math., 984 (1986), 265. doi: 10.1007/BFb0074920. Google Scholar
[43]	Quan-Fang Wang and Shin-ichi Nakagiri, Weak solutions of Cahn-Hilliard equations having forcing terms and optimal control problems,, Mathematical Models in Functional Equations, 1128 (2000), 172. Google Scholar

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