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

Asymptotics for a generalized Cahn-Hilliard equation with forcing terms

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

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.
Citation: Dimitra Antonopoulou, Georgia Karali, Georgios T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1037-1054. doi: 10.3934/dcds.2011.30.1037
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), 165-205. 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), 1-49.  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), 2880-2883. 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), 429-445. 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), 421-457.  Google Scholar

[6]

P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard Equation, Part II, Journ. of Differential Equations, 117 (1995), 165-216.  Google Scholar

[7]

D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hilliard models, in "Mathematical Methods and Models in Phase Transitions" (ed. A. Miranville), Nova Science Publishers, (2005), 1-41.  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), 258-267. 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), 1121-1124. 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), 506-518. doi: 10.1137/0148029.  Google Scholar

[11]

H. Cook, Brownian motion in spinodal decomposition, Acta Metallurgica, 18 (1970), 297-306. 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, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[13]

G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation, Nonlinear Anal., 26 (1996), 241-263. 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), 1706-1739. 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), 1421-1445. doi: 10.1137/S0036142901387956.  Google Scholar

[16]

C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rat. Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.  Google Scholar

[17]

N. Elezovic and A. Mikelic, On the stochastic Cahn-Hilliard equation, Nonlinear Anal., 16 (1991), 1169-1200. 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), 1705-1727.  Google Scholar

[19]

P. C. Fife, Models for phase separation and their mathematics, El. Journ. Diff. E., 48 (2000), 1-26.  Google Scholar

[20]

T. Funaki, The scaling limit for a Stochastic PDE and the separation of phases, Probab. Theory Relat. Fields., 102 (1995), 221-288. doi: 10.1007/BF01213390.  Google Scholar

[21]

T. Funaki, Singular limit for stochastic reaction-diffusion equation and generation of random interfaces, Acta Math. Sinica, English Series, 15 (1999), 407-438. 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 "New Trends in Stocastic Analysis" (eds. Elworthy, Kusuoka and Shigekawa), World Sci., (2000), 132-152.  Google Scholar

[23]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. 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), 19089-19101. 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), 435-479. 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, Lecture Notes-Monograph Series 26, Hayward, California, 1995.  Google Scholar

[27]

G. Karali, Phase boundaries motion preserving the volume of each connected component, Asymptotic Analysis, 49 (2006), 17-37.  Google Scholar

[28]

G. Karali and M. A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution, J. Differential Equations, 235 (2007), 418-438.  Google Scholar

[29]

M. A. Katsoulakis and D. G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution, Phys. Rev. Letters, 84 (2000), 1511-1514. 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), 765-771. 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), 1-30. 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, Pergamon, 3rd edition, 1994.  Google Scholar

[34]

J. S. Langer, Theory of spinodal decomposition in alloys, Ann. of Phys., 65 (1971), 53-86. 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), 361-375.  Google Scholar

[36]

B. Øksendal, "Stochastic Differential Equations," Springer, New York, 2003. Google Scholar

[37]

R. L. Pego, Front migration in the non-linear Cahn-Hilliard equation, Proc. R. Soc. Lond. A, 422 (1989), 261-278. 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), 1055-1078. 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), 9638-9649. 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), 829-847. 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), 1363-1384. doi: 10.1137/040605278.  Google Scholar

[42]

J. B. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Math., 984 (1986), 265-439. 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, (Kyoto, 1999), Sūrikaisekikenkyūusho Kōkyūroku 1128 (2000), 172-180.  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), 165-205. 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), 1-49.  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), 2880-2883. 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), 429-445. 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), 421-457.  Google Scholar

[6]

P. W. Bates and J. Xun, Metastable patterns for the Cahn-Hilliard Equation, Part II, Journ. of Differential Equations, 117 (1995), 165-216.  Google Scholar

[7]

D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hilliard models, in "Mathematical Methods and Models in Phase Transitions" (ed. A. Miranville), Nova Science Publishers, (2005), 1-41.  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), 258-267. 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), 1121-1124. 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), 506-518. doi: 10.1137/0148029.  Google Scholar

[11]

H. Cook, Brownian motion in spinodal decomposition, Acta Metallurgica, 18 (1970), 297-306. 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, Cambridge, 1992. doi: 10.1017/CBO9780511666223.  Google Scholar

[13]

G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation, Nonlinear Anal., 26 (1996), 241-263. 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), 1706-1739. 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), 1421-1445. doi: 10.1137/S0036142901387956.  Google Scholar

[16]

C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rat. Mech. Anal., 96 (1986), 339-357. doi: 10.1007/BF00251803.  Google Scholar

[17]

N. Elezovic and A. Mikelic, On the stochastic Cahn-Hilliard equation, Nonlinear Anal., 16 (1991), 1169-1200. 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), 1705-1727.  Google Scholar

[19]

P. C. Fife, Models for phase separation and their mathematics, El. Journ. Diff. E., 48 (2000), 1-26.  Google Scholar

[20]

T. Funaki, The scaling limit for a Stochastic PDE and the separation of phases, Probab. Theory Relat. Fields., 102 (1995), 221-288. doi: 10.1007/BF01213390.  Google Scholar

[21]

T. Funaki, Singular limit for stochastic reaction-diffusion equation and generation of random interfaces, Acta Math. Sinica, English Series, 15 (1999), 407-438. 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 "New Trends in Stocastic Analysis" (eds. Elworthy, Kusuoka and Shigekawa), World Sci., (2000), 132-152.  Google Scholar

[23]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192. 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), 19089-19101. 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), 435-479. 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, Lecture Notes-Monograph Series 26, Hayward, California, 1995.  Google Scholar

[27]

G. Karali, Phase boundaries motion preserving the volume of each connected component, Asymptotic Analysis, 49 (2006), 17-37.  Google Scholar

[28]

G. Karali and M. A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution, J. Differential Equations, 235 (2007), 418-438.  Google Scholar

[29]

M. A. Katsoulakis and D. G. Vlachos, From microscopic interactions to macroscopic laws of cluster evolution, Phys. Rev. Letters, 84 (2000), 1511-1514. 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), 765-771. 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), 1-30. 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, Pergamon, 3rd edition, 1994.  Google Scholar

[34]

J. S. Langer, Theory of spinodal decomposition in alloys, Ann. of Phys., 65 (1971), 53-86. 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), 361-375.  Google Scholar

[36]

B. Øksendal, "Stochastic Differential Equations," Springer, New York, 2003. Google Scholar

[37]

R. L. Pego, Front migration in the non-linear Cahn-Hilliard equation, Proc. R. Soc. Lond. A, 422 (1989), 261-278. 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), 1055-1078. 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), 9638-9649. 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), 829-847. 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), 1363-1384. doi: 10.1137/040605278.  Google Scholar

[42]

J. B. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Math., 984 (1986), 265-439. 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, (Kyoto, 1999), Sūrikaisekikenkyūusho Kōkyūroku 1128 (2000), 172-180.  Google Scholar

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