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July  2011, 16(1): 31-55. doi: 10.3934/dcdsb.2011.16.31

Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains

Dimitra Antonopoulou 1, and  Georgia Karali 2,

1. 

Department of Applied Mathematics, University of Crete, Heraklion, Greece
2. 

Department of Applied Mathematics, University Crete, P.O. Box 2208, 71409, Heraklion, Crete, Greece

Received  April 2010 Revised  February 2011 Published  April 2011

We consider a generalized Stochastic Cahn-Hilliard equation with multiplicative white noise posed on bounded convex domains in $R^d$, $d=1,2,3$, with piece-wise smooth boundary, and introduce an additive time dependent white noise term in the chemical potential. Since the Green's function of the problem is induced by a convolution semigroup, we present the equation in a weak stochastic integral formulation and prove existence of solution when $d\leq 2$ for general domains, and for $d=3$ for domains with minimum eigenfunction growth, without making use of any explicit expression of the spectrum and the eigenfunctions. The analysis is based on stochastic integral calculus, Galerkin approximations and the asymptotic spectral properties of the Neumann Laplacian operator. Existence is also derived for some non-convex cases when the boundary is smooth.
Keywords: convex domains, space-time white noise, convolution semigroup, Galerkin approximations., Stochastic Cahn-Hilliard, Neumann Laplacian eigenfunction basis, Green's function.
Mathematics Subject Classification: 35K55, 35K40, 60H30, 60H1.
Citation: Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31
References:
[1]

R. A. Adams, "Sobolev Spaces,", Academic Press, (1993). Google Scholar

[2]

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

[3]

N. D. Alikakos, G. Fusco and G. Karali, The effect of the geometry of the particle distribution in Ostwald Ripening,, Comm. Math. Phys., 238 (2003), 480. doi: 10.1007/s00220-003-0834-4. Google Scholar

[4]

N. D. Alikakos, G. Fusco and G. Karali, Ostwald Ripening in two dimensions - The rigorous derivation of the equations from the Mullins-Sekerka dynamics,, J. Differential Equations 205, 1 (2004), 1. Google Scholar

[5]

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

[6]

D. C. Antonopoulou, G. D. Karali and G. T. Kossioris, Asymptotics for a generalized Cahn-Hilliard equation with forcing terms,, to appear in Discrete and Cont. Dyn. Syst. A., (). Google Scholar

[7]

L. Arnold, "Stochastic Differential Equations: Theory and Applications,", Wiley and Sons, (1974). Google Scholar

[8]

P. W. Bates and P. C. Fife, The dynamics of nucleation for the Cahn-Hilliard,, SIAM J. Appl. Math., 53 (1993), 990. doi: 10.1137/0153049. Google Scholar

[9]

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

[10]

J. M. Berezanskii, "Expansions in Eigenfunctions of Selfadjoint Operators,", Translations of Mathematical Monographs, (1968). Google Scholar

[11]

D. Blömker, S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard-Cook equation,, Communications in Mathematical Physics, 3 (2001), 553. Google Scholar

[12]

D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hilliard models,, Mathematical Methods and Models in Phase Transitions (2005), (2005), 1. Google Scholar

[13]

S. C. Brenner and L. R. Scott, "The Mathematical Theory of Finite Element Methods,", Springer-Verlag, (1994). Google Scholar

[14]

J. W. Cahn, On spinodal decomposition,, Acta Metallurgica, 9 (1961), 795. doi: 10.1016/0001-6160(61)90182-1. Google Scholar

[15]

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

[16]

C. Cardon-Weber, Cahn-Hilliard stochastic equation: Existence of the solution and of its density,, Bernoulli, 5 (2001), 777. doi: 10.2307/3318542. Google Scholar

[17]

C. Cardon-Weber, Cahn-Hilliard stochastic equation: Strict positivity of the density,, Stoch. Stoch. Rep., 72 (2002), 191. Google Scholar

[18]

H. Cook, Brownian motion in spinodal decomposition,, Acta Metallurgica, 18 (1970), 297. doi: 10.1016/0001-6160(70)90144-6. Google Scholar

[19]

R. Courant and D. Hilbert, "Methods of Mathematical Physics,", Vol. \textbf{1}, 1 (1953). Google Scholar

[20]

G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation,, Nonlin. Anal. Th. Meth. Appl., 26 (1996), 241. doi: 10.1016/0362-546X(94)00277-O. Google Scholar

[21]

A. Debussche and L. Dettori, On the Cahn-Hilliard equation with logarithmic free energy,, Nonlin. Anal. Th. Meth. Appl., 24 (1995), 1491. doi: 10.1016/0362-546X(94)00205-V. Google Scholar

[22]

A. Debussche and L. Goudenège, Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections,, Prepublication - IRMAR, 29 (2009). Google Scholar

[23]

A. Debussche and L. Zambotti, Conservative stochastic Cahn-Hilliard equation with reflection,, Ann. Probab., 35 (2007), 1706. doi: 10.1214/009117906000000773. Google Scholar

[24]

J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, Invent. Math., 29 (1975), 39. doi: 10.1007/BF01405172. Google Scholar

[25]

S. D. Eidelman and N. V. Ivasisen, Investigation of the green matrix for homogeneous parabolic boundary value problem,, Trans. Moscow. Math. Soc., 23 (1970), 179. Google Scholar

[26]

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

[27]

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

[28]

L. C. Evans, "Partial Differential Equations,", American Mathematical Society, (1998). Google Scholar

[29]

P. C. Fife, "Dynamical Aspects of the Cahn-Hilliard Equations,", Barret Lectures, (1991). Google Scholar

[30]

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

[31]

A. Garsia, Continuity properties of Gaussian process with multi-dimensional time parameter,, Proc. Sixth Berkeley Symp. Math. Stat. Probab., (1972), 369. Google Scholar

[32]

L. Goudenège, Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection,, Stochastic Processes and their Applications, 119 (2009), 3516. doi: 10.1016/j.spa.2009.06.008. Google Scholar

[33]

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

[34]

A. Hassell, "Eigenvalues and Eigenfunctions of the Laplacian,", Lecture notes, (). Google Scholar

[35]

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

[36]

P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena,, Rev. mod. Phys., 49 (1977), 435. doi: 10.1103/RevModPhys.49.435. Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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

[41]

P. Kröger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space,, J. Funct. Anal., 106 (1992), 353. doi: 10.1016/0022-1236(92)90052-K. Google Scholar

[42]

J. S. Langer, Theory of spinodal decomposition in alloys,, Annals of Physics, 65 (1971), 53. doi: 10.1016/0003-4916(71)90162-X. Google Scholar

[43]

S. Maier-Paape, U. Miller, K. Mischaikow and T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square,, Rev. Mat. Complut., 21 (2008), 351. Google Scholar

[44]

B. Øksendal, "Stochastic Differential Equations,", Springer, (2003). Google Scholar

[45]

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

[46]

P. E. Protter, "Stochastic Integration and Differential Equations,", Springer-Verlag Berlin Heidelberg, (2005). Google Scholar

[47]

S. Semmes, Some aspects of calculus on non-smooth sets,, \arXiv{0709.2508v3}, (2007). Google Scholar

[48]

J. B. Walsh, An introduction to stochastic partial differential equations,, 265-439, (1180), 265. Google Scholar

[49]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen,, Math. Ann., 71 (1911), 441. doi: 10.1007/BF01456804. Google Scholar

show all references

References:
[1]

R. A. Adams, "Sobolev Spaces,", Academic Press, (1993). Google Scholar

[2]

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

[3]

N. D. Alikakos, G. Fusco and G. Karali, The effect of the geometry of the particle distribution in Ostwald Ripening,, Comm. Math. Phys., 238 (2003), 480. doi: 10.1007/s00220-003-0834-4. Google Scholar

[4]

N. D. Alikakos, G. Fusco and G. Karali, Ostwald Ripening in two dimensions - The rigorous derivation of the equations from the Mullins-Sekerka dynamics,, J. Differential Equations 205, 1 (2004), 1. Google Scholar

[5]

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

[6]

D. C. Antonopoulou, G. D. Karali and G. T. Kossioris, Asymptotics for a generalized Cahn-Hilliard equation with forcing terms,, to appear in Discrete and Cont. Dyn. Syst. A., (). Google Scholar

[7]

L. Arnold, "Stochastic Differential Equations: Theory and Applications,", Wiley and Sons, (1974). Google Scholar

[8]

P. W. Bates and P. C. Fife, The dynamics of nucleation for the Cahn-Hilliard,, SIAM J. Appl. Math., 53 (1993), 990. doi: 10.1137/0153049. Google Scholar

[9]

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

[10]

J. M. Berezanskii, "Expansions in Eigenfunctions of Selfadjoint Operators,", Translations of Mathematical Monographs, (1968). Google Scholar

[11]

D. Blömker, S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard-Cook equation,, Communications in Mathematical Physics, 3 (2001), 553. Google Scholar

[12]

D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hilliard models,, Mathematical Methods and Models in Phase Transitions (2005), (2005), 1. Google Scholar

[13]

S. C. Brenner and L. R. Scott, "The Mathematical Theory of Finite Element Methods,", Springer-Verlag, (1994). Google Scholar

[14]

J. W. Cahn, On spinodal decomposition,, Acta Metallurgica, 9 (1961), 795. doi: 10.1016/0001-6160(61)90182-1. Google Scholar

[15]

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

[16]

C. Cardon-Weber, Cahn-Hilliard stochastic equation: Existence of the solution and of its density,, Bernoulli, 5 (2001), 777. doi: 10.2307/3318542. Google Scholar

[17]

C. Cardon-Weber, Cahn-Hilliard stochastic equation: Strict positivity of the density,, Stoch. Stoch. Rep., 72 (2002), 191. Google Scholar

[18]

H. Cook, Brownian motion in spinodal decomposition,, Acta Metallurgica, 18 (1970), 297. doi: 10.1016/0001-6160(70)90144-6. Google Scholar

[19]

R. Courant and D. Hilbert, "Methods of Mathematical Physics,", Vol. \textbf{1}, 1 (1953). Google Scholar

[20]

G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation,, Nonlin. Anal. Th. Meth. Appl., 26 (1996), 241. doi: 10.1016/0362-546X(94)00277-O. Google Scholar

[21]

A. Debussche and L. Dettori, On the Cahn-Hilliard equation with logarithmic free energy,, Nonlin. Anal. Th. Meth. Appl., 24 (1995), 1491. doi: 10.1016/0362-546X(94)00205-V. Google Scholar

[22]

A. Debussche and L. Goudenège, Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections,, Prepublication - IRMAR, 29 (2009). Google Scholar

[23]

A. Debussche and L. Zambotti, Conservative stochastic Cahn-Hilliard equation with reflection,, Ann. Probab., 35 (2007), 1706. doi: 10.1214/009117906000000773. Google Scholar

[24]

J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, Invent. Math., 29 (1975), 39. doi: 10.1007/BF01405172. Google Scholar

[25]

S. D. Eidelman and N. V. Ivasisen, Investigation of the green matrix for homogeneous parabolic boundary value problem,, Trans. Moscow. Math. Soc., 23 (1970), 179. Google Scholar

[26]

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

[27]

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

[28]

L. C. Evans, "Partial Differential Equations,", American Mathematical Society, (1998). Google Scholar

[29]

P. C. Fife, "Dynamical Aspects of the Cahn-Hilliard Equations,", Barret Lectures, (1991). Google Scholar

[30]

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

[31]

A. Garsia, Continuity properties of Gaussian process with multi-dimensional time parameter,, Proc. Sixth Berkeley Symp. Math. Stat. Probab., (1972), 369. Google Scholar

[32]

L. Goudenège, Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection,, Stochastic Processes and their Applications, 119 (2009), 3516. doi: 10.1016/j.spa.2009.06.008. Google Scholar

[33]

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

[34]

A. Hassell, "Eigenvalues and Eigenfunctions of the Laplacian,", Lecture notes, (). Google Scholar

[35]

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

[36]

P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena,, Rev. mod. Phys., 49 (1977), 435. doi: 10.1103/RevModPhys.49.435. Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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

[41]

P. Kröger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space,, J. Funct. Anal., 106 (1992), 353. doi: 10.1016/0022-1236(92)90052-K. Google Scholar

[42]

J. S. Langer, Theory of spinodal decomposition in alloys,, Annals of Physics, 65 (1971), 53. doi: 10.1016/0003-4916(71)90162-X. Google Scholar

[43]

S. Maier-Paape, U. Miller, K. Mischaikow and T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square,, Rev. Mat. Complut., 21 (2008), 351. Google Scholar

[44]

B. Øksendal, "Stochastic Differential Equations,", Springer, (2003). Google Scholar

[45]

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

[46]

P. E. Protter, "Stochastic Integration and Differential Equations,", Springer-Verlag Berlin Heidelberg, (2005). Google Scholar

[47]

S. Semmes, Some aspects of calculus on non-smooth sets,, \arXiv{0709.2508v3}, (2007). Google Scholar

[48]

J. B. Walsh, An introduction to stochastic partial differential equations,, 265-439, (1180), 265. Google Scholar

[49]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen,, Math. Ann., 71 (1911), 441. doi: 10.1007/BF01456804. Google Scholar

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