# American Institute of Mathematical Sciences

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

 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.
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). [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. [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. [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. [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. [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., (). [7] L. Arnold, "Stochastic Differential Equations: Theory and Applications,", Wiley and Sons, (1974). [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. [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. [10] J. M. Berezanskii, "Expansions in Eigenfunctions of Selfadjoint Operators,", Translations of Mathematical Monographs, (1968). [11] D. Blömker, S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard-Cook equation,, Communications in Mathematical Physics, 3 (2001), 553. [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. [13] S. C. Brenner and L. R. Scott, "The Mathematical Theory of Finite Element Methods,", Springer-Verlag, (1994). [14] J. W. Cahn, On spinodal decomposition,, Acta Metallurgica, 9 (1961), 795. doi: 10.1016/0001-6160(61)90182-1. [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. [16] C. Cardon-Weber, Cahn-Hilliard stochastic equation: Existence of the solution and of its density,, Bernoulli, 5 (2001), 777. doi: 10.2307/3318542. [17] C. Cardon-Weber, Cahn-Hilliard stochastic equation: Strict positivity of the density,, Stoch. Stoch. Rep., 72 (2002), 191. [18] H. Cook, Brownian motion in spinodal decomposition,, Acta Metallurgica, 18 (1970), 297. doi: 10.1016/0001-6160(70)90144-6. [19] R. Courant and D. Hilbert, "Methods of Mathematical Physics,", Vol. \textbf{1}, 1 (1953). [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. [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. [22] A. Debussche and L. Goudenège, Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections,, Prepublication - IRMAR, 29 (2009). [23] A. Debussche and L. Zambotti, Conservative stochastic Cahn-Hilliard equation with reflection,, Ann. Probab., 35 (2007), 1706. doi: 10.1214/009117906000000773. [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. [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. [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. [27] C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation,, Arch. Rational Mech. Anal., 96 (1986), 339. doi: 10.1007/BF00251803. [28] L. C. Evans, "Partial Differential Equations,", American Mathematical Society, (1998). [29] P. C. Fife, "Dynamical Aspects of the Cahn-Hilliard Equations,", Barret Lectures, (1991). [30] P. C. Fife, Models for phase separation and their mathematics,, El. Journ. Diff. E., 48 (2000), 1. [31] A. Garsia, Continuity properties of Gaussian process with multi-dimensional time parameter,, Proc. Sixth Berkeley Symp. Math. Stat. Probab., (1972), 369. [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. [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. [34] A. Hassell, "Eigenvalues and Eigenfunctions of the Laplacian,", Lecture notes, (). [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. [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. [37] G. Karali, Phase boundaries motion preserving the volume of each connected component,, Asymptotic Analysis, 49 (2006), 17. [38] G. Karali and M. A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution,, J. Differential Equations, 235 (2007), 418. [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. [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. [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. [42] J. S. Langer, Theory of spinodal decomposition in alloys,, Annals of Physics, 65 (1971), 53. doi: 10.1016/0003-4916(71)90162-X. [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. [44] B. Øksendal, "Stochastic Differential Equations,", Springer, (2003). [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. [46] P. E. Protter, "Stochastic Integration and Differential Equations,", Springer-Verlag Berlin Heidelberg, (2005). [47] S. Semmes, Some aspects of calculus on non-smooth sets,, \arXiv{0709.2508v3}, (2007). [48] J. B. Walsh, An introduction to stochastic partial differential equations,, 265-439, (1180), 265. [49] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen,, Math. Ann., 71 (1911), 441. doi: 10.1007/BF01456804.

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##### References:
 [1] R. A. Adams, "Sobolev Spaces,", Academic Press, (1993). [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. [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. [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. [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. [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., (). [7] L. Arnold, "Stochastic Differential Equations: Theory and Applications,", Wiley and Sons, (1974). [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. [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. [10] J. M. Berezanskii, "Expansions in Eigenfunctions of Selfadjoint Operators,", Translations of Mathematical Monographs, (1968). [11] D. Blömker, S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard-Cook equation,, Communications in Mathematical Physics, 3 (2001), 553. [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. [13] S. C. Brenner and L. R. Scott, "The Mathematical Theory of Finite Element Methods,", Springer-Verlag, (1994). [14] J. W. Cahn, On spinodal decomposition,, Acta Metallurgica, 9 (1961), 795. doi: 10.1016/0001-6160(61)90182-1. [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. [16] C. Cardon-Weber, Cahn-Hilliard stochastic equation: Existence of the solution and of its density,, Bernoulli, 5 (2001), 777. doi: 10.2307/3318542. [17] C. Cardon-Weber, Cahn-Hilliard stochastic equation: Strict positivity of the density,, Stoch. Stoch. Rep., 72 (2002), 191. [18] H. Cook, Brownian motion in spinodal decomposition,, Acta Metallurgica, 18 (1970), 297. doi: 10.1016/0001-6160(70)90144-6. [19] R. Courant and D. Hilbert, "Methods of Mathematical Physics,", Vol. \textbf{1}, 1 (1953). [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. [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. [22] A. Debussche and L. Goudenège, Stochastic Cahn-Hilliard equation with double singular nonlinearities and two reflections,, Prepublication - IRMAR, 29 (2009). [23] A. Debussche and L. Zambotti, Conservative stochastic Cahn-Hilliard equation with reflection,, Ann. Probab., 35 (2007), 1706. doi: 10.1214/009117906000000773. [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. [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. [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. [27] C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation,, Arch. Rational Mech. Anal., 96 (1986), 339. doi: 10.1007/BF00251803. [28] L. C. Evans, "Partial Differential Equations,", American Mathematical Society, (1998). [29] P. C. Fife, "Dynamical Aspects of the Cahn-Hilliard Equations,", Barret Lectures, (1991). [30] P. C. Fife, Models for phase separation and their mathematics,, El. Journ. Diff. E., 48 (2000), 1. [31] A. Garsia, Continuity properties of Gaussian process with multi-dimensional time parameter,, Proc. Sixth Berkeley Symp. Math. Stat. Probab., (1972), 369. [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. [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. [34] A. Hassell, "Eigenvalues and Eigenfunctions of the Laplacian,", Lecture notes, (). [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. [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. [37] G. Karali, Phase boundaries motion preserving the volume of each connected component,, Asymptotic Analysis, 49 (2006), 17. [38] G. Karali and M. A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution,, J. Differential Equations, 235 (2007), 418. [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. [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. [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. [42] J. S. Langer, Theory of spinodal decomposition in alloys,, Annals of Physics, 65 (1971), 53. doi: 10.1016/0003-4916(71)90162-X. [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. [44] B. Øksendal, "Stochastic Differential Equations,", Springer, (2003). [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. [46] P. E. Protter, "Stochastic Integration and Differential Equations,", Springer-Verlag Berlin Heidelberg, (2005). [47] S. Semmes, Some aspects of calculus on non-smooth sets,, \arXiv{0709.2508v3}, (2007). [48] J. B. Walsh, An introduction to stochastic partial differential equations,, 265-439, (1180), 265. [49] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen,, Math. Ann., 71 (1911), 441. doi: 10.1007/BF01456804.
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