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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 |
References:
[1] | |
[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), 165-205.
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), 480-488.
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), 1-49. |
[5] |
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. |
[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), 990-1008.
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), 429-445.
doi: 10.1007/s00526-006-0012-6. |
[10] |
J. M. Berezanskii, "Expansions in Eigenfunctions of Selfadjoint Operators," Translations of Mathematical Monographs, American Mathematical Society, 1968. |
[11] |
D. Blömker, S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard-Cook equation, Communications in Mathematical Physics, 3 (2001) 553-582. |
[12] |
D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hilliard models, Mathematical Methods and Models in Phase Transitions (2005), 1-41. |
[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), 795-801.
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), 258-267.
doi: 10.1063/1.1744102. |
[16] |
C. Cardon-Weber, Cahn-Hilliard stochastic equation: Existence of the solution and of its density, Bernoulli, 5 (2001), 777-816.
doi: 10.2307/3318542. |
[17] |
C. Cardon-Weber, Cahn-Hilliard stochastic equation: Strict positivity of the density, Stoch. Stoch. Rep., 72 (2002), 191-227. |
[18] |
H. Cook, Brownian motion in spinodal decomposition, Acta Metallurgica, 18 (1970), 297-306.
doi: 10.1016/0001-6160(70)90144-6. |
[19] |
R. Courant and D. Hilbert, "Methods of Mathematical Physics," Vol. 1, Interscience Publishers, 1953. |
[20] |
G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation, Nonlin. Anal. Th. Meth. Appl., 26 (1996), 241-263.
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), 1491-1514.
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), 1706-1739.
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), 39-79.
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), 179-242. |
[26] |
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. |
[27] |
C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357
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, University of Tennessee, Spring 1991. |
[30] |
P. C. Fife, Models for phase separation and their mathematics, El. Journ. Diff. E., 48 (2000), 1-26. |
[31] |
A. Garsia, Continuity properties of Gaussian process with multi-dimensional time parameter, Proc. Sixth Berkeley Symp. Math. Stat. Probab., (1972), 369-374. |
[32] |
L. Goudenège, Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection, Stochastic Processes and their Applications, 119 (2009), 3516-3548.
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), 178-192.
doi: 10.1016/0167-2789(95)00173-5. |
[34] |
A. Hassell, "Eigenvalues and Eigenfunctions of the Laplacian," Lecture notes, Department of Mathematics, Australian National University. |
[35] |
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. |
[36] |
P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. mod. Phys., 49 (1977), 435-479.
doi: 10.1103/RevModPhys.49.435. |
[37] |
G. Karali, Phase boundaries motion preserving the volume of each connected component, Asymptotic Analysis, 49 (2006), 17-37. |
[38] |
G. Karali and M. A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution, J. Differential Equations, 235 (2007), 418-438. |
[39] |
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. |
[40] |
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. |
[41] |
P. Kröger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space, J. Funct. Anal., 106 (1992), 353-357.
doi: 10.1016/0022-1236(92)90052-K. |
[42] |
J. S. Langer, Theory of spinodal decomposition in alloys, Annals of Physics, 65 (1971), 53-86.
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), 351-426. |
[44] |
B. Øksendal, "Stochastic Differential Equations," Springer, New York, 2003. |
[45] |
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. |
[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, Lecture Notes in Math, 1180, Springer, Berlin, 1986. |
[49] |
H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann., 71 (1911), 441-479.
doi: 10.1007/BF01456804. |
show all references
References:
[1] | |
[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), 165-205.
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), 480-488.
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), 1-49. |
[5] |
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. |
[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), 990-1008.
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), 429-445.
doi: 10.1007/s00526-006-0012-6. |
[10] |
J. M. Berezanskii, "Expansions in Eigenfunctions of Selfadjoint Operators," Translations of Mathematical Monographs, American Mathematical Society, 1968. |
[11] |
D. Blömker, S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard-Cook equation, Communications in Mathematical Physics, 3 (2001) 553-582. |
[12] |
D. Blömker, S. Maier-Paape and T. Wanner, Phase separation in stochastic Cahn-Hilliard models, Mathematical Methods and Models in Phase Transitions (2005), 1-41. |
[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), 795-801.
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), 258-267.
doi: 10.1063/1.1744102. |
[16] |
C. Cardon-Weber, Cahn-Hilliard stochastic equation: Existence of the solution and of its density, Bernoulli, 5 (2001), 777-816.
doi: 10.2307/3318542. |
[17] |
C. Cardon-Weber, Cahn-Hilliard stochastic equation: Strict positivity of the density, Stoch. Stoch. Rep., 72 (2002), 191-227. |
[18] |
H. Cook, Brownian motion in spinodal decomposition, Acta Metallurgica, 18 (1970), 297-306.
doi: 10.1016/0001-6160(70)90144-6. |
[19] |
R. Courant and D. Hilbert, "Methods of Mathematical Physics," Vol. 1, Interscience Publishers, 1953. |
[20] |
G. Da Prato and A. Debussche, Stochastic Cahn-Hilliard equation, Nonlin. Anal. Th. Meth. Appl., 26 (1996), 241-263.
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), 1491-1514.
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), 1706-1739.
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), 39-79.
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), 179-242. |
[26] |
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. |
[27] |
C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357
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, University of Tennessee, Spring 1991. |
[30] |
P. C. Fife, Models for phase separation and their mathematics, El. Journ. Diff. E., 48 (2000), 1-26. |
[31] |
A. Garsia, Continuity properties of Gaussian process with multi-dimensional time parameter, Proc. Sixth Berkeley Symp. Math. Stat. Probab., (1972), 369-374. |
[32] |
L. Goudenège, Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection, Stochastic Processes and their Applications, 119 (2009), 3516-3548.
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), 178-192.
doi: 10.1016/0167-2789(95)00173-5. |
[34] |
A. Hassell, "Eigenvalues and Eigenfunctions of the Laplacian," Lecture notes, Department of Mathematics, Australian National University. |
[35] |
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. |
[36] |
P. C. Hohenberg and B. I. Halperin, Theory of dynamic critical phenomena, Rev. mod. Phys., 49 (1977), 435-479.
doi: 10.1103/RevModPhys.49.435. |
[37] |
G. Karali, Phase boundaries motion preserving the volume of each connected component, Asymptotic Analysis, 49 (2006), 17-37. |
[38] |
G. Karali and M. A. Katsoulakis, The role of multiple microscopic mechanisms in cluster interface evolution, J. Differential Equations, 235 (2007), 418-438. |
[39] |
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. |
[40] |
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. |
[41] |
P. Kröger, Upper bounds for the Neumann eigenvalues on a bounded domain in Euclidean space, J. Funct. Anal., 106 (1992), 353-357.
doi: 10.1016/0022-1236(92)90052-K. |
[42] |
J. S. Langer, Theory of spinodal decomposition in alloys, Annals of Physics, 65 (1971), 53-86.
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), 351-426. |
[44] |
B. Øksendal, "Stochastic Differential Equations," Springer, New York, 2003. |
[45] |
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. |
[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, Lecture Notes in Math, 1180, Springer, Berlin, 1986. |
[49] |
H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen, Math. Ann., 71 (1911), 441-479.
doi: 10.1007/BF01456804. |
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