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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] |
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. |
[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., (). Google Scholar |
[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). Google Scholar |
[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. 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. |
[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, (). 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. |
[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). 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. |
[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). Google Scholar |
[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. |
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. |
[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., (). Google Scholar |
[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). Google Scholar |
[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. 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. |
[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, (). 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. |
[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). 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. |
[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). Google Scholar |
[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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