The sharp interface limit for the stochastic Cahn-Hilliard equation. (English. French summary) Zbl 1391.35192

Summary: We study the \(\varepsilon\)-dependent two and three dimensional stochastic Cahn-Hilliard equation in the sharp interface limit \(\varepsilon\to0\). The parameter \(\varepsilon\) is positive and measures the width of transition layers generated during phase separation. We also couple the noise strength to this parameter. Using formal asymptotic expansions, we identify the limit. In the right scaling, our results indicate that the stochastic Cahn-Hilliard equation converge to a Hele-Shaw problem with stochastic forcing on the curvature equation. In the case when the noise is sufficiently small, we rigorously prove that the limit is a deterministic Hele-Shaw problem. Finally, we discuss which estimates are necessary in order to extend the rigorous result to larger noise strength.


35K55 Nonlinear parabolic equations
35K40 Second-order parabolic systems
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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[1] M. Alfaro, D. Antonopoulou, G. Karali and H. Matano. Formation of fine transition layers and their dynamics for the Stochastic Allen-Cahn equation. Preprint, 2016.
[2] N. D. Alikakos, P. W. Bates and X. Chen. Convergence of the Cahn-Hilliard equation to the Hele-Shaw model. Arch. Ration. Mech. Anal.128 (1994) 165-205. · Zbl 0828.35105
[3] N. D. Alikakos, X. Chen and G. Fusco. Motion of a droplet by surface tension along the boundary. Calc. Var.11 (2000) 233-305. · Zbl 0994.35052
[4] N. D. Alikakos and G. Fusco. The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions. Indiana Univ. Math. J.41 (1993) 637-674. · Zbl 0798.35123 · doi:10.1512/iumj.1993.42.42028
[5] N. D. Alikakos and G. Fusco. Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: The motion of bubbles. Arch. Ration. Mech. Anal.141 (1) (1998) 1-61. · Zbl 0906.35049
[6] N. D. Alikakos, G. Fusco and G. Karali. Motion of bubbles towards the boundary for the Cahn-Hilliard equation. European J. Appl. Math.15 (1) (2004) 103-124. · Zbl 1065.35136
[7] D. C. Antonopoulou, P. W. Bates, D. Blömker and G. D. Karali. Motion of a droplet for the stochastic mass-conserving Allen-Cahn equation. SIAM J. Math. Anal.48 (1) (2016) 670-708. · Zbl 1439.35232
[8] D. C. Antonopoulou, D. Blömker and G. D. Karali. Front motion in the one-dimensional stochastic Cahn-Hilliard equation. SIAM J. Math. Anal.44 (5) (2012) 3242-3280. · Zbl 1270.35040
[9] D. C. Antonopoulou and G. D. Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete Contin. Dyn. Syst. Ser. B 16 (1) (2011) 31-55. · Zbl 1227.35163
[10] D. C. Antonopoulou, G. D. Karali and G. T. Kossioris. Asymptotics for a generalized Cahn-Hilliard equation with forcing terms. Discrete Contin. Dyn. Syst. A 30 (4) (2011) 1037-1054. · Zbl 1222.35017
[11] D. C. Antonopoulou, G. D. Karali and E. Orlandi. A Hilbert expansions method for the rigorous sharp interface limit of the generalized Cahn-Hilliard equation. Interfaces Free Bound.16 (1) (2014) 65-104. · Zbl 1295.35031 · doi:10.4171/IFB/314
[12] P. W. Bates and J. Jin. Global dynamics of boundary droplets. DCDS-A 34 (2014) 1-17. · Zbl 1315.37049
[13] P. W. Bates, K. Lu and C. Zeng. Approximately invariant manifolds and global dynamics of spike states. Invent. Math.174 (2) (2008) 355-433. · Zbl 1157.37013
[14] P. W. Bates and J. Xun. Metastable patterns for the Cahn-Hilliard equation I. J. Differential Equations 111 (2) (1994) 421-457. · Zbl 0805.35046
[15] P. W. Bates and J. Xun. Metastable patterns for the Cahn-Hilliard equation II: Layer dynamics and slow invariant manifold. J. Differential Equations 117 (1) (1995) 165-216. · Zbl 0821.35074
[16] N. Berglund, G. Di Gesú and H. Weber. An Eyring-Kramers law for the stochastic Allen-Cahn equation in dimension two. Preprint, 2016. · Zbl 1362.60059
[17] L. Bertini, S. Brassesco and P. Buttá. Front fluctuations for the stochastic Cahn-Hilliard equation. Braz. J. Probab. Stat.29 (2) (2015) 336-371. · Zbl 1318.82033
[18] D. Blömker, S. Maier-Paape and T. Wanner. Spinodal decomposition for the Cahn-Hilliard-Cook equation. Comm. Math. Phys.223 (3) (2001) 553-582. · Zbl 0993.60061
[19] S. Brassesco, A. De Masi and E. Presutti. Brownian fluctuations of the interface in the \(D=1\) Ginzburg-Landau equation with noise. Ann. Inst. Henri Poincaré B, Calc. Probab. Stat.31 (1) (1995) 81-118. · Zbl 0822.35158
[20] G. Cagninalp and P. C. Fife. Dynamics of layered interfaces arising from phase boundaries. SIAM J. Appl. Math.48 (3) (1988) 506-518.
[21] C. Cardon-Weber. Cahn-Hilliard stochastic equation: Existence of the solution and of its density. Bernoulli 7 (5) (2001) 777-816. · Zbl 0995.60058
[22] C. Cardon-Weber. Cahn-Hilliard stochastic equation: Strict positivity of the density. Stoch. Stoch. Rep.72 (3-4) (2002) 191-227. · Zbl 1002.60050
[23] X. Chen. Spectrums for the Allen-Cahn, Cahn-Hilliard, and phase-field equations for generic interface. Comm. Partial Diff. Eqns.19 (1994) 1371-1395. · Zbl 0811.35098
[24] X. Chen. Generation, propagation, and annihilation of metastable patterns. J. Differential Equations 206 (2) (2004) 399-437. · Zbl 1061.35014
[25] H. Cook. Brownian motion in spinodal decomposition. Acta Metall.18 (1970) 297-306.
[26] G. Da Prato and A. Debussche. Stochastic Cahn-Hilliard equation. Nonlinear Anal., Theory Methods Appl.26 (2) (1996) 241-263. · Zbl 0838.60056
[27] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions, 2nd edition. Encyclopedia of of Mathematics and Its Applications 152. Cambridge University Press, Cambridge, 2014. · Zbl 1317.60077
[28] R. C. Dalang and L. Quer-Sardanyons. Stochastic integrals for SPDE’s: A comparison. Expo. Math.29 (1) (2011) 67-109. · Zbl 1234.60064
[29] N. Elezović and A. Mikelić. On the stochastic Cahn-Hilliard equation. Nonlinear Anal., Theory Methods Appl.16 (12) (1991) 1169-1200. · Zbl 0729.60057
[30] W. G. Faris and G. Jona-Lasinio. Large fluctuations for a nonlinear heat equation with noise. J. Phys. A 15 (10) (1982) 3025-3055. · Zbl 0496.60060
[31] I. Fatkullin and E. Vanden-Eijnden. Coarsening by diffusion-annihilation in a bistable system driven by noise. Preprint, 2004. · Zbl 1058.65065
[32] L. R. G. Fontes, M. Isopi, C. M. Newman and K. Ravishankar. Coarsening, nucleation, and the marked Brownian web. Ann. Inst. Henri Poincaré B, Probab. Stat.42 (1) (2006) 37-60. · Zbl 1087.60072
[33] T. Funaki. The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Related Fields 102 (2) (1995) 221-288. · Zbl 0834.60066
[34] T. Funaki. Singular limit for stochastic reaction-diffusion equation and generation of random interfaces. Acta Mathematica Sinica 15 (1999) 407-438. · Zbl 0943.60060
[35] M. E. Gurtin. Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Phys. D 92 (1996) 178-192. · Zbl 0885.35121
[36] S. Habib and G. Lythe. Dynamics of kinks: Nucleation, diffusion and annihilation. Phys. Rev. Lett.84 (2000) 1070.
[37] M. Hairer. A theory of regularity structures. Invent. Math.198 (2) (2014) 269-504. · Zbl 1332.60093
[38] M. Hairer, M. D. Ryser and H. Weber. Triviality of the 2D stochastic Allen-Cahn equation. Electron. J. Probab.17 (39) (2012) 1-14. · Zbl 1245.60063
[39] M. Hairer and H. Weber. Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions. Annales de la Faculté des Sciences de Toulouse 24 (1) (2015) 55-92. · Zbl 1319.60141
[40] P. C. Hohenberg and B. I. Halperin. Theory of dynamic critical phenomena. Rev. Modern Phys.49 (1977) 435-479.
[41] K. Kitahara, Y. Oono and D. Jasnow. Phase separation dynamics and external force field. Modern Phys. Lett. B 2 (1988) 765-771.
[42] J. S. Langer. Theory of spinodal decomposition in alloys. Ann. Phys.65 (1971) 53-86.
[43] R. L. Pego. Front migration in the non-linear Cahn-Hilliard equation. Proc. R. Soc. Lond. A 422 (1989) 261-278. · Zbl 0701.35159
[44] M. Röger and H. Weber. Tightness for a stochastic Allen-Cahn equation, stochastic partial differential equations. Analysis and Computations 1 (1) (2013) 175-203. · Zbl 1274.60207
[45] T. Shardlow. Stochastic perturbations of the Allen-Cahn equation. Electron. J. Differential Equations 47 (2000) 1-19. · Zbl 0959.60047
[46] J. Walsh. An introduction to stochastic partial differential equations. École d’Été de Probabilités de Saint Flour XIV-1984, 265-439.
[47] H. Weber. On the short time asymptotic of the stochastic Allen-Cahn equation. Ann. Inst. Henri Poincaré Probab. Stat.46 (4) (2010) 965-975. · Zbl 1210.35307
[48] H. Weber. Sharp interface limit for invariant measures of a stochastic Allen-Cahn equation. Comm. Pure Appl. Math.63 (8) (2010) 1071-1109. · Zbl 1205.82110
[49] S. Weber. The sharp interface limit of the stochastic Allen-Cahn equation. Ph.D. thesis, University of Warwick, 2014.
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