Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation

Authors: Dirk Blömker, Stanislaus Maier-Paape and Thomas Wanner
Journal: Trans. Amer. Math. Soc. 360 (2008), 449-489
MSC (2000): Primary 60H15, 35K35, 35B05, 35P10
Published electronically: August 6, 2007
MathSciNet review: 2342011
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the dynamics of a nonlinear partial differential equation perturbed by additive noise. Assuming that the underlying deterministic equation has an unstable equilibrium, we show that the nonlinear stochastic partial differential equation exhibits essentially linear dynamics far from equilibrium. More precisely, we show that most trajectories starting at the unstable equilibrium are driven away in two stages. After passing through a cylindrical region, most trajectories diverge from the deterministic equilibrium through a cone-shaped region which is centered around a finite-dimensional subspace corresponding to strongly unstable eigenfunctions of the linearized equation, and on which the influence of the nonlinearity is surprisingly small.

This abstract result is then applied to explain spinodal decomposition in the stochastic Cahn-Hilliard-Cook equation on a domain $ G$. This equation depends on a small interaction parameter  $ \varepsilon>0$, and one is generally interested in asymptotic results as  $ \varepsilon \to 0$. Specifically, we show that linear behavior dominates the dynamics up to distances from the deterministic equilibrium which can reach  $ \varepsilon^{-2+\dim G / 2}$ with respect to the $ H^2(G)$-norm.

References [Enhancements On Off] (What's this?)

  • 1. R. Aurich, A. Bäcker, R. Schubert, and M. Taglieber, Maximum norms of chaotic quantum eigenstates and random waves, Physica D 129 (1999), 1-14. MR 1690285 (2000d:81042)
  • 2. D. Blömker, Nonhomogeneous noise and $ Q$-Wiener processes on bounded domains, Stochastic Analysis and Applications 23 (2005), 255-273. MR 2130349 (2006a:60066)
  • 3. D. Blömker, S. Maier-Paape, and T. Wanner, Spinodal decomposition for the Cahn-Hilliard-Cook equation, Communications in Mathematical Physics 223 (2001), no. 3, 553-582. MR 1866167 (2002i:60114)
  • 4. Z. Brzezniak and S. Peszat, Maximal inequalities and exponential estimates for stochastic convolutions in Banach spaces, Stochastic processes, physics and geometry: New interplays, I (Leipzig, 1999), CMS Conf. Proc., vol. 28, Amer. Math. Soc., Providence, RI, 2000, pp. 55-64. MR 1803378 (2001k:60084)
  • 5. John W. Cahn, Free energy of a nonuniform system. II. Thermodynamic basis, Journal of Chemical Physics 30 (1959), 1121-1124.
  • 6. John W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, Journal of Chemical Physics 28 (1958), 258-267.
  • 7. H. Cook, Brownian motion in spinodal decomposition, Acta Metallurgica 18 (1970), 297-306.
  • 8. R. Courant and D. Hilbert, Methods of mathematical physics, Intersciences, New York, 1953. MR 0065391 (16:426a)
  • 9. G. da Prato and A. Debussche, Stochastic Cahn-Hilliard equation, Nonlinear Analysis 26 (1996), no. 2, 241-263. MR 1359472 (96k:35200)
  • 10. G. da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, no. 44, Cambridge University Press, 1992. MR 1207136 (95g:60073)
  • 11. Jonathan P. Desi, Evelyn Sander, and Thomas Wanner, Complex transient patterns on the disk, Discrete and Continuous Dynamical Systems, Series A 15 (2006), no. 4, 1049-1078. MR 2224497
  • 12. D. E. Edmunds and W. D. Evans, Spectral theory and differential operators, Oxford Science Publications, 1990. MR 929030 (89b:47001)
  • 13. M. I. Freidlin and A. D. Wentzell, Random perturbations of dynamical systems, Springer, 1998, Second edition. MR 1652127 (99h:60128)
  • 14. Marcio Gameiro, Konstantin Mischaikow, and Thomas Wanner, Evolution of pattern complexity in the Cahn-Hilliard theory of phase separation, Acta Materialia 53 (2005), no. 3, 693-704.
  • 15. C. P. Grant, Spinodal decomposition for the Cahn-Hilliard equation, Communications in Partial Differential Equations 18 (1993), no. 3-4, 453-490. MR 1214868 (94b:35147)
  • 16. Wolfgang Hackenbroch and Anton Thalmaier, Stochastische Analysis, B. G. Teubner, Stuttgart, 1994. MR 1312827 (96e:60094)
  • 17. D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, no. 840, Springer, 1981. MR 610244 (83j:35084)
  • 18. J. S. Langer, Theory of spinodal decomposition in alloys, Annals of Physics 65 (1971), 53-86.
  • 19. S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions. Part I: Probability and wavelength estimate, Communications in Mathematical Physics 195(2) (1998), 435-464. MR 1637817 (99h:35111)
  • 20. -, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics, Archive for Rational Mechanics and Analysis 151 (2000), 187-219. MR 1753703 (2001d:35083)
  • 21. S. Peszat, Exponential tail estimates for infinite-dimensional stochastic convolutions, Bull. Polish Acad. Sci. Math. 40 (1992), no. 4, 323-333. MR 1402320
  • 22. E. Sander and T. Wanner, Monte Carlo simulations for spinodal decomposition, Journal of Statistical Physics 95 (1999), no. 5-6, 925-948. MR 1712442
  • 23. -, Unexpectedly linear behavior for the Cahn-Hilliard equation, SIAM Journal on Applied Mathematics 60 (2000), no. 6, 2182-2202. MR 1763320 (2001i:35161)
  • 24. Jan Seidler and Takuya Sobukawa, Exponential integrability of stochastic convolutions, Journal of the London Mathematical Society. Second Series 67 (2003), no. 1, 245-258. MR 1942424 (2003i:60109)
  • 25. Thomas Wanner, Maximum norms of random sums and transient pattern formation, Transactions of the American Mathematical Society 356 (2004), no. 6, 2251-2279. MR 2048517 (2005a:35146)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60H15, 35K35, 35B05, 35P10

Retrieve articles in all journals with MSC (2000): 60H15, 35K35, 35B05, 35P10

Additional Information

Dirk Blömker
Affiliation: Institut für Mathematik, Templergraben 55, RWTH Aachen, 52062 Aachen, Germany
Address at time of publication: Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany

Stanislaus Maier-Paape
Affiliation: Institut für Mathematik, Templergraben 55, RWTH Aachen, 52062 Aachen, Germany

Thomas Wanner
Affiliation: Department of Mathematical Sciences, George Mason University, 4400 University Drive, MS 3F2, Fairfax, Virginia 22030

Keywords: Stochastic Cahn-Hilliard equation, pattern formation, spinodal decomposition, second decomposition stage.
Received by editor(s): April 2, 2004
Received by editor(s) in revised form: April 11, 2006
Published electronically: August 6, 2007
Additional Notes: The first author was supported by DFG-Forschungsstipendium Bl 535-5/1.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society