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Transactions of the American Mathematical Society
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
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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.


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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
Email: bloemker@instmath.rwth-aachen.de, dirk.bloemker@math.uni-augsburg.de

Stanislaus Maier-Paape
Affiliation: Institut für Mathematik, Templergraben 55, RWTH Aachen, 52062 Aachen, Germany
Email: maier@instmath.rwth-aachen.de

Thomas Wanner
Affiliation: Department of Mathematical Sciences, George Mason University, 4400 University Drive, MS 3F2, Fairfax, Virginia 22030
Email: wanner@math.gmu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04387-5
PII: S 0002-9947(07)04387-5
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.