Second phase spinodal decomposition for the Cahn-Hilliard-Cook equation
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- by Dirk Blömker, Stanislaus Maier-Paape and Thomas Wanner PDF
- Trans. Amer. Math. Soc. 360 (2008), 449-489 Request permission
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
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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
- MR Author ID: 262105
- Email: wanner@math.gmu.edu
- 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.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 360 (2008), 449-489
- MSC (2000): Primary 60H15, 35K35, 35B05, 35P10
- DOI: https://doi.org/10.1090/S0002-9947-07-04387-5
- MathSciNet review: 2342011