Front Motion in the One-Dimensional Stochastic Cahn-Hilliard Equation

  • In this paper, we consider the one-dimensional Cahn-Hilliard equation perturbed by additive noise and study the dynamics of interfaces for the new stochastic model. The noise is smooth in space and is defined as a Fourier series with independent Brownian motions in time. Motivated by the work of Bates & Xun on slow manifolds for the integrated Cahn-Hilliard equation, our analysis reveals the significant difficulties and differences in comparison with the deterministic problem. New higher order terms, that we estimate, appear due to Itô calculus and stochastic integration dominating the exponentially slow deterministic dynamics of the interfaces. We derive a first order linear system of stochastic ordinary differential equations approximating the equations of front motion. Furthermore, we prove stochastic stability for the approximate slow manifold of solutions on a very long time scale and evaluate the noise effect.

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Author:Dimitra C. Antonopoulou, Dirk BlömkerORCiDGND, Georgia D. Karali
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Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2011-05)
Year of first Publication:2011
Publishing Institution:Universität Augsburg
Contributing Corporation:Department of Applied Mathematics, University of Crete, Institute of Applied and Computational Mathematics, FORTH
Release Date:2011/03/21
Tag:Stochastische Differentialgleichung
interface motion; slow manifold; 1-D Stochastic Cahn-Hilliard; additive noise; dynamics
GND-Keyword:Cahn-Hilliard-Gleichung; Stochastische nichtlineare Differentialgleichung; Stochastische partielle Differentialgleichung
Erschienen in SIAM Journal on Mathematical Analysis, 44, 5, S. 3242-3280,
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht mit Print on Demand