The Sharp Interface Limit for the Stochastic Cahn-Hilliard Equation

  • We study the two and three dimensional stochastic Cahn-Hilliard equation in the sharp interface limit, where the positive parameter ε tends to zero, which 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 we indicate that the solutions of stochastic Cahn-Hilliard converge to a solution of a Hele-Shaw problem with stochastic forcing. 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.

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Metadaten
Author:Dimitra C. Antonopoulou, Dirk BlömkerGND, Georgia D. Karali
URN:urn:nbn:de:bvb:384-opus4-32141
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/3214
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2015-09)
Type:Preprint
Language:English
Year of first Publication:2015
Publishing Institution:Universität Augsburg
Release Date:2015/07/28
Tag:multi-dimensional stochastic Cahn-Hilliard equation; stochastic sharp interface limit; Hele Shaw problem; interface motion
GND-Keyword:Cahn-Hilliard-Gleichung; Phasenumwandlung; Stochastisches dynamisches System; Reaktions-Diffusionsgleichung; Stochastische partielle Differentialgleichung
Note:
Erschienen in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 54, 1, S. 280-298, https://doi.org/10.1214/16-aihp804
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):License LogoDeutsches Urheberrecht mit Print on Demand