Degenerate Nucleation in the Cahn-Hilliard-Cook Model

  • Phase separation in metal alloys is an important pattern forming physical process with applications in materials science, both for understanding materials structure and for the design of new materials. The Cahn-Hilliard equation is a deterministic model for the dynamics of alloys which has proven to be fundamental for the understanding of several types of phase separation behavior. However, stochastic effects occur in any physical experiment, and thus need to be incorporated into models. While white noise is one standardly chosen option, it is not immediately clear in which way the noise characteristics affect the resulting patterns. In this paper we study the effects of not necessarily small colored noise on pattern formation in a stochastic Cahn-Hilliard model in the nucleation regime. More precisely, we focus on degenerate noise which acts either on isolated eigenmodes, or with a well-defined spatial wavelength. Our studies show that the types of resulting patterns depend criticallyPhase separation in metal alloys is an important pattern forming physical process with applications in materials science, both for understanding materials structure and for the design of new materials. The Cahn-Hilliard equation is a deterministic model for the dynamics of alloys which has proven to be fundamental for the understanding of several types of phase separation behavior. However, stochastic effects occur in any physical experiment, and thus need to be incorporated into models. While white noise is one standardly chosen option, it is not immediately clear in which way the noise characteristics affect the resulting patterns. In this paper we study the effects of not necessarily small colored noise on pattern formation in a stochastic Cahn-Hilliard model in the nucleation regime. More precisely, we focus on degenerate noise which acts either on isolated eigenmodes, or with a well-defined spatial wavelength. Our studies show that the types of resulting patterns depend critically on the spatial noise frequency, and we can explain via numerical continuation methods that if this frequency is too high, then pattern formation is significantly impaired by the underlying structure of the system. In addition, we provide rigorous bounds on the nucleation time frame in the degenerate stochastic setting.show moreshow less

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Metadaten
Author:Dirk BlömkerGND, Evelyn Sander, Thomas WannerORCiDGND
URN:urn:nbn:de:bvb:384-opus4-31573
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/3157
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2015-06)
Type:Preprint
Language:English
Year of first Publication:2015
Publishing Institution:Universität Augsburg
Release Date:2015/07/02
Tag:Cahn-Hilliard-Cook systems; nucleation; stochastic partial differential equations; domain exit; degenerate noise; pattern formation
GND-Keyword:Stochastische partielle Differentialgleichung; Cahn-Hilliard-Gleichung; Keimbildung; Musterbildung
Note:
Erschienen in SIAM Journal on Applied Dynamical Systems, 15, 1, S. 459-494, https://doi.org/10.1137/15m1028844
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