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Motion of a droplet for the Stochastic mass conserving Allen-Cahn equation
Antonopoulou, Dimitra Bates, Peter W. Bloemker, Dirk Karali, Georgia D.
Antonopoulou, Dimitra
Bates, Peter W.
Bloemker, Dirk
Karali, Georgia D.
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2016-02-16
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Abstract
We study the stochastic mass-conserving Allen-Cahn equation posed on a smoothly bounded
domain of R2 with additive, spatially smooth, space-time noise. This equation describes the stochastic
motion of a small almost semicircular droplet attached to domain's boundary and moving towards a point
of locally maximum curvature. We apply It^o calculus to derive the stochastic dynamics of the center of
the droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen and Fusco [2]
for the deterministic problem. In the stochastic case depending on the scaling, the motion is driven by
the change in the curvature of the boundary and the stochastic forcing. Moreover, under the assumption
of a su ciently small noise strength, we establish stochastic stability of a neighborhood of the manifold of
boundary droplet states in the L2- and H1-norms, which means that with overwhelming probability the
solution stays close to the manifold for very long time-scales.
Citation
Antonopoulou, D., & Bates, P., & Bloemker, D., & Karali, G. (2016). Motion of a droplet for the Stochastic mass conserving Allen-Cahn equation. SIAM Journal on Mathematical Analysis, 48(1), 670-708. DOI: 10.1137/151005105
Publisher
SIAM
Journal
SIAM Journal on Mathematical Analysis
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DOI
10.1137/151005105
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Article
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en
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0036-1410