2015/01/21 by Dimitra C. Antonopoulou, Antonopoulou, Dimitra C., Peter W. Bates +5 · 1 citation
Computer Science · Materials Science · Mathematics · #35K40 #35K55 #60H15 #60H30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #Solidification and crystal growth phenomena #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1501.05288
openalex publication_date 2015/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the stochastic mass-conserving Allen-Cahn equation posed on a bounded two-dimensional domain with additive spatially smooth space-time noise. This equation associated with a small positive parameter 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 Itô calculus to derive the stochastic dynamics of the droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen and Fusco 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 sufficiently small noise strength, we establish stochastic stability of a neighborhood of the manifold of droplets in L2 and H1, which means that with overwhelming probability the solution stays close to the manifold for very long time-scales.