Motion of a droplet for the mass-conserving stochastic Allen-Cahn equation

TL;DR

This paper investigates the stochastic motion of droplets governed by a mass-conserving Allen-Cahn equation with noise, deriving their dynamics, stability, and influence of boundary curvature in a two-dimensional domain.

Contribution

It introduces a stochastic dynamic model for droplet motion on a boundary, extending deterministic invariant manifold analysis to stochastic settings with stability results.

Findings

Droplet motion is influenced by boundary curvature and stochastic forcing.

Solutions remain close to the invariant manifold for long times under small noise.

The stochastic stability of droplet configurations is established in $L^2$ and $H^1$ norms.

Abstract

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 It\^o 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 $L^2$ and $H^1$, which means that with overwhelming probability the solution stays close to the manifold for very long time-scales.