# Sharp interface limit of stochastic Cahn-Hilliard equation with singular noise

## TL;DR

This paper investigates the behavior of a stochastic Cahn-Hilliard equation with singular noise in two dimensions as the interface width parameter approaches zero, demonstrating convergence to the deterministic Hele-Shaw problem under small noise conditions.

## Contribution

It establishes the sharp interface limit of the stochastic Cahn-Hilliard equation with singular noise, linking it to the deterministic Hele-Shaw problem in the small noise regime.

## Key findings

- Solutions converge to the Hele-Shaw problem as ps approaches zero.
- Small noise ensures the stochastic solutions approximate deterministic behavior.
- Comparison with previous approximations confirms the limit behavior.

## Abstract

We study the the sharp interface limit of $\varepsilon$-dependent two dimensional stochastic Cahn-Hilliard equation driven by space-time white noise and conservative noise as $\varepsilon\to 0$. In the case when the noise is sufficiently small, by comparing the solutions to equation (1.1) with the approximation solution constructed in [ABC94], we show that the limit of the solutions is also solutions to the deterministic Hele-Shaw problem.