Improved estimates for the sharp interface limit of the stochastic Cahn–Hilliard equation with space-time white noise

Abstract

We study the sharp interface limit of the stochastic Cahn–Hilliard equation with cubic double-well potential and additive space-time white noise ${\epsilon }^{\sigma }\dot{W}$, where $\epsilon >0$ is an interfacial width parameter. We prove that, for a sufficiently large scaling constant $\sigma >0$, the stochastic Cahn–Hilliard equation converges to the deterministic Mullins–Sekerka/Hele-Shaw problem for $\epsilon \to 0$. The convergence is shown in suitable fractional Sobolev norms as well as in the $L^p$-norm for $p\in (2,4]$ in spatial dimension $d=2,3$. This generalizes the existing result for the space-time white noise to dimension $d=3$ and improves the existing results for smooth noise, which were so far limited to $p\in (2,\frac{2d+8}{d+2}]$ in spatial dimension $d=2,3$. As a byproduct of the analysis of the stochastic problem with space-time white noise, we identify minimal regularity requirements on the noise which allow convergence to the sharp interface limit in the ${\mathbb{H}}^1$-norm and also provide improved convergence estimates for the sharp interface limit of the deterministic problem.