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 sufficiently large scaling constant $\sigma >0$, the stochastic Cahn-Hilliard equation converges to the deterministic Mullins-Sekerka/Hele-Shaw problem for $\epsilon\rightarrow 0$. 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 $p\in \left(2, \frac{2d+8}{d+2}\right]$ 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.

翻译：我们研究了具有立方双阱势和加性时空白噪声$\epsilon^{\sigma}\dot{W}$的随机Cahn-Hilliard方程的尖锐界面极限，其中$\epsilon>0$为界面宽度参数。我们证明，对于足够大的标度常数$\sigma >0$，当$\epsilon\rightarrow 0$时，随机Cahn-Hilliard方程收敛到确定性Mullins-Sekerka/Hele-Shaw问题。在空间维度$d=2,3$下，该收敛性在适当的分数阶Sobolev范数以及$p\in (2, 4]$的$L^p$范数中得以证明。这一结果将现有针对时空白噪声的结果推广至$d=3$维度，并改进了现有针对光滑噪声的结果——此前在$d=2,3$空间中仅局限于$p\in \left(2, \frac{2d+8}{d+2}\right]$。作为分析时空白噪声随机问题的一个副产品，我们确定了噪声实现$\mathbb{H}^1$范数下尖锐界面极限收敛所需的最小正则性要求，并为确定性问题提供了改进的尖锐界面极限收敛估计。