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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2304.14785 |
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Table of Contents:
- We study the sharp interface limit of the stochastic Cahn-Hilliard equation with cubic double-well potential and additive space-time white noise $ε^σ\dot{W}$ where $ε>0$ is an interfacial width parameter. We prove that, for sufficiently large scaling constant $σ>0$, the stochastic Cahn-Hilliard equation converges to the deterministic Mullins-Sekerka/Hele-Shaw problem for $ε\rightarrow 0$. 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.