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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2304.14785 |
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| _version_ | 1866929220628774912 |
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| author | Baňas, Ľubomír Mukam, Jean Daniel |
| author_facet | Baňas, Ľubomír Mukam, Jean Daniel |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_14785 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Improved estimates for the sharp interface limit of the stochastic Cahn-Hilliard equation with space-time white noise Baňas, Ľubomír Mukam, Jean Daniel Probability Numerical Analysis 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. |
| title | Improved estimates for the sharp interface limit of the stochastic Cahn-Hilliard equation with space-time white noise |
| topic | Probability Numerical Analysis |
| url | https://arxiv.org/abs/2304.14785 |