Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917800512061440 |
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| author | Abels, Helmut Hurm, Christoph Moser, Maximilian |
| author_facet | Abels, Helmut Hurm, Christoph Moser, Maximilian |
| contents | We prove convergence of the nonlocal Allen-Cahn equation to mean curvature flow in the sharp interface limit, in the situation when the parameter corresponding to the kernel goes to zero fast enough with respect to the diffuse interface thickness. The analysis is done in the case of a $W^{1,1}$-kernel, under periodic boundary conditions and in both two and three space dimensions. We use the approximate solution and spectral estimate from the local case, and combine the latter with an $L^2$-estimate for the difference of the nonlocal operator and the negative Laplacian from Abels, Hurm arXiv:2307.02264. To this end, we prove a nonlocal Ehrling-type inequality to show uniform $H^3$-estimates for the nonlocal solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08596 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow Abels, Helmut Hurm, Christoph Moser, Maximilian Analysis of PDEs 45K05, 35B40, 35B10, 35K61 We prove convergence of the nonlocal Allen-Cahn equation to mean curvature flow in the sharp interface limit, in the situation when the parameter corresponding to the kernel goes to zero fast enough with respect to the diffuse interface thickness. The analysis is done in the case of a $W^{1,1}$-kernel, under periodic boundary conditions and in both two and three space dimensions. We use the approximate solution and spectral estimate from the local case, and combine the latter with an $L^2$-estimate for the difference of the nonlocal operator and the negative Laplacian from Abels, Hurm arXiv:2307.02264. To this end, we prove a nonlocal Ehrling-type inequality to show uniform $H^3$-estimates for the nonlocal solutions. |
| title | Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow |
| topic | Analysis of PDEs 45K05, 35B40, 35B10, 35K61 |
| url | https://arxiv.org/abs/2410.08596 |