Thin-film limit of the Ginzburg-Landau heat flow in a curved thin domain
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910418675433472 |
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| author | Miura, Tatsu-Hiko |
| author_facet | Miura, Tatsu-Hiko |
| contents | We consider the Ginzburg-Landau heat flow without magnetic effect in a curved thin domain under the Naumann boundary condition. When the curved thin domain shrinks to a given closed hypersurface as the thickness of the thin domain tends to zero, we show that the weighted average of a weak solution to the thin-domain problem converges weakly on the limit surface under the assumption that the initial data is of class $L^\infty$ and satisfies some conditions. Moreover, under the same assumption, we derive a limit equation by characterizing the limit function as a weak solution, and prove a difference estimate on the limit surface of an averaged weak solution to the thin-domain problem and a weak solution to the limit problem explicitly in terms of the thickness of the thin domain. We also derive a difference estimate in the curved thin domain of weak solutions to the thin-domain problem and to the limit problem, but without requiring that the initial data of the thin-domain problem is of class $L^\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14703 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Thin-film limit of the Ginzburg-Landau heat flow in a curved thin domain Miura, Tatsu-Hiko Analysis of PDEs 35B25, 35Q56, 35R01 We consider the Ginzburg-Landau heat flow without magnetic effect in a curved thin domain under the Naumann boundary condition. When the curved thin domain shrinks to a given closed hypersurface as the thickness of the thin domain tends to zero, we show that the weighted average of a weak solution to the thin-domain problem converges weakly on the limit surface under the assumption that the initial data is of class $L^\infty$ and satisfies some conditions. Moreover, under the same assumption, we derive a limit equation by characterizing the limit function as a weak solution, and prove a difference estimate on the limit surface of an averaged weak solution to the thin-domain problem and a weak solution to the limit problem explicitly in terms of the thickness of the thin domain. We also derive a difference estimate in the curved thin domain of weak solutions to the thin-domain problem and to the limit problem, but without requiring that the initial data of the thin-domain problem is of class $L^\infty$. |
| title | Thin-film limit of the Ginzburg-Landau heat flow in a curved thin domain |
| topic | Analysis of PDEs 35B25, 35Q56, 35R01 |
| url | https://arxiv.org/abs/2404.14703 |