Smooth solutions to the Schrödinger flow for maps from smooth bounded domains in Euclidean spaces into $\mathbb{S}^2$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866909976383979520 |
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| author | Chen, Bo Wang, Youde |
| author_facet | Chen, Bo Wang, Youde |
| contents | The results of this paper are twofold. One is that we show the local existence and uniqueness of very regular or smooth solution to the initial-Neumann boundary value problem of the Schrödinger flow for maps from a smooth bounded domain $Ω\subset \mathbb{R}^m$ with $m=1,2,3$ into $\mathbb{S}^2$ in the scale of Sobolev spaces. In this part, we provide a precise description of the compatibility conditions at the boundary for the initial data. The other is that we further prove that the locally smooth solution to the initial-Neumann boundary value problem of the 1-dimensional Schrödinger flow can be extended to a global smooth one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_14835 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Smooth solutions to the Schrödinger flow for maps from smooth bounded domains in Euclidean spaces into $\mathbb{S}^2$ Chen, Bo Wang, Youde Analysis of PDEs The results of this paper are twofold. One is that we show the local existence and uniqueness of very regular or smooth solution to the initial-Neumann boundary value problem of the Schrödinger flow for maps from a smooth bounded domain $Ω\subset \mathbb{R}^m$ with $m=1,2,3$ into $\mathbb{S}^2$ in the scale of Sobolev spaces. In this part, we provide a precise description of the compatibility conditions at the boundary for the initial data. The other is that we further prove that the locally smooth solution to the initial-Neumann boundary value problem of the 1-dimensional Schrödinger flow can be extended to a global smooth one. |
| title | Smooth solutions to the Schrödinger flow for maps from smooth bounded domains in Euclidean spaces into $\mathbb{S}^2$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2111.14835 |