Well-posedeness for the non-isotropic Schrödinger equations on cylinders and periodic domains
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| Format: | Preprint |
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2024
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| _version_ | 1866918264731336704 |
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| author | Nogueira, Adán J. Corcho. Marcelo Panthee, Mahendra |
| author_facet | Nogueira, Adán J. Corcho. Marcelo Panthee, Mahendra |
| contents | The initial value problem (IVP) for the non-isotropic Schrödinger equation posed on the two-dimensional cylinders and $\mathbb{T}^2$ is considered. The IVP is shown to be locally well-posed for small initial data in $H^s(\mathbb{T}\times\mathbb{R})$ if $s\geq0$. For the IVP posed on $\mathbb{R}\times\mathbb{T}$, given data are considered in the anisotropic Sobolev spaces thereby obtaining the local well-posedness result in $H^{s_1, s_2}(\mathbb{R}\times\mathbb{T})$, if $s_1\geq0$ and $s_2>\frac12$. In the purely periodic case, a particular case of the IVP is shown to be locally well-posed for any given initial data in $H^s(\mathbb{T}^2)$ if $s>\frac14$. In some cases, ill-posedness issues are also considered showing that the IVP posed on $\mathbb{T}\times \mathbb{R}$, in the focusing case, is ill-posed in the sense that the application data-solution fails to be uniformly continuous for data in $H^s(\mathbb{T}\times\mathbb{R})$ if $-\frac12\leq s<0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_01392 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Well-posedeness for the non-isotropic Schrödinger equations on cylinders and periodic domains Nogueira, Adán J. Corcho. Marcelo Panthee, Mahendra Analysis of PDEs 35Q55, 35Q60 The initial value problem (IVP) for the non-isotropic Schrödinger equation posed on the two-dimensional cylinders and $\mathbb{T}^2$ is considered. The IVP is shown to be locally well-posed for small initial data in $H^s(\mathbb{T}\times\mathbb{R})$ if $s\geq0$. For the IVP posed on $\mathbb{R}\times\mathbb{T}$, given data are considered in the anisotropic Sobolev spaces thereby obtaining the local well-posedness result in $H^{s_1, s_2}(\mathbb{R}\times\mathbb{T})$, if $s_1\geq0$ and $s_2>\frac12$. In the purely periodic case, a particular case of the IVP is shown to be locally well-posed for any given initial data in $H^s(\mathbb{T}^2)$ if $s>\frac14$. In some cases, ill-posedness issues are also considered showing that the IVP posed on $\mathbb{T}\times \mathbb{R}$, in the focusing case, is ill-posed in the sense that the application data-solution fails to be uniformly continuous for data in $H^s(\mathbb{T}\times\mathbb{R})$ if $-\frac12\leq s<0$. |
| title | Well-posedeness for the non-isotropic Schrödinger equations on cylinders and periodic domains |
| topic | Analysis of PDEs 35Q55, 35Q60 |
| url | https://arxiv.org/abs/2411.01392 |