Well- and ill-posedness of the Cauchy problem for semi-linear Schrödinger equations on the torus
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910776918278144 |
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| author | Kondo, Toshiki Okamoto, Mamoru |
| author_facet | Kondo, Toshiki Okamoto, Mamoru |
| contents | We consider the Cauchy problem for semi-linear Schrödinger equations on the torus $\mathbb T$. We establish a necessary and sufficient condition on the polynomial nonlinearity for the Cauchy problem to be well-posed in the Sobolev space $H^s(\mathbb T)$ for $s>\frac 52$. For the well-posedness, we use the energy estimates and the gauge transformation. For the ill-posedness, we prove the non-existence of solutions to the Cauchy problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04205 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well- and ill-posedness of the Cauchy problem for semi-linear Schrödinger equations on the torus Kondo, Toshiki Okamoto, Mamoru Analysis of PDEs We consider the Cauchy problem for semi-linear Schrödinger equations on the torus $\mathbb T$. We establish a necessary and sufficient condition on the polynomial nonlinearity for the Cauchy problem to be well-posed in the Sobolev space $H^s(\mathbb T)$ for $s>\frac 52$. For the well-posedness, we use the energy estimates and the gauge transformation. For the ill-posedness, we prove the non-existence of solutions to the Cauchy problem. |
| title | Well- and ill-posedness of the Cauchy problem for semi-linear Schrödinger equations on the torus |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.04205 |