Norm inflation for a higher-order nonlinear Schrödinger equation with a derivative on the circle
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915450838843392 |
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| author | Kondo, Toshiki Okamoto, Mamoru |
| author_facet | Kondo, Toshiki Okamoto, Mamoru |
| contents | We consider a periodic higher-order nonlinear Schrödinger equation with the nonlinearity $u^k \partial_x u$, where $k$ is a natural number. We prove the norm inflation in a subspace of the Sobolev space $H^s(\mathbb{T})$ for any $s \in \mathbb{R}$. In particular, the Cauchy problem is ill-posed in $H^s(\mathbb{T})$ for any $s \in \mathbb{R}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17782 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Norm inflation for a higher-order nonlinear Schrödinger equation with a derivative on the circle Kondo, Toshiki Okamoto, Mamoru Analysis of PDEs We consider a periodic higher-order nonlinear Schrödinger equation with the nonlinearity $u^k \partial_x u$, where $k$ is a natural number. We prove the norm inflation in a subspace of the Sobolev space $H^s(\mathbb{T})$ for any $s \in \mathbb{R}$. In particular, the Cauchy problem is ill-posed in $H^s(\mathbb{T})$ for any $s \in \mathbb{R}$. |
| title | Norm inflation for a higher-order nonlinear Schrödinger equation with a derivative on the circle |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.17782 |