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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2411.10975 |
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| _version_ | 1866912122649182208 |
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| author | Ban, Yingzhe Chen, Jie Zhang, Ying |
| author_facet | Ban, Yingzhe Chen, Jie Zhang, Ying |
| contents | In this paper, we study local well-posedness theory of the Cauchy problem for Schrödinger-KdV system in Sobolev spaces $H^{s_1}\times H^{s_2}$. We obtain the local well-posedness when $s_1\geq 0$, $\max\{-3/4,s_1-3\}\leq s_2\leq \min\{4s_1,s_1+2\}$. The result is sharp in some sense and improves previous one by Corcho-Linares \cite{corcho2007well}. The endpoint case $(s_1,s_2) = (0,-3/4)$ has been solved in \cite{guo2010well,wang2011cauchy}. We show the necessary and sufficient conditions for related estimates in Bourgain spaces. To solve the borderline cases, we use the $U^p-V^p$ spaces introduced by Koch-Tataru \cite{kochtataru} and function spaces constructed by Guo-Wang \cite{guo2010well}. We also use normal form argument to control the nonresonant interaction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_10975 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local well-posedness for the Schrödinger-KdV system in $H^{s_1}\times H^{s_2}$ Ban, Yingzhe Chen, Jie Zhang, Ying Analysis of PDEs In this paper, we study local well-posedness theory of the Cauchy problem for Schrödinger-KdV system in Sobolev spaces $H^{s_1}\times H^{s_2}$. We obtain the local well-posedness when $s_1\geq 0$, $\max\{-3/4,s_1-3\}\leq s_2\leq \min\{4s_1,s_1+2\}$. The result is sharp in some sense and improves previous one by Corcho-Linares \cite{corcho2007well}. The endpoint case $(s_1,s_2) = (0,-3/4)$ has been solved in \cite{guo2010well,wang2011cauchy}. We show the necessary and sufficient conditions for related estimates in Bourgain spaces. To solve the borderline cases, we use the $U^p-V^p$ spaces introduced by Koch-Tataru \cite{kochtataru} and function spaces constructed by Guo-Wang \cite{guo2010well}. We also use normal form argument to control the nonresonant interaction. |
| title | Local well-posedness for the Schrödinger-KdV system in $H^{s_1}\times H^{s_2}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.10975 |