Sharp $L^4$ Strichartz estimate for Hyperbolic Schrödinger equation on $\mathbb{R}\times \mathbb{T}$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908665170100224 |
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| author | Deng, Yangkendi Fan, Chenjie Zhao, Zehua |
| author_facet | Deng, Yangkendi Fan, Chenjie Zhao, Zehua |
| contents | We prove the sharp $L^4$ Strichartz estimate without derivative loss for the hyperbolic Schrödinger equation on $\mathbb{R}\times\mathbb{T}$, \begin{equation} \|e^{it (\partial_{x_{1}}^2-\partial_{x_{2}}^2)} ϕ\|_{L^4_{t,x_{1},x_{2}}([0,1]\times \mathbb{R} \times \mathbb{T})}\lesssim \|ϕ\|_{L_{x_{1},x_{2}}^2(\mathbb{R} \times \mathbb{T})}, \end{equation} which serves as the hyperbolic analogue of the classical result of Takaoka-Tzvetkov \cite{takaoka20012d}. The proof is based on the combination of a robust kernel decomposition method with precise measure estimates for semi-algebraic sets. As an immediate application, we establish the global well-posedness for the cubic hyperbolic Schrödinger equation on $\mathbb{R}\times\mathbb{T}$ in the $L^2$-critical space with sufficiently small initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15157 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp $L^4$ Strichartz estimate for Hyperbolic Schrödinger equation on $\mathbb{R}\times \mathbb{T}$ Deng, Yangkendi Fan, Chenjie Zhao, Zehua Analysis of PDEs Primary: 35Q55, Secondary: 35R01, 37K06, 37L50 We prove the sharp $L^4$ Strichartz estimate without derivative loss for the hyperbolic Schrödinger equation on $\mathbb{R}\times\mathbb{T}$, \begin{equation} \|e^{it (\partial_{x_{1}}^2-\partial_{x_{2}}^2)} ϕ\|_{L^4_{t,x_{1},x_{2}}([0,1]\times \mathbb{R} \times \mathbb{T})}\lesssim \|ϕ\|_{L_{x_{1},x_{2}}^2(\mathbb{R} \times \mathbb{T})}, \end{equation} which serves as the hyperbolic analogue of the classical result of Takaoka-Tzvetkov \cite{takaoka20012d}. The proof is based on the combination of a robust kernel decomposition method with precise measure estimates for semi-algebraic sets. As an immediate application, we establish the global well-posedness for the cubic hyperbolic Schrödinger equation on $\mathbb{R}\times\mathbb{T}$ in the $L^2$-critical space with sufficiently small initial data. |
| title | Sharp $L^4$ Strichartz estimate for Hyperbolic Schrödinger equation on $\mathbb{R}\times \mathbb{T}$ |
| topic | Analysis of PDEs Primary: 35Q55, Secondary: 35R01, 37K06, 37L50 |
| url | https://arxiv.org/abs/2511.15157 |