Strichartz estimates and global well-posedness of the cubic NLS on $\mathbb{T}^{2}$
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929493457764352 |
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| author | Herr, Sebastian Kwak, Beomjong |
| author_facet | Herr, Sebastian Kwak, Beomjong |
| contents | The optimal $L^4$-Strichartz estimate for the Schr{ö}dinger equation on the two-dimensional rational torus $\mathbb{T}^2$ is proved, which improves an estimate of Bourgain. A new method based on incidence geometry is used. The approach yields a stronger $L^4$ bound on a logarithmic time scale, which implies global existence of solutions to the cubic (mass-critical) nonlinear Schrödinger equation in $H^s(\mathbb{T}^2)$ for any $s>0$ and data which is small in the critical norm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_14275 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Strichartz estimates and global well-posedness of the cubic NLS on $\mathbb{T}^{2}$ Herr, Sebastian Kwak, Beomjong Analysis of PDEs Classical Analysis and ODEs The optimal $L^4$-Strichartz estimate for the Schr{ö}dinger equation on the two-dimensional rational torus $\mathbb{T}^2$ is proved, which improves an estimate of Bourgain. A new method based on incidence geometry is used. The approach yields a stronger $L^4$ bound on a logarithmic time scale, which implies global existence of solutions to the cubic (mass-critical) nonlinear Schrödinger equation in $H^s(\mathbb{T}^2)$ for any $s>0$ and data which is small in the critical norm. |
| title | Strichartz estimates and global well-posedness of the cubic NLS on $\mathbb{T}^{2}$ |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2309.14275 |