Global well-posedness for the defocusing cubic nonlinear Schrödinger equation on $\Bbb T^3$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913578768924672 |
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| author | Song, Yilin Zhang, Ruixiao |
| author_facet | Song, Yilin Zhang, Ruixiao |
| contents | In this article, we investigate the global well-posedness for the defocusing, cubic nonlinear Schrödinger equation posed on $\T^3$ with intial data lying in its critical space $H^\frac{1}{2}(\T^3)$. By establishing the linear profile decomposition, and applied this to the concentration-compactness/rigidity argument, we prove that if the solution remains bounded in the critical Sobolev space throughout the maximal lifespan, i.e. $u\in L_t^\infty{H}^\frac{1}{2}(I\times\T^3)$, then $u$ is global. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_10056 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global well-posedness for the defocusing cubic nonlinear Schrödinger equation on $\Bbb T^3$ Song, Yilin Zhang, Ruixiao Analysis of PDEs 35Q55, 35R01, 37K06, 37L50 In this article, we investigate the global well-posedness for the defocusing, cubic nonlinear Schrödinger equation posed on $\T^3$ with intial data lying in its critical space $H^\frac{1}{2}(\T^3)$. By establishing the linear profile decomposition, and applied this to the concentration-compactness/rigidity argument, we prove that if the solution remains bounded in the critical Sobolev space throughout the maximal lifespan, i.e. $u\in L_t^\infty{H}^\frac{1}{2}(I\times\T^3)$, then $u$ is global. |
| title | Global well-posedness for the defocusing cubic nonlinear Schrödinger equation on $\Bbb T^3$ |
| topic | Analysis of PDEs 35Q55, 35R01, 37K06, 37L50 |
| url | https://arxiv.org/abs/2411.10056 |