Global well-posedness for rough solutions of defocusing cubic NLS on three dimensional compact manifolds
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| Format: | Preprint |
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2024
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| _version_ | 1866929410277376000 |
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| author | Qionglei, Chen Song, Yilin Zheng, Jiqiang |
| author_facet | Qionglei, Chen Song, Yilin Zheng, Jiqiang |
| contents | In this article, we investigate the global well-posedness for cubic nonlinear Schrödinger equation(NLS) $ i\partial_tu+Δ_gu=|u|^2u$ posed on the three dimensional compact manifolds $(M,g)$ with initial data $u_0\in H^s(M)$ where $s>\frac{\sqrt{21}-1}{4}$ for Zoll manifold and $s>\frac{1+3\sqrt{5}}{8}$ for the product of spheres $\Bbb{S}^2\times\Bbb{S}^1$. We utilize the multilinear eigenfunction estimate on compact manifold to treat the interaction of different frequencies, which is more complicated compared to the case of flat torus [C. Fan, G. Staffilani, H. Wang, B. Wilson, Anal. PDE, 11 (2018), 919-944.] and waveguide manifold [Z. Zhao, J. Zheng, SIAM J. Math. Anal. 53 (2020), 3644-3660.]. Moreover, combining with the I-method adapted to the non-periodic case, bilinear Strichartz estimates along with the scale-invariant $L^p$ linear Strichartz estimates, we partially obtain the similar result of [Z. Zhao, J. Zheng, SIAM J. Math. Anal. 53 (2020), 3644-3660.] on non-flat compact manifold setting. As a consequence, we obtain the polynomial bounds of the $H^s$ norm of solution $u$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_03908 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global well-posedness for rough solutions of defocusing cubic NLS on three dimensional compact manifolds Qionglei, Chen Song, Yilin Zheng, Jiqiang Analysis of PDEs In this article, we investigate the global well-posedness for cubic nonlinear Schrödinger equation(NLS) $ i\partial_tu+Δ_gu=|u|^2u$ posed on the three dimensional compact manifolds $(M,g)$ with initial data $u_0\in H^s(M)$ where $s>\frac{\sqrt{21}-1}{4}$ for Zoll manifold and $s>\frac{1+3\sqrt{5}}{8}$ for the product of spheres $\Bbb{S}^2\times\Bbb{S}^1$. We utilize the multilinear eigenfunction estimate on compact manifold to treat the interaction of different frequencies, which is more complicated compared to the case of flat torus [C. Fan, G. Staffilani, H. Wang, B. Wilson, Anal. PDE, 11 (2018), 919-944.] and waveguide manifold [Z. Zhao, J. Zheng, SIAM J. Math. Anal. 53 (2020), 3644-3660.]. Moreover, combining with the I-method adapted to the non-periodic case, bilinear Strichartz estimates along with the scale-invariant $L^p$ linear Strichartz estimates, we partially obtain the similar result of [Z. Zhao, J. Zheng, SIAM J. Math. Anal. 53 (2020), 3644-3660.] on non-flat compact manifold setting. As a consequence, we obtain the polynomial bounds of the $H^s$ norm of solution $u$. |
| title | Global well-posedness for rough solutions of defocusing cubic NLS on three dimensional compact manifolds |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2407.03908 |