On the pointwise convergence of NLS flow on $ §^2 $
Fuente:
arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915921279320064 |
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| author | Meng, Fanfei Song, Yilin Sun, Chenmin Zhang, Ruixiao Zheng, Jiqiang |
| author_facet | Meng, Fanfei Song, Yilin Sun, Chenmin Zhang, Ruixiao Zheng, Jiqiang |
| contents | In this paper, we study the almost everywhere convergence of the cubic nonlinear Schrödinger flow to the initial data on $\mathbb S^2$,
\begin{equation*}
iu_t + Δ_g u = |u|^2u, \quad (t,x)\in\R\times §^2.
\end{equation*}
Inspired by the randomization method and the ansatz introduced by Burq, Camps, Sun, and Tzvetkov [Preprint, arXiv:2404.18229], we prove almost sure pointwise convergence almost everywhere for the nonlinear solution at very low regularity. This extends Compaan-Lucà-Staffilani [Int. Math. Res. Not. IMRN, (1) (2021), 596--647] to the spherical setting. We also provide a new necessary condition for the associated $L^p$ maximal estimate for the linear Schrödinger equation on $§^2$. More precisely, we show that the $L^p$ maximal estimate fails for $s<\frac{1}{2}-\frac{1}{2p}$ with $p\ge 2$. In the special case $p=3$, our result matches the corresponding range in the $\R^2$ case, up to the endpoint, and improves the previous result of Chen-Duong-Lee-Yan [J. Math. Pures Appl. 163 (2022), 433--449]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05851 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the pointwise convergence of NLS flow on $ §^2 $ Meng, Fanfei Song, Yilin Sun, Chenmin Zhang, Ruixiao Zheng, Jiqiang Analysis of PDEs In this paper, we study the almost everywhere convergence of the cubic nonlinear Schrödinger flow to the initial data on $\mathbb S^2$, \begin{equation*} iu_t + Δ_g u = |u|^2u, \quad (t,x)\in\R\times §^2. \end{equation*} Inspired by the randomization method and the ansatz introduced by Burq, Camps, Sun, and Tzvetkov [Preprint, arXiv:2404.18229], we prove almost sure pointwise convergence almost everywhere for the nonlinear solution at very low regularity. This extends Compaan-Lucà-Staffilani [Int. Math. Res. Not. IMRN, (1) (2021), 596--647] to the spherical setting. We also provide a new necessary condition for the associated $L^p$ maximal estimate for the linear Schrödinger equation on $§^2$. More precisely, we show that the $L^p$ maximal estimate fails for $s<\frac{1}{2}-\frac{1}{2p}$ with $p\ge 2$. In the special case $p=3$, our result matches the corresponding range in the $\R^2$ case, up to the endpoint, and improves the previous result of Chen-Duong-Lee-Yan [J. Math. Pures Appl. 163 (2022), 433--449]. |
| title | On the pointwise convergence of NLS flow on $ §^2 $ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.05851 |