On the pointwise convergence of NLS flow on ${\S }^2$

Abstract

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 ${\S }^2$. 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 ${\mathbb{R}}^2$ case, up to the endpoint, and improves the previous result of Chen-Duong-Lee-Yan [J. Math. Pures Appl. 163 (2022), 433--449].