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

TL;DR

This paper proves almost everywhere convergence of the cubic nonlinear Schrödinger flow on the sphere at very low regularity, extending prior results from Euclidean space to the spherical setting.

Contribution

It introduces a new almost sure pointwise convergence result for NLS on $ ext{S}^2$ at low regularity and establishes a necessary condition for $L^p$ maximal estimates.

Findings

Almost sure pointwise convergence for NLS on $ ext{S}^2$ at low regularity.

Failure of $L^p$ maximal estimate for $s<\frac{1}{2}-\frac{1}{2p}$, $p\ge 2$.

Results match Euclidean case for $p=3$, improving previous bounds.

Abstract

In this paper, we study the almost everywhere convergence of the cubic nonlinear Schr\"odinger flow to the initial data on $\mathbb S^2$, \begin{equation*} iu_t + \Delta_g u = |u|^2u, \quad (t,x)\in\R\times \S^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-Luc\`a-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 Schr\"odinger 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 $\R^2$ case, up to the endpoint, and improves the previous result of Chen-Duong-Lee-Yan [J. Math. Pures Appl. 163 (2022), 433--449].