Generic ill-posedness for Schrödinger equation with power-type nonlinearity on ${\mathbb{S}}^2$

Abstract

In this article, we investigate the local well-posedness of the nonlinear Schrödinger equation on the two-dimensional sphere ${\mathbb{S}}^2$: \begin{align*} i\partial_tu+Δ_{g}u=F(u). \end{align*} The nonlinearity $F(u)$ is assumed to be gauge-invariant. More presicely, there exists a function $V\in C^{\infty }(\mathbb{C},\mathbb{R})$ such that $F=\frac{\partial V}{\partial \bar{z}}$. Moreover, $V(z)$ obeys \begin{equation}\label{H-11} V(e^{iθ}z)=V(z),\,\,θ\in\Bbb R,\,\,z\in\Bbb C, |\partial_z^{k_1}\partial_{\bar{z}}^{k_2}V(z)|\leq C_{k_1,k_2}(1+|z|)^{1+α-k_1-k_2},\tag{H-1} \end{equation} for some $\hat{I}\pm \ge 3.$ The main contribution of this paper is the new lower bound of threshold of local well-posedness $s_c({\mathbb{S}}^2,\hat{I}\pm )$. Specifically, under assumption \eqref{H-11}, we prove that for $\hat{I}\pm \ge 3$, the equation is ill-posed in $H^s({\mathbb{S}}^2)$ with $s<1-\frac{2}{\hat{I}\pm -1}$ in the sense that the norm inflation occurs. Combined with the well-posedness in Yang [Sci. China Math. 58 (2015), 1023-1046], the exact threshold $s_c({\mathbb{S}}^2,\hat{I}\pm )$ for $\hat{I}\pm \ge 5$ is $1-\frac{2}{\hat{I}\pm -1}$, which matches the scaling-critical regularity as the Euclidean setting. Moreover, for $\hat{I}\pm \in [3,\frac{11}{3})$, we show that the solution map is not uniformly continuous in the range $0<s<\frac{1}{4}$ for the power-type nonlinearity $F(u)=|u{|}^{\hat{I}\pm -1}u$, which lies strictly above the scaling-invariant threshold. This provides a new characterization of the ill-posedness regime for all $\hat{I}\pm \ge 3$, extending an earlier result of Burq-Gérard-Tzvetkov [Math. Res. Lett. 9 (2002), 323-335]. Our result can also be regarded as a Schrödinger counterpart of Xia [Int. Math. Res. Not. (2021), 15533-15554].

18 pages