Global well-posedness and scattering for the defocusing energy supercritical NLS in high dimensions

Xuan Liu

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arXiv · 2608.00951

Global well-posedness and scattering for the defocusing energy supercritical NLS in high dimensions

Xuan Liu

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Abstract

We consider the defocusing energy-supercritical nonlinear Schr\"odinger equation $i\partial_{t}u+\Delta u=|u|^p u$ in dimensions $d\ge5$. Killip-Visan [Comm. Partial Differential Equations, 2010] and Li-Li [Siam J. Math. Anal., 2022] proved that for $s_c:=\frac{d}{2}-\frac{2}{p}>1$, any solution that remains bounded in the critical Sobolev space $\dot H_x^{s_c}(\mathbb{R} ^d)$ must be global and scatter. In dimensions $d \ge 8$, their results required either that $p$ be even or that $s_c < \frac{d+2-\sqrt{(d-2)^2-16}}{4}$. In this paper, we improve the upper bound on $s_c$ to $s_c<1+p$ by establishing some new nonlinear estimates. This allows us to cover all cases in which $p$ lies in the local existence range.

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Xuan Liu. 2026-08-02. Global well-posedness and scattering for the defocusing energy supercritical NLS in high dimensions. https://arxiv.org/abs/2608.00951

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