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arXiv · 2606.11376

Global well-posedness of cubic fractional Schr\"{o}dinger equation with rough data

Jiajun Wang
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Abstract

In this paper, we apply the I-method to establish global well-posedness for the fractional nonlinear Schr\"{o}dinger equation with initial data u_0 \in H^s(R^d) for s <\alpha/2, i.e., below the energy threshold. Moreover, for radial initial data, we combine a modiffed Morawetz estimate-recovered via Balakrishnan's formula-with the I-method to obtain improved results. In the same spirit, we employ the "upside-down" I-method to derive polynomial-in-time growth bounds for the higher-order Sobolev norm. The main difffculty stems from the fact that Strichartz estimates for the fractional Schr\"{o}dinger equation has a loss of derivatives, and the problem is always L^2-supercritical, thereby requiring more delicate analysis.

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Jiajun Wang. 2026-06-09. Global well-posedness of cubic fractional Schr\"{o}dinger equation with rough data. https://arxiv.org/abs/2606.11376

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