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arXiv · 2606.14384

Bounds on the growth of high Sobolev norms of solutions to the fractional nonlinear Schr\"{o}dinger equation on $\R^2$ and $\R^3$

Antonios Kapetanios·José L. Rodrigo·Vedran Sohinger
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Abstract

We prove polynomial bounds on the growth of high Sobolev norms of solutions to the defocusing fractional nonlinear Schr\"odinger equation with cubic and Hartree-type nonlinearities in two and three dimensions. Our result is based on the upside-down I-method and the method of higher modified energies. The analysis of the fractional case in higher dimensions is possible due to a higher-dimensional analogue of a resonance inequality, which allows us to control the nonresonant frequency contributions.

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Antonios Kapetanios, José L. Rodrigo, Vedran Sohinger. 2026-06-12. Bounds on the growth of high Sobolev norms of solutions to the fractional nonlinear Schr\"{o}dinger equation on $\R^2$ and $\R^3$. https://arxiv.org/abs/2606.14384

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