Global well-posedness and scattering of 3D defocusing, cubic Schrödinger equation

In this paper, we study the global well-posedness and scattering of 3D defocusing, cubic Schrödinger equation. Recently, Dodson $\href{https://dx.doi.org/10.4171/RMI/1295}{\textrm{[16]}}$ studied the global well-posedness in a critical Sobolev space $\dot{W}^{11/7,7/6}$. In this paper, we aim to show that if the initial data belongs to $\dot{H}^{\frac{1}{2}}$ to guarantee the local existence, then some extra weak space which is supercritical, is sufficient to prove the global well-posedness. More precisely, we prove that if the initial data belongs to $\dot{H}^{1/2} \cap \dot{W}^{s,1}$ for $12/13 \lt s \leqslant 1$, then the corresponding solution exists globally and scatters.

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Received 22 July 2021

Received revised 5 March 2023

Accepted 30 July 2023

Published 17 July 2024