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

Abstract

In this paper, we study the global well-posedness and scattering of 3D defocusing, cubic Schrödinger equation. Recently, Dodson [arXiv:2004.09618] 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 subcritical, 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 $12/13<s⩽1$, then the corresponding solution exists globally and scatters.

21 pages; comments are welcome!