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 $\dot H^\frac12$ 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 $12/13<s \leqslant 1$, then the corresponding solution exists globally and scatters.

Theorems & Definitions (18)

• Theorem 1.1
• Remark 1.2
• Lemma 2.1: Strichartz estimate,KT98AJM
• Lemma 2.2: Schur's test
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5: Inhomogeneous Strichartz,Fos05JHDEVil07TranAMS
• Lemma 2.6: Interaction Morawetz inequality, Iteam04CPAM
• Lemma 3.1
• proof