Global well-posedness for Schrödinger equation with derivative in

In this paper, we consider the Cauchy problem of the cubic nonlinear Schrödinger equation with derivative in Hs(R). This equation was known to be the local well-posedness for (Takaoka, 1999 [27], ), ill-posedness for (Biagioni and Linares, 2001 [1], , etc.) and global well-posedness for (I-team, 2002 [10]). In this paper, we show that it is global well-posedness in the endpoint space , which remained open previously. The main approach is the third generation I-method combined with a new resonant decomposition technique. The resonant decomposition is applied to control the singularity coming from the resonant interaction.