Global well-posedness for the derivative nonlinear Schr\"{o}dinger   equation in $H^{\frac 12} (\mathbb{R})$
  We prove that the derivative nonlinear Schr\"{o}dinger equation is globally well-posed in $H^{\frac 12} (\mathbb{R})$ when the mass of initial data is strictly less than $4\pi$. 
Zihua Guo, Yifei Wu
math.AP