Well-Posedness and Scattering for Nonlinear Schrodinger Equations with a Derivative Nonlinearity at the Scaling Critical Regularity

In the present paper, we consider the Cauchy problem of nonlinear Schrodinger equations with a derivative nonlinearity which depends only on (u) over bar and its first derivatives. The well-posedness of the equation at the scaling subcritical regularity was proved by A. Grunrock (2000). We prove the well-posedness of the equation and the scattering for the solution at the scaling critical regularity by using U-2 space and V-2 space which are applied to prove the well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (2009).