Well-posedness and scattering for nonlinear Schr\"odinger equations with a derivative nonlinearity at the scaling critical regularity

Abstract

In the present paper, we consider the Cauchy problem of nonlinear Schr\"odinger equations with a derivative nonlinearity which depends only on $\bar{u}$. The well-posedness of the equation at the scaling subcritical regularity was proved by A. Gr\"unrock (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).