Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations

Abstract

Ill-posedness is established for the initial value problem (IVP) associated to the derivative nonlinear Schrödinger equation for data in H s (R), s < 1/2. This result implies that best result concerning local well-posedness for the IVP is in H s (R), s ≥ 1/2. It is also shown that the (IVP) associated to the generalized Benjamin-Ono equation for data below the scaling is in fact ill-posed.