On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition

Abstract

It is shown that the Cauchy problem associated to the derivative nonlinear Schrödinger equation ∂t u-i∂x2 u=λ∂x ( |u| 2 u) is locally well-posed for initial data u(0)∈Hs (T), if s<1/2 and λ is real. The proof is based on a variant of the gauge transformation, introduced by Hayashi and Ozawa, adjusted to the periodic setting and sharp multilinear estimates for the gauge equivalent equation in Fourier restriction norm spaces. By the use of a conservation law, the problem is shown to be globally well-posed for s<1 and data which is small in L2.