Global well-posedness for Schrodinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces

Abstract

In this paper, we study the existence of global solutions to Schrodinger equations in one space dimension with a derivative in a nonlinear term. For the Cauchy problem we assume that the data belongs to a Sobolev space weaker than the finite energy space $H^1$. Global existence for $H^1$ data follows from the local existence and the use of a conserved quantity. For $H^s$ data with $s<1$, the main idea is to use a conservation law and a frequency decomposition of the Cauchy data then follow the method introduced by Bourgain [3]. Our proof relies on a generalization of the tri-linear estimates associated with the Fourier restriction norm method used in [1,25].

Keywords

• Nonlinear Schrodinger equation
• well-posedness.