Electron. J. Diff. Eqns., Vol. 2001(2001), No. 42, pp. 1-23.

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

Hideo Takaoka

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 H1. Global existence for H1 data follows from the local existence and the use of a conserved quantity. For Hs 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].

Submitted March 15, 2000. Published June 5, 2001.
Math Subject Classifications: 35Q55.
Key Words: Nonlinear Schrodinger equation, well-posedness.

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Hideo Takaoka
Department of Mathematics, Hokkaido University
Sapporo 060-0810, Japan
e-mail: takaoka@math.sci.hokudai.ac.jp
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