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

**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].
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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