Electronic Journal of Differential Equations,
Vol. 2001(2001), No. 42, pp. 1-23.
Title: Global well-posedness for Schrodinger equations with derivative
in a nonlinear term and data in low-order Sobolev spaces
Author: Hideo Takaoka (Hokkaido Univ., Japan)
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.