Global Well-posedness for Schrodinger Equations with Derivative in a Nonlinear Term and Data in Low-order Sobolev Spaces

Takaoka, Hideo
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Southwest Texas State University, Department of Mathematics
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].
Nonlinear Schrodinger equation, Well-posedness
Takaoka, H. (2001). Global well-posedness for Schrodinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces. <i>Electronic Journal of Differential Equations, 2001</i>(42), pp. 1-23.