Global Well-posedness for Schrodinger Equations with Derivative in a Nonlinear Term and Data in Low-order Sobolev Spaces
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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 . Our proof relies on a generalization of the tri-linear estimates associated with the Fourier restriction norm method used in [1,25].
CitationTakaoka, H. (2001). Global well-posedness for Schrodinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces. Electronic Journal of Differential Equations, 2001(42), pp. 1-23.
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