Global well-posedness for Schrodinger equations with derivative in a nonlinear term and data in low-order Sobolev spaces
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 follow...
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| Format: | Article |
| Language: | English |
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Texas State University
2001-06-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/Volumes/2001/42/abstr.html |
| Summary: | 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]. |
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| ISSN: | 1072-6691 |