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Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space $ W^{s,p}$ for $ p<2$


Author: Yi Zhou
Journal: Trans. Amer. Math. Soc. 362 (2010), 4683-4694
MSC (2010): Primary 35Q41
DOI: https://doi.org/10.1090/S0002-9947-10-05055-5
Published electronically: April 20, 2010
MathSciNet review: 2645046
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Abstract: In this paper, we consider in $ R^n$ the Cauchy problem for the nonlinear Schrödinger equation with initial data in the Sobolev space $ W^{s,p}$ for $ p<2$. It is well known that this problem is ill posed. However, we show that after a linear transformation by the linear semigroup the problem becomes locally well posed in $ W^{s,p}$ for $ \frac{2n}{n+1}<p<2$ and $ s>n(1-\frac{1}{p})$. Moreover, we show that in one space dimension, the problem is locally well posed in $ L^p$ for any $ 1<p<2$.


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Additional Information

Yi Zhou
Affiliation: School of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China – and – Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, People’s Republic of China
Email: yizhou@fudan.ac.cn

DOI: https://doi.org/10.1090/S0002-9947-10-05055-5
Keywords: Cauchy problem, nonlinear Schr\"odinger equation, local well-posedness, scaling limit.
Received by editor(s): May 28, 2008
Received by editor(s) in revised form: September 2, 2008
Published electronically: April 20, 2010
Additional Notes: The author was supported by the National Natural Science Foundation of China under grant 10728101, the 973 Project of the Ministry of science and technology of China, the doctoral program foundation of the Ministry of education of China and the “111” Project and SGST 09DZ2272900
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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