Cauchy problem of nonlinear Schrödinger equation with initial data in Sobolev space for

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider in the Cauchy problem for the nonlinear Schrödinger equation with initial data in the Sobolev space for . 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 for and . Moreover, we show that in one space dimension, the problem is locally well posed in for any .

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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.