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Local well-posedness for the $ H^2$-critical nonlinear Schrödinger equation


Authors: Thierry Cazenave, Daoyuan Fang and Zheng Han
Journal: Trans. Amer. Math. Soc. 368 (2016), 7911-7934
MSC (2010): Primary 35Q55; Secondary 35B30
DOI: https://doi.org/10.1090/tran6683
Published electronically: March 2, 2016
MathSciNet review: 3546788
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Abstract: In this paper, we consider the nonlinear Schrödinger equation $ iu_t +\Delta u= \lambda \vert u\vert^{\frac {4} {N-4}} u$ in $ \mathbb{R}^N $, $ N\ge 5$, with $ \lambda \in \mathbb{C}$. We prove local well-posedness (local existence, unconditional uniqueness, continuous dependence) in the critical space $ \dot H^2 (\mathbb{R}^N ) $.


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Thierry Cazenave
Affiliation: Université Pierre et Marie Curie and CNRS, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France
Email: thierry.cazenave@upmc.fr

Daoyuan Fang
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China
Email: dyf@zju.edu.cn

Zheng Han
Affiliation: Department of Mathematics, Hangzhou Normal University, and Department of Mathematics, Zhejiang University, Hangzhou, 311121, People’s Republic of China
Email: hanzh_0102@163.com

DOI: https://doi.org/10.1090/tran6683
Keywords: $H^2$-critical nonlinear Schr\"odinger equation, local existence, continuous dependence, unconditional uniqueness
Received by editor(s): March 20, 2014
Received by editor(s) in revised form: January 16, 2015
Published electronically: March 2, 2016
Article copyright: © Copyright 2016 American Mathematical Society