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On the 1D Cubic Nonlinear Schrödinger Equation in an Almost Critical Space

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Abstract

We consider the Cauchy problem for the one dimensional cubic nonlinear Schrödinger equation \(iu_t+u_{xx}-|u|^2u=0\). As the first step local well-posedness in the modulation space \(M_{2,p}\) (\(2\le p<\infty \)) is derived (see Theorem 1.4), which covers all the subcritical cases. Afterwards in order to approach the endpoint case, we will prove the almost global well-posedness in some Orlicz type space (see Theorem 1.8), which is a natural generalization of \(M_{2,p}\), and is almost critical from the viewpoint of scaling. The new ingredient is an endpoint version of the two dimensional restriction estimate (see Lemma 3.7).

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Acknowledgments

This work is done under the supervision of Prof. Herbert Koch. The author would like to thank him for his patient guidance and for sharing many helpful thoughts. The author also thanks Prof. Sebastian Herr for carefully reading this paper and giving a lot of valuable suggestions. The author would also like to thank a anonymous referee for pointing out several references, and other valuable comments.

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  1. Institute of Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Shaoming Guo

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  1. Shaoming Guo
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Correspondence to Shaoming Guo.

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Communicated by Hans G. Feichtinger.

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Guo, S. On the 1D Cubic Nonlinear Schrödinger Equation in an Almost Critical Space. J Fourier Anal Appl 23, 91–124 (2017). https://doi.org/10.1007/s00041-016-9464-z

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  • DOI: https://doi.org/10.1007/s00041-016-9464-z

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