Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces

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  • University of Edinburgh


In this paper, we first introduce a new function space MHθ,p whose norm is given by the lp-sum of modulated Hθ-norms of a given function. In particular, when θ<−12, we show that the space MHθ,p agrees with the modulation space M2,p(R) on the real line and the Fourier-Lebesgue space FLp(T) on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quan-tities constructed by Killip-Vi ̧san-Zhang to the modulation space and Fourier-Lebesgue space setting. By applying the scaling symmetry, we then prove global well-posedness of the one-dimensional cubic nonlin-ear Schrödinger equation (NLS) in almost critical spaces. More precisely, we show that the cubic NLS on Ris globally well-posed in M2,p(R)for any p<∞, while the renormalized cubic NLS on Tis globally well-posed in FLp(T)for any p<∞.
In Appendix, we also establish analogous global-in-time bounds for the modified KdV equation (mKdV) in the modulation spaces on the real line and in the Fourier-Lebesgue spaces on the circle. An additional key ingredient of the proof in this case is a Galilean transform which converts the mKdV to the mKdV-NLS equation.


Original languageEnglish
Pages (from-to)1-29
Number of pages29
JournalJournal of Differential Equations
Early online date16 Jan 2020
Publication statusE-pub ahead of print - 16 Jan 2020


  • nonlinear Schrödinger equation, modified KdV equation, global well-posedness, complete integrability, modulation space, Fourier-Lebesgue space