Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces

Tadahiro Oh, Yuzhao Wang

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2 Citations (Scopus)
83 Downloads (Pure)

Abstract

In this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces ℱLp(ℱLp(T) ), 1 ≤ p < ∞. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in ℱLp(ℱLp(T) ) for 1≤p≤32.

Original languageEnglish
Pages (from-to)723-762
JournalJournal d'Analyse Mathématique
Volume143
Issue number2
DOIs
Publication statusPublished - 29 Jun 2021

Bibliographical note

Funding Information:
T. Oh was supported by the ERC starting grant (no. 637995 “ProbDynDispEq”). The authors are grateful to the anonymous referee for a helpful comment that has improved the presentation of this paper.

Publisher Copyright:
© 2021, The Authors.

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

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