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Blow-Up for the 1D Cubic NLS

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Abstract

We consider the 1D cubic NLS on \(\mathbb R\) and prove a blow-up result for functions that are of borderline regularity, i.e. \(H^s\) for any \(s<-\frac{1}{2}\) for the Sobolev scale and \({\mathcal {F}} L^\infty \) for the Fourier–Lebesgue scale. This is done by identifying at this regularity a certain functional framework from which solutions exit in finite time. This functional framework allows, after using a pseudo-conformal transformation, to reduce the problem to a large-time study of a periodic Schrödinger equation with non-autonomous cubic nonlinearity. The blow-up result corresponds to an asymptotic completeness result for the new equation. We prove it using Bourgain’s method and exploiting the oscillatory nature of the coefficients involved in the time-evolution of the Fourier modes. Finally, as an application we exhibit singular solutions of the binormal flow. More precisely, we give conditions on the curvature and the torsion of an initial smooth curve such that the constructed solutions generate several singularities in finite time.

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Notes

  1. This term is controllable directly for \(\{B_j\}\in l^{2,s}\) with \(s>\frac{1}{2}\), but not for lower regularity.

  2. To avoid smallness hypothesis on \(\{\alpha _k\}_{k\in \mathbb Z}\) that would come from the linear term in the last integral term of the fixed point operator, this last integrant can be replaced by an oscilatory one in the same spirit as in §2.1 of [5].

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Acknowledgements

This research is partially supported as follows. VB is partially supported by the Institut Universitaire de France, by the French ANR project SingFlows. RL is supported by BERC program 2022-2025 and by MICINN (Spain) projects Severo Ochoa CEX2021-001142, PID2021-123034NB-I00 and by the Ramon y Cajal fellowship RYC2021-031981-I. NT is partially supported by the ANR project Smooth ANR-22-CE40-0017. LV is funded by MICINN (Spain) projects Severo Ochoa CEX2021-001142, and PID2021-126813NB-I00 (ERDF A way of making Europe), and by Eusko Jaurlaritza project IT1615-22 and BERC program. The authors would like to thank the referees for their careful reading and suggestions.

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Authors and Affiliations

  1. CNRS, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), Sorbonne Université, 75005, Paris, France

    Valeria Banica

  2. Institut Universitaire de France (IUF), Paris, France

    Valeria Banica

  3. BCAM - Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009, Bilbao, Basque Country, Spain

    Renato Lucà & Luis Vega

  4. Ikerbasque, Basque Foundation for Science, 48011, Bilbao, Basque Country, Spain

    Renato Lucà

  5. Ecole Normale Supérieure de Lyon, UMPA, UMR CNRS-ENSL 5669, 46, allée d’Italie, 69364, Lyon Cedex 07, France

    Nikolay Tzvetkov

  6. Dpto. Matemáticas UPV/EHU Apto. 644, 48080, Bilbao, Basque Country, Spain

    Luis Vega

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  1. Valeria Banica
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  2. Renato Lucà
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  4. Luis Vega
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Correspondence to Valeria Banica.

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Communicated by K. Nakanishi.

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Banica, V., Lucà, R., Tzvetkov, N. et al. Blow-Up for the 1D Cubic NLS. Commun. Math. Phys. 405, 11 (2024). https://doi.org/10.1007/s00220-023-04906-3

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