Journal of Fourier Analysis and Applications

, Volume 25, Issue 4, pp 1447–1486 | Cite as

NLS in the Modulation Space \(M_{2,q}({\mathbb {R}})\)

  • N. PattakosEmail author


We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in the modulation space \(M_{2,q}^{s}({\mathbb {R}})\), \(1\le q\le 2\) and \(s\ge 0.\) In addition, for either \(s\ge 0\) and \(1\le q\le \frac{3}{2}\) or \(\frac{3}{2}<q\le 2\) and \(s>\frac{2}{3}-\frac{1}{q}\) we show that the Cauchy problem is unconditionally wellposed in \(M_{2,q}^{s}({{\mathbb {R}}}).\) It is done with the use of the differentiation by parts technique which had been previously used in the periodic setting.


Normal form method Modulation spaces Nonlinear Schrödinger Local wellposedness Unconditional uniqueness 

Mathematics Subject Classification

35A01 35A02 35D30 35J10 



The author gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. He would also like to thank Peer Kunstmann from KIT for his helpful comments and fruitful discussions. Finally, he would like to thank the referees of the paper for their constructive criticism.


  1. 1.
    Babin, A., Ilyin, A., Titi, E.: On the regularization mechanism for the periodic Korteweg-de Vries equation. Commun. Pure Appl. Math. 64(5), 591–648 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bényi, A., Gröchenig, K., Okoudjou, K.A., Rogers, L.G.: Unimodular Fourier multipliers for modulation spaces. J. Funct. Anal. 246, 366–384 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bényi, A., Okoudjou, K.A.: Local well-posedness of nonlinear dispersive equations on modulation spaces. Bull. Lond. Math. Soc. 41(3), 549–558 (2009). ISSN 0024-6093MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Christ, M.: Power Series Solution of a Nonlinear Schrödinger Equation. Mathematical Aspects of Nonlinear Dispersive Equations, Annals of Mathematical Studies, vol. 163, pp. 131–155. Princeton University Press, Princeton (2007)zbMATHGoogle Scholar
  5. 5.
    Christ, M.: Nonuniqueness of weak solutions of the nonlinear Schrödinger equation. arXiv:math/0503366
  6. 6.
    Chung, J., Guo, Z., Kwon, S., Oh, T.: Normal form approach to global wellposedness of the quadratic derivative nonlinear Schrödinger equation on the circle. Ann. Inst. H. Poincaré Anal. Non Linéaire. 34(5), 1273–1297 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Feichtinger, H.G.: Modulation spaces on locally compact Abelian groups. Technical Report, University of Vienna, 1983, In: Proceedings of the International Conference on Wavelet and Applications, 2002, New Delhi Allied Publishers, India, pp. 99–140 (2003)Google Scholar
  8. 8.
    Gubinelli, M.: Rough solutions for the periodic Korteweg–de Vries equation. Commun. Pure Appl. Anal. 11(2), 709–733 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Guo, S.: On the \(1\)D cubic nonlinear Schrödinger equation in an almost critical space. J. Fourier Anal. Appl. 23(1), 91–124 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Guo, Z., Kwon, S., Oh, T.: Poincaré–Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS. Commun. Math. Phys. 322(1), 19–48 (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn, p. xvi+426. The Clarendon Press, New York (1979)Google Scholar
  12. 12.
    Kato, T.: On nonlinear Schrödinger equations. II. \(H^{s}\)-solutions and unconditional well-posedness. J. Anal. Math. 67, 281–306 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kwon, S., Oh, T.: On unconditional well-posedness of modified KdV. Int. Math. Res. Not. IMRN 15, 3509–3534 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Oh, T., Wang, Y.: Global wellposedness of the periodic cubic fourth order NLS in negative Sobolev spaces. Forum Math. Sigma 6, e5, 80 (2018)CrossRefzbMATHGoogle Scholar
  15. 15.
    Oh, T., Sosoe, P., Tzvetkov, N.: An optimal regularity result on the quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation. arXiv:1707.01666
  16. 16.
    Shatah, J.: Normal forms and quadratic nonlinear Klein–Gordon equations. Commun. Pure Appl. Math. 38, 685–696 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tsutsumi, Y.: \(L^{2}\) solutions for nonlinear Schrödinger equations and nonlinear groups. Funkc. Ekvac. 30, 115–125 (1987)zbMATHGoogle Scholar
  18. 18.
    Wang, B.X., Hudzik, H.: The global Cauchy problem for the NLS and NLKG with small rough data. J. Differ. Equ. 232, 36–73 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yoon, H.: Normal Form Approach to Well-posedness of Nonlinear Dispersive Partial Differential Equations. Ph.D. thesis, Korea Advanced Institute of Science and Technology (2017)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Institute for AnalysisKarlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations