Gevrey-modulation spaces and smoothing effect for the nonlinear Schrödinger equations

  • Gaku HoshinoEmail author


We study the global Cauchy problem for the nonlinear Schrödinger equations with the power type nonlinearity or the Hartree type nonlinearity, in the mass critical setting. Especially, we show the Gevrey smoothing effect for the nonlinear Schrödinger equations with data which satisfy sub-exponentially decaying condition and has sufficiently small norm. Also we show the existence of scattering state in the class of sub-exponentially decaying functions without loss of radius of convergence for sufficiently small data.



The author would like to thank the referees for their helpful comments and advices.


  1. 1.
    Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10, American Mathematical Society (2003)Google Scholar
  2. 2.
    De Bouard, A.: Analytic solution to non elliptic non linear Schrödinger equations. J. Differ. Equ. 104, 196–213 (1993)CrossRefGoogle Scholar
  3. 3.
    Bourdaud, G., Reissig, M., Sickel, W.: Hyperbolic equations, function spaces with exponential weights and Nemytskij operators. Annal. Matematica 182, 409–455 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Feichtinger, H.G.: Modulation spaces on locally compact Abelian group, Technical Report. University Vienna pp. 1–57 (1983)Google Scholar
  5. 5.
    Ginibre, J., Ozawa, T., Velo, G.: On the existence of the wave operators for a class of nonlinear Schrödinger equations. Ann. Inst. H. Poincaré, Phys. Théor. 60, 211–239 (1994)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Hayashi, N., Kaikina, E.I., Naumkin, P.I.: On the scattering theory for the cubic nonlinear Schrödinger and Hartree type equations in one space dimensions. Hokkaido Math. J. 27, 651–667 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hayashi, N., Kato, K.: Analyticity in time and smoothing effect of solutions to nonlinear Schrödinger equations. Commun. Math. Phys. 184, 273–300 (1997)CrossRefGoogle Scholar
  8. 8.
    Hayashi, N., Kato, K., Naumkin, P.I.: On the scattering in Gevrey classes for the subcritical Hartree and Schrödinger equations. Ann. Scuola Norm. Sup. Pisa CL. Sci. 4, 483–497 (1998)zbMATHGoogle Scholar
  9. 9.
    Hayashi, N., Ozawa, T.: On the derivative nonlinear Schrödinger equation. Phys. D 55, 14–36 (1992)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hayashi, N., Saitoh, S.: Analyticity and smoothing effect for the Schrödinger equation. Ann. Inst. Henri Poincaré, Phys. Théor. 52, 163–173 (1990)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hayashi, N., Saitoh, S.: Analyticity and global existence of small solutions to some nonlinear Schrödinger equations. Commun. Math. Phys. 129, 27–41 (1990)CrossRefGoogle Scholar
  12. 12.
    Hoshino, G., Ozawa, T.: Analytic smoothing effect for nonlinear Schrödinger equations in two space dimensions. Osaka J. Math. 609, 609–618 (2014)zbMATHGoogle Scholar
  13. 13.
    Hoshino, G., Ozawa, T.: Analytic smoothing effect for nonlinear Schrödinger equations with quintic nonlinearity. J. Math. Anal. Appl. 419, 285–297 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Huang, C.: On the analyticity for the generalized quadratic derivative complex Ginzburg–Landau equation. Abstr. Appl. Anal. 2014, 607028 (2014)MathSciNetGoogle Scholar
  15. 15.
    Huang, C. and Wang, B.: Analyticity for the (generalized) Navier–Stokes equations with rough initial data. arXiv:1310.2141
  16. 16.
    Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Phys. Théor. 46, 113–129 (1987)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kato, T.: Solutions to nonlinear higher order Schrödinger equations with small initial data on modulation spaces. Adv. Differ. Equ. 21, 201–234 (2016)zbMATHGoogle Scholar
  18. 18.
    Keel, M., Tao, T.: End point Strichartz estimates. Am. J. Math. 120, 955–980 (1998)CrossRefGoogle Scholar
  19. 19.
    Linares, F., Ponce, G.: Introduction to Nonlinear Dispersive Equations, 2nd edn. Springer, New York (2015)zbMATHGoogle Scholar
  20. 20.
    Nakamitsu, K.: Analytic finite energy solutions of the nonlinear Schrödinger equation. Commun. Math. Phys. 260, 117–130 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Nakanishi, K., Ozawa, T.: Remarks on scattering for nonlinear Schrödinger equations. NoDEA Nonlinear Differ. Equ. Appl. 9, 45–68 (2002)CrossRefGoogle Scholar
  22. 22.
    Ozawa, T., Yamauchi, K.: Analytic smoothing effect for global solutions to nonlinear Schrödinger equation. J. Math. Anal. Appl. 364, 492–497 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Reich, M., Reissig, M., Sickel, W.: Non-analytic superposition results on modulation spaces with subexponential weights. J. Pseudo-Differ. Oper. Appl. 7, 365–409 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ruzhansky, M., Sugimoto, M., Wang, B.: Modulation spaces and nonlinear evolution equations. Prog. Math. 301, 267–283 (2012)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Strichartz, R.S.: Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equation. Duke Math. J. 44, 705–714 (1977)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse, vol. 139. Springer, New York (1999)zbMATHGoogle Scholar
  27. 27.
    Tsutsumi, Y.: \(L^2\)-solutions for nonlinear Schrödinger equations and nonlinear groups. Funkcial. Ekvac. 30, 115–125 (1987)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Yajima, K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110, 415–426 (1987)CrossRefGoogle Scholar
  29. 29.
    Wang, B., Zhao, L., Guo, B.: Isometric decomposition operators, function spaces \(E^{\lambda }_{p, q}\) and applications to nonlinear evolution equations. J. Funct. Anal. 233, 1–39 (2006)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wang, B., Han, L., Huang, C.: Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data. Ann. I. H. Poincaré 26, 225–2281 (2009)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Osaka UniversityOsakaJapan

Personalised recommendations