Skip to main content

Space-time analytic smoothing effect for the cubic nonlinear Schrödinger equations without pseudo-conformal invariance

Abstract

We study the global Cauchy problem for the cubic nonlinear Schrödinger equations in space dimensions \(n\ge 1.\) In particular we prove the space-time analytic smoothing effect for the cubic nonlinear Schrödinger equations. If \(n=2\) then the cubic nonlinear Schrödinger equation is invariant under the pseudo-conformal transform and in this case the space-time analytic smoothing effect has been studied in Hoshino and Ozawa (Nonlinear Differ Equ Appl 23:3, 2016) by applying the generator of pseudo-conformal transform. Our main purpose of this study is to extend the result obtained in previous study.

This is a preview of subscription content, access via your institution.

References

  1. Cazenave, T.: Semilinear Schrödinger equations. In: Courant Lecture Notes in Mathematics, vol. 10. American Mathematics Society, New York (2003)

  2. Cazenave, T., Weissler, F.B.: The Cauchy problem for the critical nonlinear Schrödinger equationin $H^s$. Nonlinear Anal. 14, 807–836 (1990)

    MathSciNet  Article  Google Scholar 

  3. DeBouard, A.: Analytic solution to non elliptic nonlinear Schrödinger equations. J. Differ. Equ. 104, 196–213 (1993)

    Article  Google Scholar 

  4. Ginibre, J., Velo, G.: On a class of nonlinear Schrödinger equations. I: the Cauchy problem. J. Funct. Anal. 32, 1–32 (1979)

    Article  Google Scholar 

  5. Ginibre, J., Ozawa, T., Velo, G.: On the existence of the wave operators for a class nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré Phys. Théor. 60, 211–239 (1994)

    MathSciNet  MATH  Google Scholar 

  6. Grafakos, L., Oh, S.: The Kato-Ponce inequality. Commun. Partial Differ. Equ. 39, 1128–1157 (2014)

    MathSciNet  Article  Google Scholar 

  7. Hayashi, N., Kato, K.: Analyticity in time and smoothing effect of solutions to nonlinear Schrödinger equations. Commun. Math. Phys. 184, 273–300 (1997)

    Article  Google Scholar 

  8. Hayashi, N., Naumkin, P.I.: Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations. Am. J. Math. 120, 369–389 (1998)

    Article  Google Scholar 

  9. Hayashi, N., Ozawa, T.: Scattering theory in the weighted $L^2(\mathbb{R}^n)$ space for some Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 48, 17–37 (1988)

    MathSciNet  MATH  Google Scholar 

  10. Hayashi, N., Ozawa, T.: Smoothing effect for some Schrödinger eqations. J. Funct. Anal. 85, 307–348 (1989)

    MathSciNet  Article  Google Scholar 

  11. Hayashi, N., Saitoh, S.: Analyticity and smoothing effect for the Schrödinger equation. Ann. Inst. H. Poincaré Phys. Théor. 52, 163–173 (1990)

    MathSciNet  MATH  Google Scholar 

  12. Hayashi, N., Nakamitsu, K., Tsutsumi, M.: On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension. Math. Z. 192, 637–650 (1986)

    MathSciNet  Article  Google Scholar 

  13. Hoshino, G.: Space-time analytic smoothing effect for a system of nonlinear Schrödinger equations with non pseudo-conformally invariant interactions. Commun. Partial Differ. Equ. 42, 802–819 (2017)

    Article  Google Scholar 

  14. Hoshino, G.: Space-time Gevrey smoothing effect for the dissipative nonlinear Schrödinger equations. Nonlinear Differ. Equ. Appl. 27(32) (2020)

  15. Hoshino, G.: Space-time analytic smoothing effect for the nonlinear Schrödinger equations with nonlinearity of exponential type (submitted)

  16. Hoshino, G., Ozawa, T.: Analytic smoothing effect for nonlinear Schrödinger equation in two space dimensions. Osaka J. Math. 51, 609–618 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Hoshino, G., Ozawa, T.: Analytic smoothing effect for nonlinear Schrödinger equation with quintic nonlinearity. J. Math. Anal. Appl. 419, 285–297 (2014)

    MathSciNet  Article  Google Scholar 

  18. Hoshino, G., Ozawa, T.: Space-time analytic smoothing effect for the pseudo-conformally invariant Schrödinger equations. Nonlinear Differ. Equ. Appl. 23(3) (2016)

  19. Kato, T.: On nonlinear Schrödinger equations. II. $H^s$-solutions and unconditional well-posedness. J. Anal. Math. 67, 281–306 (1995)

    MathSciNet  Article  Google Scholar 

  20. Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier Stokes equations. Commun. Pure App. Math. 41, 891–907 (1988)

    MathSciNet  Article  Google Scholar 

  21. Keel, M., Tao, T.: Endpoint Strichartz inequalities. Am. J. Math. 120, 955–980 (1998)

    Article  Google Scholar 

  22. Linares, F., Ponce, G.: Introduction to Nonlinear Dispersive Equations, 2nd edn. Springer, New York (2015)

    Book  Google Scholar 

  23. Nakamitsu, K.: Analytic finite energy solutions of the nonlinear Schrödinger equation. Commun. Math. Phys. 260, 117–130 (2005)

    MathSciNet  Article  Google Scholar 

  24. Nakamura, M., Ozawa, T.: Low energy scattering for nonlinear Schrödinger equations in fractional order Sobolev spaces. Rev. Math. Phys. 9, 397–410 (1997)

    MathSciNet  Article  Google Scholar 

  25. Ozawa, T., Yamauchi, K.: Analytic smoothing effect for global solutions to nonlinear Schrödinger equation. J. Math. Anal. Appl. 364, 492–497 (2010)

    MathSciNet  Article  Google Scholar 

  26. Tsutsumi, Y.: $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups. Funkc. Ekvac. 30, 115–125 (1987)

    MATH  Google Scholar 

  27. Yajima, K.: Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys. 110, 415–426 (1987)

    Article  Google Scholar 

Download references

Acknowledgements

The author would like to thank anonymous referees for their valuable comments and suggestions. This work was supported by JSPS KAKENHI Grant Number 19K14570.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gaku Hoshino.

Additional information

This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hoshino, G. Space-time analytic smoothing effect for the cubic nonlinear Schrödinger equations without pseudo-conformal invariance. Partial Differ. Equ. Appl. 3, 14 (2022). https://doi.org/10.1007/s42985-022-00151-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s42985-022-00151-w

Keywords

  • Nonlinear Schrödinger equations
  • Space-time analytic smoothing effect
  • Pseudo-conformal power

Mathematics Subject Classification

  • 35Q55