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

Partial Differential Equations and Applications volume 3, Article number: 14 (2022) Cite this article

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.

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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 

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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.

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Authors and Affiliations

  1. Division of Science, Tokyo Denki University, Hatoyama, Saitama, 350-0394, Japan

    Gaku Hoshino

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  1. Gaku Hoshino
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Correspondence to Gaku Hoshino.

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

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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

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  • 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