Skip to main content
SpringerLink
Log in

Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case

  • Published:
Inventiones mathematicae Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Aubin, T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl., IX. Sér. 55, 269–296 (1976)

    MATH  MathSciNet  Google Scholar 

  2. Bahouri, H., Gérard, P.: High frequency approximation of solutions to critical nonlinear wave equations. Am. J. Math. 121, 131–175 (1999)

    MATH  Google Scholar 

  3. Berestycki, H., Cazenave, T.: Instabilité des états stationnaires dans les équations de Schrödinger et de Klein–Gordon non linéaires. C. R. Acad. Sci., Paris, Sér. I, Math. 293, 489–492 (1981)

    MATH  MathSciNet  Google Scholar 

  4. Bergh, J., Lofstrom, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Berlin, New York: Springer 1976

  5. Bourgain, J.: Global well-posedness of defocusing critical nonlinear Schrödinger equation in the radial case. J. Am. Math. Soc. 12, 145–171 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bourgain, J.: New global well-posedness results for nonlinear Schrödinger equations. AMS Colloquium Publications, 46, 1999

  7. Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10. New York: New York University Courant Institute of Mathematical Sciences 2003

  8. Cazenave, T., Weissler, F.B.: The Cauchy problem for the critical nonlinear Schrödinger equation in H s. Nonlinear Anal., Theory Methods Appl. 14, 807–836 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Colliander, J., Keel, M., Staffilani, G., Takaoke, H., Tao, T.: Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ℝ3. To appear in Ann. Math.

  10. Gérard, P.: Description du défaut de compacité de l’injection de Sobolev. ESAIM Control Optim. Calc. Var. 3, 213–233 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gerard, P., Meyer, Y., Oru, F.: Inégalités de Sobolev précisées, Séminaire sur les Équations aux Dérivées Partielles, 1996–1997, Exp. No. IV, 11. École Polytech. 1997

  12. Glassey, R.T.: On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations. J. Math. Phys. 18, 1794–1797 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  13. Grillakis, M.G.: On nonlinear Schrödinger equations. Commun. Partial Differ. Equations 25, 1827–1844 (2000)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  15. Keraani, S.: On the defect of compactness for the Strichartz estimates of the Schrödinger equations. J. Differ. Equations 175, 353–392 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Keraani, S.: On the blow up phenomenon of the critical Schrödinger equation. J. Funct. Anal. 235, 171–192 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Merle, F.: Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power. Duke Math. J. 69, 427–454 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  18. Merle, F.: On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass. Comm. Pure Appl. Math. 45, 203–254 (1992)

    MATH  MathSciNet  Google Scholar 

  19. Merle, F.: Existence of blow-up solutions in the energy space for the critical generalized KdV equation. J. Am. Math. Soc. 14, 555–578 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. Merle, F., Tsutsumi, Y.: L 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity. J. Differ. Equations 84, 205–214 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  21. Merle, F., Vega, L.: Compactness at blow-up time for L 2 solutions of the critical nonlinear Schrödinger equation in 2D. Int. Math. Res. Not. 8, 399–425 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ogawa, T., Tsutsumi, Y.: Blow-up of H 1 solution for the nonlinear Schrödinger equation. J. Differ. Equations 92, 317–330 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  23. Raphael, P.: Existence and stability of a solution blowing-up on a sphere for a L 2 supercritical nonlinear Schrödinger equation. Duke Math. J. 134(2), 199–258 (2006)

    Article  Google Scholar 

  24. Ryckman, E., Visan, M.: Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in ℝ1+4. To appear in Amer. J. Math. Preprint, http://arkiv.org/abs/math.AP/0501462

  25. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  26. Tao, T.: Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data. New York J. Math. 11, 57–80 (2005)

    MATH  MathSciNet  Google Scholar 

  27. Tao, T., Visan, M.: Stability of energy-critical nonlinear Schrödinger equations in high dimensions. Electron. J. Differ. Equ. 118, 28 (2005)

    MATH  MathSciNet  Google Scholar 

  28. Visan, M.: The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. Preprint, http://arkiv.org/abs/math.AP/0508298

  29. Weinstein, M.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87, 567–576 (1982/83)

    Google Scholar 

  30. Zhang, J.: Sharp conditions of global existence for nonlinear Schrödinger and Klein–Gordon equations. Nonlinear Anal., Theory Methods Appl. 48, 191–207 (2002)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Chicago, Chicago, IL, 60637, USA

    Carlos E. Kenig

  2. Département de Mathématiques, Université de Cergy-Pontoise, Pontoise, Cergy-Pontoise, 95302, France

    Frank Merle

Authors
  1. Carlos E. Kenig
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frank Merle
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kenig, C., Merle, F. Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case . Invent. math. 166, 645–675 (2006). https://doi.org/10.1007/s00222-006-0011-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-006-0011-4

Keywords

Search

Navigation