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Inventiones mathematicae

, Volume 166, Issue 3, pp 645–675 | Cite as

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

  • Carlos E. Kenig
  • Frank Merle
Article

Keywords

Cauchy Problem Sobolev Inequality Maximal Interval Strichartz Estimate Rigidity Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 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)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bahouri, H., Gérard, P.: High frequency approximation of solutions to critical nonlinear wave equations. Am. J. Math. 121, 131–175 (1999)zbMATHGoogle Scholar
  3. 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)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bergh, J., Lofstrom, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Berlin, New York: Springer 1976Google Scholar
  5. 5.
    Bourgain, J.: Global well-posedness of defocusing critical nonlinear Schrödinger equation in the radial case. J. Am. Math. Soc. 12, 145–171 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bourgain, J.: New global well-posedness results for nonlinear Schrödinger equations. AMS Colloquium Publications, 46, 1999Google Scholar
  7. 7.
    Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10. New York: New York University Courant Institute of Mathematical Sciences 2003Google Scholar
  8. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 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.Google Scholar
  10. 10.
    Gérard, P.: Description du défaut de compacité de l’injection de Sobolev. ESAIM Control Optim. Calc. Var. 3, 213–233 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 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. 1997Google Scholar
  12. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Grillakis, M.G.: On nonlinear Schrödinger equations. Commun. Partial Differ. Equations 25, 1827–1844 (2000)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Keel, M., Tao, T.: Endpoint Strichartz estimates. Am. J. Math. 120, 955–980 (1998)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Keraani, S.: On the defect of compactness for the Strichartz estimates of the Schrödinger equations. J. Differ. Equations 175, 353–392 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Keraani, S.: On the blow up phenomenon of the critical Schrödinger equation. J. Funct. Anal. 235, 171–192 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 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)zbMATHMathSciNetGoogle Scholar
  19. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Ogawa, T., Tsutsumi, Y.: Blow-up of H 1 solution for the nonlinear Schrödinger equation. J. Differ. Equations 92, 317–330 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 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)CrossRefGoogle Scholar
  24. 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/0501462Google Scholar
  25. 25.
    Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 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)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Tao, T., Visan, M.: Stability of energy-critical nonlinear Schrödinger equations in high dimensions. Electron. J. Differ. Equ. 118, 28 (2005)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Visan, M.: The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. Preprint, http://arkiv.org/abs/math.AP/0508298Google Scholar
  29. 29.
    Weinstein, M.: Nonlinear Schrödinger equations and sharp interpolation estimates. Commun. Math. Phys. 87, 567–576 (1982/83)Google Scholar
  30. 30.
    Zhang, J.: Sharp conditions of global existence for nonlinear Schrödinger and Klein–Gordon equations. Nonlinear Anal., Theory Methods Appl. 48, 191–207 (2002)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Carlos E. Kenig
    • 1
  • Frank Merle
    • 2
  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Département de MathématiquesUniversité de Cergy-PontoiseCergy-PontoiseFrance

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