Chinese Annals of Mathematics, Series B

, Volume 38, Issue 2, pp 579–590 | Cite as

Asymptotics and blow-up for mass critical nonlinear dispersive equations

Article
  • 69 Downloads

Abstract

The author considers mass critical nonlinear Schrödinger and Korteweg-de Vries equations. A review on results related to the blow-up of solution of these equations is given.

Keywords

Dispersive nonlinear PDE Criticality Asymptotics Blow-up Global solution Soliton 

2000 MR Subject Classification

35B40 35B44 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Berestycki, H. and Lions, P.-L., Nonlinear scalar field equations I: Existence of a ground state, Arch. Rational Mech. Anal., 82, 1983, 313–345.MathSciNetMATHGoogle Scholar
  2. [2]
    Bourgain, J., Global well-posedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Am. Math. Soc., 12, 1999, 145–171.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    Bourgain, J. and Wang, W., Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity, Ann. S. Nor. Pisa, 25, 1998, 197–215.MATHGoogle Scholar
  4. [4]
    Brezis, H. and Coron, J. M., Convergence of solutions of H-systems or how to blow bubbles, Arch. Rational Mech. Anal., 89, 1985, 21–56.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Cazenave, T. and Weissler, F., Some remarks on the nonlinear Schrödinger equation in the critical case, Nonlinear semigroups, partial differential equations and attractors, 18–29, Lecture Notes in Math., 1394, Springer, Berlin, 1989.CrossRefGoogle Scholar
  6. [6]
    Colliander, J., Keel, M., Staffilani, G., et al., Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3, Ann. of Math., 167, 2008, 767–865.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Fibich, G., Merle, F. and Raphaël, P., Proof of a spectral property related to the singularity formation for the critical NLS, Phys. D, 220, 2006, 1–13.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    Ginibre, J. and Velo, G., Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133, 1995, 50–68.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Glangetas, L. and Merle, F., A Geometrical Approach of Existence of Blow-up Solution in H1 for Nonlinear Schrödinger Equations, Publications du Laboratoire d’Analyse Numérique, Université Pierre et Marie Curie, 1995.Google Scholar
  10. [10]
    Glassey, R., On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18, 1977, 1794–1797.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    Kato, T., On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46, 1987, 113–129.MathSciNetMATHGoogle Scholar
  12. [12]
    Kenig, C., Recent developments on the global behavior to critical nonlinear dispersive equations, Proceedings of the International Congress of Mathematicians, Volume I, 326–338, Hindustan Book Agency, New Delhi, 2010.Google Scholar
  13. [13]
    Kenig, C., Ponce, G. and Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46, 1993, 527–620.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Killip, R., Tao, T. and Visan, M., The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc., 11, 2009, 1203–1258.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Killip, R. and Visan, M., The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math., 132, 2010, 361–424.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    Krieger, J., Nakanishi, K. and Schlag, W., Global dynamics away from the ground state for the energycritical nonlinear wave equation, Math. Z., 272, 2012, 297–316.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Krieger, J. and Schlag, W., Non-generic blow-up solutions for the critical focusing NLS in 1-D, Jour. Eur. Math. Soc., 11, 2009, 1–125.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    Krieger, J., Schlag, W. and Tataru, D., Renormalization and blow-up for charge one equivariant critical wave maps, Invent. Math., 171, 2008, 543–615.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Krieger, J., Schlag, W. and Tataru, D., Slow blow-up solutions for the H1(R3) critical focusing semilinear wave equation, Duke Math. J., 147, 2009, 1–53.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    Landman, M. J., Papanicolaou, G. C., Sulem, C. and Sulem, P.-L., Rate of blowup for solutions of the nonlinear Schrödinger equation at critical dimension, Phys. Rev. A, 38, 1988, 3837–3843.MathSciNetCrossRefGoogle Scholar
  21. [21]
    Lions, P.-L., The concentration-compactness principle in the calculus of variations: The limit case I and II, Rev. Mat. Ibero., 1, 1985, 45–121 and 145–201.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    Martel, Y. and Merle, F., A Liouville theorem for the critical generalized Korteweg–de Vries equation, J. Math. Pures Appl., 79, 2000, 339–425.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    Martel, Y. and Merle, F., Instability of solitons for the critical generalized Korteweg–de Vries equation, Geom. Funct. Anal., 11, 2001, 74–123.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    Martel, Y. and Merle, F., Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation, Ann. of Math., 155, 2002, 235–280.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    Martel, Y. and Merle, F., Blow-up in finite time and dynamics of blow-up solutions for the L2-critical generalized KdV equation, J. Amer. Math. Soc., 15, 2002, 617–664.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    Martel, Y. and Merle, F., Nonexistence of blow-up solution with minimal L2-mass for the critical gKdV equation, Duke Math. J., 115, 2002, 385–408.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    Martel, Y., Merle, F. and Raphaël, P., Blow-up for critical gKdV equation I: Dynamics near the soliton, Acta Math., to appear. arXiv: 1204.4625Google Scholar
  28. [28]
    Martel, Y., Merle, F. and Raphaël, P., Blow-up for critical gKdV equation II: Minimal mass solution, J.E.M.S., to appear.Google Scholar
  29. [29]
    Martel, Y., Merle, F. and Raphaël, P., Blow-up for critical gKdV equation III: Exotic regimes, Annali Scuola Norm. Sup. di Pisa, to appear. arXiv:1209.2510Google Scholar
  30. [30]
    Merle, F., Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power, Duke Math. J., 69, 1993, 427–454.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    Merle, F., Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity, Comm. Math. Phys., 129, 1990, 223–240.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    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, 1992, 203–254.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    Merle, F., Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc., 14, 2001, 555–578.MathSciNetCrossRefMATHGoogle Scholar
  34. [34]
    Merle, F., Asymptotics for critical nonlinear dispersive equations, Proceedings of the International Congress of Mathematicians, 2014, to appear.Google Scholar
  35. [35]
    Merle, F. and Raphaël, P., Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Geom. Func. Anal., 13, 2003, 591–642.CrossRefMATHGoogle Scholar
  36. [36]
    Merle, F. and Raphaël, P., On universality of blow-up profile for L2 critical nonlinear Schrödinger equation, Invent. Math., 156, 2004, 565–672.MathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    Merle, F. and Raphaël, P., The blow-up dynamics and upper bound on the blow-up rate for the critical nonlinear Schrödinger equation, Ann. of Math., 161, 2005, 157–222.MathSciNetCrossRefMATHGoogle Scholar
  38. [38]
    Merle, F. and Raphaël, P., Profiles and quantization of the blow-up mass for critical nonlinear Schrödinger equation, Commun. Math. Phys., 253, 2005, 675–704.CrossRefMATHGoogle Scholar
  39. [39]
    Merle, F. and Raphaël, P., On a sharp lower bound on the blow-up rate for the L2 critical nonlinear Schrödinger equation, J. Amer. Math. Soc., 19, 2006, 37–90.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    Merle, F., Raphaël, P. and Szeftel, J., The instability of Bourgain-Wang solutions for the L2 critical NLS, Amer. Jour. Math., 135, 2013, 967–1017.CrossRefMATHGoogle Scholar
  41. [41]
    Merle, F., Raphaël, P. and Rodnianski, I., Blow-up dynamics for smooth data equivariant solutions to the energy critical Schrödinger map problem, Invent. Math., 193, 2013, 249–365.MathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    Merle, F., Raphaël, P. and Rodnianski, I., Type IIblow up for the energy supercritical NLS, preprint.Google Scholar
  43. [43]
    Nakanishi, K. and Schlag, W., Global dynamics above the ground state energy for the cubic NLS equation in 3D, Arch. Ration. Mech. Anal., 203, 2012, 809–851.MathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    Perelman, G., On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré, 2, 2001, 605–673.MathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    Raphaël, P., Stability of the log-log bound for blow-up solutions to the critical nonlinear Schrödinger equation, Math. Ann., 331, 2005, 577–609.MathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    Raphaël, P., Blow up bubbles in Hamiltonian evolution equations: A quantitative approach, Proceedings of the International Congress of Mathematicians, 2014, to appear.Google Scholar
  47. [47]
    Raphaël, P. and Rodnianski, I., Stable blow-up dynamics for the critical co-rotational wave maps and equivariant Yang-Mills problems, Publ. Math. Inst. Hautes Etudes Sci., 115, 2012, 1–122.MathSciNetCrossRefMATHGoogle Scholar
  48. [48]
    Rodnianski, I. and Sterbenz, J., On the formation of singularities in the critical O(3) s-model, Ann. of Math., 172, 2010, 187–242.MathSciNetCrossRefMATHGoogle Scholar
  49. [49]
    Sterbenz, J. and Tataru, D., Regularity of wave-maps in dimension 2+1, Comm. Math. Phys., 298, 2010, 139–230.MathSciNetCrossRefMATHGoogle Scholar
  50. [50]
    Tao, T., Visan, M. and Zhang, X., Minimal-mass blowup solutions of the mass-critical NLS, Forum Math., 20, 2008, 881–919.MathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    Weinstein, M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87, 1983, 567–576.CrossRefMATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Université de Cergy-Pontoise, Mathématiques, CNRSCergy-PontoiseFrance
  2. 2.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance

Personalised recommendations