Japanese Journal of Mathematics

, Volume 6, Issue 2, pp 121–141 | Cite as

Critical non-linear dispersive equations: global existence, scattering, blow-up and universal profiles

Special Feature: The Takagi Lectures
  • 216 Downloads

Abstract

We discuss recent progress in the understanding of the global behavior of solutions to critical non-linear dispersive equations. The emphasis is on global existence, scattering and finite time blow-up. For solutions that are bounded in the critical norm, but which blow-up in finite time, we also discuss the issue of universal profiles at the blow-up time.

Keywords and Phrases

critical non-linear dispersive equation global existence scattering finite time blow-up universal profiles 

Mathematics Subject Classification (2010)

35L52 35Q53 35Q55 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BG.
    Bahouri H., Gérard P.: High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121, 131–175 (1999)CrossRefMATHGoogle Scholar
  2. BS.
    Bahouri H., Shatah J.: Decay estimates for the critical semilinear wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 15, 783–789 (1998)CrossRefMATHMathSciNetGoogle Scholar
  3. B1.
    Bourgain J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3, 107–156 (1993)CrossRefMATHMathSciNetGoogle Scholar
  4. B2.
    Bourgain J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3, 209–262 (1993)CrossRefMATHMathSciNetGoogle Scholar
  5. B3.
    Bourgain J.: Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc., 12, 145–171 (1999)CrossRefMATHMathSciNetGoogle Scholar
  6. BC.
    Brezis H., Coron J.-M.: Convergence of solutions of H-systems or how to blow bubbles, Arch. Rational Mech. Anal., 89, 21–56 (1985)CrossRefMATHMathSciNetGoogle Scholar
  7. CW.
    CazenaveT. Weissler F.B.: The Cauchy problem for the critical nonlinear Schrödinger equation in H s, Nonlinear Anal.,14, 807–836 (1990)CrossRefMathSciNetGoogle Scholar
  8. C-TZ1.
    Christodoulou D., Tahvildar-Zadeh A.S.: On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math., 46, 1041–1091 (1993)CrossRefMATHMathSciNetGoogle Scholar
  9. C-TZ2.
    ChristodoulouD. Tahvildar-Zadeh A.S.: On the asymptotic behavior of spherically symmetric wave maps, Duke Math. J., 71, 31–69 (1993)Google Scholar
  10. CKSTT.
    Colliander J., Keel M., Staffilani G., Takaoka H., Tao T.: Global well-posedness and scattering for the energy-critical Shrödinger equation in \({\mathbb{R}^3}\) ,. Ann. of Math. (2) 167, 767–865 (2008)CrossRefMATHMathSciNetGoogle Scholar
  11. CKM.
    Côte R., KenigC. Merle F.: Scattering below critical energy for the radial 4D Yang–Mills equation and for the 2D corotational wave map system, Comm. Math. Phys., 284, 203–225 (2008)CrossRefMATHGoogle Scholar
  12. Di.
    Ding W.Y.: On a conformally invariant elliptic equation on \({\mathbb{R}^n}\) , Comm. Math. Phys., 107, 331–335 (1986)CrossRefMATHGoogle Scholar
  13. D1.
    B. Dodson, Global well-posedness and scattering for the defocusing, L 2-critical, nonlinear Schrödinger equation when d ≥  3, preprint, 2009, arXiv:0912.2467.Google Scholar
  14. D2.
    B. Dodson, Global well-posedness and scattering for the defocusing, L 2-critical, nonlinear Schrödinger equation when d =  2, preprint, 2010, arXiv:1006.1375.Google Scholar
  15. D3.
    B. Dodson, Global well-posedness and scattering for the defocusing, L 2-critical, nonlinear Schrödinger equation when d =  1, preprint, 2010, arXiv:1010.0040.Google Scholar
  16. DM1.
    T. Duyckaerts and F. Merle, Dynamics of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP, 2008 (2008), rpn002, 67 p.Google Scholar
  17. DM2.
    Duyckaerts T., Merle F.: Dynamics of threshold solutions for energy-critical NLS, Geom. Funct. Anal., 18, 1787–1840 (2009)CrossRefMathSciNetGoogle Scholar
  18. DKM1.
    Duyckaerts T., KenigC. Merle F.: Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation, J. Eur. Math. Soc. (JEMS), 13, 533–599 (2011)CrossRefMATHMathSciNetGoogle Scholar
  19. DKM2.
    T. Duyckaerts, C. Kenig and F. Merle, Universality of the blow-up profile for small type II blow-up solutions of the energy-critical wave equation: the non-radial case, J. Eur. Math. Soc. (JEMS), to appear.Google Scholar
  20. ESS.
    L. Escauriaza, G.A. Serëgin and V. Šverák, L 3,∞-solutions of Navier–Stokes equations and backward uniqueness. (Russian), Uspekhi Mat. Nauk, 58 (2003), no. 2, 3–44; translation in Russian Math. Surveys, 58 (2003), no. 2, 211–250.Google Scholar
  21. GKP.
    I. Gallagher, G.S. Koch and F. Planchon, A profile decomposition approach to the \({{L^{\infty}_t}(L^{3}_{x})}\) Navier–Stokes regularity criterion, preprint, 2010, arXiv:1012.0145.Google Scholar
  22. GNN.
    B. Gidas, W.M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in \({\mathbb{R}^n}\) , In: Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7, Academic Press, New York, 1981, pp. 369–402.Google Scholar
  23. GV1.
    GinibreJ. Velo G.: On a class of nonlinear Schrödinger equations, J. Funct. Anal., 32, 1–71 (1979)CrossRefMathSciNetGoogle Scholar
  24. GV2.
    Ginibre J., Velo G.: Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133, 50–68 (1995)CrossRefMATHMathSciNetGoogle Scholar
  25. GM.
    L. Glangetas and F. Merle, A geometrical approach of existence of blow-up solutions in H 1 for nonlinear Schrödinger equation, Publications du Laboratoire d’Analyse Numerique, Univ. Pierre et Marie Curie, CNRS, December 1995.Google Scholar
  26. G.
    Glassey R.T.: On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18, 1794–1797 (1977)CrossRefMATHMathSciNetGoogle Scholar
  27. Gr.
    Grillakis M.G.: Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity. Ann. of Math. (2) 132, 485–509 (1990)CrossRefMATHMathSciNetGoogle Scholar
  28. Kap.
    Kapitanski L.: Global and unique weak solutions of nonlinear wave equations, Math. Res. Lett., 1, 211–223 (1994)MATHMathSciNetGoogle Scholar
  29. Ka1.
    T. Kato, On the Cauchy problem for the (generalized) Korteweg–de Vries equation, In: Studies in Applied Mathematics, (ed. V. Guillemin), Adv. Math. Suppl. Stud., 8, Academic Press, 1983, pp. 93–128.Google Scholar
  30. Ka2.
    Kato T.: On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46, 113–129 (1987)MATHGoogle Scholar
  31. K1.
    C. Kenig, Global well-posedness and scattering for the energy critical focusing nonlinear Schrödinger and wave equations, lecture notes for a mini course given at “Analyse des equations aux derivées partialles”, Evian-les-Bains, 2007, available at http://www.math.uchicago.edu/~cek.
  32. K2.
    C. Kenig, The concentration-compactness/rigidity theorem method for critical dispersive and wave equations, preprint, 2008, lectures for a course given at CRM, Bellaterra, Spain, May 2008, available at http://www.math.uchicago.edu/~cek.
  33. K3.
    C. Kenig, Recent developments on the global behavior to critical nonlinear dispersive equations, Proc. of the ICM, Hyderabad, India, 2010, to appear.Google Scholar
  34. KK.
    C. Kenig and G.S. Koch, An alternative approach to regularity for the Navier–Stokes equation in critical spaces, preprint, 2009, arXiv:0908.3349; Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear.Google Scholar
  35. KM1.
    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)MATHMathSciNetGoogle Scholar
  36. KM2.
    Kenig C., Merle F.: Global well-posedness, scatering and blow-up for the energy-critical focusing non-linear wave equation,. Acta Math. 201, 147–212 (2008)CrossRefMATHMathSciNetGoogle Scholar
  37. KM3.
    Kenig C., Merle F.: Scattering for 1/2 bounded solutions to the cubic defocusing NLS in 3 dimensions, Trans. Amer. Math. Soc., 362, 1937–1962 (2010)CrossRefMATHMathSciNetGoogle Scholar
  38. KM4.
    C. Kenig and F. Merle, Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications, preprint, 2008, arXiv:0810.4834v2; Amer. J. Math., to appear.Google Scholar
  39. KM5.
    C. Kenig and F. Merle, Radial solutions to energy supercritical wave equations in all odd dimensions, in preparation.Google Scholar
  40. KPV1.
    Kenig C., Ponce G., Vega L.: Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle. Comm. Pure Appl. Math. 46, 527–620 (1993)CrossRefMATHMathSciNetGoogle Scholar
  41. KPV2.
    Kenig C., Ponce G., Vega L.: Small solutions to nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Non Linéaire 10, 255–288 (1993)MATHMathSciNetGoogle Scholar
  42. KV1.
    R. Killip and M. Vişan, The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, preprint, 2008, arXiv:0804.1018[math.AP].Google Scholar
  43. KV2.
    R. Killip and M. Vişan, Energy-supercritical NLS: critical s-bounds imply scattering, preprint, 2008, arXiv:0812.2084.Google Scholar
  44. KV3.
    R. Killip and M. Vişan, The defocusing energy-supercritical nonlinear wave equation in three space dimensions, preprint, 2010, arXiv:1001.1761.Google Scholar
  45. KV4.
    R. Killip and M. Vişan, The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions, preprint, 2010, arXiv:1002.1756.Google Scholar
  46. KTV.
    R. Killip, T. Tao and M. Vişan, The cubic nonlinear Shrödinger equation in two dimensions with radial data, preprint, 2007, arXiv:0707.3188[math.AP]; J. Eur. Math. Soc. (JEMS), to appear.Google Scholar
  47. KVZ.
    Killip R., Vişan M., Zhang X.: The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher. Anal. PDE 1, 229–266 (2008)CrossRefMATHMathSciNetGoogle Scholar
  48. KlM.
    Klainerman S., Machedon M.: Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46, 1221–1268 (1993)CrossRefMATHMathSciNetGoogle Scholar
  49. Kr.
    Krieger J.: regularity of wave maps from \({\mathbb{R}^{2+1}}\) to H 2. Small energy. Comm. Math. Phys. 250, 507–580 (2004)MATHMathSciNetGoogle Scholar
  50. KrS.
    Krieger J., Schlag W. Concentration compactness for critical wave maps, preprint, 2009, arXiv:0908.2474.Google Scholar
  51. KST1.
    Krieger J., Schlag W., Tataru D.: Renormalization and blow up for charge one equivariant critical wave maps. Invent. Math. 171, 543–615 (2008)CrossRefMATHMathSciNetGoogle Scholar
  52. KST2.
    Krieger J., Schlag W., Tataru D.: Slow blow-up solutions for the \({H^1(\mathbb{R}^3)}\) critical focusing semilinear wave equation. Duke Math. J. 147, 1–53 (2009)CrossRefMATHMathSciNetGoogle Scholar
  53. Le.
    Levine H.A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form \({Pu_{tt}=-Au+\fancyscript{F}(u)}\) . Trans. Amer. Math. Soc. 192, 1–21 (1974)MATHMathSciNetGoogle Scholar
  54. L.
    Lions P.-L.: The concentration-compactness principle in the calculus of variations. The limit case. II. Rev. Mat. Iberoamericana 1, 45–121 (1985)MATHGoogle Scholar
  55. MM1.
    Martel Y., Merle F.: A Liouville theorem for the critical generalized Korteweg–de Vries equation. J. Math. Pures Appl. (9) 79, 339–425 (2000)MATHMathSciNetGoogle Scholar
  56. MM2.
    Martel Y., Merle F.: Stability of blow-up profile and lower bounds for the blow-up rate for the critical generalized KdV equation. Ann. of Math. (2) 155, 235–280 (2002)CrossRefMATHMathSciNetGoogle Scholar
  57. MR1.
    Merle F., Raphaël P.: On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation. Invent. Math. 156, 565–672 (2004)CrossRefMATHMathSciNetGoogle Scholar
  58. MR2.
    Merle F., Raphaël P.: Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation. Comm. Math. Phys. 253, 675–704 (2005)CrossRefMATHMathSciNetGoogle Scholar
  59. MV.
    Merle F., Vega L.: Compactness at blow-up time for L 2 solutions of the critical nonlinear Schrödinger equation in 2D. Internat. Math. Res. Notices 1998, 399–425 (1998)CrossRefMATHMathSciNetGoogle Scholar
  60. P.
    Pecher H.: Nonlinear small data scattering for the wave and Klein–Gordon equation. Math. Z. 185, 261–270 (1984)CrossRefMATHMathSciNetGoogle Scholar
  61. RR.
    P. Raphaël and I. Rodnianski, Stable blow up dynamics for the critical co-rotational Wave Maps and equivariant Yang–Mills problems, preprint, 2009, arXiv:0911.0692.Google Scholar
  62. RS.
    I. Rodnianski and J. Sterbenz, On the formation of singularities in the critical O(3) σ-model, preprint, 2008, arXiv:math/0605023; Ann. of Math. (2), to appear.Google Scholar
  63. RV.
    Ryckman E., Vişan M.: Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in \({\mathbb{R}^{1+4}}\) , Amer. J. Math. 129, 1–60 (2007)MATHGoogle Scholar
  64. S.
    R. Schoen, Recent progress in geometric partial differential equations, In: Proceedings of ICM, Berkeley, Calif., 1986, 1, Amer. Math. Soc., Providence, RI, 1987, pp. 121–130.Google Scholar
  65. Se.
    Segal I.: Space-time decay for solutions of wave equation. Advances in Math. 22, 305–311 (1976)CrossRefMATHMathSciNetGoogle Scholar
  66. SS1.
    Shatah J., Struwe M.: Well-posedness in the energy space for semilinear wave equations with critical growth. Internat. Math. Res. Notices 1994, 303–309 (1994)CrossRefMATHMathSciNetGoogle Scholar
  67. SS2.
    J. Shatah and M. Struwe, Geometric Wave Equations, Courant Lect. Notes Math., 2, New York Univ., Courant Inst. Math. Sci., New York, Amer. Math. Soc., Providence, RI, 1998, viii+153 p.Google Scholar
  68. S-TZ.
    Shatah J., Tahvildar-Zadeh A.S.: On the Cauchy problem for equivariant wave maps. Comm. Pure Appl. Math. 47, 719–754 (1994)CrossRefMATHMathSciNetGoogle Scholar
  69. ST1.
    J. Sterbenz and D. Tataru, Regularity of Wave-maps in dimension 2 + 1, preprint, 2009, arXiv:0907.3148.Google Scholar
  70. ST2.
    J. Sterbenz and D. Tataru, Energy dispersed large data wave maps in 2 + 1 dimensions, preprint, 2009, arXiv:0906.3384.Google Scholar
  71. Str.
    Strichartz R.S.: Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44, 705–714 (1977)CrossRefMATHMathSciNetGoogle Scholar
  72. St1.
    Struwe M.: Globally regular solutions to the u 5 Klein–Gordon equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15, 495–513 (1988)MATHMathSciNetGoogle Scholar
  73. St2.
    M. Struwe, Radially symmetric wave maps from (1 + 2)-dimensional Minkowski space to the sphere, Math. Z., 242 (2002), 407–414.Google Scholar
  74. St3.
    Struwe M.: Equivariant wave maps in two space dimensions. Dedicated to the memory of Jürgen K. Moser. Comm. Pure Appl. Math. 56, 815–823 (2003)MATHMathSciNetGoogle Scholar
  75. Tal.
    Talenti G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4) 110, 353–372 (1976)CrossRefMATHMathSciNetGoogle Scholar
  76. T1.
    Tao T.: Global regularity of wave maps. I. Small critical Sobolev norm in high dimension. Internat. Math. Res. Notices 2001, 299–328 (2001)CrossRefMATHGoogle Scholar
  77. T2.
    Tao T.: Global regularity of wave maps. II. Small energy in two dimensions. Comm. Math. Phys. 224, 443–544 (2001)CrossRefMATHMathSciNetGoogle Scholar
  78. T3.
    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) (electronic)MATHMathSciNetGoogle Scholar
  79. T4.
    T. Tao, Global regularity of wave maps III. Large energy from \({\mathbb{R}^{1+2}}\) to hyperbolic spaces, preprint, 2008, arXiv:0805.4666v1[math.AP].Google Scholar
  80. T5.
    T. Tao, Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class, preprint, 2008, arXiv:0806.3592[math.AP].Google Scholar
  81. TVZ.
    Tao T., Vişan M., Zhang X.: Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions. Duke Math. J. 140, 165–202 (2007)CrossRefMATHMathSciNetGoogle Scholar
  82. Tat1.
    Tataru D.: Local and global results for wave maps. I. Comm. Partial Differential Equations 23, 1781–1793 (1998)CrossRefMATHMathSciNetGoogle Scholar
  83. Tat2.
    Tataru D.: On global existence and scattering for the wave maps equation. Amer. J. Math. 123, 37–77 (2001)CrossRefMATHMathSciNetGoogle Scholar
  84. Ts.
    Tsutsumi Y.: L 2-solutions for nonlinear Schrödinger equations and nonlinear groups. Funkcial. Ekvac. 30, 115–125 (1987)MATHMathSciNetGoogle Scholar
  85. V.
    Vişan M.: The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. Duke Math. J. 138, 281–374 (2007)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© The Mathematical Society of Japan and Springer Japan 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

Personalised recommendations