The cubic nonlinear Schrödinger equation in two dimensions with radial data

Journal of the European Mathematical Society

Volumn 11, Issue 6, 2009, Pages 1203-1258

  Killip, Rowan ^a   Tao, Terence ^a   Visan, Monica ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF CHICAGO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 74549167807     PISSN: 14359855     EISSN: None     Source Type: Journal    
DOI: 10.4171/JEMS/180     Document Type: Article

References (78)

1. 2-critical nonlinear Schrödinger equation. Trans. Amer. Math. Soc. 359, 5257-5282 (2007) Zbl pre05178492 MR 2327030
2. Berestycki, H., Lions, P.-L.: Existence d'ondes solitaires dans des problèmes nonlinéaires du type Klein-Gordon. C. R. Acad. Sci. Paris Sér. A-B 288, A395-A398 (1979) Zbl 0397.35024 MR 0552061
3. Blue, P., Colliander, J.: Global well-posedness in Sobolev space implies global existence for weighted L2 initial data for L2-critical NLS. Comm. Pure Appl. Anal. 5, 691-708 (2006) Zbl 1137.35066 MR 2246003
4. Bourgain, J.: On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE. Int. Math. Res. Notices 1996, no.6, 277-304 Zbl 0934.35166 MR 1386079
5. Bourgain, J.: Refinements of Strichartz' inequality and applications to 2d-NLS with critical nonlinearity. Int. Math. Res. Notices 1998, no.5, 253-284 Zbl 0917.35126 MR 1616917
6. Bourgain, J.: Global well-posedness of defocusing 3D critical NLS in the radial case. J. Amer. Math. Soc. 12, 145-171 (1999) Zbl 0958.35126 MR 1626257
7. Bourgain, J.: Global solutions of nonlinear Schrödinger equations. Amer. Math. Soc. Colloq. Publ. 46, Amer. Math. Soc., Providence, RI (1999) Zbl 0933.35178 MR 1691575
8. Carles, R.: Critical nonlinear Schrödinger equations with and without harmonic potential. Math. Models Methods Appl. Sci. 12, 1513-1523 (2002) Zbl 1029.35208 MR 1933935
9. 2-critical case. Trans. Amer. Math. Soc. 359, 33-62 (2007) Zbl 1115.35119 MR 2247881
10. Cazenave, T., Weissler, F. B.: Some remarks on the nonlinear Schrödinger equation in the subcritical case. In: New Methods and Results in Nonlinear Field Equations (Bielefeld, 1987), Lecture Notes in Phys. 347, Springer, Berlin, 59-69 (1989) Zbl 0699.35217 MR 1030060
11. Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Math. 10, Amer. Math. Soc. (2003) Zbl 1055.35003 MR 2002047
12. Christ, M., Kiselev, A.: Maximal functions associated to filtrations. J. Funct. Anal. 179, 409-425 (2001) Zbl 0974.47025 MR 1809116
13. Colliander, J., Delort, J., Kenig, C., Staffilani, G.: Bilinear estimates and applications to 2D NLS. Trans. Amer. Math. Soc. 353, 3307-3325 (2001) Zbl 0970.35142 MR 1828607 (Pubitemid 33766675)
14. 2. Int. Math. Res. Notices 2007, no.23, art. ID rnm 090, 30 pp. Zbl 1142.35085 MR 2377216
15. Colliander, J., Keel, M., Staffilani, G., Takaoka, H., Tao, T.: Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation. Math. Res. Lett. 9, 659-682 (2002) Zbl 1152.35491 MR 1906069
16. 3. Ann. of Math. 167, 767-865 (2008) Zbl pre05578705 MR 2415387
17. 2, Discrete Contin. Dynam. Systems 21, 665-686 (2008) Zbl 1147.35095 MR 2399431
18. 1. Math. Res. Lett. 12, 357-375 (2005) Zbl 1084.35088 MR 2150890
19. 2-critical nonlinear Schrödinger equation in higher dimensions. Comm. Pure Appl. Anal. 6, 1023-1041 (2007) Zbl 1152.35106 MR 2341818
20. 2-critical nonlinear Schrödinger equation in &Rdbl;. Comm. Partial Differential Equations 33, 1395-1429 (2008) Zbl 1155.35310 MR 2450163
21. Dysthe, K. B.: Modelling a "rogue wave"-speculations or a realistic possibility? In: Rogue Waves 2000, Éditions Ifremer, Plouzané, 255-264 (2001)
22. Erdos, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate. Comm. Pure Appl. Math. 59, 1659-1741 (2006) Zbl 1122.82018 MR 2257859
23. Glassey, R. T.: On the blowing up of solution to the Cauchy problem for nonlinear Schrödinger operators. J. Math. Phys. 8, 1794-1797 (1977) Zbl 0372.35009 MR 0460850
24. Ginibre, J., Velo, G.: Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J. Math. Pures Appl. 64, 363-401 (1985) Zbl 0535.35069 MR 0839728
25. Ginibre, J., Velo, G.: Smoothing properties and retarded estimates for some dispersive evolution equations. Comm. Math. Phys. 144, 163-188 (1992) Zbl 0762.35008 MR 1151250
26. 2+1. J. Hyperbolic Differential Equations 4, 233-257 (2007) Zbl 1122.35132 MR 2329384
27. Gross, E. P.: Hydrodynamics of a superfluid condensate. J. Math. Phys. 4, 195-207 (1963)
28. Hayashi, N., Tsutsumi, Y.: Remarks on the scattering problem for nonlinear Schrödinger equations. In: Differential Equations and Mathematical Physics (Birmingham, AL, 1986), Lecture Notes in Math. 1285, Springer, Berlin, 162-168 (1987) Zbl 0633.35059 MR 0921265
29. Henderson, K. L., Peregrine, D. H., Dold, J. W.: Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation. Wave Motion 29, 341-361 (1999) Zbl 1074.76513 MR 1686798
30. Kato, T.: On nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 46, 113-129 (1987) Zbl 0632.35038 MR 0877998
31. s-solutions and unconditional wellposedness. J. Anal. Math. 67, 281-306 (1995) Zbl 0848.35124 MR 1383498
32. Keel, M., Tao, T.: Endpoint Strichartz estimates. Amer. J. Math. 120, 955-980 (1998) Zbl 0922.35028 MR 1646048
33. Kelley, P. L.: Self-focusing of optical beams. Phys. Rev. Lett. 15, 1005-1008 (1965)
34. Kenig, C.: Global well-posedness and scattering for the energy critical focusing nonlinear Schrödinger and wave equations. Lectures given at "Analyse des équations aux dérivées partielles", Evian-les-Bains (2007)
35. Kenig, C., Merle, F.: Global well-posedness, scattering, and blowup for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math. 166, 645-675 (2006) Zbl 1115.35125 MR 2257393
36. Keraani, S.: On the defect of compactness for the Strichartz estimates of the Schrödinger equations. J. Differential Equations 175, 353-392 (2001) Zbl 1038.35119 MR 1855973
37. Keraani, S.: On the blow-up phenomenon of the critical nonlinear Schrödinger equation. J. Funct. Anal. 235, 171-192 (2006) Zbl 1099.35132 MR 2216444
38. n. Arch. Ration. Mech. Anal. 105, 243-266 (1989) Zbl 0676.35032 MR 0969899
39. Merle, F.: Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equation with critical power. Duke Math. J. 69, 427-453 (1993) Zbl 0808.35141 MR 1203233
40. Merle, F.: Existence of blow-up solutions in the energy space for the critical generalized KdV equation. J. Amer. Math. Soc. 14, 555-578 (2001) Zbl 0970.35128 MR 1824989
41. Merle, F., Raphael, P.: Sharp upper bound on the blow-up rate for the critical nonlinear Schrödinger equation. Geom. Funct. Anal. 13, 591-642 (2003) Zbl 1061.35135 MR 1995801
42. 2 critical nonlinear Schrödinger equation. Invent. Math. 156, 565-672 (2004) Zbl 1067.35110 MR 2061329
43. Merle, F., Raphael, P.: The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation. Ann. of Math. 161, 157-222 (2005) Zbl pre02204253 MR 2150386
44. 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity. J. Differential Equations 84, 205-214 (1990) Zbl 0722.35047 MR 1047566
45. 2 solutions of the critical nonlinear Schrödinger equation in 2D. Int. Math. Res. Notices 1998, no.8, 399-425 Zbl 0913.35126 MR 1628235
46. Montgomery-Smith, S. J.: Time decay for the bounded mean oscillation of solutions of the Schrödinger and wave equations. Duke Math. J. 91, 393-408 (1998) Zbl 0955.35012 MR 1600602
47. Morawetz, C.: Time decay for the nonlinear Klein-Gordon equation. Proc. Roy. Soc. A 306, 291-296 (1968) Zbl 0157.41502 MR 0234136
48. Nakanishi, K.: Energy scattering for non-linear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2. J. Funct. Anal. 169, 201-225 (1999) Zbl 0942.35159 MR 1726753
49. Nawa, H.: Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power. Comm. Pure Appl. Math. 52, 193-270 (1999) Zbl 0964.37014 MR 1653454
50. Perelman, G.: On the formation of singularities in solutions of the critical nonlinear Schrödinger equation. Ann. H. Poincaré 2, 605-673 (2001) Zbl 1007.35087 MR 1852922 (Pubitemid 33570417)
51. Pitaevskii, L. P.: Vortex lines in an imperfect Bose gas. Sov. Phys. JETP 13, 451-454 (1961)
52. 1+4. Amer. J. Math. 129, 1-60 (2007) Zbl 1160.35067 MR 2288737
53. Shao, S.: Sharp linear and bilinear restriction estimates for the paraboloid in the cylindrically symmetric case. Preprint
54. Staffilani, G.: Quadratic forms for a 2D semilinear Schrödinger equation. Duke Math. J. 86, 79-107 (1997) Zbl 0872.35104 MR 1427846
55. Staffilani, G.: On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations. Duke Math. J. 86, 109-142 (1997) Zbl 0874.35114 MR 1427847
56. Stefanov, A.: Strichartz estimates for the Schrödinger equation with radial data. Proc. Amer. Math. Soc. 129, 1395-1401 (2001) Zbl 1009.35016 MR 1814165 (Pubitemid 33766189)
57. Stein, E. M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press (1993) Zbl 0821.42001 MR 1232192
58. Stein, E. M.,Weiss, G.: Fractional integrals on n-dimensional Euclidean space. J. Math. Mech. 7, 503-514 (1958) Zbl 0082.27201 MR 0098285
59. Strichartz, R. S.: Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations. Duke Math. J. 44, 705-774 (1977) Zbl 0372.35001 MR 0512086
60. Sulem, C., Sulem, P.-L.: The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse. Springer, New York (1999) Zbl 0928.35157
61. Talanov, V. I.: Self modelling wave beams in a nonlinear dielectric. Radiophysics and Quantum Electronics 9, 260-261 (1967)
62. Tao, T.: Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation. Comm. Partial Differential Equations 25, 1471-1485 (2000) Zbl 0966.35027 MR 1765155
63. Tao, T.: On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation. Dynam. Partial Differential Equations 1, 1-48 (2004) Zbl 1082.35144 MR 2091393
64. Tao, T.: Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrödinger equation for radial data. New York J. Math. 11, 57-80 (2005) Zbl 1119.35092 MR 2154347
65. Tao, T.: Nonlinear Dispersive Equations. Local and Global Analysis. CBMS Reg. Conf. Ser. Math. 106, Amer. Math. Soc., Providence, RI (2006) Zbl 1106.35001 MR 2233925
66. Tao, T.: A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations. Dynam. Partial Differential Equations 4, 1-53 (2007) Zbl 1142.35088 MR 2304091
67. Tao, T.: A pseudoconformal compactification of the nonlinear Schrödinger equation and applications. math.AP/0606254
68. Tao, T., Visan, M.: Stability of energy-critical nonlinear Schrödinger equations in high dimensions. Electron. J. Differential Equations 2005, no.118, 28 pp. Zbl pre05002687 MR 2174550
69. Tao, T., Visan, M., Zhang, X.: The nonlinear Schrödinger equation with combined power-type nonlinearities. Comm. Partial Differential Equations 32, 1281-1343 (2007) Zbl pre05204438 MR 2354495
70. Tao, T., Visan, M., Zhang, X.: Minimal-mass blowup solutions of the mass-critical NLS. Forum Math. 20, 881-919 (2008) Zbl 1154.35085 MR 2445122
71. Tao, T., Visan, M., Zhang, X.: Global well-posedness and scattering for the defocusing masscritical nonlinear Schrödinger equation for radial data in high dimensions. Duke Math. J. 140, 165-202 (2007) Zbl pre05208403 MR 2355070
72. Tsutsumi, Y.: Scattering problem for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 43, 321-347 (1985) Zbl 0612.35104 MR 0824843
73. Tzirakis, N.: The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension. Differential Integral Equations 18, 947-960 (2005) MR 2150447
74. Visan, M.: The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions. Duke Math. J. 138, 281-374 (2007) Zbl 1131.35081 MR 2318286
75. Visan, M.: The defocusing energy-critical nonlinear Schrödinger equation in dimensions five and higher. Ph.D. Thesis, UCLA (2006)
76. 2-critical focusing nonlinear Schrödinger equation in higher dimensions below the energy class. SIAM J. Math. Anal. 39, 34-56 (2007) Zbl 1135.35079 MR 2318374
77. Weinstein, M.: Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87, 567-576 (1983) Zbl 0527.35023 MR 0691044
78. Weinstein, M.: The nonlinear Schrödinger equation-singularity formation, stability and dispersion. In: The Connection between Infinite-Dimensional and Finite-Dimensional Dynamical Systems (Boulder, CO, 1987), Contemp. Math. 99, Amer. Math. Soc., Providence, RI, 213-232 (1989) Zbl 0703.35159 MR 1034501