THE MASS-CRITICAL NONLINEAR SCHRÖDINGER EQUATION WITH RADIAL DATA IN DIMENSIONS THREE AND HIGHER

Analysis and PDE

Volumn 1, Issue 2, 2008, Pages 229-266

  Killip, Rowan ^a   Visan, Monica ^b   Zhang, Xiaoyi ^c  

^a University of Chicago ^*  (United States)

^b UNIVERSITY OF CHICAGO   (United States)

^c INSTITUTE FOR ADVANCED STUDY   (United States)

Author keywords

Focusing; Mass critical; Nonlinear schr dinger equation

Indexed keywords

EID: 68349155323     PISSN: 21575045     EISSN: 1948206X     Source Type: Journal    
DOI: 10.2140/apde.2008.1.229     Document Type: Article

References (54)

1. [Bégout and Vargas 2007] P. Bégout and A. Vargas, “Mass concentration phenomena for the L2-critical nonlinear Schrödinger equation”, Trans. Amer. Math. Soc. 359:11 (2007), 5257–5282. MR 2008g:35190 Zbl 05178492
2. [Berestycki and Lions 1979] H. Berestycki and P.-L. Lions, “Existence d’ondes solitaires dans des problèmes nonlinéaires du type Klein–Gordon”, C. R. Acad. Sci. Paris Sér. A-B 288:7 (1979), A395–A398. MR 80i:35076 Zbl 0397.35024
3. [Blue and Colliander 2006] P. Blue and J. Colliander, “Global well-posedness in Sobolev space implies global existence for weighted L2 initial data for L2-critical NLS”, Commun. Pure Appl. Anal. 5:4 (2006), 691–708. MR 2007f:35265 Zbl 1137.35066
4. [Bourgain 1998] J. Bourgain, “Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity”, Internat. Math. Res. Not. 1998:5 (1998), 253–283. MR 99f:35184 Zbl 0917.35126
5. [Bourgain 1999a] J. Bourgain, Global solutions of nonlinear Schrödinger equations, American Mathematical Society Collo- quium Publications 46, American Mathematical Society, Providence, RI, 1999. MR 2000h:35147 Zbl 0933.35178
6. [Bourgain 1999b] J. Bourgain, “Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case”, J. Amer. Math. Soc. 12:1 (1999), 145–171. MR 99e:35208 Zbl 0958.35126
7. [Carles 2002] R. Carles, “Critical nonlinear Schrödinger equations with and without harmonic potential”, Math. Models Meth- ods Appl. Sci. 12:10 (2002), 1513–1523. MR 2004b:35298 Zbl 1029.35208
8. [Carles and Keraani 2007] R. Carles and S. Keraani, “On the role of quadratic oscillations in nonlinear Schrödinger equations. II. The L2-critical case”, Trans. Amer. Math. Soc. 359:1 (2007), 33–62. MR 2008a:35260 Zbl 1115.35119
9. [Cazenave 2003] T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics 10, New York Uni- versity Courant Institute of Mathematical Sciences, New York, 2003. MR 2004j:35266 Zbl 1055.35003
10. [Cazenave and Weissler 1989] T. Cazenave and F. B. Weissler, “Some remarks on the nonlinear Schrödinger equation in the subcritical case”, pp. 59–69 in New methods and results in nonlinear field equations (Bielefeld, 1987), edited by P. Blanchard et al., Lecture Notes in Phys. 347, Springer, Berlin, 1989. MR 91c:35136 Zbl 0699.35217
11. [Christ and Weinstein 1991] F. M. Christ and M. I. Weinstein, “Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation”, J. Funct. Anal. 100:1 (1991), 87–109. MR 92h:35203 Zbl 0743.35067
12. [Colliander et al. 2002] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation”, Math. Res. Lett. 9:5-6 (2002), 659–682. MR 2003j:35294 Zbl 01930988
13. [Colliander et al. 2005] J. Colliander, S. Raynor, C. Sulem, and J. D. Wright, “Ground state mass concentration in the L2- critical nonlinear Schrödinger equation below H 1”, Math. Res. Lett. 12:2-3 (2005), 357–375. MR 2006e:35300 Zbl 1084. 35088
14. [Colliander et al. 2007] J. Colliander, M. Grillakis, and N. Tzirakis, “Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on R2”, Int. Math. Res. Not. 2007:23 (2007), Art. ID rnm090. Zbl 1142.35085 arXiv math/0703606
15. [Colliander et al. 2008a] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3”, Ann. of Math..2/ 167:3 (2008), 767–865.
16. [Colliander et al. 2008b] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Resonant decompositions and the I - method for the cubic nonlinear Schrödinger equation on R2”, Discrete Contin. Dyn. Syst. 21:3 (2008), 665–686. MR 2399431 Zbl 05304859
17. [De Silva et al. 2007a] D. De Silva, N. Pavlović, G. Staffilani, and N. Tzirakis, “Global well-posedness and polynomial bounds for the defocusing L2-critical nonlinear Schrödinger equation in R”, preprint, 2007. arXiv math/0702707
18. [De Silva et al. 2007b] D. De Silva, N. Pavlović, G. Staffilani, and N. Tzirakis, “Global well-posedness for the L2 critical nonlinear Schrödinger equation in higher dimensions”, Commun. Pure Appl. Anal. 6:4 (2007), 1023–1041. MR 2341818 Zbl 05257774
19. [Fang and Grillakis 2007] Y. F. Fang and M. G. Grillakis, “On the global existence of rough solutions of the cubic defocusing Schrödinger equation in R2C1”, J. Hyperbolic Differ. Equ. 4:2 (2007), 233–257. MR 2008c:35303 Zbl 1122.35132
20. [Ginibre and Velo 1992] J. Ginibre and G. Velo, “Smoothing properties and retarded estimates for some dispersive evolution equations”, Comm. Math. Phys. 144:1 (1992), 163–188. MR 93a:35065 Zbl 0762.35008
21. [Gradshteyn and Ryzhik 2000] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed., Academic Press, San Diego, CA, 2000. Translated from the Russian. MR 2001c:00002 Zbl 0981.65001
22. [Kato 1995] T. Kato, “On nonlinear Schrödinger equations. II. Hs -solutions and unconditional well-posedness”, J. Anal. Math. 67 (1995), 281–306. MR 98a:35124a Zbl 0848.35124
23. [Keel and Tao 1998] M. Keel and T. Tao, “Endpoint Strichartz estimates”, Amer. J. Math. 120:5 (1998), 955–980. MR 2000d: 35018 Zbl 0922.35028
24. [Kenig and Merle 2006a] C. E. Kenig and F. Merle, “Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case”, Invent. Math. 166:3 (2006), 645–675. MR 2007g:35232 Zbl 1115.35125
25. [Kenig and Merle 2006b] C. E. Kenig and F. Merle, “Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation”, preprint, 2006. arXiv math/0610801
26. [Keraani 2001] S. Keraani, “On the defect of compactness for the Strichartz estimates of the Schrödinger equations”, J. Differ- ential Equations 175:2 (2001), 353–392. MR 2002j:35281 Zbl 1038.35119
27. [Keraani 2006] S. Keraani, “On the blow up phenomenon of the critical nonlinear Schrödinger equation”, J. Funct. Anal. 235:1 (2006), 171–192. MR 2007e:35260 Zbl 1099.35132
28. [Killip and Visan 2008] R. Killip and M. Visan, “The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher”, preprint, 2008. arXiv 0804.1018
29. [Killip et al. 2007] R. Killip, T. Tao, and M. Visan, “The cubic nonlinear Schrödinger equation in two dimensions with radial data”, preprint, 2007. arXiv 0707.3188
30. n”, Arch. Rational Mech. Anal. 105:3 (1989), 243–266. MR 90d:35015 Zbl 0676.35032
31. [Merle 1993] F. Merle, “Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power”, Duke Math. J. 69:2 (1993), 427–454. MR 94b:35262 Zbl 0808.35141
32. 2 concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity”, J. Differential Equations 84:2 (1990), 205–214. MR 91e:35194 Zbl 0722.35047
33. 2 solutions of the critical nonlinear Schrödinger equation in 2D”, Internat. Math. Res. Not. 1998:8 (1998), 399–425. MR 99d:35156 Zbl 0913.35126
34. [Nawa 1999] H. Nawa, “Asymptotic and limiting profiles of blowup solutions of the nonlinear Schrödinger equation with critical power”, Comm. Pure Appl. Math. 52:2 (1999), 193–270. MR 99m:35235 Zbl 0964.37014
35. 1+4”, Amer. J. Math. 129:1 (2007), 1–60. MR 2007k:35474
36. [Shao 2007] S. Shao, “Sharp linear and bilinear restriction estimates for paraboloid in the cylindrically symmetric case”, preprint, 2007. arXiv 0706.3759
37. [Staffilani 1997] G. Staffilani, “On the generalized Korteweg-de Vries-type equations”, Differential Integral Equations 10:4 (1997), 777–796. MR 2001a:35005 Zbl 0891.35135
38. [Stein 1993] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math- ematical Series 43, Princeton University Press, Princeton, NJ, 1993. MR 95c:42002 Zbl 0821.42001
39. [Stein and Weiss 1971] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathemat- ical Series 32, Princeton University Press, Princeton, N.J., 1971. MR 46 #4102 Zbl 0232.42007
40. [Strichartz 1977] R. S. Strichartz, “Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations”, Duke Math. J. 44:3 (1977), 705–714. MR 58 #23577 Zbl 0372.35001
41. [Tao 2005] T. Tao, “Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data”, New York J. Math. 11 (2005), 57–80. MR 2006e:35308 Zbl 1119.35092
42. [Tao 2006] T. Tao, “A pseudoconformal compactification of the nonlinear Schrödinger equation and applications”, preprint, 2006. arXiv math/0606254
43. [Tao 2008a] T. Tao, “Global regularity of wave maps III”, preprint, 2008. arXiv 0805.4666 [Tao 2008b] T. Tao, “Global regularity of wave maps IV”, preprint, 2008. arXiv 0806.3592
44. [Tao 2008c] T. Tao, “Global regularity of wave maps V”, preprint, 2008. arXiv 0808.0368
45. [Tao et al. 2006] T. Tao, M. Visan, and X. Zhang, “Minimal-mass blowup solutions of the mass-critical NLS”, preprint, 2006. To appear in Forum Math. arXiv math/0609690
46. [Tao et al. 2007] T. Tao, M. Visan, and X. Zhang, “Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions”, Duke Math. J. 140:1 (2007), 165–202. MR 2355070 Zbl 05208403
47. [Taylor 2000] M. E. Taylor, Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials, Mathematical Surveys and Monographs 81, American Mathematical Society, Providence, RI, 2000. MR 2001g:35004 Zbl 0963.35211
48. [Tsutsumi 1985] Y. Tsutsumi, “Scattering problem for nonlinear Schrödinger equations”, Ann. Inst. H. Poincaré Phys. Théor. 43:3 (1985), 321–347. MR 87g:35221 Zbl 0612.35104
49. [Tzirakis 2005] N. Tzirakis, “The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension”, Differ- ential Integral Equations 18:8 (2005), 947–960. MR 2006b:35306 Zbl 1075.35082
50. [Visan 2006] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, PhD Thesis, University of California, Los Angeles, 2006.
51. [Visan 2007] M. Visan, “The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions”, Duke Math. J. 138:2 (2007), 281–374. MR 2008f:35387 Zbl 1131.35081
52. [Visan and Zhang 2007] M. Visan and X. Zhang, “On the blowup for the L2-critical focusing nonlinear Schrödinger equation in higher dimensions below the energy class”, SIAM J. Math. Anal. 39:1 (2007), 34–56. MR 2008b:35284 Zbl 1135.35079
53. [Weinstein 1983] M. I. Weinstein, “Nonlinear Schrödinger equations and sharp interpolation estimates”, Comm. Math. Phys. 87:4 (1983), 567–576. MR 84d:35140 Zbl 0527.35023
54. [Weinstein 1989] M. I. Weinstein, “The nonlinear Schrödinger equation—singularity formation, stability and dispersion”, pp. 213–232 in The connection between infinite-dimensional and finite-dimensional dynamical systems (Boulder, CO, 1987), edited by B. Nicolaenko et al., Contemp. Math. 99, Amer. Math. Soc., Providence, RI, 1989. MR 90m:35188 Zbl 0703.35159