GLOBAL EXISTENCE AND UNIQUENESS RESULTS FOR WEAK SOLUTIONS OF THE FOCUSING MASS-CRITICAL NONLINEAR SCHRÖDINGER EQUATION

Analysis and PDE

Volumn 2, Issue 1, 2009, Pages 61-81

  Tao, Terence ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Nonlinear schrodinger equation; Strichartz estimates; Unconditional uniqueness; Weak solutions

Indexed keywords

EID: 79959252167     PISSN: 21575045     EISSN: 1948206X     Source Type: Journal    
DOI: 10.2140/apde.2009.2.61     Document Type: Article

References (29)

1. [Bourgain 1999] J. Bourgain, New global well-posedness results for non-linear Schrödinger equations, AMS Publications, 1999.
2. [Bourgain and Wang 1997] J. Bourgain and W. Wang, “Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25:1-2 (1997), 197–215 (1998). Dedicated to Ennio De Giorgi. MR 99m:35219 Zbl 1043.35137
3. [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
4. [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), Lecture Notes in Phys. 347, Springer, Berlin, 1989. MR 91c:35136 Zbl 0699.35217
5. [Fibich et al. 2006] G. Fibich, F. Merle, and P. Raphaël, “Proof of a spectral property related to the singularity formation for the L2 critical nonlinear Schrödinger equation”, Phys. D 220:1 (2006), 1–13. MR 2007d:35254 Zbl 1100.35097
6. [Furioli and Terraneo 2003a] G. Furioli and E. Terraneo, “Besov spaces and unconditional well-posedness for the nonlinear Schrödinger equation in H˙ s(B Rn)”, Commun. Contemp. Math. 5:3 (2003), 349–367. MR 2004m:35246 Zbl 1050.35102
7. [Furioli and Terraneo 2003b] G. Furioli and E. Terraneo, “Besov spaces and unconditional well-posedness for the nonlinear Schrödinger equation in H˙ s(B Rn)”, Commun. Contemp. Math. 5:3 (2003), 349–367. MR 2004m:35246 Zbl 1050.35102
8. [Glassey 1977] R. T. Glassey, “On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations”, J. Math. Phys. 18:9 (1977), 1794–1797. MR 57 #842 Zbl 0372.35009
9. [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
10. [Kato 1996] T. Kato, “Correction to: ‘On nonlinear Schrödinger equations. II. Hs-solutions and unconditional well-posedness’ ”, J. Anal. Math. 68 (1996), 305. MR 98a:35124b Zbl 0861.53043
11. [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
12. [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
13. [Killip et al. 2007] R. Killip, M. Visan, and X. Zhang, “The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher”, preprint, 2007, Available at arXiv:0708.0849.
14. [Killip et al. 2008a] R. Killip, D. Li, M. Visan, and X. Zhang, “The characterization of minimal-mass blowup solutions to the focusing mass-critical NLS”, preprint, 2008, Available at arXiv:0804.1124.
15. [Killip et al. 2008b] R. Killip, T. Tao, and M. Visan, “The cubic nonlinear Schrödinger equation in two dimensions with radial data”, preprint, 2008, Available at arXiv:0707.3188.
16. [Merle 1992] F. Merle, “On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlin- ear Schrödinger equation with critical exponent and critical mass”, Comm. Pure Appl. Math. 45:2 (1992), 203–254. MR 93e:35104 Zbl 0767.35084
17. [Merle and Raphaël 2005] F. Merle and P. Raphaël, “Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation”, Comm. Math. Phys. 253:3 (2005), 675–704. MR 2006m:35346 Zbl 1062.35137
18. [Merle and Raphaël 2008] F. Merle and P. Raphaël, “Blow up of the critical norm for some radial L2 super critical nonlinear Schrödinger equations”, Amer. J. Math. 130:4 (2008), 945–978. MR MR2427005
19. [Merle and Vega 1998] F. Merle and L. Vega, “Compactness at blow-up time for L2 solutions of the critical nonlinear Schrödinger equation in 2D”, Internat. Math. Res. Notices 8 (1998), 399–425. MR 99d:35156 Zbl 0913.35126
20. [Perelman 2003] G. Perelman, “Ricci flow with surgery on three-manifolds”, preprint, 2003, Available at arXiv:math.DG/ 0303109.
21. [Perelman 2007] G. Perelman, “The entropy formula for the Ricci flow and its geometric applications”, preprint, 2007, Avail- able at arXiv:math.DG/0211159v1.
22. [Scheffer 1993] V. Scheffer, “An inviscid flow with compact support in space-time”, J. Geom. Anal. 3:4 (1993), 343–401. MR 94h:35215 Zbl 0836.76017
23. [Shao 2009] S. Shao, “Sharp linear and bilinear restriction estimates for the paraboloid in the cylindrically symmetric case”, to appear in Rev. Mat. Iberoamericana.
24. [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
25. [Tao 2004] T. Tao, “On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation”, Dyn. Partial Differ. Equ. 1:1 (2004), 1–48. MR 2005j:35210 Zbl 1125.11341
26. [Tao 2006] T. Tao, “A pseudoconformal compactification of the nonlinear Schrödinger equation and applications”, preprint, 2006, Available at arXiv:math.AP/0606254.
27. [Tao et al. 2006] T. Tao, M. Visan, and X. Zhang, “Global well-posedness and scattering for the mass-critical nonlinear Schrödinger equation”, preprint, 2006, Available at arXiv:math.AP/0609692.
28. [Tao et al. 2008] T. Tao, M. Visan, and X. Zhang, “Minimal-mass blowup solutions of the mass-critical NLS”, Forum Math. 20:5 (2008), 881–919. MR MR2445122
29. [Tsutsumi 2007] Y. Tsutsumi, “Unconditional uniqueness of solution for the Cauchy problem of the nonlinear Schrödinger equation”, preprint, 2007, Available at http://tinyurl.com/c2gfkp.