Global well-posedness and scattering for the mass-critical NLS

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1. J. Bourgain “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations” Geom. Funct. Anal.3 (1993): 2, 107 – 156. Zbl0787.35097MR1209299
2. J. Bourgain “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation” Geom. Funct. Anal.3 (1993): 3, 209–262. Zbl0787.35098MR1215780
3. J. Bourgain. “Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity.” International Mathematical Research Notices, 5 (1998):253 – 283. Zbl0917.35126MR1616917
4. J. Bourgain. “Global Solutions of Nonlinear Schrödinger Equations” American Mathematical Society Colloquium Publications, 1999. Zbl0933.35178MR1691575
5. H. Berestycki and P.L. Lions, two authors Existence d’ondes solitaires dans des problèmes nonlinéaires du type Klein-Gordon, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences. Séries A et B, 288 no. 7 (1979), A395 - A398. Zbl0397.35024MR552061
6. T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$, Manuscripta Math., 61 (1988), 477–494. Zbl0696.35153MR952091
7. T. Cazenave and F. B. Weissler, two authors "The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$", Nonlinear Anal., 14 (1990), 807–836. Zbl0706.35127MR1055532
8. J. Colliander, M. Grillakis, and N. Tzirakis. “Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on ${\mathbf{R}}^2$.” International Mathematics Research Notices. IMRN, 23 (2007): 90 - 119. Zbl1142.35085
9. J. Colliander, M. Grillakis, and N. Tzirakis. “Tensor products and correlation estimates with applications to nonlinear Schrödinger equations” Communications on Pure and Applied Mathematics, 62 no. 7 (2009) : 920 - 968 Zbl1185.35250MR2527809
10. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. “Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation.” Mathematical Research Letters, 9 (2002):659 – 682. Zbl1152.35491MR1906069
11. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. “Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on ${\mathbf{R}}^3$” Communications on pure and applied mathematics, 21 (2004) : 987 - 1014 Zbl1060.35131MR2053757
12. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. “Resonant decompositions and the I-method for cubic nonlinear Schrödinger equation on ${\mathbf{R}}^2$.” Discrete and Continuous Dynamical Systems A, 21 (2007):665 – 686. Zbl1147.35095MR2399431
13. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao. “Global existence and scattering for the energy - critical nonlinear Schrödinger equation on ${\mathbf{R}}^3$” Annals of Mathematics. Second Series, 167 (2008) : 767 - 865 Zbl1178.35345MR2415387
14. J. Colliander and T. Roy, Bootstrapped Morawetz Estimates and Resonant Decomposition f or Low Regularity Global solutions of Cubic NLS on ${\mathbf{R}}^2$, preprint, arXiv:0811.1803, Zbl1252.35248MR2754279
15. B. Dodson, Global well - posedness and scattering for the defocusing $L^2$ - critical nonlinear Schrödinger equation when $d\ge 3$, preprint, arXiv:0912.2467v1, Zbl1236.35163MR2869023
16. B. Dodson, Global well - posedness and scattering for the defocusing $L^2$ - critical nonlinear Schrödinger equation when $d=1$, preprint, arXiv:1010.0040v2, Zbl06575383
17. B. Dodson, Global well - posedness and scattering for the defocusing $L^2$ - critical nonlinear Schrödinger equation when $d=2$, preprint, arXiv:1006.1375v2, Zbl1236.35163
18. B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, preprint, arXiv:1104.1114v2, Zbl1331.35316
19. P. Germain, N. Masmoudi, and J. Shatah, Global solutions for 2D quadratic Schrödinger equations, preprint, arXiv:1001.5158v1, Zbl1244.35134
20. M. Hadac and S. Herr and H. Koch “Well-posedness and scattering for the KP-II equation in a critical space” Ann. Inst. H. Poincaré Anal. Non Linéaire26 (2009): 3, 917–941. Zbl1169.35372MR2526409
21. C. Kenig and F. Merle “Global well-posedness, scattering, and blow-up for the energy-critical, focusing nonlinear Schrödinger equation in the radial case,” Inventiones Mathematicae166 (2006): 3, 645–675. Zbl1115.35125MR2257393
22. C. Kenig and F. Merle “Scattering for ${\dot{H}}^{1/2}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions,” Transactions of the American Mathematical Society362 (2010): 4, 1937 – 1962. Zbl1188.35180MR2574882
23. M. Keel and T. Tao “Endpoint Strichartz Estimates” American Journal of Mathematics120 (1998): 4 - 6, 945 – 957. Zbl0922.35028MR1646048
24. R. Killip, T. Tao, and M. Visan “The cubic nonlinear Schrödinger equation in two dimensions with radial data" Journal of the European Mathematical Society , to appear. Zbl1187.35237
25. R. Killip and M. Visan “Nonlinear Schrodinger Equations at Critical Regularity" Unpublished lecture notes , Clay Lecture Notes (2009): http://www.math.ucla.edu/ visan/lecturenotes.html. Zbl1298.35195
26. R. Killip, M. Visan, and X. Zhang “The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher" Annals in PDE , textbf1, no. 2 (2008) 229 - 266 Zbl1171.35111MR2472890
27. H. Koch and D. Tataru “Dispersive estimates for principally normal pseudodifferential operators” Communications on Pure and Applied Mathematics58 no. 2 (2005): 217 - 284 Zbl1078.35143MR2094851
28. H. Koch and D. Tataru “A priori bounds for the 1D cubic NLS in negative Sobolev spaces” Int. Math. Res. Not. IMRN16 (2007): Art. ID rnm053, 36. Zbl1169.35055MR2353092
29. H. Koch and D. Tataru, Energy and local energy bounds for the 1-D cubic NLS equation in $H^{-1/4}$, preprint, arXiv:1012.0148, Zbl1280.35137
30. M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in ${\mathbf{R}}^n$, Archive for Rational Mechanics and Analysis 105 no. 3 (1989), 243 - 266. Zbl0676.35032MR969899
31. T. Ozawa and Y. Tsutsumi, Space-time estimates for null gauge forms and nonlinear Schrödinger equations, Differential Integral Equations, 11 no. 2 (1998), 201–222. Zbl1008.35070MR1741843
32. F. Planchon and L. Vega “Bilinear virial identities and applications” Annales Scientifiques de l’École Normale Supérieure42, no. 2 (2009): 261 - 290. Zbl1192.35166MR2518079
33. T. Tao, “Nonlinear Dispersive Equations," Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006. Zbl1106.35001MR2233925
34. T. Tao and A. Vargas, A bilinear approach to cone multipliers. I. Restriction estimates, Geom. Funct. Anal., 10 no. 1 (2000), 185–215. Zbl0949.42012MR1748920
35. T. Tao, M. Visan, and X. Zhang. “The nonlinear Schrödinger equation with combined power-type nonlinearities.” Comm. Partial Differential Equations, 32 no. 7-9 (2007) :1281–1343. Zbl1187.35245MR2354495
36. T. Tao, M. Visan, and X. Zhang. “Minimal-mass blowup solutions of the mass-critical NLS.” Forum Mathematicum, 20 no. 5 (2008) : 881 - 919. Zbl1154.35085MR2445122
37. 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 Mathematical Journal, 140 no. 1 (2007) : 165 - 202. Zbl1187.35246MR2355070
38. M. E. Taylor, “Pseudodifferential Operators and Nonlinear PDE," Birkhäuser, Boston, 1991. Zbl0746.35062MR1121019
39. M. E. Taylor, “Partial Differential Equations I - III," Springer-Verlag, New York, 1996. Zbl1206.35004MR1395148
40. M. E. Taylor “Short time behavior of solutions to nonlinear Schrödinger equations in one and two space dimensions" Comm. Partial Differential Equations31 (2006): 955 - 980. Zbl1106.35104MR2233047
41. M. E. Taylor, “Tools for PDE" American Mathematical Society, Mathematical Surveys and Monographs31 Providence, RI, 2000. Zbl0963.35211MR1766415
42. M. Visan “The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions" Duke Mathematical Journal138 (2007): 281 - 374. Zbl1131.35081MR2318286
43. M. Weinstein, “Nonlinear Schrödinger equations and sharp interpolation estimates" Communications in Mathematical Physics87 no. 4 (1982/83): 567 - 576. Zbl0527.35023MR691044
44. M. Weinstein, “The nonlinear Schrödinger equation – singularity formation, stability and dispersion" The connection between infinite - dimensional and finite - dimensional dynamical systems (Boulder CO) 99 (1989): 213 - 232. Zbl0703.35159MR1034501
45. K. Yosida, “Functional Analysis" Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Volume 123, 6th Edition Springer - Verlag, Berlin, 1980. Zbl0435.46002MR617913