Global well-posedness and scattering for the mass-critical NLS
Journées équations aux dérivées partielles (2011), article no. 4, 11 p.
@article{JEDP_2011____A4_0,
author = {Dodson, Benjamin},
title = {Global well-posedness and scattering for the mass-critical {NLS}},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {4},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2011},
doi = {10.5802/jedp.76},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jedp.76/}
}
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Dodson, Benjamin. Global well-posedness and scattering for the mass-critical NLS. Journées équations aux dérivées partielles (2011), article  no. 4, 11 p. doi : 10.5802/jedp.76. http://www.numdam.org/articles/10.5802/jedp.76/

[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. | MR 1209299 | Zbl 0787.35097

[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. | MR 1215780 | Zbl 0787.35098

[3] J. Bourgain. “Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity.” International Mathematical Research Notices, 5 (1998):253 – 283. | MR 1616917 | Zbl 0917.35126

[4] J. Bourgain. “Global Solutions of Nonlinear Schrödinger Equations” American Mathematical Society Colloquium Publications, 1999. | MR 1691575 | Zbl 0933.35178

[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. | MR 552061 | Zbl 0397.35024

[6] T. Cazenave and F. B. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in ${H}^{1}$, Manuscripta Math., 61 (1988), 477–494. | MR 952091 | Zbl 0696.35153

[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. | MR 1055532 | Zbl 0706.35127

[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. | Zbl 1142.35085

[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 | MR 2527809 | Zbl 1185.35250

[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. | MR 1906069 | Zbl 1152.35491

[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 | MR 2053757 | Zbl 1060.35131

[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. | MR 2399431 | Zbl 1147.35095

[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 | MR 2415387 | Zbl 1178.35345

[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, | MR 2754279

[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, | MR 2869023

[16] B. Dodson, Global well - posedness and scattering for the defocusing ${L}^{2}$ - critical nonlinear Schrödinger equation when $d=1$, preprint, arXiv:1010.0040v2,

[17] B. Dodson, Global well - posedness and scattering for the defocusing ${L}^{2}$ - critical nonlinear Schrödinger equation when $d=2$, preprint, arXiv:1006.1375v2,

[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,

[19] P. Germain, N. Masmoudi, and J. Shatah, Global solutions for 2D quadratic Schrödinger equations, preprint, arXiv:1001.5158v1,

[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 Linéaire 26 (2009): 3, 917–941. | Numdam | MR 2526409 | Zbl 1169.35372

[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 Mathematicae 166 (2006): 3, 645–675. | MR 2257393 | Zbl 1115.35125

[22] C. Kenig and F. Merle “Scattering for ${\stackrel{˙}{H}}^{1/2}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions,” Transactions of the American Mathematical Society 362 (2010): 4, 1937 – 1962. | MR 2574882 | Zbl 1188.35180

[23] M. Keel and T. Tao “Endpoint Strichartz Estimates” American Journal of Mathematics 120 (1998): 4 - 6, 945 – 957. | MR 1646048 | Zbl 0922.35028

[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. | Zbl 1187.35237

[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.

[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 | MR 2472890 | Zbl 1171.35111

[27] H. Koch and D. Tataru “Dispersive estimates for principally normal pseudodifferential operators” Communications on Pure and Applied Mathematics 58 no. 2 (2005): 217 - 284 | MR 2094851 | Zbl 1078.35143

[28] H. Koch and D. Tataru “A priori bounds for the 1D cubic NLS in negative Sobolev spaces” Int. Math. Res. Not. IMRN 16 (2007): Art. ID rnm053, 36. | MR 2353092 | Zbl 1169.35055

[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,

[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. | MR 969899 | Zbl 0676.35032

[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. | MR 1741843 | Zbl 1008.35070

[32] F. Planchon and L. Vega “Bilinear virial identities and applications” Annales Scientifiques de l’École Normale Supérieure 42, no. 2 (2009): 261 - 290. | Numdam | MR 2518079 | Zbl 1192.35166

[33] T. Tao, “Nonlinear Dispersive Equations," Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006. | MR 2233925 | Zbl 1106.35001

[34] T. Tao and A. Vargas, A bilinear approach to cone multipliers. I. Restriction estimates, Geom. Funct. Anal., 10 no. 1 (2000), 185–215. | MR 1748920 | Zbl 0949.42012

[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. | MR 2354495 | Zbl 1187.35245

[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. | MR 2445122 | Zbl 1154.35085

[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. | MR 2355070 | Zbl 1187.35246

[38] M. E. Taylor, “Pseudodifferential Operators and Nonlinear PDE," Birkhäuser, Boston, 1991. | MR 1121019 | Zbl 0746.35062

[39] M. E. Taylor, “Partial Differential Equations I - III," Springer-Verlag, New York, 1996. | MR 1395148 | Zbl 1206.35004

[40] M. E. Taylor “Short time behavior of solutions to nonlinear Schrödinger equations in one and two space dimensions" Comm. Partial Differential Equations 31 (2006): 955 - 980. | MR 2233047 | Zbl 1106.35104

[41] M. E. Taylor, “Tools for PDE" American Mathematical Society, Mathematical Surveys and Monographs 31 Providence, RI, 2000. | MR 1766415 | Zbl 0963.35211

[42] M. Visan “The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions" Duke Mathematical Journal 138 (2007): 281 - 374. | MR 2318286 | Zbl 1131.35081

[43] M. Weinstein, “Nonlinear Schrödinger equations and sharp interpolation estimates" Communications in Mathematical Physics 87 no. 4 (1982/83): 567 - 576. | MR 691044 | Zbl 0527.35023

[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. | MR 1034501 | Zbl 0703.35159

[45] K. Yosida, “Functional Analysis" Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Volume 123, 6th Edition Springer - Verlag, Berlin, 1980. | MR 617913 | Zbl 0435.46002

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