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Journées équations aux dérivées partielles

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Benjamin Dodson
Global well-posedness and scattering for the mass-critical NLS
Journées équations aux dérivées partielles (2011), Exp. No. 4, 11 p., doi: 10.5802/jedp.76
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Bibliographie

[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, arXiv
[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 $\dot{H}^{1/2}$ 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. Article |  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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