## Analysis & PDE

• Anal. PDE
• Volume 5, Number 4 (2012), 855-885.

### Global well-posedness and scattering for the defocusing quintic NLS in three dimensions

#### Abstract

We revisit the proof of global well-posedness and scattering for the defocusing energy-critical NLS in three space dimensions in light of recent developments. This result was obtained previously by Colliander, Keel, Staffilani, Takaoka, and Tao.

#### Article information

Source
Anal. PDE, Volume 5, Number 4 (2012), 855-885.

Dates
Revised: 12 May 2011
Accepted: 11 June 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731247

Digital Object Identifier
doi:10.2140/apde.2012.5.855

Mathematical Reviews number (MathSciNet)
MR3006644

Zentralblatt MATH identifier
1264.35219

#### Citation

Killip, Rowan; Vişan, Monica. Global well-posedness and scattering for the defocusing quintic NLS in three dimensions. Anal. PDE 5 (2012), no. 4, 855--885. doi:10.2140/apde.2012.5.855. https://projecteuclid.org/euclid.apde/1513731247

#### References

• J. Bourgain, “Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case”, J. Amer. Math. Soc. 12:1 (1999), 145–171.
• J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb R\sp 3$”, Comm. Pure Appl. Math. 57:8 (2004), 987–1014.
• J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in $\mathbb R\sp 3$”, Ann. of Math. $(2)$ 167:3 (2008), 767–865.
• B. Dodson, “Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d = 1$”, preprint, 2011.
• B. Dodson, “Global well-posedness and scattering for the defocusing, $L^{2}$-critical, nonlinear Schrödinger equation when $d = 2$”, preprint, 2011.
• B. Dodson, “Global well-posedness and scattering for the defocusing, $L\sp {2}$-critical nonlinear Schrödinger equation when $d\geq3$”, J. Amer. Math. Soc. 25:2 (2012), 429–463.
• J. Ginibre and G. Velo, “Smoothing properties and retarded estimates for some dispersive evolution equations”, Comm. Math. Phys. 144:1 (1992), 163–188.
• M. Keel and T. Tao, “Endpoint Strichartz estimates”, Amer. J. Math. 120:5 (1998), 955–980. http://www.ams.org/mathscinet-getitem?mr=2000d:35018MR 2000d:35018
• 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.
• S. Keraani, “On the blow up phenomenon of the critical nonlinear Schrödinger equation”, J. Funct. Anal. 235:1 (2006), 171–192.
• R. Killip and M. Vi\commaaccentsan, “Nonlinear Schrödinger equations at critical regularity”, preprint, Clay Mathematics Institute Summer School, Zürich, 2008, http://www.claymath.org/programs/summer_school/2008/visan.pdf.
• R. Killip and M. Vi\commaaccentsan, “The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher”, Amer. J. Math. 132:2 (2010), 361–424.
• R. Killip, T. Tao, and M. Vi\commaaccentsan, “The cubic nonlinear Schrödinger equation in two dimensions with radial data”, J. Eur. Math. Soc. $($JEMS$)$ 11:6 (2009), 1203–1258.
• J. E. Lin and W. A. Strauss, “Decay and scattering of solutions of a nonlinear Schrödinger equation”, J. Funct. Anal. 30:2 (1978), 245–263.
• C. S. Morawetz, Notes on time decay and scattering for some hyperbolic problems, Regional Conference Series in Applied Mathematics 19, Society for Industrial and Applied Mathematics, Philadelphia, 1975.
• K. Nakanishi, “Scattering theory for the nonlinear Klein–Gordon equation with Sobolev critical power”, Internat. Math. Res. Notices 1999:1 (1999), 31–60.
• E. Ryckman and M. Vi\commaaccentsan, “Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in $\mathbb R\sp {1+4}$”, Amer. J. Math. 129:1 (2007), 1–60.
• 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.
• 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.
• T. Tao, M. Vi\commaaccentsan, 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.
• T. Tao, M. Vi\commaaccentsan, and X. Zhang, “Minimal-mass blowup solutions of the mass-critical NLS”, Forum Math. 20:5 (2008), 881–919.
• M. Vi\commaaccentsan, “The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions”, Duke Math. J. 138:2 (2007), 281–374.
• M. Vi\commaaccentsan, “Global well-posedness and scattering for the defocusing cubic nonlinear Schrödinger equation in four dimensions”, Internat. Math. Res. Notices 2012:5 (2012), 1037–1067.
• A. Zygmund, Trigonometric series, I, II, 3rd ed., Cambridge University Press, Cambridge, 2002. http://www.ams.org/mathscinet-getitem?mr=2004h:01041MR 2004h:01041