## Analysis & PDE

• Anal. PDE
• Volume 1, Number 2 (2008), 229-266.

### The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher

#### Abstract

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation $iut+Δu=±|u|4∕du$ for large spherically symmetric $Lx2(ℝd)$ initial data in dimensions $d≥3$. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.

#### Article information

Source
Anal. PDE, Volume 1, Number 2 (2008), 229-266.

Dates
Revised: 20 August 2008
Accepted: 23 September 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513797961

Digital Object Identifier
doi:10.2140/apde.2008.1.229

Mathematical Reviews number (MathSciNet)
MR2472890

Zentralblatt MATH identifier
1171.35111

#### Citation

Killip, Rowan; Visan, Monica; Zhang, Xiaoyi. The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher. Anal. PDE 1 (2008), no. 2, 229--266. doi:10.2140/apde.2008.1.229. https://projecteuclid.org/euclid.apde/1513797961

#### References

• \bibmarginparIs this the paper you cited? P. Bégout and A. Vargas, “Mass concentration phenomena for the $L\sp 2$-critical nonlinear Schrödinger equation”, Trans. Amer. Math. Soc. 359:11 (2007), 5257–5282.
• H. Berestycki and P.-L. Lions, “Existence d'ondes solitaires dans des problèmes nonlinéaires du type Klein–Gordon”, C. R. Acad. Sci. Paris Sér. A-B 288:7 (1979), A395–A398.
• P. Blue and J. Colliander, “Global well-posedness in Sobolev space implies global existence for weighted $L\sp 2$ initial data for $L\sp 2$-critical NLS”, Commun. Pure Appl. Anal. 5:4 (2006), 691–708.
• J. Bourgain, “Refinements of Strichartz' inequality and applications to $2$D-NLS with critical nonlinearity”, Internat. Math. Res. Notices 5 (1998), 253–283.
• J. Bourgain, Global solutions of nonlinear Schrödinger equations, American Mathematical Society Colloquium Publications 46, American Mathematical Society, Providence, RI, 1999.
• J. Bourgain, “Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case”, J. Amer. Math. Soc. 12:1 (1999), 145–171.
• R. Carles, “Critical nonlinear Schrödinger equations with and without harmonic potential”, Math. Models Methods Appl. Sci. 12:10 (2002), 1513–1523.
• R. Carles and S. Keraani, “On the role of quadratic oscillations in nonlinear Schrödinger equations. II. The $L\sp 2$-critical case”, Trans. Amer. Math. Soc. 359:1 (2007), 33–62.
• T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics 10, New York University Courant Institute of Mathematical Sciences, New York, 2003.
• 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), edited by P. Blanchard et al., Lecture Notes in Phys. 347, Springer, Berlin, 1989.
• F. M. Christ and M. I. Weinstein, “Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation”, J. Funct. Anal. 100:1 (1991), 87–109.
• \bibmarginparpage numbers corrected J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation”, Math. Res. Lett. 9:5-6 (2002), 659–682.
• J. Colliander, S. Raynor, C. Sulem, and J. D. Wright, “Ground state mass concentration in the $L\sp 2$-critical nonlinear Schrödinger equation below $H\sp 1$”, Math. Res. Lett. 12:2-3 (2005), 357–375. http://www.emis.de/cgi-bin/MATH-item?1084.35088Zbl 1084.35088
• \bibmarginparIs this the paper you cited? J. Colliander, M. Grillakis, and N. Tzirakis, “Improved interaction Morawetz inequalities for the cubic nonlinear Schrödinger equation on $\mathbb R\sp 2$”, Int. Math. Res. Not. IMRN 23 (2007), Art. ID rnm090.
• 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.
• J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb R\sp 2$”, Discrete Contin. Dyn. Syst. 21:3 (2008), 665–686.
• \bibmarginparupdated publishing info? D. De Silva, N. Pavlović, G. Staffilani, and N. Tzirakis, “Global well-posedness and polynomial bounds for the defocusing $L^{2}$-critical nonlinear Schrödinger equation in $\R$”, preprint, 2007.
• D. De Silva, N. Pavlović, G. Staffilani, and N. Tzirakis, “Global well-posedness for the $L\sp 2$ critical nonlinear Schrödinger equation in higher dimensions”, Commun. Pure Appl. Anal. 6:4 (2007), 1023–1041.
• \bibmarginparIs this the paper you cited? Y. F. Fang and M. G. Grillakis, “On the global existence of rough solutions of the cubic defocusing Schrödinger equation in $\mathbf R\sp {2+1}$”, J. Hyperbolic Differ. Equ. 4:2 (2007), 233–257.
• J. Ginibre and G. Velo, “Smoothing properties and retarded estimates for some dispersive evolution equations”, Comm. Math. Phys. 144:1 (1992), 163–188.
• I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, 6th ed., Academic Press, San Diego, CA, 2000. Translated from the Russian.
• T. Kato, “On nonlinear Schrödinger equations. II. $H\sp s$-solutions and unconditional well-posedness”, J. Anal. Math. 67 (1995), 281–306.
• 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.
• \bibmarginparupdated publishing info? C. E. Kenig and F. Merle, “Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation”, preprint, 2006.
• S. Keraani, “On the defect of compactness for the Strichartz estimates of the Schrödinger equations”, J. Differential Equations 175:2 (2001), 353–392.
• 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. Visan, “The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher”, preprint, 2008.
• R. Killip, T. Tao, and M. Visan, “The cubic nonlinear Schrödinger equation in two dimensions with radial data”, preprint, 2007.
• M. K. Kwong, “Uniqueness of positive solutions of $\Delta u-u+u\sp p=0$ in ${\bf R}\sp n$”, Arch. Rational Mech. Anal. 105:3 (1989), 243–266.
• F. Merle, “Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power”, Duke Math. J. 69:2 (1993), 427–454.
• F. Merle and Y. Tsutsumi, “$L\sp 2$ concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power nonlinearity”, J. Differential Equations 84:2 (1990), 205–214.
• F. Merle and L. Vega, “Compactness at blow-up time for $L\sp 2$ solutions of the critical nonlinear Schrödinger equation in 2D”, Internat. Math. Res. Notices 8 (1998), 399–425.
• H. Nawa, Comm. Pure Appl. Math. 52:2 (1999), 193–270.
• E. Ryckman and M. Visan, “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.
• S. Shao, “Sharp linear and bilinear restriction estimates for paraboloid in the cylindrically symmetric case”, preprint, 2007.
• G. Staffilani, “On the generalized Korteweg-de Vries-type equations”, Differential Integral Equations 10:4 (1997), 777–796.
• E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton University Press, Princeton, NJ, 1993.
• E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series 32, Princeton University Press, Princeton, N.J., 1971.
• 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, “A pseudoconformal compactification of the nonlinear Schrödinger equation and applications”, preprint, 2006.
• T. Tao, “Global regularity of wave maps III”, preprint, 2008.
• T. Tao, “Global regularity of wave maps IV”, preprint, 2008.
• T. Tao, “Global regularity of wave maps V”, preprint, 2008.
• T. Tao, M. Visan, and X. Zhang, “Minimal-mass blowup solutions of the mass-critical NLS”, preprint, 2006. To appear in Forum Math.
• 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 Math. J. 140:1 (2007), 165–202.
• M. E. Taylor, Tools for PDE. Pseudodifferential operators, paradifferential operators, and layer potentials, Mathematical Surveys and Monographs 81, American Mathematical Society, Providence, RI, 2000. http://www.emis.de/cgi-bin/MATH-item?0963.35211Zbl 0963.35211
• Y. Tsutsumi, “Scattering problem for nonlinear Schrödinger equations”, Ann. Inst. H. Poincaré Phys. Théor. 43:3 (1985), 321–347.
• N. Tzirakis, “The Cauchy problem for the semilinear quintic Schrödinger equation in one dimension”, Differential Integral Equations 18:8 (2005), 947–960.
• M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, PhD Thesis, University of California, Los Angeles, 2006.
• M. Visan, “The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions”, Duke Math. J. 138:2 (2007), 281–374.
• M. Visan and X. Zhang, “On the blowup for the $L\sp 2$-critical focusing nonlinear Schrödinger equation in higher dimensions below the energy class”, SIAM J. Math. Anal. 39:1 (2007), 34–56.
• M. I. Weinstein, “Nonlinear Schrödinger equations and sharp interpolation estimates”, Comm. Math. Phys. 87:4 (1982/83), 567–576.
• M. I. Weinstein, “The nonlinear Schrödinger equation–-singularity formation, stability and dispersion”, pp. 213–232 in The connection between infinite-dimensional and finite-dimensional dynamical systems (Boulder, CO, 1987), edited by B. Nicolaenko et al., Contemp. Math. 99, Amer. Math. Soc., Providence, RI, 1989.