Maximizers for the Strichartz norm for small solutions of mass-critical NLS
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, p. 427-476

Consider the mass-critical nonlinear Schrödinger equations in both focusing and defocusing cases for initial data in L 2 in space dimension N. By Strichartz inequality, solutions to the corresponding linear problem belong to a global L p space in the time and space variables, where p=2+4 N. In 1D and 2D, the best constant for the Strichartz inequality was computed by D. Foschi who has also shown that the maximizers are the solutions with Gaussian initial data.

Solutions to the nonlinear problem with small initial data in L 2 are globally defined and belong to the same global L p space. In this work we show that the maximum of the L p norm is attained for a given small mass. In addition, in 1D and 2D, we show that the maximizer is unique and obtain a precise estimate of the maximum. In order to prove this we show that the maximum for the linear problem in 1D and 2D is nondegenerated.

Published online : 2018-08-07
Classification:  35Q55,  35P25,  35B50,  35B45
@article{ASNSP_2011_5_10_2_427_0,
     author = {Duyckaerts, Thomas and Merle, Frank and Roudenko, Svetlana},
     title = {Maximizers for the Strichartz norm for small solutions of mass-critical NLS},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {2},
     year = {2011},
     pages = {427-476},
     zbl = {1247.35142},
     mrnumber = {2856155},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2011_5_10_2_427_0}
}
Duyckaerts, Thomas; Merle, Frank; Roudenko, Svetlana. Maximizers for the Strichartz norm for small solutions of mass-critical NLS. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 2, pp. 427-476. http://www.numdam.org/item/ASNSP_2011_5_10_2_427_0/

[1] J. Bourgain. Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity, Internat. Math. Res. Notices (1998), 253–283. | MR 1616917 | Zbl 0917.35126

[2] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), 145–171. | MR 1626257 | Zbl 0958.35126

[3] P. Bégout and A. Vargas, Mass concentration phenomena for the L 2 -critical nonlinear Schrödinger equation, Trans. Amer. Math. Soc. 359 (2007), 5257–5282. | MR 2327030 | Zbl 1171.35109

[4] R. Carles, Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Models Methods Appl. Sci. 12 (2002), 1513–1523. | MR 1933935 | Zbl 1029.35208

[5] R. Carles, Rotating points for the conformal nls scattering operator, Dyn. Partial Differ. Equ. 6 (2009), 35–51. | MR 2517827 | Zbl 1191.35270

[6] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in 3 , Ann. of Math. 167 (2008), 767–865. | MR 2415387 | Zbl 1178.35345

[7] P. Constantin and J.-C. Saut, Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J. 38 (1989), 791–810. | MR 1017334 | Zbl 0712.35022

[8] T. Cazenave and F. B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H s , Nonlinear Anal. 14 (1990), 807–836. | MR 1055532 | Zbl 0706.35127

[9] T. Duyckaerts and F. Merle, Scattering norm estimate near the threshold for energy-critical focusing semilinear wave equation, Indiana Univ. Math. J. 58 (2009), 1971–2002. | MR 2542985 | Zbl 1179.35189

[10] K.-J. Engel and R. Nagel, “One-parameter Semigroups for Linear Evolution Equations”, Vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000, with contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. | MR 1721989 | Zbl 0952.47036

[11] D. Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc. (JEMS) 9 (2007), 739–774. | MR 2341830 | Zbl 1231.35028

[12] J. Holmer and S. Roudenko, A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm. Math. Phys. 282 (2008), 435–467. | MR 2421484 | Zbl 1155.35094

[13] D. Hundertmark and V. Zharnitsky, On sharp Strichartz inequalities in low dimensions, Int. Math. Res. Not. 2006, Art. ID 3408, 18 pages. | MR 2219206 | Zbl 1131.35308

[14] S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal. 235 (2006), 171–192. | MR 2216444 | Zbl 1099.35132

[15] 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 (2006), 645–675. | MR 2257393 | Zbl 1115.35125

[16] R. Killip, T. Tao and M. Visan, The cubic nonlinear schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS) 11 (2009), 1203 –1258. | MR 2557134 | Zbl 1187.35237

[17] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145. | Numdam | MR 778970 | Zbl 0541.49009

[18] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, I, Rev. Mat. Iberoamericana 1 (1985), 145–201. | MR 834360 | Zbl 0704.49005

[19] F. Merle and L. Vega, Compactness at blow-up time for L 2 solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices (1998), 399–425. | MR 1628235 | Zbl 0913.35126

[20] U. Niederer, The maximal kinematical invariance groups of Schrödinger equations with arbitrary potentials, Helv. Phys. Acta 47 (1974), 167–172. | MR 366263

[21] O. Rey, The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990), 1–52. | MR 1040954 | Zbl 0786.35059

[22] A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen and R. K. Bullough, Similarity solutions and collapse in the attractive gross-pitaevskii equation, Phys. Rev. E 62 (2000), 6224–6228. | MR 1796440

[23] P. Sjölin, Convergence properties for the Schrödinger equation, Rend. Sem. Mat. Fis. Milano 57 (1989), 293–297. | MR 1017858 | Zbl 0699.35015

[24] R. S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705–714. | MR 512086 | Zbl 0372.35001

[25] V.I. Talanov, Focusing of light in cubic media, JETP Lett. 11, 199–201.

[26] 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 (electronic). | MR 2154347 | Zbl 1119.35092

[27] T. Tao, Spacetime bounds for the energy-critical nonlinear wave equation in three spatial dimensions, Dyn. Partial Differ. Equ. 3 (2006), 93–110. | MR 2227039 | Zbl 1145.35089

[28] T. Tao, A pseudoconformal compactification of the nonlinear Schrödinger equation and applications, New York J. Math. 15 (2009), 265–282. | MR 2530148 | Zbl 1184.35296

[29] T. Tao and M. Visan, Stability of energy-critical nonlinear Schrödinger equations in high dimensions, Electron. J. Differential Equations (2005), pages No. 118, 28 pp. (electronic). | MR 2174550 | Zbl 1245.35122

[30] 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 (2007), 165–202. | MR 2355070 | Zbl 1187.35246

[31] T. Tao, M. Visan and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math. 20 (2008), 881–919. | MR 2445122 | Zbl 1154.35085

[32] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874–878. | MR 934859 | Zbl 0654.42014

[33] W.-M. Wang, Pure point spectrum of the Floquet Hamiltonian for the quantum harmonic oscillator under time quasi-periodic perturbations, Comm. Math. Phys. 277 (2008), 459–496. | MR 2358292 | Zbl 1144.81018