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 $p=2+\frac{4}{N}$. 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.
@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}, pages = {427--476}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 10}, number = {2}, year = {2011}, mrnumber = {2856155}, zbl = {1247.35142}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2011_5_10_2_427_0/} }
TY - JOUR AU - Duyckaerts, Thomas AU - Merle, Frank AU - Roudenko, Svetlana TI - Maximizers for the Strichartz norm for small solutions of mass-critical NLS JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2011 SP - 427 EP - 476 VL - 10 IS - 2 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2011_5_10_2_427_0/ LA - en ID - ASNSP_2011_5_10_2_427_0 ER -
%0 Journal Article %A Duyckaerts, Thomas %A Merle, Frank %A Roudenko, Svetlana %T Maximizers for the Strichartz norm for small solutions of mass-critical NLS %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2011 %P 427-476 %V 10 %N 2 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2011_5_10_2_427_0/ %G en %F 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://archive.numdam.org/item/ASNSP_2011_5_10_2_427_0/
[1] J. Bourgain. Refinements of Strichartz’ inequality and applications to $2$D-NLS with critical nonlinearity, Internat. Math. Res. Notices (1998), 253–283. | MR | Zbl
[2] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), 145–171. | MR | Zbl
[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 | Zbl
[4] R. Carles, Critical nonlinear Schrödinger equations with and without harmonic potential, Math. Models Methods Appl. Sci. 12 (2002), 1513–1523. | MR | Zbl
[5] R. Carles, Rotating points for the conformal nls scattering operator, Dyn. Partial Differ. Equ. 6 (2009), 35–51. | MR | Zbl
[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 ${\mathbb{R}}^{3}$, Ann. of Math. 167 (2008), 767–865. | MR | Zbl
[7] P. Constantin and J.-C. Saut, Local smoothing properties of Schrödinger equations, Indiana Univ. Math. J. 38 (1989), 791–810. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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 | Zbl
[11] D. Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc. (JEMS) 9 (2007), 739–774. | EuDML | MR | Zbl
[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 | Zbl
[13] D. Hundertmark and V. Zharnitsky, On sharp Strichartz inequalities in low dimensions, Int. Math. Res. Not. 2006, Art. ID 3408, 18 pages. | MR | Zbl
[14] S. Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal. 235 (2006), 171–192. | MR | Zbl
[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 | Zbl
[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 | Zbl
[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. | EuDML | Numdam | MR | Zbl
[18] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, I, Rev. Mat. Iberoamericana 1 (1985), 145–201. | EuDML | MR | Zbl
[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 | Zbl
[20] U. Niederer, The maximal kinematical invariance groups of Schrödinger equations with arbitrary potentials, Helv. Phys. Acta 47 (1974), 167–172. | MR
[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 | Zbl
[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
[23] P. Sjölin, Convergence properties for the Schrödinger equation, Rend. Sem. Mat. Fis. Milano 57 (1989), 293–297. | MR | Zbl
[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 | Zbl
[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). | EuDML | MR | Zbl
[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 | Zbl
[28] T. Tao, A pseudoconformal compactification of the nonlinear Schrödinger equation and applications, New York J. Math. 15 (2009), 265–282. | EuDML | MR | Zbl
[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 | Zbl
[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 | Zbl
[31] T. Tao, M. Visan and X. Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math. 20 (2008), 881–919. | MR | Zbl
[32] L. Vega, Schrödinger equations: pointwise convergence to the initial data, Proc. Amer. Math. Soc. 102 (1988), 874–878. | MR | Zbl
[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 | Zbl