Consider the mass-critical nonlinear Schrödinger equations in both focusing and defocusing cases for initial data in in space dimension . By Strichartz inequality, solutions to the corresponding linear problem belong to a global space in the time and space variables, where . In and , 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 are globally defined and belong to the same global space. In this work we show that the maximum of the norm is attained for a given small mass. In addition, in and , 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 and 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/
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