Maximizers for the Strichartz norm for small solutions of mass-critical NLS

TL;DR

This paper investigates the maximizers of the Strichartz norm for small solutions of the mass-critical nonlinear Schrödinger equation, establishing existence, uniqueness, and precise estimates in 1D and 2D.

Contribution

It proves the existence and uniqueness of maximizers for small initial data in 1D and 2D, and provides precise estimates of the maximum Strichartz norm.

Findings

Maximizers exist for small initial data in 1D and 2D.

Maximizers are unique in 1D and 2D.

The maximum linear problem in 1D and 2D is nondegenerate.

Abstract

Consider the mass-critical nonlinear Schr\"odinger 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.