The cubic nonlinear Schr\"odinger equation in two dimensions with radial data

TL;DR

This paper proves global well-posedness and scattering for the 2D mass-critical nonlinear Schrödinger equation with radial data, showing that solutions behave predictably over time and blowup solutions concentrate mass.

Contribution

It establishes the first comprehensive results on global behavior and blowup concentration for the 2D radial mass-critical NLS, including focusing and defocusing cases.

Findings

Global well-posedness and scattering for radial data in 2D

Blowup solutions in the focusing case must concentrate at least the ground state mass

Partial results for non-radial cases and other dimensions

Abstract

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^2 u$ for large spherically symmetric L^2_x(\R^2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry.