The mass-critical nonlinear Schr\"odinger equation with radial data in dimensions three and higher

TL;DR

This paper proves global well-posedness and scattering for the mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, extending understanding of solution behavior in these settings.

Contribution

It establishes the first large-data global well-posedness and scattering results for radial solutions in dimensions d ≥ 3 for the mass-critical NLS.

Findings

Global well-posedness for large radial data in d ≥ 3

Scattering results for solutions in these dimensions

In the focusing case, blowup solutions must concentrate at least the ground state mass

Abstract

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schr\"odinger equation $iu_t + \Delta u = \pm |u|^{4/d} u$ for large spherically symmetric L^2_x(R^d) initial data in dimensions $d\geq 3$. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.