Global well-posedness and scattering for the higher-dimensional energy-critical non-linear Schrodinger equation for radial data

TL;DR

This paper proves that spherically symmetric solutions to the energy-critical defocusing nonlinear Schrödinger equation in dimensions three and higher are globally well-posed and scatter, with improved bounds on spacetime norms, extending previous lower-dimensional results.

Contribution

It generalizes global well-posedness and scattering results to higher dimensions for radial data, with new bounds on spacetime norms and handling of low-power non-linearities.

Findings

Solutions exist globally and scatter in all dimensions n ≥ 3.

Bounds on spacetime norms are of exponential type in the energy.

Addresses technical challenges in dimensions n ≥ 6 due to low non-linearity power.

Abstract

In any dimension $n \geq 3$, we show that spherically symmetric bounded energy solutions of the defocusing energy-critical non-linear Schr\"odinger equation $i u_t + \Delta u = |u|^{\frac{4}{n-2}} u$ in $\R \times \R^n$ exist globally and scatter to free solutions; this generalizes the three and four dimensional results of Bourgain and Grillakis. Furthermore we have bounds on various spacetime norms of the solution which are of exponential type in the energy, which improves on the tower-type bounds of Bourgain. In higher dimensions $n \geq 6$ some new technical difficulties arise because of the very low power of the non-linearity.