Scattering above energy norm of solutions of a loglog energy-supercritical Schrodinger equation with radial data

TL;DR

This paper proves scattering for solutions of a loglog energy-supercritical Schrödinger equation with radial data in certain Sobolev spaces, extending previous methods to high dimensions.

Contribution

It extends Bourgain's, Grillakis's, and Tao's scattering techniques to a loglog energy-supercritical setting with radial solutions in dimensions 3 and 4.

Findings

Proves scattering for solutions in H^k with radial data.

Extends existing scattering methods to a new supercritical regime.

Handles the loglog correction in the nonlinearity.

Abstract

We prove scattering of $\tilde{H}^{k} $ solutions of the loglog energy-supercritical Schrodinger equation $i \partial_{t} u + \triangle u = |u|^{\frac{4}{n-2}} u \log^{c} {(\log{(10+|u|^{2})})}$, $0 < c < c_{n}$, $n={3,4}$, with radial data $u(0):=u_{0} \in \tilde{H}^{k} $, $k>n/2$. This is achieved, roughly speaking, by extending Bourgain's argument (see also Grillakis) and Tao's argument in high dimensions.