Almost Sure Scattering of the Energy Critical NLS in $d>6$

TL;DR

This paper proves that for high-dimensional energy-critical nonlinear Schrödinger equations with randomized initial data, solutions exist globally and scatter almost surely, extending previous results to dimensions greater than six.

Contribution

It extends almost sure global well-posedness and scattering results for the energy-critical NLS to dimensions greater than six with randomized initial data.

Findings

Almost sure global well-posedness in high dimensions

Scattering results for randomized super-critical data

Extension of previous 4D results to $d>6$

Abstract

We study the energy-critical nonlinear Schr\"{o}dinger equation with randomised initial data in dimensions $d>6$. We prove that the Cauchy problem is almost surely globally well-posed with scattering for randomised super-critical initial data in $H^s(\mathbb{R}^d)$ whenever $s>\max\{\frac{4d-1}{3(2d-1)},\frac{d^2+6d-4}{(2d-1)(d+2)}\}$. The randomisation is based on a decomposition of the data in physical space, frequency space and the angular variable. This extends previously known results of Spitz in dimension 4. The main difficulty in the generalisation to high dimensions is the non-smoothness of the nonlinearity.