Almost sure scattering for the nonradial energy-critical NLS with arbitrary regularity in 3D and 4D cases

TL;DR

This paper proves that for the defocusing energy-critical nonlinear Schrödinger equations in 3D and 4D, solutions with non-radial initial data in any Sobolev space almost surely scatter, regardless of symmetry, size, or regularity.

Contribution

It establishes almost sure scattering for non-radial data in any Sobolev space for energy-critical NLS in 3D and 4D, without symmetry or size restrictions.

Findings

Almost sure scattering for non-radial data in H^s for all s in R.

No symmetry, size, or regularity restrictions needed.

Results apply to energy-critical NLS in 3D and 4D.

Abstract

In this paper, we study the defocusing energy-critical nonlinear Schr\"odinger equations $$ i\partial_t u + \Delta u = |u|^{\frac{4}{d-2}} u. $$ When $d=3,4$, we prove the almost sure scattering for the equations with non-radial data in $H_x^s$ for any $s\in\mathbb{R}$. In particular, our result does not rely on any spherical symmetry, size or regularity restrictions.