Almost Sure Scattering at Mass Regularity for Radial Schr\"odinger Equations

TL;DR

This paper establishes almost sure global well-posedness and scattering for radial nonlinear Schrödinger equations in higher dimensions using a Gaussian measure, extending previous results to a broader class of nonlinearities and dimensions.

Contribution

It constructs a natural Gaussian measure supporting almost every initial data for global solutions and proves scattering in higher dimensions for certain nonlinearities.

Findings

Almost sure global well-posedness for radial NLS in dimensions 2 to 10.

Construction of a Gaussian measure supported on radial functions.

Scattering results for supercritical nonlinearities in higher dimensions.

Abstract

We consider the radial nonlinear Schr\"odinger equation $i\partial_tu +\Delta u = |u|^{p-1}u$ in dimension $d\geqslant 2$ for $p\in \left(1,1+\frac{4}{d}\right]$ and construct a natural Gaussian measure $\mu_0$ which support is almost $L^2_{\text{rad}}$ and such that $\mu_0$ - almost every initial data gives rise to a unique global solution. Furthermore, for $p>1+\frac{2}{d}$ and $d\in\{2, \dots, 10\}$ the solutions constructed scatters in a space which is almost $L^2$. This paper can be viewed as the higher dimensional counterpart of the work of Burq and Thomann, in the radial case.