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

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.