Almost Sure Scattering at Mass Regularity for Radial Schrödinger Equations

Abstract

We consider the radial nonlinear Schrödinger equation $i{\partial }_{t}u+\Delta u=|u{|}^{p-1}u$ in dimension $d⩾2$ for $p\in \left(1,1+\frac{4}{d}\right]$ and construct a natural Gaussian measure ${\mu }_0$ which support is almost $L_{\text{rad}}^2$ 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.

arXiv commentComputations corrected, the main result is valid up to d=10

Cited by in corpus (1)

• Probabilistic local well-posedness for the Schrödinger equation posed for the Grushin Laplacian