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.

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

Cited by in corpus (1)

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