Almost sure scattering for the energy-critical NLS with radial data below H1(R4)

Abstract

We prove almost sure global existence and scattering for the energy-critical nonlinear Schrödinger equation with randomized spherically symmetric initial data in $H^s({\mathbb{R}}^4)$ with $\frac{5}{6}<s<1$. We were inspired to consider this problem by the recent work of Dodson–Lührmann–Mendelson, which treated the analogous problem for the energy-critical wave equation.

Keywords:

• Almost sure scattering
• energy-critical NLS
• supercritical data

MATHEMATICS SUBJECT CLASSIFICATION:

• 35Q55

Acknowledgements

The authors thank the anonymous referee for their careful reading of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

R. K. was supported by NSF grant DMS-1600942. J. M. was supported in part by NSF DMS-1400706. M. V. was supported by NSF grant DMS-1500707.