Critical well-posedness and scattering results for fractional Hartree-type equations

TL;DR

This paper establishes scattering results for a mass-critical fractional Schrödinger equation with Hartree nonlinearity, using bilinear estimates and analyzing flow map regularity in a three-dimensional radial setting.

Contribution

It introduces sharp bilinear estimates for free solutions and extends them to perturbations, providing new well-posedness and scattering results for fractional Hartree equations.

Findings

Scattering established for small initial data in critical space

Bilinear estimates proved for free solutions and perturbations

Unboundedness of third order flow map derivative in super-critical range

Abstract

Scattering for the mass-critical fractional Schr\"odinger equation with a cubic Hartree-type nonlinearity for initial data in a small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data is established. For this, we prove a bilinear estimate for free solutions and extend it to perturbations of bounded quadratic variation. This result is shown to be sharp by proving the unboundedness of a third order derivative of the flow map in the super-critical range.