Scattering results for Dirac Hartree-type equations with small initial data

TL;DR

This paper proves small data scattering for the Dirac equation with cubic Hartree nonlinearity, utilizing advanced estimates and null-structure, establishing near-optimal results in the subcritical range.

Contribution

It introduces a novel approach combining localized Strichartz estimates and null-structure analysis to achieve scattering results for Dirac-Hartree equations in the subcritical regime.

Findings

Proved small data scattering in the full subcritical range.

Identified the limitations of iteration methods in supercritical regimes.

Demonstrated the near-optimality of the results through failure analysis.

Abstract

We consider the Dirac equation with cubic Hartree-type nonlinearity derived by uncoupling the Dirac-Klein-Gordon systems. We prove small data scattering result in full subcritical range. Main ingredients of the proof are the localized Strichartz estimates, improved bilinear estimates thanks to null-structure hidden in Dirac operator and $Up,Vp$ function spaces. We apply the projection operator and get a system which of linear part is the Klein-Gordon type. It enables us to exploit the null-structures in equation. This result is shown to be almost optimal by showing that iteration method based on Duhamel's formula over supercritical range fails.