Small data scattering for a cubic Dirac equation with Hartree type nonlinearity in $ \R^{1+3}$

TL;DR

This paper proves global well-posedness and scattering for a cubic Dirac equation with Hartree nonlinearity in 3+1 dimensions for small initial data in a subcritical Sobolev space, using advanced harmonic analysis techniques.

Contribution

It establishes almost critical well-posedness and scattering results for the Dirac-Hartree equation with novel use of $U^p$, $V^p$ spaces and bilinear null-form estimates.

Findings

Global existence and scattering for small initial data in $H^s$, $s>0$

Almost critical well-posedness in the $L^2$ space

Application of advanced harmonic analysis tools to Dirac equations

Abstract

We prove that the initial value problem for the Dirac equation $ \left ( -i\gamma^\mu \partial_\mu + m \right) \psi = \left(\frac{e^{- |x|}}{|x|} \ast ( \overline \psi \psi)\right) \psi \quad \text{in } \ \R^{1+3} $ is globally well-posed and the solution scatters to free waves asymptotically as $t \rightarrow \pm \infty$, if we start with initial data that is small in $H^s$ for $s>0$. This is an almost critical well-posedness result in the sense that $L^2$ is the critical space for the equation. The main ingredients in the proof are Strichartz estimates, space-time bilinear null-form estimates for free waves in $L^2$, and an application of the $U^p$ and $V^p$-function spaces.