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

Abstract

We prove that the initial value problem for the Dirac equation $(-i\gamma^\mu \partial_\mu + m) \psi= ( \frac{e^{- |x|}}{|x|} \ast ( \overline \psi \psi)) \psi \quad \text{in } \ \mathbb{R}^{1+3}$ is globally well-posed and the solution scatters to free waves asymptotically as $t \rightarrow \pm \infty$ if we start with initial data that are 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.