Long-time Behavior of Solutions to Cubic Dirac Equation with Hartree Type Nonlinearity in ℝ1+2

In this paper we study the long-time behavior of solutions to the Dirac equation $$\begin{equation*} \big ( -i\gamma^\mu \partial_\mu + m \big) \psi= \left(V \ast ( \overline \psi \psi)\right) \psi \ \ \textrm{in } \ {\mathbb{R}}^{1+2},\end{equation*}$$where $V$ is the Yukawa potential in ${\mathbb{R}}^{2}$. It is proved that if $m>0$ and the initial data is small in $H^s({\mathbb{R}}^2)$ for $s>0$, the corresponding initial value problem is globally well posed and the solution scatters to free waves asymptotically as $t \rightarrow \pm \infty $. The main ingredients in the proof are Strichartz estimates and space-time $L^2$-bilinear null-form estimates for free waves.