Small data scattering of 2d Hartree type Dirac equations

In this paper, we study the Cauchy problem of 2d Dirac equation with Hartree type nonlinearity c(|⋅|^−γ⁎〈ψ,βψ〉)βψ with c∈R∖{0}, 0<γ<2. Our aim is to show the small data global well-posedness and scattering in H^s for s>γ−1 and 1<γ<2. The difficulty stems from the singularity of the low-frequency part |ξ|^−(2−γ)χ_{|ξ|≤1} of potential. To overcome it we adapt U^p−V^p space argument and bilinear estimates of [27,25] arising from the null structure. We also provide nonexistence result for scattering in the long-range case 0<γ≤1.