Small data scattering of 2d Hartree type Dirac equations

Abstract

In this paper, we study the Cauchy problem of 2d Dirac equation with Hartree type nonlinearity $c(|\cdot {|}^{-\gamma }\ast ⟨\psi ,\beta \psi ⟩)\beta \psi$ with $c\in \mathbb{R}\setminus \{0\}$, $0<\gamma <2$. Our aim is to show the small data global well-posedness and scattering in $H^s$ for $s>\gamma -1$ and $1<\gamma <2$. The difficulty stems from the singularity of the low-frequency part $|\xi {|}^{-(2-\gamma )}{\chi }_{\{|\xi |\le 1\}}$ of potential. To overcome it we adapt $U^p-V^p$ space argument and bilinear estimates of \cite{yang, tes2d} arising from the null structure. We also provide nonexistence result for scattering in the long-range case $0<\gamma \le 1$.

arXiv comment25 page, no figure, no table

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