Low regularity well-posedness of Hartree type Dirac equations in 2, 3-dimensions

Abstract

<p style='text-indent:20px;'>In this paper, we consider the Cauchy problem of <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimension Hartree type Dirac equation with nonlinearity <inline-formula><tex-math id="M2">\begin{document}$ c|x|^{-\gamma} * \langle \psi, \beta \psi\rangle $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ c\in \mathbb R\setminus\{0\} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ 0 &lt; \gamma &lt; d $\end{document}</tex-math></inline-formula>(<inline-formula><tex-math id="M5">\begin{document}$ d = 2,3 $\end{document}</tex-math></inline-formula>). Our aim is to show the local well-posedness in <inline-formula><tex-math id="M6">\begin{document}$ H^s $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M7">\begin{document}$ s &gt; \frac{\gamma-1}2 $\end{document}</tex-math></inline-formula> with mass-supercritical cases(<inline-formula><tex-math id="M8">\begin{document}$ 1 &lt; \gamma&lt;d $\end{document}</tex-math></inline-formula>) and mass-critical case(<inline-formula><tex-math id="M9">\begin{document}$ {\gamma} = 1 $\end{document}</tex-math></inline-formula>) via bilinear estimates and angular decomposition for which we use the null structure of nonlinear term effectively. We also provide the flow of Dirac equations cannot be <inline-formula><tex-math id="M10">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula> at the origin for <inline-formula><tex-math id="M11">\begin{document}$ H^s $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M12">\begin{document}$ s &lt; \frac{\gamma-1}2 $\end{document}</tex-math></inline-formula>.</p>