Local well-posedness for the Maxwell-Dirac system in temporal gauge

Abstract

<p style='text-indent:20px;'>We consider the low regularity well-posedness problem for the Maxwell-Dirac system in 3+1 dimensions:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \partial^{\mu} F_{\mu \nu} &amp; = - \langle \psi, \alpha_{\nu} \psi \rangle \ \\ -i \alpha^{\mu} \partial_{\mu} \psi &amp; = A_{\mu} \alpha^{\mu} \psi \, , \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ F_{\mu \nu} = \partial^{\mu} A_{\nu} - \partial^{\nu} A_{\mu} $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M2">\begin{document}$ \alpha^{\mu} $\end{document}</tex-math></inline-formula> are the 4x4 Dirac matrices. We assume the temporal gauge <inline-formula><tex-math id="M3">\begin{document}$ A_0 = 0 $\end{document}</tex-math></inline-formula> and make use of the fact that some of the nonlinearities fulfill a null condition. Because we work in the temporal gauge we also apply a method, which was used by Tao for the Yang-Mills system.</p>