Improved well-posedness results for the Maxwell-Klein-Gordon system in 2D
Authors: Pecher Hartmut 
Published: June 2021

<p style='text-indent:20px;'>The local well-posedness problem for the Maxwell-Klein-Gordon system in Coulomb gauge as well as Lorenz gauge is treated in two space dimensions for data with minimal regularity assumptions. In the classical case of data in <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-based Sobolev spaces <inline-formula><tex-math id="M2">\begin{document}$ H^s $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ H^l $\end{document}</tex-math></inline-formula> for the electromagnetic field <inline-formula><tex-math id="M4">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> and the potential <inline-formula><tex-math id="M5">\begin{document}$ A $\end{document}</tex-math></inline-formula>, respectively, the minimal regularity assumptions are <inline-formula><tex-math id="M6">\begin{document}$ s &gt; \frac{1}{2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ l &gt; \frac{1}{4} $\end{document}</tex-math></inline-formula>, which leaves a gap of <inline-formula><tex-math id="M8">\begin{document}$ \frac{1}{2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \frac{1}{4} $\end{document}</tex-math></inline-formula> to the critical regularity with respect to scaling <inline-formula><tex-math id="M10">\begin{document}$ s_c = l_c = 0 $\end{document}</tex-math></inline-formula>. This gap can be reduced for data in Fourier-Lebesgue spaces <inline-formula><tex-math id="M11">\begin{document}$ \widehat{H}^{s, r} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ \widehat{H}^{l, r} $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M13">\begin{document}$ s&gt; \frac{21}{16} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ l &gt; \frac{9}{8} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M15">\begin{document}$ r $\end{document}</tex-math></inline-formula> close to <inline-formula><tex-math id="M16">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>, whereas the critical exponents with respect to scaling fulfill <inline-formula><tex-math id="M17">\begin{document}$ s_c \to 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M18">\begin{document}$ l_c \to 1 $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M19">\begin{document}$ r \to 1 $\end{document}</tex-math></inline-formula>. Here <inline-formula><tex-math id="M20">\begin{document}$ \|f\|_{\widehat{H}^{s, r}} : = \| \langle \xi \rangle^s \tilde{f}\|_{L^{r'}_{\tau \xi}} \, , \, 1 &lt; r \le 2 \, , \, \frac{1}{r}+\frac{1}{r'} = 1 \, . $\end{document}</tex-math></inline-formula> Thus the gap is reduced for <inline-formula><tex-math id="M21">\begin{document}$ \phi $\end{document}</tex-math></inline-formula> as well as <inline-formula><tex-math id="M22">\begin{document}$ A $\end{document}</tex-math></inline-formula> in both gauges.</p>