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In this paper, we consider the Cauchy problem of local well-posedness of theChern-Simons-Dirac system in the Lorenz gauge for $B^{\frac14}_{2,1}$ initialdata. We improve the low regularity well-posedness, compared to Huh-Oh\cite{huhoh} and Okamoto \cite{oka}, by using the localization of space-timeFourier side and bilinear estimates given by Selberg \cite{selb}, whereas theauthors of \cite{huhoh, oka} used global estimates of \cite{danfoselb}. Then weshow the Dirac spinor flow of Chern-Simons-Dirac system is not $C^2$ at theorigin in $H^s$ if $ s < \frac14$. From this point of view, the space$B_{2,1}^\frac14$ can be regarded as a critical space for the localwell-posedness. We apply the argument for failure of smoothness to the Diracequation decoupled from Chern-Simons-Dirac system and show the flow is not$C^3$ in $H^s, s < 0$.

Well-posedness in a critical space of Chern-Simons-Dirac system in the Lorenz gauge