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

TL;DR

This paper establishes local well-posedness of the Chern-Simons-Dirac system in a critical Besov space, improving previous results by using refined Fourier analysis and bilinear estimates, and analyzes the flow's smoothness failure at low regularity.

Contribution

It improves low regularity well-posedness results for the Chern-Simons-Dirac system in the Lorenz gauge by employing localized Fourier analysis and bilinear estimates, identifying the critical space.

Findings

Well-posedness in the critical space $B^{1/4}_{2,1}$ is established.

Flow is not $C^2$ at the origin in $H^s$ for $s<1/4$.

Flow is not $C^3$ in $H^s$ for $s<0$.

Abstract

In this paper, we consider the Cauchy problem of local well-posedness of the Chern-Simons-Dirac system in the Lorenz gauge for $B^{\frac14}_{2,1}$ initial data. We improve the low regularity well-posedness, compared to Huh-Oh \cite{huhoh} and Okamoto \cite{oka}, by using the localization of space-time Fourier side and bilinear estimates given by Selberg \cite{selb}, whereas the authors of \cite{huhoh, oka} used global estimates of \cite{danfoselb}. Then we show the Dirac spinor flow of Chern-Simons-Dirac system is not $C^2$ at the origin 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 local well-posedness. We apply the argument for failure of smoothness to the Dirac equation decoupled from Chern-Simons-Dirac system and show the flow is not $C^3$ in $H^s, s < 0$.