Almost Optimal Local Well-Posedness of the Chern-Simons-Dirac System in the Coulomb Gauge

Abstract

In this paper, we study the local well-posedness of Chern-Simons-Dirac system in the Coulomb gauge for initial data in $H^s(\mathbb R^2)$ for $s>0$. The novelty of this paper is to prove almost critical regularity by using the bilinear estimates of wave type localized in a thickened null cone, given by \cite{selb} via null structure. We also prove the Dirac spinor flow of Chern-Simons-Dirac system cannot be $C^3$ at the origin in $H^s$ if $s<0$.