Almost critical regularity of non-abelian Chern-Simons-Higgs system in the Lorenz gauge

In this paper we consider a Cauchy problem on the self-dual relativistic non-abelian Chern-Simons-Higgs model, which is the system of equations of $\mathfrak{su}(n)\, (n \ge 2)$-valued matter field $\phi$ and gauge field $A$. Based on the frequency localization as well as the null structure we show the local well-posedness in Sobolev space $H^{s+\frac12} \times H^s$ for $s>\frac14$. We also prove that the solution flow map $(\phi(0), A(0)) \mapsto (\phi(t), A(t))$ fails to be $C^2$ at the origin of $H^s \times H^\sigma$ when $\sigma < \frac14$ regardless of $s \in \mathbb R$. This means the regularity $H^s$, $s>\frac14$ is almost critical.