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

Abstract: 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 $H^{s+\frac12} \times H<sup>s$ for $s&gt;\frac14$. We also prove that the solution flow map $(\phi(0), A(0)) \mapsto (\phi(t), A(t))$ fails to be $C<sup>2$ at the origin of $H^s \times H<sup>\sigma$ when $\sigma &lt; \frac14$ regardless of $s \in \mathbb R$. This means the regularity $H<sup>s$, $s&gt;\frac14$ is almost critical.