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 $\varphi$ and gauge field $A$. Based on the frequency localization as well as the null structure we show the local well-posedness in Sobolev space $H^{s+\frac{1}{2}}\times H^s$ for $s>\frac{1}{4}$. We also prove that the solution flow map $(\varphi (0),A(0))\mapsto (\varphi (t),A(t))$ fails to be $C^2$ at the origin of $H^s\times H^{\sigma }$ when $\sigma <\frac{1}{4}$ regardless of $s\in \mathbb{R}$. This means the regularity $H^s$, $s>\frac{1}{4}$ is almost critical.

References in corpus (2)

• The Boson star equation with initial data of low regularity
• Finite energy global well-posedness of the Chern-Simons-Higgs equations in the Coulomb gauge