Almost critical regularity of non-abelian Chern-Simons-Higgs system in the Lorenz gauge
英文摘要：
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 $ϕ$ 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 $(ϕ(0), A(0)) \mapsto (ϕ(t), A(t))$ fails to be $C^2$ at the origin of $H^s \times H^σ$ when $σ< \frac14$ regardless of $s \in \mathbb R$. This means the regularity $H^s$, $s>\frac14$ is almost critical.
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