Low regularity solutions to the 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 sl(n, C) (n >= 2)-valued matter field phi and gauge field A. Based on the frequency localization as well as the null structure we show local well-posedness in the Sobolev space Hs+ 1/2 x H-s for s > 1/4. We also prove that the solution flow map (phi(0), A(0)) -> (phi(t), A(t)) fails to be C-2 at the origin of H-s xH(sigma) when sigma < max(0, s - 1) regardless of s is an element of R.