A Remark on Unconditional Uniqueness in the Chern-Simons-Higgs Model

The solution of the Chern-Simons-Higgs model in Lorenz gauge with data for the potential in $H^{s-1\mathrm{/}2}$ and for the Higgs field in $H^s\times H^{s-1}$ is shown to be unique in the natural space $C([0,T];H^{s-1/2} \times H^s \times H^{s-1})$for$s \ge 1$,where$s=1$ corresponds to finite energy. Huh and Oh recently proved local well-posedness for $s>3\mathrm{/}4$, but uniqueness was obtained only in a proper subspace $Y^s$ of Bourgain type. We prove that any solution in $C([0,T];H^{1\mathrm{/}2}\times H^1\times L^2)$ must in fact belong to the space $Y^{3\mathrm{/}4+ϵ}$, hence it is the unique solution obtained by Huh and Oh.