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

TL;DR

This paper proves unconditional uniqueness of solutions to the Chern-Simons-Higgs model in the natural energy space, extending previous results by showing solutions must belong to a specific function space where uniqueness is guaranteed.

Contribution

It establishes unconditional uniqueness in the natural energy space for the Chern-Simons-Higgs model, improving upon prior results that required solutions to be in a subspace.

Findings

Uniqueness holds in the natural energy space for the model.

Solutions in a broader space are shown to belong to the subspace where uniqueness is known.

The result extends the class of solutions for which uniqueness is guaranteed.

Abstract

The solution of the Chern-Simons-Higgs model in Lorenz gauge with data for the potential in $H^{s-1/2}$ 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 $s > 3/4$, but uniqueness was obtained only in a proper subspace $Y^s$ of Bourgain type. We prove that any solution in $C([0,T];H^{1/2} \times H^1 \times L^2)$ must in fact belong to the space $Y^{3/4+\epsilon}$, hence it is the unique solution obtained by Huh and Oh.