Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge

In this paper, we address the problem of local well-posedness of the Chern-Simons-Dirac (CSD) and the Chern-Simons-Higgs (CSH) equations in the Lorenz gauge for low regularity initial data. One of our main contributions is the uncovering of a null structure of (CSD). Combined with the standard machinery of $X^{s,b}$ spaces, we obtain local well-posedness of (CSD) for initial data $a_{\mu },\psi \in H_x^{1\mathrm{/}4+ϵ}$. Moreover, it is observed that the same techniques applied to (CSH) lead to a quick proof of local well-posedness for initial data $a_{\mu }\in H_x^{1\mathrm{/}4+ϵ}$, $(\phi, \partial_{t} \phi) \in H^{3/4+\epsilon}{x} \times H^{-1/4+\epsilon}{x}$, which improves the previous result of Selberg-Tesfahun (2012).