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

TL;DR

This paper proves local well-posedness for low regularity initial data in the Chern-Simons-Dirac and Chern-Simons-Higgs equations using null structures and $X^{s,b}$ spaces, improving previous results.

Contribution

It uncovers a null structure in the CSD equations and applies $X^{s,b}$ space techniques to establish low regularity well-posedness, extending prior work.

Findings

CSD equations are well-posed for initial data in $H^{1/4+\epsilon}$.

CSH equations are well-posed with initial data in $H^{1/4+\epsilon}$ and related spaces.

Improves previous regularity thresholds established by Selberg-Tesfahun (2012).

Abstract

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 $a_{\mu}, \psi \in H^{1/4+\epsilon}_{x}$. Moreover, it is observed that the same techniques applied to (CSH) lead to a quick proof of local well-posedness for initial data $a_{\mu} \in H^{1/4+\epsilon}_{x}$, $(\phi, \partial_{t} \phi) \in H^{3/4+\epsilon}_{x} \times H^{-1/4+\epsilon}_{x}$, which improves the previous result of Selberg-Tesfahun (2012).