Local well-posedness of Yang-Mills equations in Lorenz gauge below the energy norm

TL;DR

This paper establishes local well-posedness for Yang-Mills equations in Lorenz gauge with initial data below the energy norm, extending previous results and proving unconditional uniqueness in the energy class.

Contribution

It extends the local well-posedness results for YM-LG to data below the energy norm and proves unconditional uniqueness in the energy class.

Findings

Well-posedness for data in $H^s\times H^{s-1}$ with $s$ below 1

Unconditional uniqueness in the classical energy space

Identification of null structures in bilinear terms

Abstract

We prove that the Yang-Mills equations in the Lorenz gauge (YM-LG) is locally well-posed for data below the energy norm, in particular, we can take data for the gauge potential $A$ and the associated curvature $F$ in $H^s\times H^{s-1}$ and $H^r\times H^{r-1}$ for $s=(\frac67+,-\frac1{14} +)$, respectively. This extends a recent by Selberg and the present author on the local well-posedness of YM-LG for finite energy data (specifically, for $(s, r)=(1-, 0)$). We also prove unconditional uniqueness of the energy class solution, that is, uniqueness in the classical space $C([-T, T]; X_0)$, where $X_0$ is the energy data space. The key ingredient in the proof is the fact that most bilinear terms in YM-LG contain null structure some of which uncovered in the present paper.