Low regularity local well-posedness for the Yang-Mills equation in Lorenz gauge

TL;DR

This paper establishes local well-posedness for the Yang-Mills equation in Lorenz gauge with low regularity initial data in higher dimensions, extending previous results by leveraging null structures and advanced harmonic analysis techniques.

Contribution

It proves local well-posedness for the Yang-Mills equation with minimal regularity assumptions in dimensions four and three, utilizing null structures and recent analytical methods.

Findings

Well-posedness for data in low regularity Sobolev spaces

Extension of results to higher dimensions (n ≥ 3)

Use of null structure to handle nonlinear terms

Abstract

We prove that the Yang-Mills equation in Lorenz gauge in the (n+1)-dimensional case is locally well-posed for data of the gauge potential in $H^s$ and the curvature in $H^r$ , where $s >\frac{n}{2}-\frac{7}{8}$ , $r > \frac{n}{2}-\frac{7}{4}$ , if $n \ge 4$, and $ s > \frac{3}{4}$ , $ r > - \frac{1}{8}$ , if $n=3$. The proof is based on the fundamental results of Klainerman-Selberg [KS] and on the null structure of most of the nonlinear terms detected by Selberg-Tesfahun [ST] and Tesfahun [Te].