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

Abstract

We prove that the Yang-Mills equation in Lorenz gauge in the (n+1)-dimensional case, $n \ge 4$, is locally well-posed for data of the gauge potential in $H^s$ and the curvature in $H^r$, where $s > \frac{n}{2}-\frac{7}{8}$, $r > \frac{n}{2}-\frac{7}{4}$, respectively. The proof is based on the fundamental results of Klainerman-Selberg [7] and on the null structure of most of the nonlinear terms detected by Selberg-Tesfahun [16] and Tesfahun [19].