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

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 $s=(\frac{6}{7}+,-\frac{1}{14}+)$, 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.

26 pages

References in corpus (3)

• Finite energy global well-posedness of the Yang-Mills equations on ${\mathbb{R}}^{1+3}$: An approach using the Yang-Mills heat flow
• Gauge choice for the Yang-Mills equations using the Yang-Mills heat flow and local well-posedness in $H^1$
• Null structure and local well-posedness in the energy class for the Yang-Mills equations in Lorenz gauge