A Note on the smoothness of flow maps for the Yang-Mills system in the Lorenz gauge

Abstract

We study the failure of the smoothness of flow maps for the $(1+3)$ dimensional Yang-Mills system in the Lorenz gauge by Knapp type counterexamples. This shows a gap between the scaling critical regularity exponents and the best attainable regularity via Picard's iteration in the Yang-Mills system under the Lorenz gauge condition.