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

TL;DR

This paper investigates the limitations of smoothness in the flow maps of the 3+1 dimensional Yang-Mills equations in the Lorenz gauge, revealing a gap between critical and attainable regularities through counterexamples.

Contribution

It provides Knapp type counterexamples demonstrating the failure of smoothness, highlighting a gap between critical and achievable regularity in the Yang-Mills system.

Findings

Flow maps are not smooth at certain regularities.

A gap exists between critical and attainable regularity.

Counterexamples illustrate the failure of smoothness.

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.