The gauge-invariant I-method for Yang-Mills

Abstract

We prove global well-posedness of the $ 3d $ Yang-Mills equation in the temporal gauge in $ H^{\sigma} $ for $ \sigma > \frac{5}{6} $. Unlike related equations, Yang-Mills is not directly amenable to the method of almost conservation laws (I-method) in its Fourier and global version. We propose a modified energy which: 1) Is gauge-invariant and easy to localize 2) Provides local gauges which give control of local Sobolev norms (through an Uhlenbeck-type lemma for fractional regularities) 3) Is slightly smoother in time compared to the classical I-method energy for related systems. The spatial smoothing is realized via the Yang-Mills heat flow instead of the multiplier $I$. Due to the temporal condition and its finite speed of propagation, the local gauge selection is compatible with recent initial data extension results. Therefore, smoothened energy differences can be partitioned into local pieces whose (appropriately extended) bounds can be square summed. After revealing the null structure within the trilinear integrals, these can be estimated using known methods. In an appendix we show how an invariant modified energy for Maxwell-Klein-Gordon can extend previous results to regularities $ \sigma > \frac{5}{6} $.