Finite energy global well-posedness of the Yang–Mills equations on R 1 + 3 : An approach using the Yang–Mills heat flow

Abstract

In this work, we propose a novel approach to the problem of gauge choice for the Yang–Mills equations on the Minkowski space ${\mathbb{R}}^{1+3}$. A crucial ingredient is the associated Yang–Mills heat flow. As this approach avoids the drawbacks of previous approaches, it is expected to be more robust and easily adaptable to other settings. Building on the author’s previous results, we prove, as the first application of our approach, finite energy global well-posedness of the Yang–Mills equations on ${\mathbb{R}}^{1+3}$. This is a classical result first proved by Klainerman and Machedon using local Coulomb gauges. As opposed to their method, the present approach avoids the use of Uhlenbeck’s lemma and hence does not involve localization in space-time.