GAUGE CHOICE FOR THE YANG–MILLS EQUATIONS USING THE YANG–MILLS HEAT FLOW AND LOCAL WELL-POSEDNESS IN H1

Abstract

We introduce a novel approach to the problem of gauge choice for the Yang–Mills equations on the Minkowski space ℝ1+3, which uses the Yang–Mills heat flow in a crucial way. As this approach does not possess the drawbacks of the previous approaches, it is expected to be more robust and easily adaptable to other settings. As a first application, we give an alternative proof of the local well-posedness of the Yang–Mills equations for initial data [Formula: see text], which is a classical result of Klainerman and Machedon (1995) that had been proved using a different method (local Coulomb gauges). The new proof does not involve localization in space–time, which had been the key drawback of the previous method. Based on the results proved in this paper, a new proof of finite energy global well-posedness of the Yang–Mills equations, also using the Yang–Mills heat flow, is established in a companion article.