GAUGE CHOICE FOR THE YANG-MILLS EQUATIONS USING THE YANG-MILLS HEAT FLOW AND LOCAL WELL-POSEDNESS IN H~1

摘要

We introduce a novel approach to the problem of gauge choice for the Yang-Mills equations on the Minkowski space R~(1+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 (A_i,E_i) ∈ (H_x~1 ∩ L_x~3) × L_x~2, 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.