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In this work, we introduce a novel approach to the problem of gauge choicefor the Yang-Mills equation on the Minkowski space $\mathbb{R}^{1+3}$, whichuses the Yang-Mills heat flow in a crucial way. As this approach does notpossess the drawbacks of the previous approaches, it is expected to be morerobust and easily adaptable to other settings.  As the first demonstration of the `structure' offered by this new approach,we will give an alternative proof of the local well-posedness of the Yang-Millsequations for initial data in $(\dot{H}^{1}_{x} \cap L^{3}_{x}) \timesL^{2}_{x}$, which is a classical result of S. Klainerman and M. Machedon thathad been proved using a different method (local Coulomb gauges). The new proofdoes not involve localization in space-time, which had been the key drawback ofthe previous method. Based on the results proved in this paper, a new proof offinite energy global well-posedness of the Yang-Mills equations, also using theYang-Mills heat flow, is established in a companion article.

Gauge choice for the Yang-Mills equations using the Yang-Mills heat flow and local well-posedness in $H^{1}$