1210.1558 (Sung-Jin Oh)

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

Sung-Jin Oh

In this work, we introduce a novel approach to the problem of gauge choice for the Yang-Mills equation on the Minkowski space $\mathbb{R}^{1+3}$, which uses the Yang-Mills heat flow in a crucial way. As this approach does not possess the drawbacks of the previous approaches (as in Klainerman-Macheon (1995) and Tao (2003)), it is expected to be more robust 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-Mills equations for initial data in $(\dot{H}^{1}_{x} \cap L^{3}_{x}) \times L^{2}_{x}$, which is a classical result of S. Klainerman and M. 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 the companion article Oh (2012).

View original: http://arxiv.org/abs/1210.1558