Finite Energy Solutions of the Yang-Mills Equations in ℝ 3+1

Abstract

Yang-Mills equations in R3+1 is well-posed in the energy norm. This means that for an appropriate gauge condition, we construct local, unique solutions in a time interval which depends only on the size of the energy norm of the data. Since the energy norm is left invariant by the Yang-Mills flow the local solution is automatically extended to the entire space-time. Thus our results, which settle a problem stated in [Str], imply the well-known, fundamental, regularity result of Eardley and Moncrief [E-M]. That result, proved in the temporal gauge, requires a higher degree of smoothness for the data. Our main result, proved also in the temporal gauge, allows us to extend the concept of solutions to arbitrary finite energy initial data. The solutions are automatically unique in the class of solutions obtained by our procedure. Moreover, the global regularity proof given by Eardley and Moncrief depends in an essential way on the specific properties of the fundamental solution of the wave equation in the flat Minkowski space-time R3+1, namely the strong Huygens Principle. Indeed due mainly to this fact their proof does not seem to extend to general curved space-times. We have reasons to hope that the very different approach we take here will resolve this difficulty. The basic ingredients of our method are: 1. The introduction of appropriate local Coulomb gauges adapted to the causal structure of the equations. 2. An appropriate method of localizing the new space-time estimates for