General properties of the Yang-Mills equations in physical space

摘要

With the formulation of the gauge group as a Banach-Lie group of suitable Sobolev type, the Cauchy problem for the Yang-Mills equation in physical space-time reduces rigorously to the case of the temporal gauge. In this gauge there exist spatially global strong solutions for given data for field and potential that are L2 together with one or two derivatives (respectively). Regarding global existence in time, there is strong unicity, strong existence unless the potential becomes unbounded, and existence of a quasi-solution for arbitrary finite-energy Cauchy data. The variety of solutions of the equations is endowed with a canonical symplectic structure. This structure is degenerate to an extent precisely reflecting gauge-invariance and is conformally invariant.