The cauchy problem for the Yang-Mills equations

AbstractThe Yang-Mills equation in Minkowski space with given initial data is considered. The gauge group is formulated in terms of Sobolev-Banach-Lie group, and the Cauchy problem for the equations thereby reduced to the temporal (hamiltonian) gauge. Given data for which are square-integrable over have respectively one and two derivatives which are square-integrable over space, a strong solution exists throughout space in a nontrivial time interval. If the initial data are infinitely differentiable in L2, the solution may be represented as a C∞ function on space-time satisfying the equations in the elementary sense. Strong solutions which agree at one time and have square-integrable derivatives as earlier agree throughout their regions of definition. For arbitrary finite-energy Cauchy data, there exists a quasi-solution (weak limit of solutions of truncated equations) which is global on Minkowski space. The solutions may take values in an arbitrary separable Hilbert Lie algebra