On the global solution of the Cauchy problem for the Yang–Mills–Higgs equations

In this paper we deal with the Yang–Mills–Higgs equations in the temporal gauge in 4-dimensional Minkowski space-time. We prove that the Cauchy problem is globally and uniquely soluble provided the initial data lie in the appropriate local Sobolev spaces. Our results apply to any compact gauge group and any invariant positive Higgs self-coupling of degree $\leq4$. The spontaneously broken symmetry is admitted. The initial configuration may have an arbitrary magnetic charge and we prove it's conservation in time.