For which gauge groups is the initial-value problem for the classical Yang-Mills equations known to be uniquely solvable up to gauge?
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Yang-Mills equations have gauge symmetries, so any solution can be changed to another solution by a gauge transformation in the future of an initial data surface thus violating uniqueness, as the standard argument goes. Perhaps you had in mind "uniquely solvable up to gauge".
Eardley, Douglas M.; Moncrief, Vincent, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties, Commun. Math. Phys. 83, 171-191 (1982). ZBL0496.35061. Here's the abstract of the paper:
Abstract. In this paper and its sequel we shall prove the local and then the global existence of solutions of the classical Yang-Mills-Higgs equations in the temporal gauge. This paper proves local existence uniqueness and smoothness properties and improves, by essentially one order of differentiability, previous local existence results. Our results apply to any compact gauge group and to any invariant Higgs self-coupling which is positive and of no higher than quartic degree.
Eardley, Douglas M.; Moncrief, Vincent, The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. II. Completion of proof, Commun. Math. Phys. 83, 193-212 (1982). ZBL0496.35062.
Abstract. In this paper we complete the proof of global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space by showing that an appropriate norm of the solutions cannot blow up in a finite time. A key step in the proof is the demonstration that the $L^\infty$ norm of the curvature is bounded a priori. Our results apply to any compact gauge group and to any invariant Higgs self-coupling which is positive and of no higher than quartic degree.
See also the references in this answer by Willie Wong.
Update: It seems that compactness does play some role in long time existence.
Yang, Yisong, Blow-up of the $SU(2,\mathbb{C})$ Yang-Mills fields, J. Math. Phys. 31, No. 5, 1237-1239 (1990). ZBL0702.53077.
Abstract: It is remarked that, if the gauge group is not compact, finite−time blowing−up Yang–Mills fields over the Minkowski spaces may exist. The example taken is $G=SU(2,\mathbb{C})$. It is shown that blow−up can occur in an arbitrarily small time interval when the initial data of the fields are suitably chosen.
The paper by Yang has exactly one citation on Google Scholar. So I have no idea how the study of Yang-Mills with a non-compact gauge group has progressed since then.
answered Aug 3 at 16:28
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$\begingroup$ Thats the kind of answer I was looking for. What about the noncompact case? Is it essentially different? It should combine features of Maxwell's equations and compact YAng-Mills.... The DOI of the second link is wrong. $\endgroup$ Commented Aug 4 at 11:54
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$\begingroup$ @ArnoldNeumaier For non-compact gauge groups, there seem to be counter examples to long time existence. $\endgroup$ Commented Aug 4 at 13:28
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2$\begingroup$ I don't know if this may be of interest to the OP, but it should also be remarked that the result holds in practically${}^*$ the same generality for any globally hyperbolic 4-dim. space-time, see P. T. Chrusciel and J. Shatah, Global Existence of Solutions of the Yang-Mills Equations on Globally Hyperbolic Four-Dimensional Lorentzian Manifolds, Asian J. Math. 1 (1997) 530-548. (${}^*$ - pure Yang-Mills, with no Higgs coupling. It's also assumed that the Lie algebra of the gauge group has a faithful rep. as a matrix subalgebra) $\endgroup$ Commented Aug 4 at 18:04