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This is the second part in a four-paper sequence, which establishes theThreshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for thehyperbolic Yang--Mills equation in the $(4+1)$-dimensional space-time. Thispaper provides the key gauge-dependent analysis of the hyperbolic Yang--Millsequation.  We consider topologically trivial solutions in the caloric gauge, which wasdefined in the first paper arXiv:1709.08599 using the Yang--Mills heat flow. Inthis gauge, we establish a strong form of local well-posedness, where the timeof existence is bounded from below by the energy concentration scale. Moreover,we show that regularity and dispersive behavior of the solution persists aslong as energy dispersion is small. We also observe that fixed-time regularity(but not dispersive) properties in the caloric gauge may be transferred to thetemporal gauge without any loss, proving as a consequence small data globalwell-posedness in the temporal gauge.  The results in this paper are used in the subsequent papers arXiv:1709.08604,arXiv:1709.08606 to prove the sharp Threshold Theorem in caloric gauge in thetrivial topological class, and the dichotomy theorem in arbitrary topologicalclasses.

The hyperbolic Yang--Mills equation in the caloric gauge. Local well-posedness and control of energy dispersed solutions