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

TL;DR

This paper establishes local well-posedness and energy dispersion control for the hyperbolic Yang--Mills equation in the caloric gauge, forming a key part of a larger program to analyze solution behavior in 4+1 dimensions.

Contribution

It provides the first strong local well-posedness results for the hyperbolic Yang--Mills equation in the caloric gauge, linking regularity and dispersive properties to energy dispersion.

Findings

Proved local well-posedness with existence time bounded by energy concentration scale.

Showed regularity and dispersive behavior persist under small energy dispersion.

Transferred fixed-time regularity from caloric to temporal gauge, enabling small data global results.

Abstract

This is the second part in a four-paper sequence, which establishes the Threshold Conjecture and the Soliton Bubbling vs.~Scattering Dichotomy for the hyperbolic Yang--Mills equation in the $(4+1)$-dimensional space-time. This paper provides the key gauge-dependent analysis of the hyperbolic Yang--Mills equation. We consider topologically trivial solutions in the caloric gauge, which was defined in the first paper arXiv:1709.08599 using the Yang--Mills heat flow. In this gauge, we establish a strong form of local well-posedness, where the time of existence is bounded from below by the energy concentration scale. Moreover, we show that regularity and dispersive behavior of the solution persists as long as energy dispersion is small. We also observe that fixed-time regularity (but not dispersive) properties in the caloric gauge may be transferred to the temporal gauge without any loss, proving as a consequence small data global well-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 the trivial topological class, and the dichotomy theorem in arbitrary topological classes.