Vol. 2, No. 2, 2020

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The hyperbolic Yang–Mills equation in the caloric gauge: local well-posedness and control of energy-dispersed solutions

Sung-Jin Oh and Daniel Tataru

Vol. 2 (2020), No. 2, 233–384
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 of the sequence 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 properties of the solution persist 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.

We use the results in this paper in subsequent papers to prove the sharp threshold theorem in caloric gauge in the trivial topological class, and the dichotomy theorem in arbitrary topological classes.

Keywords
hyperbolic Yang-Mills, energy critical, caloric gauge, small energy dispersion, regularity, scattering
Mathematical Subject Classification 2010
Primary: 35L70, 70S15
Milestones
Received: 23 November 2018
Revised: 29 January 2020
Accepted: 8 March 2020
Published: 22 May 2020
Authors
Sung-Jin Oh
Department of Mathematics
University of California
Berkeley, CA
United States
School of Mathematics
Korea Institute for Advanced Study
Seoul
South Korea
Daniel Tataru
Department of Mathematics
University of California
Berkeley, CA
United States