Finite energy global well-posedness of the Yang-Mills equations on \(\mathbb{R}^{1+3}\): an approach using the Yang-Mills heat flow. (English) Zbl 1325.35180

Oh, Sung-Jin

Duke Math. J. 164, No. 9, 1669-1732 (2015).

The purpose of this article is to study the behaviour of the energy for a generalization of Maxwells equation, the Yang-Mills equations. After presenting a system of equations for the Yang-Mills tension field, a gauge is chosen, which does not involve localization in space-time, and which is also suitable for large input values. Then the Yang-Mills heat flow is defined as the solution of a certain tensor PDE. The main result of the paper is to show that the energy controls the growth of a certain norm as \(| t | \to \pm \infty\), which implies that this norm is always finite. The proof consists of several steps and uses Hölders, Sobolev, Gagliardi-Nirenberg, energy integral, Bochner-Weitzenböck-type, Young’s, Minkowski, Gronwall, triangle and Kato’s inequalities, Duhamel’s principle, Bianchis identity, weak maximim principle, Sobolev embedding, and a bootstrap argument. Finally, a list of the dependence to a companion paper, and a list of symbols are given. A monomental paper.

Reviewer: Thomas Ernst (Uppsala)

MSC:

35Q40	PDEs in connection with quantum mechanics
70S15	Yang-Mills and other gauge theories in mechanics of particles and systems
81T13	Yang-Mills and other gauge theories in quantum field theory
35Q60	PDEs in connection with optics and electromagnetic theory

Keywords:

Yang-Mills equations; Yang-Mills heat flow; gauge choice; finite energy; global well-posedness

References:

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