Finite energy global well-posedness of the Yang-Mills equations on \(\mathbb{R}^{1+3}\): an approach using the Yang-Mills heat flow
type: Item
instance of: scholarly article
title: Finite energy global well-posedness of the Yang-Mills equations on \(\mathbb{R}^{1+3}\): an approach using the Yang-Mills heat flow
zbMATH Open document ID: 1325.35180
DOI: 10.1215/00127094-3119953
author: Sungjin Oh
published in: Duke Mathematical Journal
publication date: 2015-07-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1210.1557
full work available at URL: https://projecteuclid.org/euclid.dmj/1434377459
review text: 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.
reviewed by: Thomas Ernst
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 70S15
Mathematics Subject Classification ID: 81T13
Mathematics Subject Classification ID: 35Q60
zbMATH DE Number: 6464859
zbMATH Keywords: Yang-Mills equations
zbMATH Keywords: Yang-Mills heat flow
zbMATH Keywords: gauge choice
zbMATH Keywords: finite energy
zbMATH Keywords: global well-posedness
MaRDI profile type: Publication
OpenAlex ID: W2166643479
arXiv ID: 1210.1557
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description: scientific article