Finite energy global well-posedness of the Yang-Mills equations on $\mathbb{R}^{1+3}$: An approach using the Yang-Mills heat flow

TL;DR

This paper introduces a new approach using the Yang-Mills heat flow to establish finite energy global well-posedness of the Yang-Mills equations on Minkowski space, avoiding previous limitations and localization techniques.

Contribution

It presents a novel gauge choice method based on Yang-Mills heat flow, providing a more robust and adaptable framework for analyzing Yang-Mills equations.

Findings

Proves finite energy global well-posedness of Yang-Mills equations on R^{1+3}

Avoids use of Uhlenbeck's lemma and space-time localization

Offers a more robust approach compared to previous methods

Abstract

In this work, along with the companion work Oh (2012), we propose a novel approach to the problem of gauge choice for the \emph{Yang-Mills equations} on the Minkowski space $\mathbb{R}^{1+3}$. A crucial ingredient is the associated \emph{Yang-Mills heat flow}. As this approach does not possess the drawbacks of the previous approaches (as in Klainerman-Machedon (1995) and Tao (2003)), it is expected to be more robust and easily adaptable to other settings. Building on the results proved in the companion article Oh (2012), we prove, as one of the first applications of our approach, finite energy global well-posedness of the Yang-Mills equations on $\mathbb{R}^{1+3}$. This is a classical result first proved by S. Klainerman and M. Machedon (1995) using local Coulomb gauges. As opposed to their method, the present approach avoids the use of Uhlenbeck's lemma (1982), and hence does \emph{not} involve localization in space-time.