CHAPTER 1

Introduction

In this work we investigate the global in time regularity of the Yang-Mills

equations on high dimensional Minkowski space with compact matrix gauge group

G. Specifically, we show that if a certain gauge covariant Sobolev norm is small, the

so called critical regularity norm

˙

H

n−4

2

A

, and the dimension satisfies n 6, then if

the initial data is regular a global solution exists and remains regular for all times.

This is in the same spirit as the recent result [8] for the Maxwell-Klein-Gordon

system, as well as earlier results for high dimensional wave-maps (see [11], [6], [9],

and [7]). Our approach shares many similarities with those works, whose underlying

philosophy is basically the same. The idea is to introduce Coulomb type gauges in

order to treat a specific potential term as a quadratic error. To achieve this for the

Yang-Mills system we employ a non-abelian variant of the remarkable parametrix

construction contained in [8], in conjunction with a version of the Uhlenbeck lemma

[13] on the existence of global Coulomb gauges. In the case of high dimensional

wave-maps, Coulomb gauges can be used to globally “renormalize” the equations in

such a way that the existence theory can be treated directly via Strichartz estimates.

For the case of Yang-Mills, as is the case with the Maxwell-Klein-Gordon system,

the corresponding renormalization procedure is necessarily more involved because

it needs to be done separately for each distinct direction in phase space. In the

present work, the parametrix which achieves the renormalization can be viewed as a

certain kind of Fourier integral operator with G-valued phase. The construction and

estimation of such an operator relies heavily on elliptic-Coulomb theory, primarily

due to the diﬃculty one faces from the fact that at the critical regularity the G-

valued phase function cannot be localized within a neighborhood of any given point

on the group (if you like, there is a logarithmic twisting of the phase group elements

as one moves around in physical space; fortunately the group is compact so this

doesn’t lead to unbounded behavior).

1.1. A Description of the Problem

To get things started we first give gauge covariant description of the equations

we are considering. The (hyperbolic) Yang-Mills equations arise as the evolution

equations for a connection on the bundle V = Mn × g, where Mn is some n

(spatial) dimensional Minkowski space, with metric g := (−1, 1,..., 1) in inertial

coordinates (x0,xi), and g is the Lie algebra of some compact matrix group G. We

endow V with the Ad(G) gauge structure: If φ is any section to V over M, then a

connection assigns to every vector-field X on the base

Mn

a derivative which we

denote as DX , such that the following Leibniz rule is satisfied for every scalar field

f:

DX (fφ) = X(f)φ + fDX φ .

1