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
, 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
a derivative which we
denote as DX , such that the following Leibniz rule is satisfied for every scalar field
DX (fφ) = X(f)φ + fDX φ .
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