1.1. A DESCRIPTION OF THE PROBLEM 3

(1.3)–(1.4). The first is conservation. From the Lagrangian nature of the field

equations (1.3)–(1.4) we have the tensorial conservation law:

Qαβ[F ] =

Fαγ,Fβγ

−

1

4

gαβ Fγδ,F

γδ

,

∇αQαβ[F

] = 0 ,

where ∇ is the covariant derivative on

Mn.

In particular, contracting Q with the

vector-field T = ∂t we arrive at the following constant of motion for the system

(1.3)–(1.4):

(1.6)

Rn

Q00 dx =

1

2

Rn

(

|E|2

+ |F

|2

)

dx .

The second main aspect of the system (1.3)–(1.4) is that of scaling. If we perform

the transformation:

(1.7)

(x0,xi) (λx0,λxi)

,

on

Mn,

then an easy calculation shows that:

D λD , F

λ2F

. (1.8)

If we now define the homogeneous gauge covariant (integer) Sobolev spaces:

(1.9) F

2

˙

H

s

A

:=

|I|=s

DI

F

2

L2(Rn)

,

where for each multiindex I = (i1,...,in) we have that DI = D∂1

i

x1

. . . D∂n

i

xn

is the

repeated covariant differentiation with respect to the translation invariant spatial

vector-fields {∂x1 , . . . , ∂xn }, then for

even2

spatial dimensions the norm

˙

H

n−4

2

A

is in-

variant with respect to the scaling transformation (1.8). In particular, the conserved

quantity (1.6) is invariant when n = 4 and this is called the critical dimension.

It can be shown that in dimensions n 4 the Cauchy problem for (1.3)–(1.4)

with smooth initial data will in general not be well behaved unless one imposes

size control on the critical regularities sc =

n−4

2

. In other words, for n 4 one

can construct (large) initial data in such a way that some higher norm of the type

(1.9) will fail to be bounded at later times, even though it was initially (see [2] and

[3]). Since these norms are gauge covariant, this type of singularity development

corresponds to intrinsic geometric breakdown of the equations, and is not an artifact

of poorly chosen local coordinates (gauge) on V . Going in the other direction, it

is expected that if the critical norm

˙

H

n−4

2

A

is suﬃciently small, then regular initial

data will remain regular for all times. This can be seen as a preliminary step toward

understanding the general picture of large data solutions in the critical dimension

n = 4. It is also an interesting problem in its own right.

A central diﬃculty in the demonstration of critical

˙

H

n−4

2

A

regularity for the

Yang-Mills system is to construct a stable set coordinates on the bundle V such

that the Christoffel symbols of D are well behaved. We will do this for dimensions

n 6 via (spatial) Coulomb gauges. Unfortunately, this preliminary gauge con-

struction is far from suﬃcient to close the critical regularity argument as it turns

2For

odd spatial dimensions the above discussion needs to be modified somewhat because

we do not make an attempt to define fractional powers of the spaces

˙

H

s

A

. Instead, in the case of

odd dimensions one can simply start with the equations in a Coulomb gauge and use the usual

(fractional) Sobolev spaces

˙

H

s

instead of

˙

H

s

A

.