Global regularity for the Yang-Mills equations on high dimensional Minkowski space

111 Pages·2013·1.073 MB·English

by  Joachim Krieger, Jacob Sterbenz

MEMOIRS
of the
American Mathematical Society
Volume 223 • Number 1047 (first of 5 numbers) • May 2013
Global Regularity for
the Yang-Mills Equations on
High Dimensional Minkowski Space
Joachim Krieger
Jacob Sterbenz
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Number 1047
Global Regularity for
the Yang-Mills Equations on
High Dimensional Minkowski Space
Joachim Krieger
Jacob Sterbenz
May2013 • Volume223 • Number1047(firstof5numbers) • ISSN0065-9266
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Contents
Chapter 1. Introduction 1
1.1. A Description of the Problem 1
1.2. Some Basic Notation 4
Chapter 2. Some Gauge-Theoretic Preliminaries 7
Chapter 3. Reduction to the “Main a-Priori Estimate” 13
3.1. The Comparison Principle and Coulomb Form of the Equations 13
3.2. Local Existence in the Coulomb Gauge 18
3.3. The Main A-priori Estimate for the Curvature 20
Chapter 4. Some Analytic Preliminaries 25
4.1. Fourier Analytic Notation 25
4.2. A Besov Calculus 26
4.3. Microlocal Angular Decompositions 29
4.4. Additional Notational Conventions 30
Chapter 5. Proof of the Main A-Priori Estimate 31
5.1. Function Spaces 31
5.2. Proof of the Critical A-Priori Estimate 32
Chapter 6. Reduction to Approximate Half-Wave Operators 39
6.1. A Further Reduction 41
Chapter 7. Construction of the Half-Wave Operators 43
7.1. Construction of the Gauges 43
7.2. A Preliminary Estimate for the Modified Potentials 45
7.3. The Div-Curl System for the Gauge Transformations 46
7.4. The Differentiated Parametrix 47
Chapter 8. Fixed Time L2 Estimates for the Parametrix 49
8.1. The “Smooth/Small” Decomposition of the TT∗ Kernel 49
8.2. Bounds for the “Smooth” Portion of the Kernel 51
8.3. Bounds for the “Small” Portion of the Kernel 62
8.4. Proof of the Fixed-Time Accuracy Estimate 74
Chapter 9. The Dispersive Estimate 77
Chapter 10. Decomposable Function Spaces and Some Applications 81
10.1. Decomposable Estimates for the Connection 85
10.2. Proof of the Square Sum Strichartz Estimates 89
10.3. Proof of the Differentiated Strichartz Estimates 91
iii
iv CONTENTS
Chapter 11. Completion of the Proof 93
Bibliography 99
Abstract
This monograph contains a study of the global Cauchy problem for the Yang-
Millsequationson(6+1)andhigherdimensionalMinkowskispace,whentheinitial
data sets are small in the critical gauge covariant Sobolev space H˙(n−4)/2. Regu-
A
larity is obtained through a certain “microlocal geometric renormalization” of the
equations which is implemented via a family of approximate null Cro¨nstrom gauge
transformations. The argument is then reduced to controlling some degenerate el-
liptic equations in high index and non-isotropic Lp spaces, and also proving some
bilinear estimates in specially constructed square-function spaces.
ReceivedbytheeditorDecember12,2005,andinrevisedform,November7,2007.
ArticleelectronicallypublishedonOctober4,2012;S0065-9266(2012)00566-1.
2010 MathematicsSubjectClassification. Primary35L70;Secondary70S15.
Key wordsand phrases. wave-equation,Yang-Millsequations,criticalregularity.
ThefirstauthorwassupportedinpartbyNSFGrantDMS-0401177.
ThesecondauthorwassupportedinpartbyanNSFPostdoctoralFellowship.
Affiliations at time of publication: Joachim Krieger, Bˆatiment des Math´ematiques, EPFL,
Station 8, CH-1015 Lausanne, Switzerland, email: joachim.krieger@epfl.ch; and Jacob Sterbenz,
DepartmentofMathematics,UniversityofCalifornia,SanDiego,LaJolla,California92093-0112,
email: [email protected].
(cid:2)c2012 American Mathematical Society
v

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,weshowthatifacertaingaugecovariantSobolevnormissmall,the
so called critical regularity norm H˙ n−24, and the dimension satisfies n(cid:2)6, then if
A
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]). Ourapproachsharesmanysimilaritieswiththoseworks,whoseunderlying
philosophy is basically the same. The idea is to introduce Coulomb type gauges in
order to treata 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
constructioncontainedin[8],inconjunctionwithaversionoftheUhlenbecklemma
[13] on the existence of global Coulomb gauges. In the case of high dimensional
wave-maps,Coulombgaugescanbeusedtoglobally“renormalize”theequationsin
suchawaythattheexistencetheorycanbetreateddirectlyviaStrichartzestimates.
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
presentwork,theparametrixwhichachievestherenormalizationcanbeviewedasa
certainkindofFourierintegraloperatorwithG-valuedphase. Theconstructionand
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-
valuedphasefunctioncannotbelocalizedwithinaneighborhoodofanygivenpoint
onthegroup(ifyoulike,thereisalogarithmictwistingofthephasegroupelements
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 algebraof 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 D , such that the following Leibniz rule is satisfied for every scalar field
X
f:
D (fφ) = X(f)φ+fD φ .
X X
1
2 1. INTRODUCTION
We may also endow the fibers g of V with an Ad(G) invariant metric (cid:2)·,·(cid:3) which
respects the action of D. That is, one has the formula:
(1.1) d(cid:2)φ,ψ(cid:3) = (cid:2)Dφ,ψ(cid:3)+(cid:2)φ,Dψ(cid:3) .
The curvature associated to D is the g valued two-form F which arises from the
commutation of covariant derivatives and is defined via the formula:
D D φ−D D φ−D φ = [F(X,Y),φ] .
X Y Y X [X,Y]
We say that the connection D satisfies the Yang-Mills equations if its curvature is
a (formal) stationary point of the following Maxwell type functional:
(cid:2)
1
(1.2) L[F] = −4 (cid:2)Fαβ,Fαβ(cid:3) DVMn .
Mn
The Euler-Lagrange equations of (1.2) read:
(1.3) DβF = 0 .
αβ
Furthermore, from the fact that F arises as the curvature of some connection D
the “Bianchi identity” is satisfied:
(1.4) D F = 0.
[α βγ]
From now on we will refer to the system (1.3)–(1.4) as the first order Yang–Mills
equations (FYM).
OuraimistostudytheCauchyproblemfortheFYMsystem. Todescribethis
in a geometrically invariant way, we split the connection-curvature pair (F,D) in
the following way: Foliating M by the standard Cauchy hypersurfaces t = const.,
we decompose:
(F,D) = (F,D)⊕(E,D ) ,
0
where (F,D) denotes the portion of (F,D) which is tangent to the surfaces t =
const. (i.e. the induced connection), and (E,D ) denotes respectively the interior
0
product of F with the foliation generator T =∂ , and the normal portion of D. In
t
inertial coordinates we have:
E = F , D = D .
i 0i 0 ∂t
OntheinitialCauchyhypersurfacet=0wecallaset(F(0),D(0),E(0))admissible
Cauchy data1 if it satisfies the following compatibility condition:
(1.5) DiE (0) = 0 .
i
We define the Cauchy problem for the Yang-Mills equation to be the task of con-
struction a connection (F,D) which solves (1.3), and has Cauchy data equal to
(F(0),D(0),E(0)).
In order to understand what the appropriate conditions on the initial data
should be, it is necessary to consider the following two basic features of the system
1Note that the set of initial data (F(0),D(0),E(0)) is overdetermined because the initial
curvature F(0) depends completely on the initial connection D(0). Also note while it is not
completely obvious at first that the set of initial data uniquely determines a solution (F,D) to
(1.3)–(1.4), this is the case. In particular, it is not necessary to specify the normal derivative
D0(0) initially as long as one knows the initial normal curvature E(0). This is a consequence
ofthe constraintequation (1.5) as will be demonstratedshortly(see equations (3.19) and(3.20)
below,andthediscussionfollowing).
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:
1
Q [F] = (cid:2)F ,F γ(cid:3)− g (cid:2)F ,Fγδ(cid:3) ,
αβ αγ β 4 αβ γδ
∇αQ [F] = 0 ,
αβ
where ∇ is the covariant derivative on Mn. In particular, contracting Q with the
vector-field T = ∂ we arrive at the following constant of motion for the system
t
(1.3)–(1.4):
(cid:2) (cid:2)
(cid:3) (cid:4)
1
(1.6) Q dx = |E|2+|F|2 dx .
00 2
Rn Rn
The second main aspect of the system (1.3)–(1.4) is that of scaling. If we perform
the transformation:
(1.7) (x0,xi) (cid:3) (λx0,λxi) ,
on Mn, then an easy calculation shows that:
(1.8) D (cid:3) λD , F (cid:3) λ2F .
If we now define the homogeneous gauge covariant (integer) Sobolev spaces:
(cid:5)
(1.9) (cid:6)F (cid:6)2 := (cid:6)DIF (cid:6)2 ,
H˙s L2(Rn)
A
|I|=s
where for each multiindex I = (i ,...,i ) we have that DI =Di1 ...Din is the
1 n ∂x1 ∂xn
repeated covariant differentiation with respect to the translation invariant spatial
vector-fields{∂x1,...,∂xn},thenforeven2 spatialdimensionsthenormH˙An−24 isin-
variantwithrespecttothescalingtransformation(1.8). Inparticular,theconserved
quantity (1.6) is invariant when n=4 and this is called the critical dimension.
It can be shown that in dimensions n (cid:2) 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 s = n−4. In other words, for n (cid:2) 4 one
c 2
can construct (large) initial data in such a way that some higher norm of the type
(1.9) will fail tobebounded at latertimes, eventhough itwas initially (see [2] and
[3]). Since these norms are gauge covariant, this type of singularity development
correspondstointrinsicgeometricbreakdownoftheequations,andisnotanartifact
of poorly chosen local coordinates (gauge) on V. Going in the other direction, it
is expected that if the critical norm H˙ n−24 is sufficiently small, then regular initial
A
datawillremainregularforalltimes. Thiscanbeseenasapreliminarysteptoward
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−24 regularity for the
A
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 (cid:2) 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
wedonotmakeanattempttodefinefractionalpowersofthespacesH˙s. Instead,inthecaseof
A
odd dimensions one can simply start with the equations in a Coulomb gauge and use the usual
(fractional)SobolevspacesH˙s insteadofH˙s.
A