The Cauchy problem for coupled Yang-Mills and scalar

ANNALES DE L’I. H. P., SECTION A

J. GINIBRE

G. VELOThe Cauchy problem for coupled Yang-Mills andscalar fields in the Lorentz gaugeAnnales de l’I. H. P., section A, tome 36, no 1 (1982), p. 59-78<http://www.numdam.org/item?id=AIHPA_1982__36_1_59_0>

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59

The Cauchy problem for coupled Yang-Millsand scalar fields in the Lorentz gauge

J. GINIBRE

G. VELO

Laboratoire de Physique Theorique et Hautes Energies (*),Universite de Paris-Sud, 91405 Orsay, France

Istituto di Fisica A. Righi, Universita di Bolognaand I. N. F. N., Sezione di Bologna, Italy

Ann. Inst. Henri Poincaré,

Vol. XXXVI, n° 1, 1982,

Section A :

Physique théorique.

RESUME.2014 On étudie le probleme de Cauchy pour un champ deYang Mills et un champ scalaire classiques couples de façon minimale,dans l’espace temps de dimension n + 1, en jauge de Lorentz. On demontrel’existence et l’unicité de solutions dans des intervalles de temps petitspour n quelconque, aussi bien dans des espaces locaux que globaux. Endimension deux d’espace temps, les solutions precedentes peuvent etreetendues a des temps quelconques par la methode des estimations a priori.En dimension trois d’espace temps, nos estimations ne donnent que desresultats partiels sur le probleme d’existence globale.

ABSTRACT. - We study the Cauchy problem for minimally coupledclassical Yang-Mills and scalar fields in n + 1 dimensional space-timein the Lorentz gauge. We prove the existence and uniqueness of solutionsfor small time intervals and for any n, both in local and global spaces.In space time dimension two, the previous solutions can be extended toall times by the method of a priori estimates. In space time dimensionthree, our estimates yield only partial results on the global existenceproblem.

(*) Laboratoire associe au Centre National de la Recherche Scientifique.

Annales de l’Institut Henri Poincaré-Section A-Vol. XXXVI, 0020-2339/1982/59/$ 5,00/© Gauthier-Villars

60 J. GINIBRE AND G. VELO

1. INTRODUCTION

The initial value problem for classical coupled Yang-Mills and scalarfields has been considered recently by several authors [2 ]- [8 ], [10 ]. In parti-cular in a previous paper [6 ], to the introduction of which we refer formore details, we have studied this problem in the so-called temporalgauge. The main results of [6] ] include the existence and uniqueness ofsolutions in small time intervals for arbitrary space-time dimension andthe existence and uniqueness of global solutions in space-time dimension1 + 1 and 2 + 1. All these results hold both in global and local spaces.In the latter case the initial data and the solutions are required to satisfyonly local regularity conditions in space, but no restriction on theirbehaviour at infinity. Remarkably enough all these results can be provedwithout using the elliptic constraint that generalizes Gauss’s law.

In this paper we take up the same problem in the Lorentz gauge In addition to its intrinsic interest, this gauge condition is imposed as aconsequence of the Lagrange equations if the Yang-Mills field is massive.In this gauge the situation seems to be less favourable than in the temporalgauge : we are still able to prove existence and uniqueness of solutions ofthe Cauchy problem in small time intervals for arbitrary space-time dimen-sion ; however we are able to prove existence and uniqueness of globalsolutions only for space-time dimensions 1 + 1. Furthermore, even in

that case, the proof makes explicit use of the elliptic constraint mentionedabove. The additional problems posed by the Lorentz gauge as comparedwith the temporal gauge can best be seen by considering pure Lorentzgauges. If the gauge group is SU(2), the Lorentz condition becomes the

equation of motion for the O(4) non linear a-model. More generally, onecan study the Cauchy problem for the O(N) non linear a-model as well asfor the so-called CP(N) and GC(N, p) models. This will be done in a sub-sequent paper. In all these models, in the same way as for the Yang-Millsfield in the Lorentz gauge, our proof of global existence works in dimen-sion 1 + 1 and breaks down in higher dimensions.

In the same way as in the case of the temporal gauge treated in [6 ],the finite propagation speed and the presence of an elliptic constraint,which produces long range effects in the massless case, lead naturally tostudy the problem in local spaces, where only local regularity conditionsare imposed on the initial data and the solutions.The methods used here are similar to the ones used in [6 ]. The local

problem in global spaces is treated by the method of [9 ]. The local theoryin local spaces relies on Section 3 of [6 ]. The global existence problemboth in global and local spaces is treated by the standard method of a prioriestimates. The need to use the elliptic constraint complicates the proof:

Annales de l’Institut Henri Poincaré-Section A

61THE CAUCHY PROBLEM FOR COUPLED YANG-MILLS AND SCALAR FIELDS

it requires a local version of these estimates and an iterative restrictionand extension procedure to construct the global solution.The results of this paper have been announced in [5 ]. For technical

reasons they will be derived here in a slightly different formalism (seeRemark 2 .1 ).The paper is organized as follows. In Section 2 we introduce the notation,

choose the dynamical variables, write the equations in suitable form, andtreat the local problem in global spaces for arbitrary space-time dimension.In Section 3 we extend the previous treatment to the theory in local spaces.In Section 4 we study the problem of global existence, both in global andlocal spaces.

2. THE CAUCHY PROBLEM IN GLOBAL SPACESFOR SMALL TIME INTERVALS

In this section we begin our study of the initial value problem for theclassical Yang-Mills field minimally coupled to a scalar field, in the Lorentzgauge = 0. We first introduce some notation. The Yang-Mills potentialA,(t, x) is a function from n + 1 dimensional space-time to the Lie algebra ~of a compact Lie group G. The corresponding field is

where [ , ] denotes the commutator in ~. We assume the existenceof a non degenerate positive definite bilinear form in ~ denoted by ( . , . ) ,invariant under the adjoint representation of the group. The scalar field

x) belongs to a unitary representation of G in a finite dimensionalvector space ~ . We also denote by ( -, -) the invariant scalar productin ff and we use the same notation for an element of ~ and for its represen-tative in ~ . We shall write ( B, B )> = ~ B 12. We use the same notation D~for the covariant derivative in ~ where D~ + e [All’. ] and in ~ whereD~ = a~ + We use the metric gl1v with goo = 1, gii = - 1, gl1v = 0for ~u ~ v.The field equations are the variational equations associated with the

Lagrangian density

where V is a real CC1 function defined in (~ + with V(0) = 0. The equationsare

Vol. XXXVI, n° 1-1982.


and V’ is the derivative of V. We shall work in the first order formalismand choose as dynamical variables the quantities Ao, A=={ A~, 1

and ~={~,1~/~}(~~, has to be thought of as Du~). The variational equations and the Lorentzcondition can be rewritten as the following system of equations of motion

supplemented by the constraints

Here Jo and J = {J j, 1 j n ~ are supposed to be expressed in termsof ~, and 1/1 j; R is the 1 x n matrix operator with entries R( = 2014 ~and RA is the n x n(n - 1)/2 matrix operator with entries

The system of equations (2.4) and (2. 5) can be written more compactly as

where



and T~, and can be read directly from (2 . 4) and (2 . 5). The fields u~and uq, take values in the finite dimensional vector spaces

respectively, so that u takes values in 1/ = 1/A EB The Cauchy problemfor the equation (2.9) can be transformed into the integral equation

where uo is the initial condition,

with cv = (RR*)1/2 = (- ~)1~2, = and is given by asimilar formula (see (2.13) of [6 ]).

REMARK 2.1. - In the Lorentz gauge there is a large flexibility in thechoice of the dynamical variables. For instance one could take the quan-tities Ao, A = {A~ 1 # j # n ) , B == { B~ 1 ~’ ~ } (to be thoughtof as B~ = DoA), 4> and (to be thought of as = Do4», therebyobtaining the equations

supplemented by the constraint (2.7). In (2.13), (2.14) and (2.7), F~," J~and 03C8j --_ are supposed to be expressed in terms of the dynamicalvariables. In the choice leading to (2.13), (2.14), as well as in the previouschoice leading to (2.4), (2.5), the Lorentz condition has been used as the

Vol. XXXVI, n° 1-1982. 3


equation of motion for Ao. Another, more standard choice, consists insubstituting the Lorentz condition also into the variational equationKo = 0, thereby converting it into an evolution equation for Ao. Onecan then choose as dynamical variables A~ (0 ~ ~ ~ n), B~ (0 ~ ~ ~ n)(to be thought of as BJ1 = and The equations of motion for Alland B~ become

while the equations for 03C6 and are still given by (2.14). These equationsmust be supplemented by the constraints = 0 and Ko = 0 (at t = 0).Up to a minor difference in the definition of B 11’ this last formalism is theone used in [5 ]. It has the well-known drawback that the energy is boundedfrom below only if the constraints are satisfied.We now define the spaces in which we shall look for solutions of the

integral equation (2.11). For any integer k, we define the space ~k ofthose u such that, componentwise, Fllv and t/1 Ji belong to the Sobolev space

while All and 03C6 belong to the Sobolev space Hk - More precisely, for any u 4>, 03C8 } we define the norm ~ u IIin ~’‘ as follows:



Here a is a mutiindex of space derivatives denotes the norm

in Lq --_ When equipped with the norm (2.16), %k is a Hilbertspace which is a direct sum of usual Sobolev spaces. The norm (2.16) isequivalent to the simpler looking norm

However the choice of (2.16) is better adapted to the free evolution U(t)as will be seen in Lemma 2.1 below. For brevity we have not appendedan index to the norm ~ ~ ~ . . Furthermore, from now on we shall omitthe I I when appearing in an LP norm.

In order to prove the existence of solutions of (2 .11 ) we need the follow-ing properties of U(t).

LEMMA 2.1. - For any integer k, U(t) is a (bounded) strongly continuousone-parameter group in %k and, for any t E R, for any u E U(t) satisfiesthe following estimate

where

Proof - The proof runs in the same way as that of Lemma 2 .1 of [6 ],after noticing that, for any k, Mk(U(t)u) and Pk(U(t)u) are constant in time,while

We can now prove the existence of local solutions of (2.11). For anyinterval I and any Banach space ~ we denote by E8) the space of conti-nuous functions from I to and, for any positive integer I, we denoteby ~) the space of I-times continuously differentiable functionsfrom I to ~. For compact I, ~) is a Banach space when equippedwith the Sup norm.

PROPOSITION 2 .1. - Let k = [n/2 + 1 ] ([/L] ] is the integral part of ;w)~V E ~k + 1 ( ~ + ) and Then, there exists a To > 0, depending onI , such that (2.11) has a unique solution u e W( [ - To, To ],

Proof - The proof runs in the same way as that of Proposition 2.1of [6 ] after noticing that the multiplication by a function of Hk is a boundedoperator both in H~ and in H’‘ -1. Q. E. D.Under additional regularity assumptions on the potential V and on the

initial data one can prove additional regularity properties of the solutions.

PROPOSITION 2.2. - = [n/2 + 1 ], let and let

uo E Let I be an interval of ~ containing the origin and let u E fk)be a solution of (2.11). Then u E ~l(I, ~~~ - ~) for any 1, 0 ~ I ~ k’.



Proof - The proof is similar to that of Proposition 2 . 2 of [6], butslightly more complicated because of the choice of the spaces. We shalltherefore present the estimates in some detail. First we prove that ffk’)by induction. Let therefore u E for some 1, k 1 ~ k’. Wewant to show that u E or equivalently that, for any multiindex x,with a == 1 - belongs to ffk). Now v03B1 E Kk - 1) and v°‘satisfies the equation

where a labels the various components of u, and g" is a polynomial in thespace derivatives of u of order at most 1 - k - 1 and in the derivativesof f with respect to u of order at most I - k. We want to show that (2.27)considered as a linear integral equation for vrx has a unique solution bothin and ). This will prove that ~03B1u E and completethe induction. For this purpose it will be sufficient to show that

belongs to ffm) if v" belongs to ffm) for m = k, k - 1, and that

belongs to ffk). We consider the terms in h" and g" containingonly A and F : the terms containing 4> and g/, including those coming fromthe potential V, can be treated in the same way. Now h" contains terms ofthe type A/" and Fa", where a" and f " are the A and F componentsof By the induction assumption we know that componentwise

2). We take a" E and f" E with m being either k or k - 1, and we have to show that Aa" belongsto Hm) and that and Fa" belong to H"~~ ~). This is a conse-quence of the Sobolev inequalities, which imply that multiplication bya function in Hk is a bounded operator in Hr for - k S r k and thatthe product of two functions in Hk -1 lies in Hk - 2, with continuity. Similarly,the terms with A and F in .g" have the form and with

1J (xj I + = 1 - k. We have to show that for Hl-I)and 2), these terms lie in Hk) and respectively.This again follows from the Sobolev inequalities as above.

The end of the proof is the same as that of Proposition 2 . 2 of [6 ] withappropriate changes. Q. E. D.

So far we have considered the system (2.4) and (2.5) without takinginto account the constraints (2. 6), (2.7) and (2. 8). We now show that theconstraints are preserved in time.



Let I be an interval of ~ containing the origin and let u E be a

solution of (2.11). Let Q be an open subset of possibly [R" itself. Let uosatisfy (2. 6) (resp. (2. 8)) in Q at t = 0. Then u(t) satisfies (2.6) (resp. (2. 8))in Q for all If in addition uo satisfies (2. 7) in Q at t = 0, then u(t)satisfies (2. 7) in Q for all t ~ I.

Proof - From Proposition 2.2 it follows that the quantities

and f1j - belong to componentwise. Furthermore,by (2.4) and (2.5), they satisfy

with initial conditions = 0 and = 0 in Q. From the fact that

Ao E H~) and that multiplication by a function in Hk is a boundedoperator in Hk - 2(S~~ it follows that = 0 and = 0 in S2 for all times.

Similarly, from Proposition 2.2 it follows that

componentwise. From (2.4), (2.5) and the previous constraints it followsthat

in Hk- ~(Q). Using again the fact that multiplication by Ao E Hk is a boundedoperator in Hk - 2(S~), we see that K° satisfies (2. 30) in Hk - 2(SZ). Togetherwith the initial condition K° = 0 in Q at t = 0, (2.30) implies that K°vanishes in Q for all times. Q. E. D.

3. THE CAUCHY PROBLEM IN LOCAL SPACESFOR SMALL TIME INTERVALS

In this section we extend the theory developed in Section 2 to a theoryin local spaces. The general framework is described in Section 3 of [6] towhich we refer for details. For the convenience of the reader we recall afew definitions. We call dependence domain any open subset Q x [Rn

such that, for any (t, x) E Q, Q contains the set

The sections of Q at fixed time t are denoted by



and are open. For any open ball Q = B(x, R) with center x and radius Rand for any t E R, we define

with the convention that B(x, R) is empty if R ~ 0. We also define

which is a dependence domain for t > 0.We next define the local spaces. Let k be an integer, k a 1. In Section 2

we have defined %k as a direct sum of usual Sobolev spaces Hm withm = k or m = k - 1. For any open (respectively bounded open) set Q ci we define (respectively as the corresponding direct sumof the Sobolev spaces (respectively The space is aHilbert space with norm

where and Pk,n are defined by formulas similar to (2.17), (2.18),(2.19) where however the L2-norms are now taken in For Q = [?",we shall denote by For suitable dependence domains Q,we shall be interested in Kkloc-valued solutions of equation (2.11) in Q inthe sense of Section 3 of [6] (see especially (3.17) of [6 ]).The crucial property of the free evolution is the following local version

of Lemma 2.1.

LEMMA 3.1. - For any integer k, k a 1, U(t) is a strongly continuousone-parameter group in Kkloc and, for any t e R for any open ball Q c fR",for any u E the following estimate holds

where is defined by (2.25).

Proof - The proof runs in the same way as that of Lemma 4.1 of [6 ].One first proves that, for any smooth u, for any non negative integer kand any open ball Q,

,

From there on the proof proceeds as that of Lemma 2 .1 of [6 ]. Q. E. D.We can now state the local existence result that follows from the general

theory presented in Section 3 of [6 ].

PROPOSITION 3.1. - Let k = [n/2 + 1 ], and Then there exists a dependence domain Q with Q(0) = [?" and a (unique)



Kkloc-valued solution of the equation (2.11) in Q (in the sense of Section 3of [6 ]).The regularity result of Section 2 can also be extended to the theory

in local spaces. We denote by rn the operator in of restriction to Q,namely the operator of multiplication by the characteristic function of Q.

PROPOSITION 3. 2. = [n/2 + 1 ], let V E ~k~(f~ + ), let Q bea dependence domain, let uo E Kk’loc(Q(0)) and let u be a Kkloc-valued solutionof (2.11) in Q. Then, for any t > 0 and any open ball Q in [R" with closure

contained in Q(t), rgzu E [0, t ], for any 1, 0 ~ 1 ~ k’. In parti-cular u is a Kk’loc-valued solution of (2.11) in Q.

REMARK 3.1. - For 1 = k’, the statement of Proposition 3.2 involvesthe space yet undefined. The fact that E ~~(Q)) meansthat each component of rQA and belongs to L~(Q)) and thateach component of r03A9F and rQt/J belongs to H-1(0)), where H-1(0)is defined as in [11 (p. 213).The proof of Proposition 3.2 is essentially the same as that of Propo-

sition 4 . 2 of [6 ] and will be omitted. -

Finally the constraints are locally preserved in time also in the localtheory.

PROPOSITION 3 . 3. - Let k = [n/2 + 1 ], let V E let Q be depen-dence domain, let u° E and let u be a Kkloc-valued solutionof (2.11) in Q. Let T > 0 and let Q be an open ball in ~n with closurecontained in Q(T). Let uo satisfy (2.6) (resp. (2 . 8)) in Q. Then u(t) satisfies (2.6)(resp. (2. 8)) in Q for all t E [0, T ]. If in addition uo satisfies (2. 7) in Q, thenu(t) satisfies (2. 7) in Q for all t E [0, T].

Proof - Identical with that of Proposition 2.3. Q. E. D.

4. EXISTENCE OF GLOBAL SOLUTIONS

In this section we prove the existence of global solutions of (2 .11 ) forn = 1 and make some comments on the case n = 2. The proof relies ona priori estimates of the solutions in xk. As mentioned in the introduction,the derivation of some of these estimates requires the use of the ellipticconstraint K° = 0 (see (2.2)). As a consequence the method of proof ofglobal existence in local spaces used in Proposition 5 . 3 of [6 ] no longerworks since it is based on a cut off procedure which is not compatiblewith the constraint. We shall therefore use a different method which

requires in particular a local version of the basic estimates. These localestimates will be derived in Propositions 4.1 and 4. 2 below. Their deriva-tion requires integration by part in truncated cones of the type (3.3).



For this purpose we need smooth approximations of the solutions of (2 .11 ).We therefore begin by introducing a regularization procedure.Let h be a real positive even function in 1

= 1 and

let ~"~M = mnh(xm). Together with the original problem associatedwith the Lagrangian density (2.1), we also consider the similar problemobtained by formally replacing p) in (2 .1 ) by V( htm~ * ~ ~ p) where *denotes convolution in !R". The corresponding variational equationsare (2.2) and

where

They are not compatible in general since the current is no longer conserved.However, if one drops the equation K° = 0 and chooses the dynamicalvariables and the gauge condition as in Section 2, the remaining systemcan be written as

supplemented by the constraints (2.6) and (2.8), where j(m) is obtainedfrom f by replacing ~V’( ~ ~ 12) by h~m~ ~ g~rn~(~). We shall approximatesolutions of (2.11) with uo E ffk for suitable k by solutions of the equation

LEMMA 4.1. - Let k = [n/2 + 1 ], and Then :

(1) There exists a To > 0, depending on II u0 I 1 but independent of m,such that (2.11) has a unique solution M6~([- To, To ], ffk) and (4 . 4)has a unique solution E ~( [ - To, To ],

(2) Let I be a closed interval of R containing the origin and let Kk)be a solution of (2.11). Then, for m sufficiently large (possibly dependingon I and uo), there exists a unique solution of (4. 4) in ffk) and tends to u in %k) as m tends to infinity.

(3) Let I be an interval of R containing the origin and let u E ffk)be a solution of (4 . 4). Then, for any I ~ 1, u E 1(I,

Proof (1) The proof is the same as that of Proposition 2.1. The unifor-mity in m follows from the fact that the basic estimates can be taken uniformin

(2) The proof is similar to that of the continuity of the solution of (2.11)with respect to the initial data [9 ], with the additional complication thatthe equation itself depends on m. One first proves the statement for smalltime intervals, as considered in part (1), and then extends it to the wholeof I by splitting I into a finite union of such intervals. That the second step



is at all possible follows from the existence of the solution u. We omitthe details.

(3) The statement follows by differentiating (4.4) once in which

yields (4. 3), and then by successive differentiation of (4. 3). The limitationto follows from the assumption on V. Q. E. D.

In all this section we shall assume that the potential satisfies the condition

for some a ~ 0. The first estimate we shall use is the local version of theenergy conservation. For any open ball Q in [Rn, for any

we define the local energy

where II. denotes the norm in Lq(Q), 1 q oo. We also define

If u depends on t, we denote by and Ea(t) the corresponding quantitiesassociated with u(t).

PROPOSITION 4.1. - Let k = [n/2 + 1 ], let satisfy thecondition (4. 5), let Uo E ffk, let I be an interval of [R containing the originand let u E ~(1, ffk) be a solution of (2 .11 ). Then, for any open ball Q in [R"and any t e I, u satisfies the estimates

where

Proof - By Lemma 4.1, for sufficiently large m, (4.4) has a uniquesolution E 1( [0, t ], ~’l) for any l ~ 1. In particular



is 1 of space-time componentwise. We now define

and

Using the field equations we obtain

where

and

We next integrate (4.14) in _the truncated cone Q(Q, t) (see (3 . 3)) andapply Gauss’s theorem. Since 0~ is outgoing on the side surface of Q(Q, t),we obtain

Now it follows from Lemma 4.1 that, when m tends to infinity, all termsin (4.17) have well-defined limits (in particular x) tends to zero).so that (4.17) becomes J

Similarly, for any re [0, t ], we have

Let

Applying Schwarz’s inequality to the last term on the RHS of (4.18) andincreasing the integration domain in the RHS of (4.19) we obtain

and



Substituting (4.22) into (4.21) we obtain

from which (4.8) and (4.11) follow by an elementary computation. Theestimate (4.9) then follows from (4.22). In order to prove (4.10) we firstnotice that from the field equations it follows that satisfies the relation

Integrating (4.24) over Q(Q, t), applying Gauss’s theorem, letting m tendto infinity and using (4.8), we finally obtain (4.10). Q. E. D.The previous proposition holds for any space-time dimension. We

now concentrate on the case n = 1 and estimate the components of Ain

PROPOSITION 4.2. - Let n = 1, let V E ~2(f~+) satisfy the condition (4.5),let Q = B(x, R) = (x - R, x + R) be an open interval in R with R > 1,let uo E $’1 satisfy the constraints (2 . 7) and (2 . 8) in Q. Let I be an intervalof R containing the origin and let be a solution of (2 .11 ).Then, for any satisfies the estimate

for some estimating function Po. independent of t.

and

We shall prove the result by deriving an a priori estimate on the quantity

This will follow from an integral inequality which will be derived by theuse of the cut off procedure of Lemma 4.1. By this lemma, for sufficientlylarge m, (4 . 4) has a unique solution E ~2( [0, t ], for any t ~ 1.

In particular, = (A~B F~m~, 1/1(m») is CC2 of space-time component-wise. By a straightforward computation, using the equations of motion,we obtain



where

and Kbm) is the quantity Ko computed for M~B We next integrate (4.28)on the truncated cone Q(Q, t) (see (3.3)) and apply Gauss’s theorem.Since B~(A~m~) is outgoing on the side surface of Q(Q, t), we obtain

We now let m tend to infinity. From Lemma 4.1 part (2), it follows thatand a~(u~m~) tend to eo{A) and respectively in CC( [0, t ],

We now show that Kbm) tends to zero in ~([0, t ], so that its contri-bution to the RHS of (4.30) tends to zero. For this purpose we first considerthe constraint (2 . 8) : we note that the quantity D lm~~~m~ _ ~, lm~ satisfiesthe equation

, ,, , ,,_B , . /_--" -

with the initial condition

in Q. From this and from Lemma 4 .1, part (2), it follows that tends to zero in ~( [0, t ], H 1 (SZ)) when m tends to infinity. We next considerthe quantity where is the current JI1 computed for t~. Fromthe equations of motion it follows that, for any C E rg,

It follows from the previous argument and from Lemma 4.1, part (2),that the first term in the RHS of (4. 33) tends to zero in CC( [0, t ], L 2(Q));the second term also tends to zero in CC( [0, t ], L~(Q)) by Lemma 4.1,part (2). We finally turn to From the equations of motion it followsthat

and therefore

with



in Q. In particular tends to zero in From this, from (3.34)and the fact that tends to zero in ~( [0, t ], L~(Q)), it follows that Kbm)also tends to zero in CC( [0, t ], Taking the limit m - oo in (4.30), we obtain therefore

The quantity 17(u) contains terms of various types and we now estimatetheir contributions to the space integral at time ~ in (4.36). For brevitywe omit the time ~ and the spacetime indices; all space integrals and LPnorms are taken in Q~(r). We obtain

We estimate the L CYJ norms through the local one dimensional Sobolevinequality (covariant or non covariant)

’

where L is the length of the interval under consideration, in the presentcase L = 2(R - i) ~ 2(R - t) ~ 2. We then estimate the norms 111/1112,t!~!~ !)~!t2 by Proposition 4.1 and obtain from (4 . 36) asublinear inequality for the function of t defined by the L. H. S. The resultfinally follows by a straightforward application of Gronwall’s inequality.

Q. E. D.For n = 1, Propositions 4.1 and 4.2 provide an a priori control of F

and 1/1 in L2 (componentwise) and of A in In order to complete thea priori control of u in r.~’ 1, which is the relevant space for n = 1, it sufficesin addition to estimate 4> in Hi. This is done easily by using the equationsof motion and the fact that 4> E L2 and A E Hi, so that the norms definedwith ordinary and covariant derivatives are equivalent (see Lemma 5.6of [6 ]).

REMARK 4.1. - In Propositions (4.1) and (4.2) we have given a localversion of the estimates needed to prove the existence of global solutions.As mentioned at the beginning of this section, this local form of the estimates



is necessary to establish global existence in local spaces. If one is interestedin solutions in global spaces, one needs these estimates only in globalform, namely with Q replaced by In this case the proof simpler (comparewith Lemma 5.1 of [6 ]).We are now in a position to prove the global existence result in the

case n = 1. We first state the result in global spaces.

PROPOSITION 4. 3. - Let n = 1. let V E CC2(lR +) satisfy the condition (4. 5),let uo E %1 satisfy the constraints (2. 7) and (2. 8). Then the equation (2.11)has a unique solution u in CC(lR, Xl) and u satisfies the constraints (2.7)and (2.8) for all times.

Proof - The result follows by standard arguments from Proposition 2 .1and from the estimates of Propositions 4.1 and 4.2 in the global formdescribed in Remark 4.1. Q. E. D.The global theory in local spaces requires a more careful treatment.

In the proof we shall need the fact that, for any open ball Q = B(x, R),there exists an extension in which is a bounded map from gk(Q) to ~’‘such that = u for all u E and that

for all Furthermore, one can choose Ci(Q) independent of xand uniformly bounded for R ~ 1 (see Sect. 3 of [6 ] and [1 ]) : Clfor R ~ 1. In what follows we shall make such a choice. We can now statethe global existence result.

PROPOSITION 4 . 4. - Let n = 1, let V E ~2(1R +) satisfy the condition (4. 5),let uo E satisfy the constraints (2. 7) and (2. 8). Then the equation (2 .11)has a unique solution u in and u satisfies the constraints (2.7)and (2.8) for all times.

Proof - We first prove that for any open interval Q = B(0, R) = (- R, R),there exists a (unique) K1loc-valued solution uR of (2.11) in the truncateddouble cone Ai(R) * Q(Q, R - 1) u Q(Q, - R + 1) (see (3. 3)), in thesense that the restriction of uR to Q(Q, R - 1) is a K1loc-valued solutionof (2.11) in Q(Q, R - 1) and that a similar property, defined by an obvioussymmetry, holds in Q(Q, - R + 1). For this purpose, we first note thatif u is a K1loc-valued solution of (2.11) in Ai(R), then by Propositions 4.1and 4.2, u satisfies an estimate of the type

for some estimating function y~ uniformly in t for 0 ~ ~ ~ R - 1. Asa consequence, for any t with 0 # R - 1,



It follows now from Proposition 2.1 and its proof that, for any s,0 x R - 1, the equation

has a unique solution in To, s + To ], Jf~), where To can be takenindependent of s because of (4.40). We choose such a To and constructnow a solution of (2 .11 ) in Ai(R). For brevity we consider only the caseof positive time. Let tj = jTo, 0 ~’ ~ [(R - l)/To] ] and Ij = t~+ 1 ].Applying Proposition 2.1, we define v J E %1) as the solution of (4 . 41 )with s = tj and with u(s) replaced by uo for j = 0 and by for j l.Clearly, if we define u(t) = v j( t) for t e 1~, then the restriction t~ of u toQ(Q, R - 1 ) is a K1loc-valued solution of (2 .11 ) in Q(Q, R - 1 ). Combiningthis construction with the similar one in Q(Q, - R + 1), we obtain theannounced solution ~ in Ai(R).The end of the proof consists in taking an increasing sequence { Rn }

tending to infinity and gluing together the solutions uRn in constructedas above. The argument is the same as in the proof of Proposition 5.3of [6] to which we refer for more details. Q. E. D.We conclude this section with some comments on the case of dimen-

sion n = 2. In this case (as well as for n = 3), the relevant space is %2 sothat one needs to control F and 03C8 in H1 (componentwise) and A and 03C6in H2. In order to estimate F and 03C8 in it is natural to consider thequantity Ei 1 as in [6 ] (see (5 . 2) of [6 ]). For n = 2 and if the Yang-Millsfield is massless (K = 0), Ei 1 which is gauge invariant, can be controlledexactly in the same way as in the temporal gauge, both globally (seeLemma 5 . 2 of [6 ]) and locally. However, even in the massless case, theproof of Proposition 4.2 breaks down for n = 2 and we are unable tocontrol A in If the Yang-Mills field is massive, the difficulty occursalready at the level of Ei, and we are unable to control Ei in that case.In all cases, the last step, namely the control of A in H2, knowing that F,

and A are in can be done easily.

ACKNOWLEDGMENTS

One of us (G. V.) is grateful to K. Chadan for the kind hospitality atthe Laboratoire de Physique Theorique in Orsay, where part of this workwas done.

REFERENCES

[1] R. ADAMS, Sobolev Spaces, Academic Press, New York, 1975.[2] D. CHRISTODOULOU, C. R. Acad. Sci. Paris, t. 293, 1981, p. 139-141.[3] Y. CHOQUET-BRUHAT, D. CHRISTODOULOU, Existence of global solutions of the

Yang-Mills-Higgs and spinor field equations in 3 + 1 dimensions, C. R. Acad. Sci.Paris, t. 293, 1981, p. 195-200, and Ann. Scient. E. N. S. (in press).


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[4] D. EARDLEY, V. MONCRIEF, The global existence of Yang-Mills-Higgs field in4 dimensional Minkowski space. I. Local existence and smoothness properties, II.

Completion of the proof, Commun. Math. Phys. (in press).[5] J. GINIBRE, G. VELO, Phys. Lett., t. B99, 1981, p. 405-410.

[6] J. GINIBRE, G. VELO, The Cauchy problem for coupled Yang-Mills and scalar fieldsin the temporal gauge, Commun. Math. Phys. (in press).

[7] R. T. GLASSEY, W. A. STRAUSS, Commun. Math. Phys., t. 81, 1981, p. 171-188.[8] R. KERNER, Ann. IHP, t. 20, 1974, p. 279-284.[9] I. E. SEGAL, Ann. Math., t. 78, 1963, p. 339-364.

[10] I. E. SEGAL, J. Funct. Anal., t. 33, 1979, p. 175-194.

[11] F. TRÈVES, Basic linear partial differential equations, Academic press, New York,1975.


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The Cauchy problem for coupled Yang-Mills and scalar