Nonlinear Partial Differential Equations in Geometry and Physics

£(t) = £(0) . This provides us only with an L2 bound for the derivatives of
~(( the scalar product in N. Since we are in 1 + 1 dimensions T has only the components Too, TOl = T lO , Tn. Moreover,

since T is trace-less in Ml+l, we have Too = Tn. Now recall that T verifies the divergence equation o,6Ta,6 = O. Hence,

OtTOO = OxTOl OtTOl = oxToo and therefore Too is a solution of the linear wave equation in Ml+l, DToo = O. One can now easily check that T remains bounded for any t > 0 provided that Too,otTOO are bounded at t = O. Since Too = ~(lot
On the Regularity of Classical Field Theories in Minkowski Space-Time R3+ 1

43

4. The problem of global regularity for the Yang-Mills equation in Minkowski space-time, for sufficiently regular initial data, was solved in a beautiful paper by Eardley-Moncrief [E-M]. The proof required an insightful observation concerning the structure of the nonlinear terms of the Yang-Mills equations expressed in the Cronstrom gauge. It also uses in an essential way the explicit form of the fundamental solution of the wave equation in the fiat Minkowski space-time. Finally, in view of the same simple minded scaling analysis we can easily check that, relative to the total ADM mass 12 , the Einstein field equations are supercritical 13 . On the other hand the reduced Einstein equations are critical. General Conjecture. (i) The interesting14 subcritical field theories are regular for all smooth data 15 . (ii) Under reasonable restrictions the critical field theories are regular for all smooth data. (iii) "Sufficiently small" solutions to the supercritical field theories are regular. The general, large, solutions develop singularities in finite time. The first two parts of the conjecture have been, so far, verified only for the critical and subcritical scalar wave equation, for the Yang-Mills equations in ]R3+1, and for the spherically symmetric and equivariant Wave Maps equations with certain geometric assumptions of the target manifold. It is believed by many that the scalar nonlinear wave equations are regular even in the supercritical cases. This is based on the fact that numerical calculations fail to produce large amplitudes. It is however possible that the break-down phenomenon is unstable and thus impossible to detect by numerical calculations. The regularity of solutions for small initial conditions has been verified in the case of supercritical Wave Maps 16 and in the case of the Einstein vacuum equations. In these Lectures I will present our attempts to attack the Conjecture, or rather the regularity part of it, from the local point of view the "well posedness" of the initial value problem. According to this, if we can establish that the LV.P. is well posed in the energy norm 17 then global regularity is automatically implied by Energy conservation. 12)

The total ADM mass is the positive conserved quantity for the Einstein equations playing the role of total energy.

13)

We assign to the space-time metric the scale L O and remark that the ADM mass

E

= 1~7r }~~ for L: (Oigij - OJ 9i') NJ da

has the "supercritical" scale L 1.

',J

14)

What I mean by interesting may be difficult to define. It should include most well known examples of relativistic field theories such as the ones discussed above. In what follows I will omit the word.

15)

This means that solutions which are smooth in the past cannot become singular.

16)

see discussion below

17)

This means that we have local existence, uniqueness and continuous dependence of the data in the energy norm. See also the more precise definition given below.

s. Klainerman

44

Conjecture 1. For all subcritical field theories and, in an appropriate sense, for critical field theories the 1. V. P. is well posed in the energy norm.

The problem of well posedness is that of local in time existence, uniqueness and continuous dependence for the development of an initial data set. The classic results, based on energy estimates and Sobolev inequalities, apply to very large classes of non-linear wave equations but require too much differentiability on the initial data. Take, for example, the case of the Wave Maps equations 18 . Consider initial data sets (¢o, ¢d such that,

11(¢0, ¢dIIH8(Rn)

= II¢(O, ·)IIHs+l(Rn) + 118t ¢(0, ·)IIHs(Rn) ::; ~.

(2.1)

The classical local existence result reads as follows: Proposition 2.1 Let So be a fixed exponent > ~. There exists a time T > 0, depending only on So and~, such that any initial data (¢o, ¢l) verifying (2.1) with s = So admit a unique development defined in a time slab [0, T] x Rn which verifies, for t ::; T, and s 2': so, all

°: ;

11¢(t, ·)IIHs+l(Rn) + 118t ¢(t, ·)IIHs(Rn) ::; C(II¢(O, ·)IIHs+l(Rn) + 118t ¢(0, ·)IIHS(Rn)) (2.2) Definition. In what follows we will refer to a result of the type discussed in Proposition 2.1. by stating that the 1. V. P is well posed in Hl+so . According to the Proposition the solution can be extended as long as the norm (2.3) remains finite. Moreover the solution preserves the HS regularity of the initial conditions for all s 2': so. Let T* = T*(so,~) be the supremum of the values of T for which the norm in (2.3) remains finite. This is called the life-span of solutions corresponding to all initial data verifying (2.1.) The Proposition 2.1 asserts in fact that, T*(s,~)

>

°

(2.4)

for all 8 > ~ and ~ > 0. Similar results hold for the Yang-Mills and Einstein-Vacuum equations. For example, in the case of the Yang-Mills equations in M1+3 relative to the Lorentz gauge, the classical result of local existence and uniqueness requires that the initial data set (A,E) be in H so(R3 )19, i.e.

II(A, E)IIH o(R3) S

with 18) 19)

80

=

IIAIIH

8

o+1(R3)

+ IIEIIH o(R3) S

::;

~,

(2.5)

> ~.

For simplicity throughout the discussion we shall assume that the entire manifold

N is covered by one coordinate chart.

In fact it suffices to work with local H S spaces.

On the Regularity of Classical Field Theories in Minkowski Space-Time R3 +1

45

For the Einstein-Vacuum equation the classical local existence result, obtained by Y. Chocquet-Bruhat in wave coordinates, requires the metric g to belong, locally, to HSo and the second fundamental form k to HSO-l, 80 > ~. In all the above examples the classical local existence result, which is based only on energy estimates and Sobolev inequalities, requires one more derivative 20 than what we hope to be the optimal result. The statement below is a more general and version of Conjecture l. Conjecture 2. For the field theories of scaling exponent 8 the initial value problem is, in an appropriate sense, locally well posed in the Sobolev spaces Hs+1, 8 > S. Moreover, in a weak sense, the I. v.p is well posed in HH1.

Let us interpret Conjectures 1 and 2 in the case of the Wave Maps equations. Define the number, 8e = inf{8 E R, such that there exists ~

> 0 for

whichT*(8,~)

> o.}

(2.6)

To prove Conjecture 2 one would have to verify that this "optimal regularity exponent" is equal to the scaling exponent defined before, i.e. 8e

and that,

n-2

=8=--

2

n-2

T*(-2-'~)

> O.

(2.7)

However the statement of Proposition 2.1 would have to be modified in the critical case 8

= 8e = 8 =

n-2 -2-'

To see this consider the blow-up example provided by Shatah [Shl. Indeed let ¢( t, x) be the solution, constructed by Shatah, which breaks down at t = T*. Then, in view of the scaling properties of (W.M.), the maps ¢)..(t, x) = ¢(At, AX) are also solutions of (W.M). It is easy to check that for all A :::: 1,

with C a positive constant independent of A. On the other hand, each ¢).. breaksdown at precisely T; = T*. Thus for large A the life span of the solutions, n-2 corresponding to a set of initial data bounded in the H-2- norm, tends to zero. This shows that in fact (2.2) cannot hold for all ~ > O. For large ~ we can still have local, and even global solutions, without the estimate 21 (2.2.)

t

20)

In fact it is slightly more than a derivative.

21)

See the results of Shatah-Struwe [Sh-Stru2] and Kapitanski [Ka2] for the critical nonlinear scalar wave equation.

46

S. Klainerman

A precise formulation of Conjectures 1 and 2 should contain a correct formulation of well posedness in the critical case s = Se. The formulation should also take into account the geometric properties of the equations, indeed without a proper geometric interpretation of the Sobolev spaces HS the statements of Conjectures 1 and 2 are meaningless 22 • According to Conjecture 2 the optimal local existence result for the YangMills equations requires data (A, E) for which the norm II(A, E)IIHS(R3) < 00 ,s ~ - ~, defined in (2.5), is finite. In the case of the Einstein field equations the optimal local existence result should be expressed in terms of a norm defined in terms of some Sobolev type norm on (g, k) involving no more than ~ derivatives of g and ~ derivatives of k. As we have discussed above the classical local existence theorem requires one more derivative than Conjecture 2. Gaining back this derivative is an extremely challenging task requiring, on one hand, the development of new analytic methods and, on the other hand, a deep understanding of the geometric structure of the nonlinear field theories. However one may ask, in the supercritical case, whether Conjecture 2 has anything to do with the general problem of break-down. Leaving aside the development of new methods which, I believe, are and will be generated in the process of gaining the above mentioned derivative, Conjecture 2 is also connected with the regularity of small data for supercritical field theories. Here is in fact the following more precise reformulation of part (iii) of the General Conjecture. Conjecture 3. For supercritical field theories "small data" have globally regular and "asymptotically free" developments.

By small data we mean smallness of an appropriate global, weighted, Sobolev norm. By asymptotically free we mean that the corresponding solutions behave, for large time, like solutions of the underlying linear problem. The Conjecture states the fact that the developments corresponding to small perturbations of the trivial initial data set remain close, in an appropriate sense, to the trivial solution. We state below two results of this type. The first concerns the Wave Maps equations: Theorem 2. Consider the Minkowski space-time M1+ n , n ~ 2, and the initial smooth maps fo, h defined form Rn with values in a sufficiently small neighborhood of a point p in the target manifold N. There exists a global smooth map ¢ : M1+n ---t N, verifying the initial conditions ¢(O,·) = fo, 8d)(O,') = h. Moreover the image of the map ¢ remains concentrated in a small neighborhood ofp·

The result was originally proved by Kovalyov [Ko]. It was later extended by Sideris lSi] to the case of maps which remain close to a geodesic in N. 22)

This is the case of the geometric theories such as Wave Maps, Yang Mills, Einstein equations etc.

On the Regularity of Classical Field Theories in Minkowski Space-Time R3+ 1

47

In the case of the Einstein equations Conjecture 3 is precisely the problem of stability of the Minkowski space-time. Namely the Minkowski space-time M3+1 is a special solution of (E-V) free of singularities. If an initial data set 'Ho, g, k is flat 23 its development is precisely M3+ 1 . It is thus natural to ask what happens to the developments of initial data sets which are small perturbations of a flat initial data set. This problem is important from two points of view. First, all attempts 24 to find explicit, non-flat, solutions of (E-V) equations, in the asymptotically flat regime, have lead to singular space-times. Second, any asymptotically flat initial data set can be interpreted, outside a sufficiently large relatively compact set K, as a small perturbation of the flat initial data set. Thus the study of the global stability of Minkowski space-time is also the study of the asymptotic properties of the development of any asymptotically flat initial data set outside the future set of a sufficiently large set K c 'H. The problem of the stability of Minkowski space-time has been recently addressed in my joint work with D. Christodoulou [Ch-K12]. The result which we were able to prove asserts the following, Theorem 3 [Ch-K12]. Any S.A.F.25 initial data set which satisfies, in addition, a Global Smallness Assumption, leads to a unique, smooth solution of the EinsteinVacuum equations, which is a geodesically complete development of the initial data. Moreover, this development is globally asymptotically fiat, by which we mean that its Riemann curvature tensor approaches zero26 on any causal or space-like geodesic, as the corresponding affine parameter tends to infinity. The global smallness assumption requires that an appropriate weighted L2_ norm of up to 2 derivatives of the curvature tensor of g and 3 derivatives of k are small. The smallness assumption in Theorem 2 requires an appropriate weighted L 2 -norm of more than n + 1 derivatives of the initial data. Both results are clearly not optimal. We expect that the optimal result should require no more derivatives than needed in the problem of local well posedness. This is the content of the following stronger version of Conjecture 3. Conjecture 4. There exists an appropriate global L2 norm, involving the minimum number of derivatives of the data for which the I. V.P. is locally well posed, whose smallness implies global regularity and some weak version of "asymptotic freedom".

= O.

23)

In other words 9 is the euclidean metric and k

24)

In particular this is the case for the two parameter family of stationary solutions, called the Kerr family, which are all singular with the exception of the trivial member of the family - the Minkowski space-time.

25)

We note that the precise fall-off conditions of the initial data set are in fact given in £2- weighted norms.

26)

Our result gives precise information on the rate of decay of different components of the curvature tensor.

48

S. Klainerman

Of course we expect that if the "smallness condition" is not satisfied solutions could break-down. In the case of Wave Maps we do have the singularity results of Shatah [Sh]. However we are very far from understanding how the singularities form in general. In the case of General Relativity we are still in a very primitive stage of understanding how black holes and singularities form. We have the famous incompleteness theorem of Penrose which asserts that if an initial data set of a space-time verifying the Einstein field equations (with very general assumptions on the energy-momentum tensor T) has a trapped sphere 27 then some outgoing null geodesics normal to S must be future-incomplete. It is nevertheless not at all clear how trapped surfaces form, if they form at all from regular initial conditions, except in spherical symmetric situations. Unfortunately, however, the (E- V) equations do not allow interesting asymptotically flat solutions which are spherically symmetric 28 Thus at the present time we have no results concerning the formation of black holes for the (E- V) equations. On the other hand there are non-trivial spherically symmetric solutions for the Einstein field equations coupled with an additional matter field, such as a scalar field. The program of analyzing the general spherically symmetric solutions of the coupled Einstein-scalar wave equation was initiated and carried out with remarkable success by D. Christodoulou, see [Ch2], [Ch3]. The analogue of Conjecture 3 in the case of sub critical and critical field theories is, Conjecture 5. For critical and subcritical field theories all initial data, with regular behavior at space-like infinity, behave asymptotically free.

Finally the question of analyzing the global regularity features of the developments of arbitrary initial dat set is a distant goal even in the case of the simplest supercritical Field Theories. In the case of General Relativity this is essentially the problem of the so called "Cosmic Censorship" conjecture of Penrose. Loosely speaking the conjecture asserts that, generically, there are no singularities outside black holes or, in in other words, there exist no stable naked singularities. In the remaining sections of this paper I will present some recent results obtained in collaboration with M. Machedon concerning the problem of well posedness. Our results are part of an ongoing program to try to settle Conjectures 1 and 2 for the case of the Yang-Mills and Wave Maps equations. The main results we will discuss are: Theorem 4. Under some technical restrictions which will be discussed in section 4 the initial value problem for the Wave Maps equations in ffi,n+l is locally well posed in the space H~+E.

27)

i.e. a space-like sphere S on H with a compact filling such that the outgoing null normals to S are everywhere converging

28)

Except, of course, for the Schwarzschild space-time itself.

On the Regularity of Classical Field Theories in Minkowski Space-Time R 3+ 1

49

Theorem 5 [KI-Ma21. Any locally HI finite energy initial data set in R3 admits a unique, global, admissible, generalized2 9 development3° in the temporal gauge. Moreover, the admissible solutions preserve any additional H S nyu/llrilll that the initial data may have. In particular Theorem 5 implies the global regularity result of EardleyMoncrief [E-MI. In section 3 below I shall give a discussion of the main new techniques needed in the proof of Theorem 5. The point of departure here remains the classical energy estimates. The novelty consists in isolating the quadratic terms involving derivatives in the nonlinear terms of the Yang-Mills equations and observing that, in the Coulomb gauge, there are subtle cancellations which can be taken into account in the L2-space-time norm. These leads us to the "Null estimates" presented in Proposition 3.3. as well as Proposition 3.5. We first show how these estimates help to improve the optimal "well posed" exponent for a general class of systems, see (3.9.), verifying the "Null Condition". I then show how to use them for the Yang-Mills equations. In section 4, I make a brief presentation of the main ideas in the proof of the null estimates. Finally in section 5, I will indicate the proof of Theorem 4. This requires a somewhat different point of view than that of section 4 by circumventing the classical energy inequalities alltogether and working instead directly in space-time norms. 3. Energy estimates and the Problem of Optimal Local Well Posedness In this section I will present the main estimates needed in the proof of Theorem 5. These estimates are to be combined with the classical energy estimates. Let us start by considering general systems of wave equations of the form: O¢

= N(¢, 8¢)

(3.1)

We consider two cases:

(I) (II) 29)

30)

N(¢,8¢) =¢·8¢+ cubic (¢) N(¢,8¢) = f(¢)8¢· 8¢ A generalized solution of the Yang-Mills equations is defined as a class of gauge equivalent connection I-forms for which there exists a sufficiently regular representative A with a well defined, locally integrable, curvature F[A] which verifies (Y-M) in the sense of distributions. We say that a generalized solution A of the Yang-Mills equation, in the temporal gauge, is admissible if

and it can be approximated, in the corresponding topology, by smooth solutions.

50

S. Klainerman

The type (I) equations can be viewed as a caricature of the Yang-Mills equations. Recall, see equation (1.3), that (Y-M) take in fact the form (I) in the Lorentz gauge. The type (II) equation can be viewed as a general class of nonlinear wave equations resembling those satisfied by Wave Maps. The classical local existence result requires data ¢(O,·) E H s o+ 1, 8t ¢(0,·) E HSO where So > ~ for equations of type (II) and So > ~ - 1 for equations of type (I). The proof of these results rests on the standard energy inequality for solutions to the inhomogeneous wave equation, O¢=F. (3.2) We have,

and, more generally for Sobolev norms, 1IIIs

118¢(t, ·)lls 'S 118¢(0, ·)lls

=

+

1IIIHS(lRn)

lot IIF(t', ·)llsdt'.

(3.4)

Applying the inequality (3.4) to the systems of type (II) we infer that, as long as 1¢100 remains bounded, and for any s 2': 0, (3.5)

In the standard proof for local existence, one uses the Sobolev inequality, 18¢(t', ·)100 'S CI18¢(t', ·)II~+E for any E > 0. Thus, very crudely,

r ID¢(t', ·)Ioodt' 'S Ct o-e:;t'-e:;t sup 118¢(t', ·)11 "+E.

Jo

(3.6)

2

Combining the estimates (3.5) with (3.6) we easily derive the classical local existence theorem stated in Proposition 2.1. In the case of equations of type (I) the same argument proves a local existence result requiring one less derivative of the initial data. More precisely, Theorem 3.1. FOT equations of type (1) the 1. v.P. is "well posed,,31 in H~+E while fOT equations of type (II) the 1. v.P. is well posed in H~+l+E.

Can we do better? The energy estimates (3.3~3.6) seem solid32 , indeed it is well known that (3.3) is the only available inequality, for n 2': 2, in which the first derivatives of ¢ are estimated in terms of F alone, without loss of derivatives. This 31)

Recall that this requires data ¢(O,·) E H~+<,at¢(O,·) E H~-l+E.

32)

see however Remark 2 below.

On the Regularity of Classical Field Theories in Minkowski Space-Time R 3+ 1

51

fact is crucial for us, since the nonlinear terms contain derivatives. On the other hand the estimate (3.6) is clearly wasteful. We lose a lot of information by giving up the time integration on the left hand side. To do better we shall make use the following estimate, for solutions of (3.2.).

Proposition 3.1. Consider the inhomogeneous wave equation (8.2) in ~n+l. Consider first the case n ~ 3. For every E > 0 there exists a constant C for which we have the estimate, (i)

(I

For n

= 2 we have on the other hand 33

(ii)

t

(

I¢(t',

rt ID

Jo

')I~dt')1/2 .:; C(IID¢(O, ')11 ,,;3+, + 1 / 4 ¢(t',

lt

')I~dt')1/4.:; C(IID¢(O, ')11, +

IIF(t',

')11 n;3+,dt').

rt iIF(t', ')IIE dt').

Jo

The proof of Proposition 3.1 , for n ~ 3 is an immediate consequence of the Sobolev inequalities and the following form of the Strichartz-Brenner inequality (see [P]),

Proposition 3.2. Consider the inhomogeneous wave equation (8.2.) in ~n+l, n ~ 3. For every E > 0 there exists a constant C for which we have the estimate,

Remark 1. We remark that for n = 3 the sharp form of the inequality stated in Proposition 8.1 is valid provided that ¢is sphericallY'.lJllilI/t I,.i(·. Indeed in that case we have,

This is a straightforward application of the Hardy-Littlewood maximal function inequality applied to the specific form of the solution. This result is probably true also for any n ~ 3. The inequality fails in general however. This has been shown in (Kl-Mal) in the case of dimension n = 3.

Proposition 3.2 allows us to prove the following sharper version of the local existence theorem (see [Po-Si] also [Kl-Ma3]).

Theorem 3.2. For general systems of type (I) the 1. v.P. is well posed in

33)

3

for n

~

for n

=2

For a proof of the inequality (ii) below see [P]' [G-V], [Kal], [L-S].

52

S. Klainerman

In the case of systems of type (II) the I. V.P. is well posed in for n 2: 3 for n = 2 In the case of dimension n = 3 the results of Theorem 3.2 are sharp 34, in general. Indeed H. Lindblad [L] has proved that the particular case of an equation of type (I) O¢ = ¢at ¢ and respectively equation of type (II) O¢ = (at ¢)2

is not well posed35 in Hl, respectively H2. In view of the negative results of Lindblad we have to give up the full generality of systems (I), (II). To see what additional restrictions we should consider we have to investigate more closely the actual structure of the field theories we are interested in. To start with consider the local coordinate form of the Wave Maps equations in Minkowski space:

(3.7) Observe that the most dangerous term, quadratic in the first derivatives, m/1va/1¢baV¢c is special. We have, m/1va/1¢baV¢c = QO(¢b,¢c) where Qo is the null quadratic form (see [KIl], [KI2]) , Qo(¢, 'lj;)

= m/1Va/1¢av'lj; = -at¢at'lj; +

L ai¢ai'lj;

(3.8a)

The other null quadratic forms introduced in [KIl], [ChI], [KI2] were, (3.8b) Suppose we consider equations of type (II) where we only allow quadratic interactions of the type (3.8a), (3.8b). Would that make a difference concerning well-posedness? The question was raised by M. Machedon and I in [KI-Mal] where we proved the following result: Theorem 3.3. Consider systems of equations of type (II) of the form,

O¢I

+L

f 5,K(¢)B}K(a¢J,a¢K)

(3.9)

J,K

34)

In dimension n = 2 the result is also sharp. However in higher dimension we expect that the sharp result, for general systems, can be lowered by an additional ~ in dimension n = 4 and by ~ for n ~ 5.

35)

They are however well posed in these spaces for spherically symmetric solutions.

On the Regularity of Classical Field Theories in Minkowski Space-Time R3+ 1

53

where the B)K are any of the null forms (3. 8a-3. 8b}. For such systems the l. V.P . . n+l zs well posed zn H-2- .

The key new estimate which allows us to prove Theorem 3.2 is given by the following: Proposition 3.3. Consider the solutions ¢, 'ljJ of inhomogeneous wave equations in Minkowski space-time ]Rn+1 , O¢=F,

O'ljJ=G

with initial conditions, ¢(O,x) = fo(x),

8t ¢(0,x)

'ljJ(0, x) = go(x),

8t 'ljJ (0, x) = gl(X).

= JI(x)

Then, for any of the null forms (3.8a}-(3.8b) we have, T r r IQ(¢, 'ljJ)12 dxdt Jo I~n

~ c(ll\7 ntl foIIL2(lRn) + II\7 n;-l JIII£2(lRn) + lT II\7 n;-l F(t, ')II£2(lRn) dt)2 . (11\7goll£2(lRn) + IlgIil£2(lRn) + lT IIG(t, ')II£2(lRn)

(3.10)

dtf

Moreover, in dimensions n = 2,3 the estimate is false if we replace the null forms (3.8a}-(3.8b) by arbitrary quadratic forms.

Remark 2. The estimate (3.10) can be used in conjunction with the energy estimates (3.4) to prove Theorem 3.3. The method has however an obvious flaw. This is due to the inconsistency between the norm JoT IIF(t, ')IIL2(Rn)dt which appears in the energy inequality (3.3) and the norm (J~ IIF(t, ')lli2(Rn)dt) ~ needed to apply (3.10). Thus, while the method provides better results than the Strichartz-Brenner inequalities, we cannot expect it to yield the optimal results expected, according to Conjecture 2. To overcome this difficulty we will have to give up the energy inequalities (3.3-3.5) and consider instead the L2 space-time framework suggested by (3.10). We shall discuss this appproach in section 5.

It is more difficulty to see what is the special structure of the Yang-Mills equations which allows one to improve the general results of Theorem 3.1 for equations of type (I). To do that we need to use the gauge covariance of the equations to our advantage. Thus, while relative to the Lorentz gauge the YangMills equations don't seem to exhibit any special structure, we shall see that the Coulomb gauge allows us to recast the equations in a form similar to (3.9). For convenience we rewrite the equations (Y-M) in the form,

(Y-M')

s. Klainerman

54

where

J",

= - [AJl, F"'Jl].

Differentiating (Y-M') we notice that,

which is the law of conservation of charge. In view of the Coulomb gauge condition

setting a = 0 in (Y-M'), we derive Jo = ciFoi = -~Ao

+ 8 i [Ao, Ai]

Hence, since Jo = - [Ai, Ei] = - [Ai, 80Ai - 8iAO ~Ao

=

+ [Ao, A;)),

-Jo + 8 i [Ao, Ai]

= 2[8iAo, A;) + [Ai,80Ai] + [Ai, [Ao,Ai]] Setting a

=

i in (Y-M') we derive in the same fashion,

Ji

= -8tFiO + 8j Fij = -DAi - 8t8iAo - 8dAi' Ao] + 8j [Ai, Aj]

where 0 denotes the D'Alembertian operator 0 = 80i8Oi = -8; DAi = -8t8iAo - 8 t [Ai, Ao]

or, since J i = [Ao, 8iAo - 80Ai DAi

+ 8j [Ai, Aj] -

+ Do.

Hence,

Ji

+ [Ai, Ao]] - [Aj,8i Aj - 8jAi + [Ai, Aj]],

+ 8t8iAo = - 2 [Aj, 8jAi] + [A j , 8i Aj] + [8tAo, Ai] + 2 [Ao, 8tAi] - [Ao,8iAo] - [Aj, [Aj, Ai]] + [Ao, [Ao,A;)].

We are therefore looking for solutions A o, A of the following system of differential equations,

= 2 [8iAo, A;) + [Ai,80Ai] + [Ai, [Ao,Ai]] (3.l1a) DAi +8t8iAo = -2[Aj,8jAi] + [A j ,8iAj ] + [8tAo,A;] +2[Ao,8tAi] ~Ao

- [Ao,8iAo] - [Aj, [Aj,Ai]]

+ [Ao,

[Ao, Ai]]

(3.l1b) (3.l1c)

Let P denote the projection operator on divergence free vectorfields, i.e.

On the Regularity of Classical Field Theories in Minkowski Space-Time R3+ 1

55

Using it we can rewrite

OA = VB i + Bi + Gi VB i = P[Aj,oiAj], Gi = P ([OtAO' Ai]

- [Ao,oiAo]

(3.12)

= -2P[Aj ,OjAi]

Bi

+ 2 [Ao, OtAi]

+ [Ao,

[Ao, Ai]] - [Aj, [Aj, Ai]])

In view of the energy inequality (3.3) to control IloA(t, ')II£2(JRn) we need to control I~ IIVB(t', ')IIL2(JRn)dt', I~ IIB(t', ')IIL2(JRn)dt' and I~ IIG(t', ·)IIL2(JRn)dt'. The last term can be estimated by standard methods. To estimate B we need the following: Proposition 3.4. Consider the following system of wave equations in ]R3+1 ,

subject to the initial conditions at t = 0,

with diva(o)

A(O, .)

=

a(O),

OtA(O, .)

=

¢(O, .)

= 4'(0),

,Ot¢(O, .)

= 4'(1)

= diva(l) = 0.

a(1)

Assume also that the spatial divergence of F, div F,

is zero. Then,

Jr

J10,TJ xJR3

:::; C

i[Aj,8j ¢] i2dxdt

lT + lT

(II'va(o) II £2 (JR3) + Ila(1) II £2 (JR3) +

. (1IV'4'(o) II £2 (JR3)

+ 114'(1)IIL2(JR3)

IIF(t, ')II£2(JR3) dt) 2.

(3.13)

Ilf(t, ')II£2(JR3) dt) 2

for any data a(O), a(l), 4'(0),4'(1), and F, f which verify our assumptions and for which the right hand side of the above inequality is finite. To prove the Proposition observe that, since F is divergence free we can write F = V' x F, div F = 0. Simmilarily the data a(O) = \7 x 0,(0), a(1) = \7 x 0,(1). Hence A = \7 x A where A verifies the equations,

with the new initial data 0,(0),0,(1), Now [Aj,8j ¢] =Eiab [8aA,8i ¢]. Observe that the last term is a linear combination of the null forms Qij (¢, A) defined in (3.8b).

s. Klainerman

56

The result is now an immediate application of Proposition 3.3 applied to the wave equations satisfied by A and ¢. To estimate VB we observe that Eiab Ba (V B)b =Eiab [BaAj, BiAj]. Hence VB can also be expressed in terms of the null forms Qij(A, A). We can thus estimate it using the following refinement of Proposition 3.3. Proposition 3.S. Under the same assumptions as in Proposition 3.3 , in ]R1+3 we have,

(3.14)

The estimates (3.13 ) and (3.14) provide the main new analytic tool in proving Theorem 5 stated in section 2. Nevertheless there are some further serious difficulties to overcome. The main one concerns the use of the global Coulomb condition Vi Ai = 0. It is well known that, for large data, the global Coulomb gauge leads to fundamental difficulties. In our paper [KI-Ma2] we circumvent this problem by considering local Coulomb gauges adapted to finite past causal domains. This requires an appropriate boundary condition. The condition has to be chosen in such a way as to be able to localize the estimates (3.13-3.14). One also needs a method of passing from the estimates in the Coulomb gauge, which are local, to the temporal gauge in which the solutions can be globally extended. 4. Proof of the Null Estimates

In this section I shall sketch the proof of the Null estimates of the propositions 3.3, 3.5. We first observe that it suffices to consider the homogeneous case, D¢=O,

(4.1)

D~=O

subject to the standard initial value problem,

¢(O,x) = O,Bt¢(O,x) ~(O, x) = O,Bt~(O, x)

= f(x) = g(x).

The general case follows then by a simple application of the Duhamel's principle. Observe that, ¢ = (¢+ - ¢-) where,

t

¢±(t, x)

= (27f)-3

J

e±it(lel+ix·e)

IR3

~~~) d~

(4.2)

On the Regularity of Classical Field Theories in Minkowski Space-Time R3+ 1

57

with }(O the Fourier transform of f. Let ¢ denote the space-time Fourier transform i.e., ¢(~, T) = J ¢(x, t)e-i(x.t;+tr) dxdt. Then, 1I~1+3

Proposition 3.3, resp. 3.5, is an immediate consequence of part (i), resp (iii), of the following: Theorem 4.1. Let ¢,7j; be solutions of Sobolev norms,

4.1.

1

IlflIIi" = (J 1~12Slf(~Wd~) 2 A

Let}[8 be the standard homogeneous .

i)

For all dimensions n

2 2 and all null forms 3.8a, 3.8b

ii)

For all dimensions n

2 3 and all null forms 3.8a, 3.8b

iii)

For all dimensions n

2 3 and all null forms Qij,

Before proving Theorem 4.1 we shall first review the proof of the classical Strichartz inequality in L 4(JR1+3). The proof uses a method similar in spirit to that of [Ca-Sj]. Proposition 4.1. Consider the homogeneous wave equation in JRl+n, O¢ = 0, subject to the standard initial value problem, ¢(O,·) = 0, Ot¢(O,·) = f. The classical, isotropic36 Strichartz inequality in dimensions n = 3 reads as follows:

To prove the Proposition we first observe that it suffices to estimate ¢+. Indeed 11¢IIL4 ::; 11¢+IIL4 + 11¢-IIL4. Now 11¢+lli4 = 11(¢+)21Iu = cll¢+ * ¢+llu 36)

The general Strichartz, or Strichartz-Brenner, inequality refers to LP,q norms with different exponents for the space and time variables.

58

S. Klainerman

To prove the proposition it thus suffices to prove the following,

(4.5) Now,

where

l' =

I~I-! j(~) E L2(JRn). It thus suffices to prove that, for all L2 functions f,

II /

8(T

-11]1-1~ -1]1) Ij(~ -1]}1 I~

:S cllflli2

j(1]{1 d1]11

- 1]1'2 11]1'2

(4.6)

L2(drdYJ)

We can prove it by using Cauchy-Schwartz with respect to the measure 8(T -

11]1 - I~ -1]I)d1],

The first integral above is bounded uniformly in T, ~ (see Lemma 4.1 below). Integrating the second one with respect to T, ~ proves the estimate (4.6 ) and thus the Proposition. Lemma 4.1. The integral

is bounded uniformly for all values of (T,~) E JRH3,

ITI 2

I~I

The proof of Theorem 4.1 is similar. We start by writing,

The symbol b of the corresponding null form is given by, (4.7a)

On the Regularity of Classical Field Theories in Minkowski Space-Time

R3+1

59

in the case of Qo,

(4.7b) in the case of Qij and,

(4.7c) in the case of QOi. The major difference between estimating null forms and the proof of Proposition 4.1 is that we now have to consider both Q(¢+,'I/J+) and Q(¢+,'I/J-). We write:

JJ

Q(¢+,'I/J±)(T,~)

= =

b( T - A, ~ - TJ, A, TJ)f(~ - TJ)g(TJ) 8( - A- I~ - 1)8(A ~ I l)dAd I~ _ TJIITJI T TJ TJ TJ (J'±(~ - TJ TJ) , I~ _ TJII~I f(~ - TJ)g(TJ)8(T -I~ - TJI ~ ITJI)dAdTJ

J

(4.8)

where,

(4.9) Lemma 4.2 Consider the null forms (3.8a, 3.8b) their full symbols (4. 7a-4. 7c) and reduced symbols defined by (4· g).

(i)

In the case of the null form Qo we have,

and

(ii)

+ ITJI-I~ + TJI) (I~I + ITJI) :::; c(l~ + TJI-II~I-ITJII) (I~ + 171)

I(J'+(~, TJ)I :S c(I~1

(4. lOa)

I(J'-(~,TJ)I

(4.1Ob)

In the case of the null form Qij we have, (J'±(~, TJ) = ~iTJj

- ~jTJi'

and,

I(J'±(~, TJ) :S cl~l! ITJI! I~ + TJI! nAn ( (I~I + ITJI - I~ + TJI ; I~ + TJI - II~I -

ITJII) ) (4.lOc)

(iii) In the case of the null form QOi we have,

and

1

1

I(J'+(~, TJ)I :S CI~121TJ12 (I~I I(J'_(~, TJ)I

+ ITJI-I~ + TJI) 2 (I~I + ITJI) 2

- ' - '

:::; cl~121TJ12 (I~ + TJI-II~I-ITJII) 1

1

(I~ + TJI) 2

-'-'

2

(4.lOd) (4.lOe)

s. Klainerman

60

The proof of the estimates (4.1 Oa-4.1 Ob) are an immediate consequence of the identities 2(1~111]1-~ .1]) = (I~I + 11]1-1~ +1]I)(I~1 + 11]1 + I~ +1]1) and 2(1~111]1 +~ .1]) = (I~ + 1]1 -II~I -11]11) (I~ + 1]1 + II~I -11]11)· To prove the estimate 4.10c we shall make use of the identity, I~ x 1]1 2 = L,i<j(~i1]j - ~j1]i)2 = 1~1211]12 - (~.1])2. Hence,

I~ x 1]1 :::; (1~111]1 - ~ .1]) ~ (1~111]1 + ~ .1]) ~ Now, if I~I + 11]1 -I~ + 1]1 :::; I~ + 1]1-11~1-11]1 then I~I, 11]1 :::; I~ + 1]1· Hence, 1

1

1

1

I~ x 1]1 :::; (1~111]1 + ~ .1]) 2 J2 (I~I + 11]1-1~ + 1]1) 2 (I~I + 11]1 + I~ + 1]1) 2 1

1

1

:::; WII1]1 + ~ .1]) 21~ + 1]1 2 (I~I + 11]1-1~ + 1]1) 2 :::;

2~ I~I ~ 11]1 ~ I~ + 1]1 ~ (I~I +

11]1

-I~ + 1]1) ~

On the other hand if I~I + 11]1 - I~ + 1]1 ;::: I~ + 1]1 - II~I - 11]11 we have, 1

1

1

1

I~ x 1]1 :::; (1~111]1- ~ .1])2 J2 (I~ + 1]1-11~1-11]11)2 (I~ + 1]1 + 11~1-11]11) 2 :::;

2~1~1~11]1~1~+1]1~(1~+1]1-11~1-11]11)~

For simplicity I shall only indicate how to prove the estimates (4.3b) and (4.3c) of Theorem 4.1 for the special case of the null form Qij and dimension n = 3. These are in fact the estimates needed in applications to the Yang-Mills equations. The other estimates are proved roughly in the same way. According to (4.8) and Lemma 4.2 we have:

Q(r,~) :::; cllrl-I~II~ J1~1~11]1~1~ -1]1~8(r -I~ -1]1 =f 11]1/~~ =~~f~,1]) d1] :::; cllrl

-I~II ~ I~I ~

J

8( r -

I~ -1]1 =f 11]1) j(~ - 1]:g(~) d1] 1~-1]1211]12

Therefore,

(4.11) where <1>, Ware both solutions of the homogeneous wave equation, 0<1> = Ow = 0 with initial data, A

(O,~)

= 0, 8t (0, 0 = 1~12 f(~)

~(O,O = 0,

A

lA.

8t~(0,~) = 1~I~g(~)

1

E H-2

E

ir~,

provided that f, g E L2 (JR3). Therefore the estimate (4.3b), in this case, is now an immediate consequence of Proposition 4.1. The estimate (4.3c) follows from (4.3b)

On the Regularity of Classical Field Theories in Minkowski Space-Time R 3+ 1 in the region I~I 2': 1;1. On the other hand, the region I~I

61

S 1;1 is disjoint of the

support of Q(¢+,1fJ-), we therefore only need to estimate Q(¢+,V!+) for I~I S 1;1. This region can be handled by proceeding precisely as in the proof of Proposition 4.1.

5. The Proof of Theorem 4 As we have mentioned in Remark 2 of section 3 the L2 space-time estimates of Proposition 3.3 don't combine well with the energy estimates (3.3-3.4). In this section we discuss a different approach, see [Kl-Ma4], based entirely on space-time estimates. The approach is similar to that used by Bourgain [B] and Kenig-PonceVega [Ke-Po-Ve] for the Korteweg-de Vries equation. Consider the equations (3.9),

O¢I with the

B} K

+ L f 5,K(¢)B}K(8¢J,8¢K) = 0

(5.1 )

J.K

any of the null forms (3.8a-3.8b), n

(5.2a) ;=1

(5.2b)

In the particular case when only the null form Qo is allowed to appear in (5.1), we may say that the corresponding equations are of "Wave Maps" type. Indeed the equations satisfied by Wave Maps ¢ defined from the Minkowski space-time lR n +1 to a Riemannian manifold M, take precisely that form when expressed relative to a system of local coordinates in M for which the f's arc the corresponding Christoffel symbols. We consider the space-time norms,

(5.3) where,

(5.3a) Consider also the homogeneous weights, (5.3b) The reason these norms are natural is that they capture the optimal gain of regularity of the solution of O¢ = F. Indeed, if we take X to be a smooth cut-off

s. Klainerman

62 function in time, we have

-!.

for all 8 ~ The norms are also compatible with the null forms (5.2a-5.2b). This can be seen already from the following, Proposition 5.1.

1]) = TA -

(i)

The symbol bo(T,~; \

(ii)

The symbol bij (T,~; A1]) =

~i1]j

~

-

(iii) The symbol bOi(T,~;A,1]) = T1]i -

. 1] of Qo verifies the estimate,

~j1]i

A~i

of Qij verifies the estimate,

ofQoi verifies the estimate,

Iboi 1::; 11]luL (T,~) + 1~luL (A, 1]) + cl~l! 11]1! (I~I + 11]1)! (W_(T,~)! + w_(\ 1])! + W_(T + A,~ + 1])!) for T . A ::::: 0 and,

Iboi 1::; 11]lw- (T,~) + 1~lw_ (A, 1]) + cl~l! 11]1! (I~ + 1]1)! (W_(T,~)! + W_(A, 1])! + W_(T + \~ + 1])!) for T . A ::;

o.

The proof of part i) follows directly from the identity,

The proof of part ii) follows from 4.lOc as well as the following, Lemma 5.2. Let W = W(T,~;A,1]) be the maximum of the weights W_(T,~), W_(A, 1]), W_(T+\~+1]). Then (i) if T, A::; 0 or T, A::; 0, then HI~I + 11]1 -I~ + 1]1) ::; W(T,~; \ 1]) (ii) if T ::::: 0, A ::; 0 and T + A ::::: 0 or T ::; 0, A ::::: 0 and T + A ::; 0 then ~ (11]1 + I~ + 1]1 -I~I) ::; W( T,~; A, 1]) (iii) if T ~ 0, A ::; 0 and T + A ::; 0 or T ::; 0, A ~ 0 and T + A ~ 0 ~ (I~I + I~ + 1]1-1~1) ::; W(T,~: A,1])

On the Regularity of Classical Field Theories in Minkowski Space-Time R 3 +1

63

Proof· (i)

Assume 7,>' 2': O. If either of IT -I~II or I>. -11711 is greater or equal, then ~ (I~I + 1171 - I~ + 171) we are done. Thus, assume the opposite. Then, I~I

+ 1171-1~ -171

= (I~I-

<

(ii)

2

T)

+ (1171- >.) + T + >. -I~ + 171

3(I~I + 1171 -

I~ + 171)

+ IT + >. -

I~ + 1711

Hence IT + >. -I~ + 1711 is greater than ~ (I~I + 1171-1~ + 171) and we are done. The case T, >. ::; 0 is proved in the same manner. Assume 7 2': 0, >. ::; 0 and T + >. 2': o. Ifeither IT -I~II or I>. + 11711 is greater or equal ~ (1171 + I~ + 171 -I~I) we are done. Assume the opposite. Then,

1171 + I~ -17I-I~1 = (1171 + >.) + (T -IW + I~ + 171- T - >. 2

< 3(1171 + I~ + 171-1171) + IT + >. -I~ + 1711 i.e. 17+ >. -I~ +1711 > HI171 + I~ +17I-I~I) , which proves the desired inequality. All other cases are proved in the same manner.

Corollary 1. Let W = W (T, ~; >., 17) be defined as before. Then, (i) if T . >. 2': 0,

(ii)

if T

.

>. ::; 0,

The main results of this section are contained in the following,

Theorem 5.1. Consider the space-time norms (5.3.) and functions cP,7/J defineJ37

in JR3+1. (i) For the null form Q = Qo and anJJ8 s > ~,

N S ,-1/2(Q(cP,7/J)) ::; CNs +1,1/2(cP)Ns +1,1/2(7/J)

(ii)

(5.4)

For any space-time functions (cPi)i=l, ... ,k+1 and any polynomial P(cPl, ... , cPk-t) we have, for every s > ~, N s ,-1/2 (P(cPl"'" cPk-t)QO(cPk, cPk+d) :::; CNs +1,1/2(cPd ..... N s +1,1/2(cPk+d·

(iii) The estimate (i) fails for the null forms Q = Qo:{3 (cP, 7/J), 0 :::; a < (3 :::; 3 and all s in the range ~ < s < 1.

37)

For which the norms N s + 1,1/2(¢), N s + 1,1/2('!f;) are well defined and finite.

38)

For IR1+2 the corresponding result holds for s > O.

s. Klainerman

64

As application we use the estimates (i), (ii) of Theorem 5.1 to prove the following result concerning non-linear equations of the type (1-3.) Theorem 5.2 The initial value problem,

¢(O,x) = fo(x),

8t ¢(0,x) = !1(x)

for non-linear wave equations of the form,

O¢I + 'Lrj,K(¢)QO(¢J,¢K) = 0, J,K

where rj K(¢) are real analytic in ¢ = (¢I, ... ¢N), is well posed for fo E H~+E,

!1

E

H!+E.

The result of Theorem 5.2 is optimal. Indeed this can easily be seen by considering the scalar equation O¢ = Qo(¢, ¢) which can be solved exactly, see

[Kl-Ma1]. Proof of Theorem 5.1. The main ideas in the proof of Theorem 5.1 are the following. Using duality, the estimate (i) is equivalent to

11

wt w ::::! log(l +

W_)QH(T,~)dTd~::; CNs ,1/2(¢)Ns ,1/2('l/J)IIHII£2(IR1+3).

On the other hand, introducing, 1

_

F = wtHw~ log(l + w_)¢ 1

_

G = wtHw~ log(l + w_)'l/J and writing,

where b( T,~; A, ry) is the symbol corresponding to the null forms, we can reexpress the inequality (5.4) in the form,

I=11

b(T-A,~-ry;A,ry) W+(T,~) W±(T,~)W±(T - A,~ - ry)W±(A,ry) Wt+1(T - A,~ - ry)wt+l(A,ry) log(l+w_(T,~))

log(l+w_(T-A,~-ry))log(l+w_(A,ry))

F(

T-

A

,~-ry

)G() A,ry

H( T, ~)dTd~dAdry ::; CIIFIIIIGIIIIHII, (5.5) where IIII denotes the L2 norm in JR.1+3. The symbol b of the null form is given by, b = bo(T,~; A, ry)

in the case of Qo.

=

-T A

+ ~ . ry

On the Regularity of Classical Field Theories in Minkowski Space-Time R3+ 1

65

After a change of variables,

Next, we make use of part (i) of Proposition 5.1,

Therefore,

I:S;h+h+h where,

U=ITI-I~I=±T-I~I

v

=

1.\1 - 1171

=

±.\ - 1171

(5.6)

thus T = ± (I ~ 1 + u), .\ = ± (117 1 + v). We can thus reduce the integral to one ofthe type which we have analyzed in [Kl-Ma1J. The integrals h h are treated in the same manner after a simple change of variables. More precisely our estimates depend on the following Lemmas.

s. Klainerman

66

Lemma 5.3. Consider the integral,

and Lemma 5.4. Consider the integral,

Ja,b(f, g, H) 1

11~111J1 =f ~ ·1J12

=

f f (1+11~1±11J11+1~ +

1J1)a(l

+ 11JI)b

f(~)g(1J)H(I~1 ± 11J1,~ + 1J)d~d1J

where f, 9 E L2(JR3), H E L2(JR1+3). Then, for any a, b ~ 0, a ~::;a::;l

+b

~ 2, and

Both Lemmas are used to treat the "hyperbolic" regions lui, Ivl small. On the other hand, in the "elliptic" regions where lui or Ivl are large we use Lemma 5.5. Consider the integral,

for any functions f, g, h E L2 (JR 3 ) and numbers a, b ~

o.

Then, if a + b > ~,

Part (ii) of Theorem 5.1 is proved by similar means. On the other hand part (iii) is somewhat surprising in view of the fact that Proposition 5.1 is valid for all null forms. Our counterexample depends on one hand on the lack of an identity such as the one used in Proposition 5.1 (i), and on the other hand on a subtle interaction between the '+' and '-' regions of our foliation (5.6).

On the Regularity of Classical Field Theories in Minkowski Space-Time R3+ 1

67

6. Conclusions The methods presented in sections 3-5 have allowed us to make significant progress towards the resolution of the Conjectures 1 and 2 for semilinear Field Theories such as Yang-Mills and Wave Maps in Minkowski space-time. I will end my Lectures by pointing out what are the main unresolved issues. (i)

As mentioned above the estimate 5.4 is false for all other null forms -I Qo. Nevertheless the norms 5.3 seem hard to avoid. One has some freedom to play with the hyperbolic exponent 8; instead of estimating Ns , _12 (Q(cp,'l/J)) as in 5.4 one can try instead a norm N s ,t5(Q(cp,'l/J)) with -~ < 8 < O. This idea was recently used by Yi Zhou [Z]. Modifying appropriately our proof of Theorem 5.1 he was able to improve the result of Theorem 3.3 by gaining an additional in both ffi.2+1 and ffi.3+1. While his result in ffi.2+1 is sharp the result in ffi.3+1 is still off by Do we have to give up altogether on the norms 5.3? Recently Machedon and I [KI-Ma 6] were able to show that, in a slightly different situation when the estimates corresponding to 5.3 fail, the nonlinear problem admits nevertheless solutions in the spaces Ns,_l for exponents s arbitrarily close to the optimal one. The method uses to a bigher degree the nonlinear character of the equations. We hope that similar techniques can be applied for the Yang-Mills equations.

i

(ii)

i.

The methods discussed above will, I believe, prove Conjecture 2 for all exponents s > s. To obtain results for the optimal exponent s = s we need, on one hand, to suitably modify the norms 5.3. and, on the other hand, to use more of the geometric properties of the equations at hand.

(iii) The case of quasilinear Field Theories, such as the Einstein field equations, is completely open. So far the best "well posedness" results available are those derived by classical energy estimates. Bibliography [B]

J. Bourgain, "Fourier Transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations", Geometric and Functional Anal. 3 (1993) 107-156, 209-262. [Bour] J.P. Bourguignon, "Stabilite par deformation non-lineaire de la met rique de Minkowski" Seminaire N. Bourbaki 1990--1991 June 1991. [Br1] C. Bruhat, "Theoreme d'existence pour certain systemes d'equations aux derivees partielles nonlineaires", Acta Matematica 88 (1952) 111 225. [Br2] C. Bruhat, "Un theoreme d'instabilite pour certains equations hyperboliques nonlineaires" c.R. Acad. Sci. Paris 276A (1973) pp. 281. [Ca-Sj] L. Carlesson-P. Sjolin "Oscillatory integrals and the multiplier problem for the disk" St. Math. 44 (1972) 287-299. [ChI] D. Christodoulou "Global solutions of nonlinear hyperbolic equations for small data" Comm. Pure Appl. Math. 39 (1986 ) 267-282.

68

S. Klainerman

[Ch2] D. Christodoulou, "A Mathematical Theory of Gravitational Collapse" Comm. Math. Ph. 109 (1987) p. 613. [Ch3] D. Christodoulou, "The Formation of Black Holes and Singularities in Spherically Symmetric Gravitational Collapse" Comm. Pure AppI. Math. 44 (1991) 339-373. [Ch-Kll] D. Christodoulou-S. Klainerman, "Asymptotic Properties of Linear Field Equations in Minkowski Space-Time" Comm.Pure AppI.Math. 43 (1990) 137-199. [Ch-KI2] D. Christodoulou-S. Klainerman The nonlinear stability of the Minkowski space-time. Princeton Univ. Press 1993. [Ch-Za] D. Christodoulou-A. Shadi Tahvildar-Zadeh, "Regularity of Spherically Symmetric Harmonic Maps of the 2 + 1 dim. Minkowski Space." Duke Math. J. 71 (1993) 31-69. [E-M] D. Eardley-V. Moncrief "The Global Existence of Yang-Mills-Higgs fields in M3 +l" C.M.P. 83 (1982) 171-212. [Ge] R. Geroch, Asymptotic structure of space-time P. Esposito and L. Witten (eds.), Plenum, New York, 1976. [G-V] J. Ginibre-G. Vela "The global Cauchy problem for the nonlinear Klein-Gordon Equations" Math. Z. 180(1985), 487-505. [Grl] M. Grillakis "Regularity and Asympt. Behavior of the Wave Eq. with a critical nonlinearity" Ann. of Math. 132 (1990) , 485-509. [Gr2] M. Grillakis "Classical solutions for the Equivariant Wave Maps in 1+2 dimensions" preprint. [Gu] C. Gu, "On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space", Comm. Pure ApI. Math. 33 (1980) 727-737. [Jo] K. Jorgens, Math. Z., "Das Angfangswertproblem im Grossen fur eine Klasse nichtlinearer Wellengleichungen 77, 295-307 (1961). [Kal] L. V. Kapitansky "Some generalizations of the Strichartz-Brenner inequality" Leningrad Math. Jour. 1 (1990), no. 3, , 693-726. [Ka2] L. V. Kapitansky "Global and unique weak solutions of nonlinear wave equations" Math. Res. Letters 1, 211-223 (1994). [Kll] S. Klainerman "Long-time behavior of solutions to nonlinear wave equations" Proc. of Int. Congress for Mathematicians, Warsaw 1982. [KI2] S. Klainerman "The null condition and global existence to nonlinear wave equations" Lectures on AppI. Math., 23 (1986) 293-326. [KI-Ma1] S. Klainerman-M. Machedon "Space-Time estimates for null forms and the local existence theorem", Comm. Pure AppI. Math. 46 (1993), 1221-1268 [KI-Ma2] S. Klainerman-M. Machedon "Finite energy solutions of the Yang-Mills equations in RHl." Annals of Math. 142 (1995), 39-119 [KI-Ma3] S. Klainerman-M. Machedon "On the regularity properties of the Wave Equation" Physics on Manifolds, 1994 Kluwer Academic Publishhers edited by M. Flato, R. Kerner, A. Lichnerowicz. [KI-Ma4] S. Klainerman-M. Machedon "Smoothing Estimates for Null Forms and Applications" Duke Math J. 81 (1995) 99-133. [KI-Ma5]

S. Klainertnan-M. Maehedon "Retnark on an extension of Striehart
inequalities" , preprint. [KI-Ma6] S. Klainerman-M. Machedon "Hyperbolic Sobolev Norms and optimal local existence for a class of non-linear wave equations", in preparation.

On the Regularity of Classical Field Theories in Minkowski Space-Time RHI

69

[Ko] M. Kovalyov "Long time behavior of a system of non-linear wave equations" Comm. P.D.E. 12 (1987) 471-50l. [Ke-Po-Ve] C. Kenig, G. Ponce, L. Vega, "The Cauchy problem for the Korteweg-De Vries equation in Sobolev spaces of negative indices" Duke Math. Journal 71 No l. pp. 1-21 (1994). [1] H. Lindblad "Counterexamples to local existence for semilinear wave equations" To appear in Amer. J. Math. (1996). [L-S] H. Lindblad-C.D. Sogge "On Existence and Scattering with minimal regularity for semi linear wave equations" J. Punet. Anal. 130 (1995), 357-426. [P] H. Pecher, "Nonlinear Small Data Scattering for the Wave and Klein-Gordon Equation", Math. Z., 185 (1984) 261-270. [Po-Si] G. Ponce-T. Sideris, "Local Regularity of Nonlinear Wave Equations in Three Space Dimensions", Comm. P.D.E 18 (1993), 169-177. [S-Y] R. Schoen and S.-T. Yau, "On the proof of the positive math conjecture in general relativity" Comm. Math. Physics 65(1) (1979) 45-76. [Sh] J. Shatah "Weak solutions and development of singularities in the SU(2) -(J" model" Comm. Pure Appl. Math. 41 (1988) 459-469. [Sh-Za1] J. Shatah-A. Shadi Tahvildar-Zadeh "Non-uniqueness and development of singularities for harmonic maps of the Minkowski space" [Sh-Za2] J. Shatah-A. Shadi Tahvildar-Zadeh "Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds" Comm. Pure Appl. Math. 45 (1992) 947-971. [Sh-Stru1] J. Shatah-M. Struwe "Regularity results for nonlinear wave equations" Annals of Math. (2) 138 (1993), 503-518. [Sh-Stru2] J. Shatah-M. Struwe "Well Posedness in the energy space for Semilinear Wave Equations with critical growth" Int. Math. Res. Notices 7 (1994) 303-309. lSi] T. Sideris "Global existence of harmonic maps in Minkowski space" Comm. Pure Appl. Math. 42 (1989) 1-13. [Str] W. Strauss "Nonlinear Wave Equations" Regional Conference series in mathematics, nr. 73. [Stru] M. Struwe "Semilinear wave equations" Bull. Amer. Math. Soc. (N.S.) 26(1992), 53-85. [Z] Yi Zhou "Local Existence with optimal regularity for Nonlinear wave equations" preprint.

Progress in Nonlinear Differential Equations and Their Applications, Vol. 29 © 1997 Birkhiiuser Verlag Basel/Switzerland

Static and Moving Vortices in Ginz burg-Landau Theories FANG-HUA LIN

Courant Institute, New York University

ABSTRACT.

Lecture 1: After a brief introduction to the history of the theory of superconductivity, I shall describe the main mathematical result due to Bethuel-Brezis-Helein and recent improvements and simplifications. In particular the asymptotic behavior of distributions of vortices for static solutions or approximate solutions. Lecture 2: The aim of this lecture is to demonstrate the important role played by the so-called renormalized energy in the study of vortices. I shall explain the connection between the critical points renormalized energy and solutions to Ginzburg-Landau equations. Using this we can show certain histeresis phenomena for the phase transition near the lower critical magnetic field. Lecture 3: Here I shall start with the Gor'kov-Eliashberg equation for the evolution of Ginzburg-Landau. The global existence of classical solutions and their asymptotic behavior follow from existencing results. We are particularly interested in the dynamical properties of vortices. Several well-known speculations should be verified. It is obvious that three lectures can hardly cover any topic with relative completeness in this fascinating and fast growing subject. Nonetheless I shall try to give a rough state of the art survey, and mention various unsolved, certainly more challenging problems.

Introduction

This write-up covers the author's three Lectures given in March of 1995 at the University of Tennessee, Knoxville for the Barrett Lectures series. The purpose of these lectures was to give a brief description of some rigorous mathematical work concerning the static and moving vortices of solutions to Ginzburg-Landau equations arising in the theory of superconductivity. This short presentation was relatively complete and comprehensive at that time. Since then, the literature on the subject related to these discussions has almost doubled. I have no intention here to give a complete survey. Indeed, it would not be very wise to do so on a subject that is still fast growing. Instead, I shall simply write up, with some

72

Fang-Hua Lin

necessary details, what was covered in these three lectures. I should, however, take this opportunity to make a few comments on some closely related research which has been done since then in the area covered by each lecture. It was an honor for me to give a Barrett Lecture series. I would like to thank the Department of Mathematics, University of Tennessee for giving me the opportunity. In particular, I would like to thank the organizers, Professor G. Baker and Professor A. Freire for their assistance and warm hospitality. Lecture 1.

1. Background and Models The response of a superconducting material to an externally imposed magnetic field is most conveniently described by the diagram below, which shows the minimum energy state of the superconductor as a function of Ho, the applied magnetic field, and the dimensionless material parameter K, (known as the Ginzburg-Landau parameter). The parameter K, determines the type of superconducting material. For K, < ~, type-I superconductors, there is a critical magnetic field Hc below which the material will be the superconducting state, but above which it will be in the normal state. For K, > ~, type-II superconductors, there is a third state known as the mixed (or vortex) state. The mixed state consists of many normal filaments embedded in a superconducting matrix. Each of these filaments carries with it a quantized amount of magnetic flux, and is circled by a vortex of superconducting current. Thus these filaments are often known as vortex lines. One of the most challenging problems to mathematicians working on the superconductivity models is the understanding of vortex phenomena [GO] in type-II superconductors, which include the recently discovered high-temperature superconductors. The transition from the normal state to the mixed state takes place by a bifurcation as the magnetic field is lowered through some critical value H c2 . The critical field Hc l , on the other hand, is calculated so that at this field the energy of the wholly superconducting solution becomes equal to the energy of the single vortex filament solution for an infinite superconductor. The vortex structures have been studied extensively on the mezoscale using the well-known Ginzburg-Landau models of superconductivity [DMG, COH]. The existence of vortex-like solutions for the full nonlinear Ginzburg-Landau equations have been substantiated through many arguments ranging from asymptotic analysis to numerical simulations. However, it remains to be justified by rigorous mathematical analysis. Much progress has been made in recent years on establishing a mathematical framework for a rigorous description of both the static and dynamic properties of the vortex solutions; in particular, as the coherence length tends to zero (K, goes to infinity), various results have been obtained. From a technological point of view, this is of interest since recently discovered high critical temperature superconductors are known to have large values of K" say K, in excess of 50.

73

Static and Moving Vortices in Ginzburg-Landau Theories

Ho

Ho

Normal

Normal

He2

He Hcl Superconducting

K,

T

1

< v'2

Vortex

Meissner

Te K,

T 1

Te

> v'2

Figure 1: The diagram of various states for type-II superconductors The vortex structures can be set in motion by a variety of mechanisms, including thermal fluctuations and applied voltages and currents. Unfortunately, such vortex motion in an applied magnetic field induces an effective resistance in the material, and thus a loss of superconductivity. Therefore, it is a crucial and rather difficult issue to understand the dynamics of these vortex lines. At the same time, one is interested in studying mechanisms that can pin the vortices at a fixed location i.e., prevent their motion. Various mechanisms have been advanced by physicists, engineers and mathematical scientists. For example, normal (nonsuperconducting) impurities in an otherwise superconducting material sample are believed to provide sites at which vortices are pinned. Likewise, regions of the sample that are thin relative to other regions are also believed to provide pinning sites. These mechanisms have been introduced into the general Ginzburg-Landau framework [DGPj to derive various variants of the original Ginzburg-Landau models of superconductivity. Numerical simulation clearly suggests the pinning effect. Before addressing the above problems, we introduce the notations and the models that will be used in the lectures. The starting point of our study is the phenomenological model due to Ginzburg and Landau for superconductivity in isotropic, homogeneous material samples. Let D be a smooth bounded domain in R 3 , occupied by the superconducting material. By ignoring the effect of the region exterior to the sample, the steady state model can be stated as a minimization problem of the free energy functional:

Q(7/J,A)

L{in

+ al1f!1 2 + ~17/J14 + 2~s I (in\? + e;

+ ~; h . (h -

2H) } dD ,

A)~r (1.1)

74

Fang-Hua Lin

where f n is the free energy density of the non-superconducting state in the absence of a magnetic field, 'l/J is the (complex-valued) superconducting order parameter, A is the magnetic vector potential, h = (1/ JLs) curl A is the magnetic field, H is the applied magnetic field, a and b are constants whose values depend on the temperature and such that b > 0, ms is the mass and es is the charge of the superconducting charge carriers which is twice the electronic charge, e, c is the speed of light, JLs is the permeability, 27r1i is Planck's constant. It can be rewritten in nondimensionalized form:

where", is the so-called Ginzburg-Landau parameter. The functional 9('l/J, A) has an interesting gauge invariance property and the minimization of 9 in appropriate functional spaces gives the following system of nonlinear differential equations that are referred to as the Ginzburg-Landau equations.

(1.3) curl curl A =

J...2",

('l/J"V'l/J* - 'l/J*"V'l/J) -1'l/J12 A

(1.4)

m n

along with natural boundary conditions curIA 1\ n = h 1\ n

on

r

(1.5)

= an

and

(1.6) where n is the exterior normal to the boundary r. The time-dependent Ginzburg-Landau model is often described by the Gor'kov-Eliashberg evolution equation: rt

~~

aA

at

+

i 17 '" if>'l/J +

+ "Vif> +

r

(~ "V + A

curl curl A =

'l/J - 'l/J + 1'l/J12 'l/J = 0,

_J...2",

}

(1. 7)

('l/J*"V'l/J - 'l/J"V'l/J*) - AI'l/J12 ,

Here if> denotes the (real) scalar electric potential, 17 is a relaxation parameter and 'l/J* denotes the complex conjugate of'l/J. The system is supplemented by the initial and boundary conditions:

'l/J(x, 0)

=

'l/Jo(x),

A(x,O)

=

A(x),

x

E

n;

(1.8)

75

Static and Moving Vortices in Ginzburg-Landau Theories

(~V+A)~.n

0,

curl A 1\ n

hl\n,

(1.9)

if· n = 0, on a0,. Note that (1.7) is Gauge-invariant, in the sense that if (~, A, ~) is a solution, then so it (~x, AX, ~x), where ~x

= ~eiKx, Ax = A + Vx,

x

=

~

aX . at

(cf. [D])

For type-II superconductors, ~ > 1/J2. The minimizers of 9 are believed to exhibit vortex structures. Numerical experiments have shown that for large values of ~ and moderate field strengths, the number of vortices could be exceedingly large even for small sample size in actual physical scale. Thus, resolving the vortex phenomenon using the full Ginzburg-Landau equations remains computationally intensive. Various simplications have been made to reduce the complexity. For thin films of super conducting material, a two-dimensional model has been developed [DG, CDG] that can account for thickness variations through an averaging process. The model is given by the following minimization problem:

g~(~) =

1 n

a(x)

(I(V ~ iAu) ~12 + ~ 2f

(1

~ 11/)12)2)

dx

(1.10)

where 0, denotes the platform of the film, a(x) measures the relative thickness of the film and Au is a prescribed vector potential due to the normal (to film) component of the applied field. The role of the Ginzburg-Landau parameter ~ is assumed by the parameter f('==' 1/4 It has been proved that for fixed f, the minimizers ofthe above problem, along with the prescribed vector potential Au provide the leading order approximation to the solution of the three dimensional problem [CDG]. The creation and interaction of vortices based on the above model is connected with the prescribed magnetic potential. The number of vortices cannot be prescribed a priori, independently of Ao. In order to simplify the analysis further, a simpler problem, in which the number of vortices is prescribed and the magnetic potential is ignored, can be studied. By rescaling the spatial variables, one may consider the minimization of the following functional:

F:(~) = .In a(x) (IV~12 + 2!2 (1 ~ 1~12)2) dx .

(1.11)

with the boundary condition: ~(x) =

g(x)

for

x E a0, .

(1.12)

Where 9 is smooth with Ig(x) 1 = 1, x E a0,. We shall simply consider here the variational problem (1.11)-(1.12) with a(x) = 1.

76

Fang-Hua Lin

2. The Work of Bethuel-Brezis-Helein and Others

Let n be a smooth, bounded, connected domain in ]R2. We consider the variational problem: minimize (2.1) for u E H~ (n, ]R2) = {v E HI (n, ]R2) : vlan = g}. Consider the case that deg(g, an) == d = 0, see [BBH], and hence there is a smooth map g* : n ---t §1 with g* = 9 on an. In particular, Vc

=

min {Ec(u) : u E H~

::; C(n, g) <

(n, ]R2)}

00.

We now give a different proof of the main result shown first in [BBH]. By a theorem of C. B. Morrey, there is a map Uo : n---t 51 which minimizes I\7U1 2 dx over the set H~ (n, 51) = {U E HI (n, 51) : U = 9 on an}. Moreover, Uo is smooth. When n is simply connected, a simple lifting argument shows that Uo = ei¢o. Here rPo is the harmonic extension of rP, ei ¢ = 9 on an. Here we want to show a uniform estimate for the minimizers Us of (2.1) for < f < 1 under the hypothesis that deg(g, an) = 0. To do so , we note first that

In

°

(2.2) For any sequence fi ---t 0, there is a subsequence of UCi which converges weakly in HI and strongly in L2 to some U* E Hl(n, 51). Moreover, U* = 9 on an, and In l\7U*1 2 dx ::; In l\7Uol2 dx. By the minimizing property of Uo, we see U* again is a minimizer of I\7UI 2 dx over H~ (n, 51). In particular U* is smooth. Moreover, it follows from (2.2) that Uc , converges strongly to U*. We therefore obtain the following

In

Lemma 1. For any fO > 0, there is an ro Uc is a minimizer of {2.1} then

r

JnnB(x,ro)

for all x E

[1\7Uc I2

+

>

°

depending on

2\ (IU I2 - 1)2] c

an

dx ::;

and 9 such that if

fO

(2.3)

f

n provided 0< f ::; f*(ro, fO)'

Proof. Let F denote the set of energy minimizing maps over the set H~ (n, 51). Then it is easy to see that F is compact in Hl(n,5 1 ). Moreover, by Morrey's theorem, one has for any fO > 0, U* E F

r

JnnB(x,r)

for all x E nand

I\7U* 12 dx ::; fo/2

°< r ::; ro provided that ro is chosen to be suitably small.

(2.4)

77

Static and Moving Vortices in Ginzburg-Landau Theories

Now we apply the convergence argument above to conclude that (2.3) is valid for all minimizers Ue whenever 0 < c ~ c*. Note that (1/c 2) In(lUel 2 _1)2 dx -+ 0 as c -+ 0+. Theorem 1. Let Ue be a minimizer of (2.1) overthe set H~(n, q with deg(g, an)

=

O. Then

(2.5) for all 0 < c < 1. Proof. It is obvious, by the maximum-principle, that IUel ~ 1 on n. Thus IUel(lIUe I2) 1/c2 ~ 1/c2 ~ l/c; whenever c ~ c*. It follows that (2.5) is true whenever c ~ c*. For 0 < c < c*, we need the following lemmas. Lemma 2. Let ee (u)

= ~ [1V'uI2 + 2!2 (l u l2 - 1) 2], then ilee(Ue) ~ -3 [e;(Ue) + ee(Ue:)]

Proof. A direct computation yields the conclusion. Lemma 3. There is a positive number

7]0 E

(0, 1) such that, if IB,o(o) ee(Ue:) dx ~

7]0, then

Proof. Look at the function v(x) = (ro - Ixl)2 ee:(Ue), and set K2 = v(xo) = v(x),O'o = ro -Ixol· We claim: if 7]0 is small enough, then K ~ 1. For otherwise, we may consider Ve(y) = Ue: (xo +~) defined on BK(O). It is easy to see that e(Ve) =

maxBro(O)

2

a2

e(Ue:) , ee:(Ve) ~ jfxee(Ue), e(Ve) ~ 4 on B K / 2(0) and ee(Ve)(O) = 1. Since ilee(Ve)+3 [e~(Ve) + ee:(Vc)] ~ 0, sUPB ,/4 ee:(Ve) ~ GdBl/2 ee(Ve) dy = GlIB ( ) ee(Ue ) dx ~ G l 7]o < 1, by choosing 7]0 suitably small. Thus we ao/2K Xo ~~

obtain a contradiction. Hence K ~ 1, in particular ee(Ue) ~ ro2 on B ro/2 (0). The conclusion of Lemma 3 follows. In the proof above, we have implicitly assumed that Bro (0) c n. If Br(JO) n an of- r/>, we refer to [eLl for the details. Finally, we consider We: = ~ (1 - lUe 12) ~ 0,

2 { 2c2 ilwe - We: ~ -41V'UeI ~ -G1 , We

= 0 on an.

for all

0
~

c*.

(2.6)

Fang-Hua Lin

78

Let Xo E n such that wc(xo) = maxxEo wc(x) thus wc(xo) :::; C 1. The latter implies that

> O. Then

~wc(xo)

:::; 0, and

for a E (0, 1) follows. Next we shall describe the main result in [BBH]. Here deg(g, n) = d > O. Theorem [BBH]. Let En 1 0, and {Ucn } be a sequence of minimizers of (2.1). Then, by taking subsequences if necessary, one has the following:

x a e t. h a ( x, ) 1 (-n\{ a1, ... ,ad} ) , U* () X = TIdj=l Ix::::a~1 in C,<> ~ha = 0 in n, U* = g on an; (ii) There is an EO > 0 such that, for 0 < E < EO, Uc has exactly d distinct zero's ai, ... ,ad' and each zero is of degree 1. Moreover, the d-tuple point a = (a1,"" ad) is a global minimum of the renormalized energy W (g, n, b), bE nd, (see next lecture for details concerning the renormalized energy);

(1')

(iii)

Ucn (X)

---t

2"'" ~

(IUcn 2 _1)2 1

E2

---t"

n

l'VUcn l2 log ..l..

s; va) ,

6

j=l

2 ---t

~

7r 6

s; va)

,

j=l

IOn

in the sense of distributions; (iv)

1

Ec(Uc) = 7rd log -

E

+

"(d

+

min W (g,

bEnd

n, b) +

0(1) .

Remarks (a) The above statements were shown in [BBH] under the additional assumption that n is star-shaped. The key conclusion following from this assumption is that the quantity

is bounded by C(g, n). Using the approach in [Bt], one can drop this additional assumption. Indeed, the estimate

(2.8) also follows from [Bt]. Later in [FP] an elegant approach showed also this estimate without using the star-shaped property of n. (b) The proof of (iii) follows easily from (i) and the result of [BCP]. In fact, one can combine the proof of [BCP] and the classical result of E. Heinz to conclude that d

Jac(UcJ

---t

7r

L j=l

Daj .

Static and Moving Vortices in Ginzburg-Landau Theories

79

(c) Some more refined estimates due to P. Mironescu et al. were obtained recently. In particular, they showed 1

EE(UE) = 7l'dlog -

E

+O(E,B),

+ 1'd +

.

mm W(g,

bEnd

for some

(3

n, b)

(2.9)

> O.

Here l' is a universal positive constant. (d) If n is star-shaped, it is also shown in [BBH] that results similar to the above are valid for arbitrary solutions, which are not necessarily minimizing. Without the additional assumption that n is star-shaped the conclusions fail in general. Under the assumption that the energy is bounded above by 7l'd log ~ + C, for some C (independent of E), we have more precise conclusions, cf. [L3]. We also constructed solutions which have both positive and negative degree zeros in [L3]. (e) Similar results for the full Ginzburg-Landau energy functionals (with magnetic field) have been obtained in [BR]. See also [DL] under a more physical boundary condition, and with an applied magnetic field. (f) Let a E En be one of the zero's of UE • By Theorem [BBH], we may assume, for a sequence En 10, aEn ----t ao, UEn ----t U* in CI,Q(BRo(ao) \ {ao}). Moreover, U* = I~=~O I ei h in B Ro (ao), for a smooth harmonic function h defined on B Ro (ao). By [BMR], one may assume Vn(x) = UEJa En VR> 0, Vk 2': 0, where F satisfies

- fl.F = (1 -

r

and

JlR 2

(1

1F12) F

on

]R2,

_1F12)2

dx = 27l' .

with

+ EnX)

----t

F(x) in Ck(BR(O)),

F(O) = 0

(2.10)

Moreover, F is locally energy minimizing in ]R2. Entire solutions of (2.10) have the property (cf. [BMR]) flR2 (1 -IFI2)2 dx = 27l'd2, for d = 0,1,2, ... ,. Moreover, when d < 00, F(re iO ) ~ eid(O+Oo), as r ----t 00, for some eo E R (cf. [S]). It is still an open problem to classify all entire solutions of (2.10) with flR2 (1IIFI2)2 dx < 00, even for the case that F is locally energy minimizing. In the latter case, one can show (cf. [S]) that F (rei 0) ~ ei (0+00), as r ----t 00. It was shown later by Shafrir, and also independently in personal communications of the author and H. Brezis that lim

n~oo

IWE (x) n

F (x - aE n En

)

ei (h(x)-h(ao)) I

L

00

(BRo(ao))

= 0.

(2.11)

It is an open problem, however, if (2.11) is true when the L oo norm is replaced by HI-norm.

Fang-Hua Lin

80

(g) One of the most challenging problem remaining to understand is the uniqueness of energy minimizers. Let us look at a simple situation. One considers ::

{ -flUe Ue(x)

=

(1

-IUe I2 )

X,

Ue in B 1 , x E 8B 1

(2.12) .

There is a solution of the form fe(r)e iIJ such that fe(r) - t 1, as c - t 0+, for all r > O. It is not known if == fe(r) e iIJ is the only solution of (2.12). Recently, Lieb and Loss [LiL] and Mironescu [M] showed independently that is a stable (and strictly stable) solution, and hence it is locally energyminimizing. T. C. Lin [Lin] showed further that the linearized equation at has exactly two small positive eigenvalues corresponding to the translation invariance of (2.10) and the other eigenvalues are strictly positive (independent of c). There is also an interesting result shown in [CK] that the entire solutions F of (2.10) are of the form F = fe(r) eiIJ (up to translation and rotation) whenever IIR2 (1 - IF12)2 dx < CXl and that either

u2

u2

u2

div F

=0

or curl F

=0

in ]R2 .

There is good reason to believe the solution of (2.12) is unique (for small c). First of all, the Renormalized energy (defined in the next lecture) has a unique critical point, the origin, which is also the global minimum. Thus, by [BBH], any minimizer Ue of (2.1) tends to I~I as c - t 0+. Here we want to show the latter fact is true even for solutions of (2.12). To see this, we note first, by Pohozaev's identity, that

r

JaB l

(8;e)2 + JrBl ul/

e

12 (1-IU I2)2 c

=

27r.

(IV 12_1)2 Hence, by the general convergence result in [BBH, Chapter X], one has e2 - t 27rOa , for some a E B 1 . Moreover, a is a critical point of the renormalized energy W(g, B, b), bE BI, where g(x) = I~I . A simple computation shows that W(g, B, b) = -log(1-lbI 2), and thus a = Q. It, therefore, follows that Ue(x) - t I~I as c - t 0+. €

Proof of Theorem [BBH} (Sketch of ideas). We should sketch the ideas involved in the proof of the first statement (i). The statements in (iii) are easy consequences of the convergence result (i). The proof of the fact that the d-tuple point a = (aI, ... , ad) is a global minimum of W(g, n, b), bEnd follows from the following: 1. A direct consequence of a comparison argument shows that

min {EE:(U) : u E H~ (0, ]R2)} = EE:(UE:) 1 < 7rdlog - + d"( + min W(g, n, b) c bE f!d Here 0(1)

-t

0 as c - t 0+;

+

0(1) .

Static and Moving Vortices in Ginzburg-Landau Theories

81

2. The convergence result in part (i) and another comparison argument yield that 1 Ee:(Ue:) 2 7rdlog - + &"y + W(g, 0, ae:) + 0(1) , E

where ae: = (a!, ... , a~) are d-tuple points consisting of zero's of Ue:. (cf. Lecture 2). Thus the key step in proving Theorem [BBH] is to prove part (i). Now we sketch its proof. We shall simply assume 0 is star-shaped (this assumption is unnecessary, by [St],[FP]).

Step I. First, Ee:(Ue:) ::; 7rd log ~ + C(g, 0) by a direct comparison. Next, the Maximum principle implies that 1Ue:I(x) ::; 1, x E O. Then by a simple scaling (by looking at the equation satisfied by Ue:(E x)) one deduces l'VUe:(x)1 ::; ~, x E O. Also a Pohozaev-type identity yields:

n

Step II. One covers the set Se: = {x EO: IUe: (x) I ::;

=

Thus, there are balls, Be:(xj), j

by balls of radius E.

U Be:(x)) and Nc

1, ... , Ne:, such that Se: ~

j=l

= 1, ... , Ne: are pairwise disjoint. Since IUe:(xj) I ::; ~, I'VUe: I ::;~, one sees that JB 5(X fx (IUc I2 - 1)2 dx 2 > 0, for some positive Co, for each j = 1, ... , Ne:. This latter fact, and

that Be:/5(Xj),j

E

Co

/

J

)

estimates in Step I imply that Ne: ::; N(g, 0).

Step III. Let U : Ar, R = {x E and

JAr,R

fx

~) where d = deg ( U, 8B p

Proof, We write U 1. l'J!ol2 4

p2

]R2 :

r ::; Ixl ::; R}

(IUI 2 _1)2 dx::; K. Then ~ JAr,R

=

,

~ r ::; p ::; R, U(x)

fei(dO+'J!(p,e)),

=

---t

]R2,

r 2

E,

satisfy lUI 2 ~

l'VUl 2dx 2 7rd2 log ~-C(K, d),

~

IUI(x)'

then l'VU1 2 2

f21~+712 > ~~

+~ d, \II p2 e'

Thus -1 2

1

Ar,

R

+2d

2 l'VUl 2 dx 2 -d 2

r

}) Ar,R

1

Ar, R

2dx p

2

+ -d

r

2

(12 - 1) \II dx + ~ e 4})Ar, R P2

1 Ar,

f2 - 1 --2R

l\Il el2 P2

P

dx

dx

f2

+

82

Fang-Hua Lin

d2'

[L.u' dxr (f ~

- 4d2

J

_I)'

(j2 - 1)2 dx p2

+ ~

4

J \[J~ p2

dP)'"

dx

d 2 c:K I / 2 (4nc:- 2)1/2 _ C(d,K) 2 C(K, d) . Step IV. By a simple grouping and induction argument, one can use the results in Step I, Step II and Step III to conclude that, as c: -+ 0, the sets {Xj }f=l converge (in Hausdorff distance) to {aj}1=1, for some d distinct points aI, ... , ad in D.

Moreover, deg

(Ue, aBp(aj)) = 1, for each j, and p > 0 (suitably small) whenever

c: is sufficiently small. Also

J

IV'Ue I2 dx ::; C(p). The final estimate

n\U:=l Bp (aj) will imply the conclusion of part (i) in Theorem [BBH]. We refer to [St] and [BBH] for details. D By employing the same arguments as above, one can show the following somewhat more general statement:

4. Let U be a minimizer of the functional (2.1) in a Lipschitz domain D with U = 9 on aD. Suppose that

Lemma

for a constant K. Then, for all sufficiently small c: > 0 (depending on K and D) we have Ee(U) < C(K, D) whenever deg(g, aD) = 0, and 1

Ee(U) > n Idllog - - C(K, D) . c:

if deg(g, aD) = d i- o.

5. With the same hypothesis as in Lemma 4, suppose deg(g, D) = O. Then IU(x)1 ~ ~ in D whenever 0 < c: ::; c:(K, D), for some positive c:(K, D). In general, let v E HI(D, ]R.2) with v = 9 on aD and lV'vl ::; Then Ee(v) ~ min {E",(u) : ulan = g} + co(K, D) for some positive co(K, D) whenever Iv(O)1 :::; ~ for a point QE D. Lemma

q..

The proof of Lemma 5 follows from Theorem 1 and its proof.

Static and Moving Vortices in Ginzburg-Landau Theories

83

3. Some Generalizations To end this lecture, we would like to present two general results proved in [L]. Both of these results will play an important role in the next two lectures. For this purpose, we introduce the following: Definition 1. Let 0 C ]R2 be a smooth, bounded domain, and let g : 80 -> §1 be a smooth map of degree d > O. We say a map U : 0 -> ]R2 belongs to the class S (co, K, c, g, n) if U E H~ (0, ]R2) and . 1 (1) Ec(U):S 7fdlog - + K, c (ii) for Xo E n with IU(xo)1 < ~, then IU(x)1 < 4"3 whenever x E 0, Ix-xol :S COCo We have the following structure theorem concerning U E S (co, K, c, g, 0). The proof given in [L] is somewhat more complicated, but it is useful for other purposes. Here we shall give a simplified proof of the following: Structure Theorem. There are two positive numbers co, ao depending only on co, K, g and 0 such that, for any 0 < c < co, U E S( co, K, c, g, 0), there are No disjoint balls Bj of radius cD:J, j = 1, ... ,No with the following properties: (i) ao:S aj :S 1, for j = 1, ... , Nc! and No :S N(co, K, g, 0).

(il)

Th,

.,d {x

En, 1,,(x)1 < n i., ,",,(ained in [! n

(,9 llj)

(iii) The estimates c"'J Ja(B) no) ec(u):S C(co, ao, K, g, 0), j = 1, ... ,NE ! are

valid. In particular, the degrees dj

=

deg

(I~I' 8(Bj nO)) are well-defined.

Here ec(u) = lV'ul 2 + 2!2 (lul2 _1)2. (iv) There are exactly d balls, say B l , ... ,Bd such that the corresponding degrees dj , j = 1, ... , d! are not zero. Moreover, each dj equals 1, for j = 1. ... , d. Suppose Xl, ... ,Xd be centers of balls B 1 , ... ,Bd, then min {dist(xj, (0), Ix; - Xj I, i oF j, i,j = 1, ... ,d} 2:: b* = b* (co, K, g, 0) > 0 . (v)

If B j n 0 is scaled by a factor of size':::' CCi)! the resulting domain has diameter one and is uniformly Lipschitz (independent of j and c).

Proof. Let u E S (co, K, c, g, 0), then

Let 1£ be a minimizer of the energy functional

Fang-Hua Lin

84

By Lemma 4, 1 7fd log - - C(O, g) c Thus

Ee(u) - E e(u) =

8~2

< E cCyJ < E e(u).

In (1 _luI 2)2 dx :::; K + C(O, g).

Using the last fact, and noting that if Xo E {x EO: lu(x)1 :::; ~}, then

as u E S (co, K, c, g, 0), we see, from the proof of Theorem [BBH] that the set {x EO: lu(x)1 :::; can be covered by Be(xj), j = 1, ... , Me, Me < N(co, K, g, 0). Now we want to find (for any c > 0 suitably small) at most Ne balls B j , of radius COj , covering the same set and satisfying the additional conditions:

n

(I) aj E Here

0:0

[0:0'

l],

for

j = 1, ... , Ne :::; N(co, K, g, 0).

is a positive constant which may depend on N e .

n is contained in On UBj; and the balls c-ojl3 B j Ne

(II) The set {xEO:lu(x)l:::;

j=l

(with same center as B j and radius c o ]13 ) are pairwise disjoint, for j 1,2, .. . ,Ne . To prove these two properties (I) and (II), we need the following:

=

Covering Lemma. Let B 1 , B 2 , ... , B N be N balls in R2, each with radius no larger than c(\ for some 0: E (0, 1/4). Then there are a positive number 0:0 (depending only on 0: and N), and balls B j of radius co] , for j = 1, ... ,Ne :::; N, such that (I) and (II) above are valid provided that c is ·'lI.ffir-itlltl.l/ small. N

Proof. Let A =

UB

j .

We are going to prove the covering Lemma by induction

j=l

on the number of connected components of A. If A is connected, then we simply take a1 = and a ball B 1 of radius COl 2: 2Nc (this inequality will be valid whenever c is suitably small) such that A c B 1 . The conclusion of the covering Lemma follows automatically. Suppose that the conclusions of the covering lemma are true whenever the number of the connected components of A is 1 :::; k :::; N -1. Moreover, these a/s satisfy aj 2: ; , for each j.

J'

Q

85

Static and Moving Vortices in Ginzburg-Landau Theories

We want to show the covering lemma is true when the number of the connected components of A is k + 1 :::; N, and that each aj in the lemma can be chosen to be not less than 3k~1 whenever E is small enough. For this purpose, we let AI' ... ' Ak+l be connected components of A. Without loss of the generality, we may assume that the diameter of A is larger than 3(k + n 1) E3k+l • For, otherwise, we may simply choose a ball B of radius :::; E3k+l that covers A entirely (whenever E is small enough), and then the conclusion of the covering lemma is obvious. Now we let x', x" E A be such that IX' - x"I = diameter of A ;::: 3(k + ~

r

1) E3~+1. We may find a Po E (0, 3(k + 1) E3 +1 ) such that the boundary oBr(x' ) of the ball Br(x' ) will not intersect any of A/s, for j = 1, ... , k + 1, and for any r E [po - E 3~+1 , Po + E 3~+1 ] . Then it is obvious that A n Bpo (x') = A', and A" = A '" A', each contains some of AI, ... Ak+l . We may apply the induction step to both A' and A" to conclude that A = A' u A" can be covered by at most _ _ 20: N balls B j of radius Ea'], aj ;::: 3'7.. Now since dist(A' , oBpo(x' )) ;::: pk+l, and 2n

since dist (A", oBpo(x' )) ;::: E3 k +1 , the conclusions of the covering lemma follows. This completes the induction argument. D Now we can apply Fubini's Theorem again, to find balls B j (with the same center as Bj ) of radius Eo.j, Dj E [aj/3, ,aj] such that (i), (ii) and (iii) of the Structure Theorem are valid. Part (iv) and (v) follow from the same proof as in D Step IV of Theorem [BBH]. We shall call Xl, ... ,Xd in the statement (iv) of the Structure Theorem the essential zero's of u. It is then clear that Xj'S are well-defined up to errors which are not larger than 2Eo. o . The latter statement means: if Yj, j = 1, ... ,d, are another possible choice of essential zero's of u as those xl's in the statement (iv) above, then IXj - Yj I :::; 2Eo. o, for j = 1, ... ,d . Compactness Theorem. Let UE E S(co, K, E, g, 0). Then, for any En 10, there is a d x-b . h( ) subsequence of {UEn } that converges to a map of the form I1j =1 I;-b; I e' x weakly in Hl~c (0 \ {bl, ... , bd }). Moreover, IlhllHl~c :::; C(co, K, g, 0) and W(g, 0, b) :::; C(co, K, g, 0). The last statement implies, in particular, that

min { Ibi

-

bjl, dist (b i , (0), i

-f. j,

i, j = 1, ... ,d};::: (\(co, K, g, 0) .

Proof. For UE E S(co, K, E, g, 0), we let B j , j Structure Theorem. We replace UEon each B j by

= 1, ... , NE be balls as in the

UEwhich minimizes IB

]

eE(v) dx

i

with v = UE on oBj, for j = d+ 1, ... ,NE • Thus, in particular, IUEI ;::: on Bj , for j = d + 1, ... , NE (see Lemma 5). We will denote the resulting map defined on o by UE •

86

Fang-Hua Lin

Let 8 E (c ao , 8.), where ao, 8. are given in the Structure Theorem, and d

suppose that the set

U8Btj(xj) does not intersect the set U B j . Note that ~

j=1 j=l+d the latter assumption holds outside of a set of 8 E (c ao , 8.) whose measure is smaller than Nc; c ao . For such a 8, we have 8 7l'dlog - - C('\, K) c

where

Ac;

Here the first inequality is true since (by lemma 4)

and

see Step III in the proo~ of Theorem [BBH]. In the second inequality of (*) we have used Lemma 5 for Uc; on each B j , d + 1::::: j ::::: N c . Since Uc; E S(co, K, c, g, D), we deduce from (*) that

where Dtj = D \ U1=1 Btj(xj). Now, for a sequence of Cn 1 0, we may assume, without loss of generality, that Xj -+ bj as Cn 10. Note Xj may also depend on c. We may also assume, by taking a subsequence if necessary, that ucJ x) -+ u· (x) weakly in HI~c ("0\ {b 1 , ... , bd } ) and strongly in L 2 (D). The conclusion that

u·(x) = with h(x) E Hl(D) follows.

II d

j=1

x - bj

Ix -

eih(x)

bjl D

Static and Moving Vortices in Ginzburg-Landau Theories

87

Lecture 2. The Role of the Renormalized Energy 1. Renormalized Energy We start with the definition of the renormalized energy for the functionals (2.1) ----t §l be of Lecture 1. Let n be a bounded smooth domain in ]R2, and let 9 : a smooth map of degree d > O. To any d distinct points al, ... , ad in n, we can associate a canonical harmonic map Ua : 0\ {al, ... , ad} ----t §l, which is smooth on 0\ {al, ... , ad} and such that U a = 9 on the degrees of U a : aBp(aj) ----t §l are all equal to 1, for j = 1, ... , d, and for all small p> o. One can write

on

on,

(1.1 ) for some harmonic function ha defined on determined (mod· 27r) by the requirement A simple computation shows that

n. Ua

The value of ha on

= 9 on on.

on is uniquely

(1.2) +O(p),

and

p----tO+.

The function W(a), a End, which depends on 9 and n, is called the renormalized energy, see [BBH, Chapter I]. It is easy to check that W (a) possesses the following property:

W(a) =

+00

(1.3)

if either one of ai E on, for some i E {I, ... ,d} or ai = aj, for some i =I j. Otherwise W(a) is locally analytic. A rather important connection between the above renormalized energy and the Ginzburg-Landau energy functionals

0<

C

< 1,

(1.4)

defined on H~(n) = {u E Hl(n,]R2) : ulao = g}, is described in the previous lecture. Here we are interested in the following question (cf. [BBH], open problem #6]): Q. Let a = (al, ... , ad) be a nondegenerate critical point of the renormalized energy W(-). Is there a sequence of critical points uSn of Eon' n = 1,2, ... , such that Cn ----t 0, and that U cn (x) ----t ua(x) in C1,a(O\{al, ... , ad}) as n ----t oo'? In [L], the author obtained a positive answer to the above question provided that a is a nondegenerate local minimum of W (.) on nd. In fact, the case a is a degenerate local minimum of We) was also treated in [L]. The main result of

88

Fang-Hua Lin

[L] was later improved by del Pino and Felmer, [DF]. Here we shall give another simple proof, based on some arguments in [L], of their improved result which can be stated as follows. Theorem A. Let K be a open subset with compact closure in Od\ {b E Od: bi = bj , for some i i- j}. Let a E K be such that min W > W(a) = min W. Then there is 8K

K

an co> 0 depending on g, 0 and K such that there is a family of local minimizers, Uc;, 0 < e ::; co, of Ec; with the following property: for any sequence en ---+ 0, one may find a subsequence of {UC;n} converging to Ub, Ub(X) =

d

I1

j=l

1~=:jleihb(X), the J

canonical harmonic map associated with the point b = (bI, ... , bd), for some point bE Ka. Here Ka = {b E K, W(b) = W(a)}. The main goal of this lecture is to establish various minimax solutions of the Euler-Lagrange equations associated with energy functionals Ec;, for small positive e's. As a particular consequence we shall prove the following. Theorem B. Let a = (aI, ... ,ad) be a nondegenerate critical point of the renormal-

ized energy W(·). Then there is an co> 0 depending only on g, 0 (and maybe also the point a) such that one may construct a family of critical points, Uc;, 0 < e ::; co, of Ec; with the property that, Uc; ---+ u a , in Cl~~ (0\ {aI, ... ,ad}), as e ---+ 0+. Here U a is given in (1.1). Theorem B, therefore, gives an affirmative answer to the question posed by Bethuel-Brezis-Helein. Recently, we learned from the announcement [AB] that Almeida and Bethuel had used topological arguments to construct some nonminimal solutions of the Ginzburg-Landau equations. Both the proof of existence of mountain-pass type critical points in [L] and that of existence of nonminimal solutions in [AB] used global information on the level sets of the renormalized energy W(·). The key point in establishing of Theorem B is to localize the nontrivial topological information of level sets of W (.) near a critical point in Od, and to transfer it to the functionals Ec; on H~ (0). It still remains as an interesting open problem to calculate the Morse indices of these saddle points. There are two important implications from the results stated above. The first one is that it reduces a variational problem in infinite dimensions, for which the numerical computations are rather unstable for e small, to a finite dimensional problem. The second is that it may shed some light into the prediction of Abrikosov on the lattice structures of vortices in type II superconductors. Of course, in the latter situation, one also has to look at the applied field. See [BR] and [DL] for some related discussions. Here we simply describe a result in [DL] concerning the renormalized energy when there is an applied field. With proper scaling, we focus on the following form of the G-L functional

Static and Moving Vortices in Ginzburg-Landau Theories

89

In this nondimensionalization, one may view

E as proportional to ~ and H as proportional to /'i, times the (nondimensionalized) applied field. For n = 2, let o E R2 be a bounded Lipschitz domain, H = Ho a constant field. Following the discussion in [BBH, BRJ, we now formulate the renormalized energy: let

= ei(/>b(x) = II d

'ljJ

x - hj eih Ix - hjl

j=1

for some points h

= (hI, h2, ... , h d )

E Od and

a/: = 0 on a~. Let

Bp = UJ=IBp(h j

) .

Choose the gauge div A = 0 in 0 and A· n = 0 on that

a~.

We may define (, such

A = V.l( in 0 , (= 0 on

ao .

Now, consider

Note

r

In\B p d

div ((. V.l¢b) dO

L 27f((hj) + O(p) j=1

So,

90

Fang-Hua Lin

Minimizing the term involving function on 0 \ {b 1 , b 2 , ... , b d },

r

(Ph

~ l\7rPbl 2dO -

In\B p 2

we see that

d1l'10g

rPb is a multi-valued harmonic

~ = gn(b) + O(p) P

Minimizing the terms involving (, we can choose ( to satisfy d

211' LDbJ

in 0

j=l

(

=

0

Ho

~(

on 00 on 00

So, d

211' {;((bj ) =

-1 (1\7(12 + 1~(12)

ian ~~ df.

dO

+ Ho

df

+ Ho O(p)

Since Ho is a constant, we get

r

In\B p

Ho

6.( dO =

Ho

r

aa(

Jan n

Thus,

Now, let us define (

=

(b

+ (Ho

where

j=l

(

0

on 00

6.(

0

on 00

and (Ho = H O(l with

o o

in 0 on 00

1 on 00 Then

1 1

1\7(12 dO

H5!o1\7(11 2dO + !o1\7(bI2dO-2H0!o !:,.(l(b dD

16.(12 dO

H5116.(112 dO + 1 16.(bI 2dO + 2Ho 1 6.(1 ~(b dO

91

Static and Moving Vortices in Ginzburg-Landau Theories

and

In

(ILl (Ll(b - (b) dO d

-211"

L

(1 (bj

)

j=l

So,

H61n (1\7(11 2+ ILl(112) dO

+

1 o

d

(1\7(bI 2+ ILl(bI 2) dO - 211" H o L (l(bj ) j=l

Therefore, 1

Wo(b, Ho) = "2 H6 C(O)

d

+ 11" HO?=

(1 (bj )

+ go(b) + O(p)

(1.5)

)=1

where C(O) is a constant and go(b) has the property - (b)

go

=

{+oo

b i = b j for some i bE ao d

-00

i= j

,

and otherwise, it is a smooth function in Od. It is then easy to show that Wo(b, Ho) has a local minimum inside Od whenever Ho ~ Ho(d, 0). Moreover, one can establish a result similar to Theorem A. 2. A Technical Result We construct a natural embedding of Od\{b E Od: bi = bj , for some i i= j}. Let bl , b2 , ... , bd be d distinct points in O. Then we can choose a small p (p can be chosen to be uniform for any compact subset of A == Od\{b E 0: bi = bj for some i i= j}), so that there is a map Wc,b satisfying

1

Ec(Wc,b) = 11"dlog f

+ 'Yd + W(b) + O(p) + 0(1).

Here 0(1) is a quantity which goes to zero as c ~ 0 (uniformly on any compact subset of A), O(p) ~ 0 as p ~ 0 (uniformly on any compact subset of A), and 'Y is a universal constant. The map Wc,b can be chosen so that Wc,b(X) = Ub(X) for Ix-bjl ~ p, uo(x-bj ) for Ix-bjl ::; ~,and that on Bp(bj )\Bp/2 (b j ), Wc,b is the canonical harmonic map with admissible boundary conditions, for j = 1, ... , d,. (uo is a minimizer of fB p / 2 (O) ec(u) with the boundary condition I~I on aBp/ 2 (0)). It is easy to see, for a given compact subset K of A, one may find such a p > 0,

92

Fang-Hua Lin

such that for any b E K, one may construct We,b as above. Then b E K ---> We,b gives a continuous embedding of K into H~(O). Next, we want to construct a projection from the mapping class S(E, co, K, g, 0) to its essential zeros which lie in the set A COd.

Lemma 1. Let D be a compact topological submanifold with boundary in ]R2d, and let h: D ---> H~(O) be a continuous map such that h(D) C S(E,co,K,g,O). Then there is a constant'\ = '\(co, K, g, 0), and a continuous map II: h(D) ---> W.x = {b E Od: W(b) ::; '\} whenever 0 < E ::; EO. Moreover, for any p E D, II(h(p)) is at most 4E~o distant from b(p) where b(p) = (b 1 (p), ... , bd (p)), and bj (p) 's are essential zero's of h(p) E S(E, co, K, g, 0). Proof. For a point p ED, we let b1 (p), ... , bd(p) be a choice of essential zero's ofthe map h(p) E S(E,co,K,g,O). As 0 < E::; EO, the points b1 (p), ... ,bd(p) are defined up to errors of, at most, 2E"'O (cf. Structure Theorem). Since h: D ---> H~(O) is continuous, one may find an open ball Vp around p, such that, for any q E Vp, b1 (p), ... , bd(p), can be also viewed as essential zero's of the map h(q). Therefore, we have an open cover {Vp,p E D} of D. Since D is compact, we may find a finite sub cover {Vpj,j = 1, ... ,m}. Moreover, by the Besicovitch covering lemma, we m

may also assume that LXv: . (p) ::; c(d), Vp E D. j=l PJ Let {¢j} be a partition of unity subordinate to the cover {VpJ }. That is each

¢j is smooth and nonnegative, with support ¢j C VPJh ' and

m

L ¢j =

j=l

1. We let

m

II(h(p)) = LBj¢j(p), j=l Then II is obviously continuous on h(D). Moreover, since each p E D, there are at most c(d) number of Vpj's such that p E Vpj . By our construction, if p E Vpj n Vpk , then IBk - Bj I ::; 4E~o. The conclusion of the Lemma 7 follows from the latter 0 estimate and the Compactness Theorem.

3. Proof of Theorem A First let us consider the case that a is a nondegenerate minimum of W(·). We may

assume the set K in the Theorem A is given by bi

= bj , for some

i

#- j},

d

I1

j=l

BRo(aj)

c:,;;;

A == Od\{b E Od:

and so that Ka = {a}. The reason we can do so is

that K contains a set of form

d

I1 B Ro (aj),

j=l Theorem A will not be affected.

for some Ro, and the conclusion of the

Static and Moving Vortices in Ginzburg-Landau Theories

93

We introduce

v

{n E HN1), luI2

~ on fl \j~' BR"(aj), and

C~I' aBRo (a j ))

degree

= 1, for j = 1,2, ... , d}

and

C(O) and U has exactly one zero in each BRa (aj), for j = 1,2, ... ,d}.

{u E V:

Va

U

E

It is clear that V is weakly closed in H~ (n). One can also show that V is connected, though we will not need this. Therefore the inf Ee (u) is achieved. uEV

We claim: inf Ee (u) is also achieved and that is equal to inf Ee (u). uEVa

uEV

To show the claim, we let Un E Va be a minimizing sequence. Let Xl, ... , xd be zero's of Un. Since Ee( un) :S ndlog ~ +W(a)+'Yd+O e (1) is obviously true by the explicitly construction in Section 1, we may find balls Be"'] (xj), for j = 1,2, ... ,d (and for a given n), such that Cci] JaB" (xn) ee( Un) :S C(O:o, a, g, n) (cf. Structure E]

Theorem). Moreover, as

on the set

minimizes

fl \kBe", (x'J)

J

n\

d

U B."J (x;)

Un

]

l

E Va, det(~,aBe"](xj)) = 1. Replace

SO

that.

Un

~ Un on 8 [fl\~, B,", (xl')

by

Un

and that

u"

Un

ee(u)dx.

J=l

Without loss of generality, we may assume that xj --+ aj as n --+ by the proof of [L, Theorem A], one has that Un converges as c --+ 0, n the map

_II Ix d

Uii -

j=l

00.

Then to

--+ 00,

_

1 ,Ci(t=i\{aj - }) - Ieiha(x)·In Cloc H aI, ... , ad . x-a'J

Moreover, the proof there also showed that

Since a E

d

I1

j=l

BRo(aj), and a is a nondegenerate minimum of W in

d

I1

j=l

BRo(aj),

one sees that, for sufficiently small c, the limit aj of xj as n --+ 00 (for the fixed c) satisfies laj - ajl < ~. Thus, for n large enough, one may assume Be"] (xj) ~

BRo/2(aj)'

94

Fang-Hua Lin

The new minimizing sequence Un which is given by Un on each Bc;"'i (xj) and

Un on

n\ j=lU Bc;"'i (xj) is also in Va. As Un d

may assume UnlaBRo(aJ) and Ua;laBRo(ai) are Let

-t

Ua;, as n

e 1,a

- t 00

and c

-t

0, we

close for small c, and large n.

d

u~ be the map which is equal to Un on n\ j~l B Ro (aj) and which minimizes

IB RO (0) ec;(u) on each BRo(aj) (with respect to its Dirichlet boundary condition). aJ From the fact that Ua;laBRo(aJ) is a diffeomorphism from aBRo(aj) to 8 1 (for a suitable small Ro, and we may assume Ro is so chosen at the beginning), u~ has a unique zero in B Ro (aj) of degree 1 (cf. [BCP]). Finally the minimizing sequence {u~}, (for a sufficiently small c, but fixed) of Ec; on Va is also equicontinuous for all n. Moreover the limit, as n - t 00, of u~ is obviously also in Va (cf. [BCP]). We have, therefore, proved that the inf Ec;(u) is achieved. It is obvious that Va

inf Ee(u) :S inf Ee(u). This completes the proof of the claim. V

Va

Next we let U e be a minimizer of Ec; in V. Let Xj E BRo(aj) be such that lue(xj)1 :S ~ in the Lebesgue sense, for j = 1, ... ,d. As before, we may find balls Be"'i, aD :S aj :S 1, such that

We claim that deg (~, aBe"'J (Xj)) = 1, for all j = 1,2, ... , d, whenever c is small enough. As in the proof of [[L], Theorem A], it suffices to show degree

(1~:I,aBe"'J(Xj)) #0.

Indeed, when c is small enough, one has Iuel > ~ on each aBe"'J (Xj), j = 1, ... , d. Assume, for some j, deg

(~, aBe"'i (Xj))

= O. Then we may replace

Ue

inside

the ball Be"'i (Xj) by an energy minimizer ue with the same Dirichlet boundary condition. Such a minimizer ue satisfies IUe I 2: ~ inside Bc;"'i (Xj). Thus if we let u; be the map coinciding with U e on n\Bc;"'i (Xj) and equal to ue on Be"'i (Xj), then u; E V. Moreover Ee(u;) < Ee(u e) as lue(xj)1 :S ~ in the Lebesgue sense. This contradicts to the minimality of U e . Therefore we may assume each deg (~,aBe"'J (Xj)) = 1. Now we follow the first part of the argument to show as c - t 0, Xj - t aj, and U e - t Ua in Cl~;cn\{al, ... ,ad}). in el~;cn\{al'''' ,ad})' Then Ug E Va follows from [Bep] again. It is easy to see from the above argument that if Uc; is a minimizer in either

Va or V of the energy Ee, then IU e I >

~

in

d

n\ j ~1 B Ro (aj ). Therefore U e is a local

95

Static and Moving Vortices in Ginzburg-Landau Theories

minimizer of Ee in H~ ([2). This completes the proof of Theorem A in the case that a is a nondegenerate minimum of W. In the general case, we may define function spaces

{u E H~([2):

V(K)

deg

lui::::: 4:3 on [2 \

(I~I ,8BRo (b j )) = 1,

jtd BRo(b d

j ),

for j = 1, ... ,d,

for some point b = (b l , ... ,bd) E K}. Here Ro > 0 has to be chosen suitably small (which may depend on K). Again one can show V(K)is weakly closed and infv(K) Ee(u) = Ee(u e), for some U e E V (K). One can also define

v;,(K)

=

{u E V(K): U is continuous on 0" and there is a point b E K such that U vanishes only at bl , b2 , ... , bdl.

One can use the exactly same proof as above to show that inf Ee(u) = inf Ee(u).

Vc(K)

V(K)

Moreover, if U e is a minimizer of Ee in either Vc(K) or V(K), then U e vanishes exactly at points (ai, ... , a~D

d

---+

Ka as c ---+ 0+. Also Iuel > ~ on [2 \ j~l B Ro / 2 (aj),

and thus ue's are local minimizers of Ee over H~([2). This completes the proof of Theorem A. 4. Proof of Theorem B We consider a nondegenerate critical point a = (al, ... , ad) of the renormalized energy W (.) defined on [2d. That is, the function W satisfies the following at the point a: V'W(a) = 0 and V' 2 W(a) has no zero eigenvalues. (4.1)

Thus we may find a 2d dimensional ball B8o (a) centered at a and of radius 80 , and a local orthonormal coordinate system (XI, •.. ,X2d) on Blio(a) such that, in this coordinate system, a becomes the origin (Xl, ... ,X2d) = (0, ... ,0), and that k

W(x) = -

2d

L AiX; + L i=l

AiX;

+ W(a) + 0(lxI 2 ),

i=k+l

for some positive numbers AI, ... , A2d, and some k, 0

~

k

~

2d.

(4.2)

96

Fang-Hua Lin

Note that the case k = 0 means a is a local nondegenerate minimum of WO. For this case we have already established Theorem A in the previous section. Thus we may assume k 2 1. If k = 2d, then a is a local nondegenerate maximum of W (. ). One may choose a suitable 00 > 0 so that

-01 =

max (W(b) - W(a)), bE8B6o (a)

Moreover, let D = B6o(a), and let 8 = {b End: W(b) 2 W(a) - 01/2}. Then aD and 8 are homotopic ally linked in nd. That is, aD n 8 = O. Moreover, one can easily verify that maxW(h(D)) 2 W(a), for any hE C(D, B6o+u(a)) with hl8D = id8D· Suppose now 1 ::; k ::; 2d - 1. We set

D

= {x E JR2d:

Ixl

< 00, Xi = 0, for i 2 k + I},

and let DJ.. = {x E JR2d: Ixl < 00, Xi = 0, for i ::; k}, where coordinate system chosen so that (4.2) is true. Then

(Xl, ... , X2d)

is the

01 = min{ min (W(a) - W(x)), min(W(x) - W(a))} xE8D 8D~ is positive whenever 00 is suitably small. Next we introduce a diffeomorphism h from the open ball {x E JR2d: Ixl < 00 - ~} to §2d\ {N}. Here §2d is the standard 2d-dimensional sphere, and N is the north pole. We also assume h maps {x E JR2d: 00 - (Jo/4 ::; Ixl ::; Do} to N so that h becomes continuous from the closed ball {x E JR2d: Ixl ::; Do} to §2d. We note that the image of D under h becomes a k-dimensional sphere in §2d. We choose (Jo > 0 so small that on the set {x E JR2d: dist(x, aDJ..) ::; 2(Jo} the function W is bounded from below by W(a) - 201/3. We also note that under the map h, the image of the set Y = {x E DJ..: Ixl = 00 - (Jo} is a 2d - k - 1 dimensional sphere. Moreover, these two spheres h(Y) and h(D) are linked in §2d. We shall now embed the k-dimensional ball D, k 2 1, into 8(10, Co, K, g, n) as in Section 1. Thus, for each p, we have a map wp E 8(10, Co, K,g, n) such that 1 Ec(wp ) = 7fdlog -

10

and that

+ "(d + W(p) + 0(1),

(4.3)

Static and Moving Vortices in Ginzburg-Landau Theories

97

where CI is a constant depending only on g, D and the point a, Co may be chosen so that it will depend only on CI. The constant K in the definition of the class S(c, Co, K, g, D) can be chosen to be

K

=

max

bEB6o+<70(a)

W(b).

(4.5)

Moreover, the quantity o( 1) goes to zero uniformly on D as c --t 0+. We remark that the embedding I: p E D --t wp E S(c,co,K,g,O) c H~(D) is continuous, and that wp vanishes exactly at d points PI,P2, ... ,Pd so that P = (PI, ... ,Pd)' From now on we shall always assume c to be so small that the quantity 0(1) in (4.3) is bounded by oI/4 in absolute value. Let uE(t,p,') be the unique solution of the problem:

au 1 at = ~U + c2 (1 -

lui

2.

)u

m0 x R+

(4.6)

with initial and boundary conditions:

uE(O,p,x) uE(t,p, x)

x E 0,

wp(x),

g(x),

for x E aD,

(4.7) (4.8)

It is then obvious that each uE(t,p,') E S(c,co,K,g,O). Moreover, the map --t uE(t, p, .) E S(c, Co, K, g, D) is continuous (with respect to the HI norm metric on H~(D)) for each fixed t 2: O. Indeed, the latter follows from the unique solvability of (4.6)-(4.8). From the energy decreasing property of the flow (4.6) and our choice of the initial data wp , we deduce, from (4.3), that

pED

1

< ndlog - + I'd + W(p) + 0(1) E

1 c

< ndlog -

(4.9)

+ I'd + W(a) + OI/4.

Hence we may apply both structural and compactness theorem in Section 1 to all maps uE(t,p, .), for t 2: 0, P E D. In particular, it follows, when E :::; co, that the essential zeros of uE(t,p, .), say bdt,p), ... , bd(t,p) are well-defined (up to possible errors that are not greater than 4c ao ). Moreover, the point b( t. p) = (bl(t,p), ... ,bd(t,p)) cannot lie in the set {x E JR2d: dist(x,aD~) :::; 20"0} for, otherwise, one deduce from the proof of [[L], Theorem A] that EE(UE(t,p,') 2: ndlog ~ + I'd + W(b(t,p)) + 2~1 + 0(1) 2: ndlog ~ + I'd + W(a) + ~. The latter contradicts to (4.9). Next we observe, from [[L2], Theorem 4.5], that for any t, t' E R+ with It - t'l :::; 1, (4.10) Ib(t,p) - b(t',p)1 :::; 8c ao + OE(1). Here OE (1) is again a quantity which goes to zero (uniformly independent of t and --t 0 where b(t',p) = (b 1 (t',p), ... ,bd(t',p)), and bj(t',p), for j = 1, ... ,d

t') as c

Fang-Hua Lin

98

are essential zeros of Uc: (t', p, .) (well-defined up to a possible error of eno ). Therefore we may choose e to be sufficiently small that the right-hand side is not larger than ~. For tn = n, n = 1,2, ... , we may define continuous map IIn: {uc:(n,p, .),p E D} --+ WK = {b End: W(b) ::; K + I'd} as in Lemma 1. Moreover IIIn(uc:(n,p,·)IIn+l(uc:(n + 1,p, ·))1 ::; ao/8, and IIn(uc:(n,p, .)) is at most 4e no distance away from essential zeros of Uc: (n, p, .). Now we define a continuous map, f: D x R+ --+ nd as follows:

f(p,O) f(p, n) f(P, t)

\f E D, IIn(uc:(n,p, .)),

p,

\fp E D, n = 1,2, ... ,

(t - k)f(p, k + 1) + (k + 1 - t)f(p, k).

Note that f(p, t) is at most !?f distance away from essential zeros of uc:(p, t, .). We also note that, by the energy decreasing property f(p, t) ~ Boo-ao/4(a) for all t 2:: 0 and p E aD. (Note p E aD achieves the minimum for W(·) on Boo(a)). We already noted at the beginning of this section that h(Y) and h(D) are linked in §2d. One may also extend h outside the ball {x E JR2d: Ixl ::; Do} by

simply setting h(x) = h( ~), for Ixl > Do. Now we consider h(Y) and ht(D), t 2:: 0, where ht = h 0 f(·, t). It is clear ho = hand ht is a family of continuous maps from D to §2d (continuous in t). Moreover ht(aD) = N, for each t 2:: o. Finally, since f(p, t) ~ Y, we see that h(Y) n ht(D) = ¢ for each t 2:: O. Therefore h(Y) and ht(D) are linked in §2d for each t 2:: o. From the latter fact, one concludes that for n = 1,2, ... , there is a Pn E D such that f (Pn, n) E D.L. Then, it follows again from the proof of Theorem A in [L] that, 1 e

Ec:(uc:(n,Pn, .)) 2:: 7rdlog -

+ I'd + W(a) + 0c:(1),

for n = 1,2,... .

(4.11)

Since Ec: (uc: (n, Pn, .)) ::; Ec: (uc: (0, Pn, .)), and by our choice of the initial data wp (.), p E D, one obtains that IPn - al ::; De, for n = 1,2, .... Here De is a quantity which goes to zero when e --+ 0+. Without loss of generality, we may assume lim Pn = P n-+oo exist. (By taking a subsequence if necessary; here e is fixed.) Then, on the one hand, one has

and, on the other hand, by the continuity of the map p fixed p,

--+

ue(t,p, .), for each

Here we used the energy decreasing property of the flow (4.6)-(4.8) for each Pn.

99

Static and Moving Vortices in Ginzburg-Landau Theories

From (4.12), (4.13) and (4.11), we have, for this p, that

IEc; (uc; (t,p, .)) - (ndlog ~ + W(a) +

')'d) I ~ Tie -+ 0, as c -+ 0+, for any t ~ O.

(4.14) Moreover, for this particular p, and for a fixed if small c, one may take the time limit to obtain ue(oo,p,·) = lim uc;(t,p,·) so that Ee(ue(oo,p, .)) = ndlog 1. +

W(a)

+ ')'d + 0c;(1).

e

t->oo

Finally, we can apply Corollary 5.5 of [L2] to conclude that uc;(x) = uc:(oo,p,x) has the property that

Here we have also used the property that a is the only critical point in

Bfjo

(a) of

W(·). Remark 1. It is not hard to see that the above proof of Theorem B can also be applied to some cases where a may be a degenerate critical point of W(·). But, on the other hand, certain nontrivial topological information on the level sets of W(·) near the critical point a is necessary for the above arguments. Remark 2. In the above proof, we have implicitly assumed that 1 ~ k ~ 2d - 1. When k = 2d, the set Y will be empty. But when k = 2d, we follow the above proof to show that ht(D) is always the whole sphere S2d. Thus, for n = 1,2, ... , one may always find apn ED such that f(n,Pn) = a. From the latter fact, one immediately obtains the same conclusion that IPn - al ~ be. Let Pn, -+ p, as ni -+ 00 (here c is fixed), then

IEe(Ue(t,p, .)) - (ndlog ~ + W(a) +

')'d) I ~ be,

for all t 2: O. The conclusion of Theorem B, in this case, again follows from Corollary 5.5 of [L2]. Remark 3. The argument of Theorem B can be simply modified for the case dj = ±1 in [L3]. Suppose (a, d) is the nondegenerate saddle point or local maximum point of the renormalized energy W (cf. [BBH]) , where a = (al,"" ad+2£), d = (d l , ... ,dd+2£), d j = 1 for j ::; d + £; -1 for j > d + £, and £ E N is a constant. Then there is still a linking structure in Bfjo (a), where bo > 0 is small. We also have the embedding (cf. Section 1), the gradient flow with intrinsic energy estimates (cf: [L], [L2]) , and the associated projection lemma which is similar to Lemma 1. Hence we obtain the same result as Theorem B for this case. Note: The problem for general higher degree case Idj I > 1 is still open.

100

Fang-Hua Lin

Lecture 3. The Dynamical Law of Ginzburg-Landau Vortices 1. Gor'kov-Eliashberg Equation Before we examine the dynamics of vortices, we have to look at the Gor'kovEliashberg evolution equation: 7)

au

at aA

at

+

i7)fi:u+

+

\7

C;' +A)2 u -u+ lul u = ° 2

+ curl curl A = -~ (u*\7u 2fi:

u\7u*) - Alul 2 ,

}

(1.1 )

in 0 x R+ , 0 C ]R3 . In system (1.1), u is the complex order parameter, A is the magnetic (real) potential, fi:, 7) are positive constants. Here u* denotes the complex conjugate of u and 0 denotes a smooth bounded domain in ]R3, is a scalar electric potential. The system (1.1) is supplemented by the initial and boundary conditions:

A(x, 0) = Ao(x),

u(x, 0) = uo(x),

(~\7+A)u.n .

0, (Ho = 0),

E'n

n

= 0,

(1.3) on

ao.

n

is the exterior unit normal along ao. Note that (1.1)-(1.3) is gauge-invariant, in the sense that if (u, A, solution, then so is (u x ' Ax, x), where Here

(1.2)

0,

curlA;\ n

(~~ + \7 )

x E 0;

Ux

= u ei K x,

Ax = A + \7X ,

x

= -

ax at .

(cf.

<1»

is a

[D], [GEl)

The global existence, uniqueness (up to gauge transformations) of classical solutions of (1.1)-(1.3) have been studied by various authors, [D], [LR], [CHJ]. In [TW], the long time behavior, in particular, the existence of the global attractor is also investigated. Here we shall sketch the proof of the asymptotic stability result, which shows that, as t ---+ 00, (1.1)-(1.3) has a unique asymptotic limit up to gauge transformations. Next we shall consider the vortex motion for the following nonlinear equation:

au {

at

u(x, t) u(x, 0)

1

D.u + 2" u(l - lul 2 ) in 0 x R+ E g(x) for xEan, t>O, Uo (x) for x En.

(1.4)

Here n is a two-dimensional, smooth bounded domain, E is a positive parameter, u : n x R+ --. ]R2, g: an --. §l. The system (1.4) can be viewed as a

101

Static and Moving Vortices in Ginzburg-Landau Theories

simplified model for (1.1) (cf. [N]). A system similar to (1.4) also appears in a canonical way when one expands a large class of second order dissipative systems about bifurcation points, [K], [BKP]. It serves, therefore, as one of the fundamental models in the study of the dynamics of non-equilibrium patterns [PZM], [RS]. The dynamics of the vortices in the limit E --7 0 can be considered within the framework of a general program initiated by J. Neu [N], and later extended, and improved by many others [RSJ, [PRJ, [E]. They formally used the method of matched asymptotic expansions to derive equations of motion for vortices. To leading order in

(IO~

the equations are of the form:

E ),

. d ai (t) __ n . w- ( ) dt v a, 9 a ,

i = 1,2, ... d.

m2

(1.5)

The constants mi are called mobilities of the vortices. One of the key facts they have derived is mi '::::' 11ogEI. In fact, it is derived in [N], [E] that mi == log~, for all i. We want to give a rigorous proof of this dynamical law. We shall not address here the hydrodynamical limit of the above dynamical law for an infinite number of vortices. Some discussions may be found in recent articles [CRS] and [E2]. 2. Uniqueness of Asymptotic Limit As in [D], one may choose the so-called zero electric potential gauge for system (1.1)(1.3). This amounts to solving

ax =

= -div

and

(2.1 )

at t = O.

A on fI with 'VX' n

= -A· n

on

afl.

(2.2)

Thus, in this gauge,

As shown in [D] and [TW] , the flow

dv

-

dt

= - grad

E(v),

v(O) = Va ,

(2.4)

= (u, A) has a global classical solution. Our main result concerning (2.4) is the following

V

Theorem 1. Voo

= lim v(t) t->oo

exists.

Fang-Hua Lin

102

To describe the idea, we start with the O.D.E.

{

~~

=

x(O)

= Xo.

-grad f(x),

x E ]RN

(2.5)

We assume f E C 2(B1)' V' f(O) = 0 and Xo is close to O. Case (i). If A = V'2 f(O) is positive definite, then X(t) -70 (at exponential rate) as t -7 +00.

!it

Ixl 2 = _2(V'2 f(x) . x, x) ::; -2A Ix1 2. Here we shall assume Proof. Calculate Ixl(t) ::; 00, and (V'2 f(x)) 2 AI, whenever Ixl ::; 00. Thus Ix(t)1 ::; Ix(O)le- At , 'It> 0, and Ix(t)1 ::; Ixol + tlx(O)1 = Ixol + ItV' f(xo)l· We shall always assume Xo is so close to the origin that Ixol + tlV' f(xo)1 ::; 00. Then the assumption Ix(t)1 ::; 00 is true for all t > 0, and thus x(t) -70 at the exponential rate as t -7 +00. Case (ii). det (A) #- O. Then one has that

-~(f(x) -

1

IV' f(x)llxl

1

IV' f(x)llxl

f(0)?/2

"2 (f(x) _ f(0))1/2'

and that d --(f(0) - f(x)) 1/2

"2 (f(0) _ f(X))1/2'

dt

dt

if f(x)

> f(O)

if f(x) < f(O).

(2.6)

(2.7)

We obtain the following: Either there is aTE (0, 00) such that

f(x)(T) ::; f(O) - 00

(for some 00> 0)

Or lim x(t) t---t+oo

= Xoo exists.

Indeed, if x is close to 0, then

IV' f(x)1 If(x) - f(0)l1/2

> ~ Amin -

2 Amax

=

C(A) > 0 .

Here Amin and Amax are the minimum and maximum, respectively, eigenvalues of (AT A)1/2. Therefore

roo

io

20 1/ 2

Ix(t)1 dt ::; C(~) ,

by (2.6), (2.7), whenever f(x)(t) 2 f(O) - 00 for all t 2 O. In such a case the conclusion limt--->+oo x(t) = Xoo follows (in fact Xoo = 0) as for the case (i).

103

Static and Moving Vortices in Ginzburg-Landau Theories

Case (iii). det (A) = 0 and f(x) is real analytic in B 1 . We have the following well-known estimate (cf. [SL2]). There are two positive constants eo, (To E (0, 1) depending on f such that

If(x) - f(O)loO ::; Igrad f(x)1

whenever x E Bao (0),

(2.8)

V' f(O) = o. Then, as for the case (ii), one has: either there is aTE (0, (0) such that

f(T) ::; f(O) - 80 or

lim x(t) =

t->oo

Xoo

exists. In [SL2], Simon considered the case where

1M F(x, u, V'u) dx,

E(u) = {

it u(O)

= =

- grad E(u)

uo c::: 0,

(2.9)

and

== M(u)

M(O) =

(2.10)

o.

where F is assumed to be analytic in both u and V'u for u, V'u near O. Here M is a compact manifold without boundary. Suppose L is elliptic, Lu = dds Is=o M(sv). Then either there is aTE (0, (0) such that E(u(T)) ::; E(u(O)) - 80 , for some 80 > 0, or U oo

= lim u(t)

exists.

t->oo

The key point is to show

IE(u) - E(O)IOO ::; II grad E(u)llL2

(2.11)

for u near zero, and for some eo E (0, 1) (independent of u). (2.11) plays the same role as (2.8). We can show (2.11) is valid for our functional E(u, A) which is gauge invariant, degenerate (in a sense, not coercive). The conclusion of the theorem follows as for the case (iii) of (2.5). We refer to [DL] for more details. 3. Vortex Motion Equations Let us consider the following model problems

AE

at

U,(x, 0) U,(x, t)

1 b..UE + 2" U, (1 E

U~(x),

g(x),

2 -IUEI)

xEn

xE

an.

m

n x R+

,

(3.1) (3.2) (3.3)

104

Fang-Hua Lin

Here A€ is a constant depending only on following are true for the initial data in (3.2).

E

> O. We shall always assume the (HI)

U~(x) ~ Uo(x)

II d

=

j=l

x - bj eih(x)

Ix -

bjl

weakly

(H2)

in Hl~c (0 \ {b 1 , ... , bd }) for some h E Hl(n) and some d points b1 , ... , bd in n. Here d = degree of g : an ~ §2, d > O. Our first result says vortices will not move (or move very slowly) in any finite time, when E --t 0+, for solutions of (1.4) with initial data satisfying (Hl)-(H2). Theorem 2. Assume A€ == 1. Then, as

0, solutions U€(x, t) of (3.1}-(3.3) converge, for any t ~ 0, to Uo(x, t) = I1~=1 I~::::~~I ei h(x, t), weakly in Hl~c (0 \ {b1 , ... , bd }). Moreover, the function h satisfies

ah at h(x, 0) d

II

j=l

x - bj ei h(x, t)

Ix -

bjl

E --t

b.h m nx R+, h(x), xEn, g(x),

for x

(3.4)

E an.

We shall not give the proof of this theorem (cf. [L]). Instead, we point out that one of the key points in the proof of the above theorem is to locate the vortices in the flow. We used a quantity of the form (3.5) for some properly chosen p(x), to achieve this purpose. The proof involves some crucial lower energy bounds, obtained in [BBH] and [St], and measurement of changes of the stated quantity (3.5). The proof can be generalized to obtain the following:

and A€/ log ~ --t 0, as E --t 0, then the solutions, u€(x, t) of (3.1}-(3.3), converge to I1~=1 I~::::~~I eihb(x), weakly in Hl~c (0\ {bl, ... ,bd }), where b.hb = 0 in n.

Corollary 1. If A€

--t 00,

The corollary implies the mobilities of vortices are not less than the scale of log ~. Our next result shows mobilities cannot be larger than the scale log ~ either.

Static and Moving Vortices in Ginzburg-Landau Theories

Theorem 3. If AE/ log ~

°

105

+00 as E -+ 0, then any possible weak limits of U, (-, t), are given by rr1=1 I~::::~I eiha(x). Here 6..h a = in n, and

°

-+

for any t > a = (al, ... , ad) is a critical point of the renormalized energy Wg(a) defined by: (cf. Lecture 2 and (BBH)) for (Xl, ... ,Xd) E nd, d

Wg(Xl, ... , Xd) = -

L log IXi - xjl- L R(xj) i#j

+

2~.1ao (11+ ~ log Ix -Xii) gAg,. ° in

where

6..R {

(3.6)

j=l

=

n

{}R

(~

{}

d

on - '" A g, - a"

log

Ix -- Xjl) .

Finally we have the following dynamical law. Theorem 4. If A,

= log ~ ,

Moreover, 6..h a (t) (x) =

then,

°

in

n,

da(t) dt

a(O)

as

E

-+

0,

and

- grad Wg(a(t)), b == (b l

, ... ,

bd )

(3.7)

.

Here a(t) = (al (t), ... , ad(t)). Thus we conclude that there are three time regimes. When 0 :S t :S 8(log ~), there is no vortices motion, but the phase function evolves with time according to the standard heat equation. For 8 (log ~) :S t :S M log ~, 8 < < 1, M > > 1, the vortices motion obeys the law given by the O.D.E. in the statement of Theorem 4, and the phase function is a simple adjustment so that it will be harmonic in the domain and that it matches up the proper boundary conditions. When t > M log ~, both the location of vortices and the corresponding phase functions become static. Note that L. Simon's theorem also implies this (see Section 2 of this lecture) but for much larger time, say t » ~. Indeed, by using Theorem 3, Theorem 4 and the proof of Corollary 5.5 in [L], one can deduce the somewhat stronger statement

106

Fang-Hua Lin

that, in Theorem 3, as E --t 0, the limit of UE(x, t) ----+ n~=l I~:::~!I eiha(x) is unique for all t > O. Now we sketch the proof of Theorem 4. It is in the same spirit as [L2], but with a somewhat different approach. We consider a family of Radon measures 1

[2

(2

1

/LE(t)="2 IVUEI (.,t)+2E2 IUEI (·,t)-l for each t ~ O. Proof of Theorem

)2]

dx 7rlog~'

4·

Step I. We show there is a sequence of Ei 1 0, such that /LEi (t) ----' /L(t), as i --t 00, for each t ~ O. Here ----' denotes the weak convergence for measures. To do so, we let ¢ E CJ(lR2 ), and calculate

_inr ¢21 aUE 12 . _ 1 dx at 10g2 ~ -2

<

In ¢2 ./LE(t)

l

n

+ C(¢)

< C(¢) [II/LE(O)II +

auE ¢ v¢ VUE .

at

In 1a~E 12

1 dx log E

--1

1

--dx log ~

K~(t)]

In

Here C(¢) = 11¢1I~1' KE(t) = I~ 1~12 ~d! and II/LE(O)II denotes the total measure of /LE (0) . Thus the function EE(¢' t) - C(¢) [II/LE(O)II . t + KE(t)] is monotone nonincreasing for t E [0,00). Here EE(¢, t) = /LE(t). We choose a countable dense subset {¢j}~l of CJ(lR2 ), and for each j, we may find a sequence of E'S tending to zero such that the corresponding sequence of functions 'T]t (t) == EE(¢j, t) - C(¢j) [II/LE(O)II . t + KE(t)] pointwise converges to a monotone nonincreasing function 'T]j(t), for each t ~ O. Now we use the diagonal sequence to obtain a sequence of En 10 such that 'T]L (t) pointwise converges to a function 'T](t) , for each t ~ O. It is then easy to see that /LEn (t) ----' /L(t) for each t ~ 0, and for a Radon measure /L(t). Here we also note that II/LE(O)II--t d as E --t 0+.

In ¢2

Step II. The Radon measure obtained in Step I is of the form: /L(t) = ~;=1 8aJ (t) , for some d distinct points a1(t), ... ,ad(t) in n. Moreover, min{lai(t) - aj(t)l, dist (ai(t), an), i -:J j, i,j = 1, ... , d} ~ 8*. This is simply a consequence of the Compactness Theorem in Lecture 1. See also Theorem [BBH], part (iii). Step III. The d-tuple point aCt) = (a1 (t), ... , ad(t)) is continuously dependent on t E

[0, 00),

Static and Moving Vortices in Ginzburg-Landau Theories

107

Indeed, one calculates

1:t EE(¢' t)1

L

~ C(¢) 1a~E 2. lo~2 E

2

llrz aul2 t

< C(¢)

-aE

•

.

('XJ

Jo

lt2[1 lau rz

EI

2

at

r aaUE

Jrz

1

t

r

1 log

-2

2

t,

dx

+

.

.[L lo~' 1IO~ -

E

1/2 [ r1au 12 1 ]1/2 r I log1] [JrzIVU ~ dx Jrz atE . log ~ dx

+ C(¢)

Using the fact that

1

12 .

E

dx 2

~

log E

]1/2 dt

. - 1 dx log ~

+ C(¢)V7r (d + 1) .

dx dt

< C(K, g, [2), and hence,

~ It2 _t11 1/ 2 . VC(K, g, [2),

one can deduce the continuity of a(t) by various proper choices of ¢ E C6([2).

Step IV. Proof of (3.7). We first note it suffices to show (3.7) for 0 ~ t ~ 8, for some 8 = 8(g, [2, K) > O. In fact, we simply choose 8 = ~~ . For 0 ~ t ~ 8, we have then aj(t) c B6(aj(O)) , aj(O) = bj , for j = 1, ... ,d. For j = 1,2, ... ,d, R E [Ro/2, RoJ, Ro = we multiply (3.1), with AE = log~, by VUE and use

6:,

integration by parts in BR(b j

)

to obtain

aUE . VU

av

E

On the other hand, one calculates, with e E= ~ [IVUEI2 + 2~2 (IUEI2 - 1)2], that

Fang-Hua Lin

108

Therefore, by integrating with respect to R E [~, Ro], and with respect to the variable t, one has fE(t) = gE(t) + hE(t), where:

hE(t) 2

+ Ro log E1

r

IGE(u)ldx::;

) BRo \BRo/2 (b j

)

}

r

BRo \ BRo/2 (b j

)

[4\ E

(IU I2 _1)2 + ~ E

IVUEI2 ]dX

< C(K, g, D) . Hence gE(t) is also lipschitz continuous on [0, 8] (independent of E). By Step II, fEn (t) -----+ 7r(aj(t) - bj ) and hence gE n(t) -----+ 7r(aj(t) - bj ). Finally we use a strong convergence theorem (cf. [L] Theorem 4.5) to conclude that

1

BRo \BR o/ 2 (b j

GEJu) dx converges to

~RoVaj Wg(a(t)).

Therefore we obtain the

)

identity:

7r(aj(t) -aj(O)) =

-7r

lot V aj Wg(a(t))dt.

This is exactly (3.7). Since the measure f-l(t) obtained in Step I and Step II is completely identified by (3.7), we see f-lE(t) -----+ L~=l 8aJ (t) follows. D Remarks. (a) Recently, Jerrard and Soner [JS] proved the same dynamical law as (3.7) for the case that the initial data may have both +1 and -1 degree vortices.

Of course (3.7) is then verified before the possible anhilation of vortices occurs. On the other hand, it is also possible that vortices of ± 1 degree may persist for all time, and the limiting static configurations may even be locally energy minimizing.

(cf. [L3]) (b) Recently the author also derived rigorously the dynamic law for vortices under pinning effects. The idea can be generalized to the higher dimensions to study vortex-submanifold, in particular, filaments dynamics.

Static and Moving Vortices in Ginzburg-Landau Theories

109

(c) It is a challenging open problem to study the dynamics of vortices for Schrodinger's equations:

{

. au,

at

t --

UE(x, 0) U,(x, t)

1

flUE + 2"U' o E U, (x),

g(x) ,

(1

-IU,1 2 ),

In

0 x R ,

xEO xEO.

The formal asymptotic analysis was carried out in [N] and [E]. There are also related questions in fluid dynamics. Similarly, one can also look at the corresponding hyperbolic equations. Many formal arguments there need to be justified. (d) From the physics point of view, it will be very important to understand the dynamics of vortices under magnetic fields and applied currents. It is also important to study problems in the entire space and to study the stability of solutions. References [AB] L. Almeida and F. Bethuel, Methodes topologiques pour l'equation de GinzburgLandau, C. R. Acad. Sci. Paris, t. 320, Serie I, (1995), 935~939. [AS] N. Andre and I. Shafrir, Asymptotic behavior for the Ginzburg-Landau functional with weight, part I, II, preprints. [BBH] F. Bethuel, H. Brezis and F. Helein, Ginzburg-Landau vortices, Birkhiiuser, Boston, (1994). [BCP] P. Bauman, N. Carlson and D. Phillips, On the zeros of solutions to GinzburgLandau type systems, SIAM J. Math. Anal. 24 (1993), 1283~1293. [BKP] K. Bodenschats, W. Pesch and L. Kramer, Structure and dynamics of dislocations in anisotropic pattern forming systems, Phys. D 32 (1988), 135~ 145. [BMR] H. Brezis, F. Merle and T. Riviere, Quantization effects for -.0.u = u(l lul 2) in ]R2, Arch. Rat. Mech. Anal. 126 (1994), 35~58. [BR] F. Bethuel and T. Riviere, Vortices for a variational problem related to superconductivity. [CDG] S. Chapman, Q. Du and M. Gunzburger, A variable thickness model for superconductivity thin films, preprint. [CET] Zhiming Chen, C.M. Elliot and Q. Tang, Justification of a two-dimensional evolutionary Ginzburg-Landau superconductivity model, preprint. [CHJ] Z. M. Chen, K. H. Hoffmann and L. S. Jiang, On the Lawrence-Doniach model for layered superconductors, preprint. [CHO] J. Chapman, S. Howison and J. Ockendon, Mareoscopic models for superconductivity, SIAM Review 34 (1992), 529~560. [CK] S. Chanillo and M. Kiessling, Symmetry of solutions of Ginzburg-Landau equations, preprint. [CRS] S. Chapman, J. Rubinstein and M. Schatzman, A mean field model of superconducting vortices, Euro. J. Appl. Math. (to appear). [D] Q. Du, Global existence and uniqueness of solutions of the time-dependent GinzburgLandau models for Superconductivity, to appear in Applicable Analysis.

110

Fang-Hua Lin

[De] De Gennes, Superconductivity of metals and alloys, Addison-Wesley Publishing Company. [Di] S. Ding and Z. Liu, Pinning of vortices for a variational problem related to the superconducting thin film having variable thickness, preprint. [DF] M. del Pino and P. L. Felmer, Local minimizers for Ginzburg-Landau energy, preprint (1995). [DG] Q. Du and M. D. Gunzburger, A model for superconducting thin films having variable thickness, Phys. D (Nonlinear Phenomena) 69 (1993), 215-23l. [DGP] Q. Du, M. D. Gunzburger and J. Peterson, Analysis and approximation of Ginzburg-Landau models for superconductivity, SIAM Review 34 March (1992). [DL] Q. Du and F. H. Lin, Ginzburg-Landau vortices, pinning and hysterisis, preprint. [E] E. Weinan, Dynamics of vortices in Ginzburg-Landau theories, with applications to superconductivity, Phys. D 77 (1994), 383-404. [E2] __ , Dynamics of vortex liquid in Ginzburg-Landau theories, with applications to superconductivity, Phys. Review B. vol. 50 #2 (1994), 1126-1135. [F] H. Federer, Geometric Measure Theory, Springer-Verlag, Heidelberg-Berlin-New York, (1969). [FP] P. C. Fife and L. A. Peletier, On the location of defects in stationary solutions of the Ginzburg-Landau equations in ~2, Quart. Appl. Math. (to appear). [GE] L. Gor'kov and G. Eliashberg, Generalization of the Ginzburg-Landau equations for nonstationary problems in the case of alloys with paramagnetic impurities, Soviet Phys. JETP 27 (1968), 328-334. [GL] V. Ginzburg and L. Landau, On the theory of superconductivity, Zh. Eksper. Teoret. Fiz. 20 (1950),1064-1082. [English translation in Men of Physics: L. Landau, I (D. ter. Haar ed.), Pergamon, New York and Oxford, (1965), 138-167.] [GO] M. Guzburger and J. Ockendon, Mathematical models in superconductivity, SIAM News, November and December 1994. [HL] R. Hardt and F. H. Lin, Singularities for p-energy minimizing unit vector fields on planar domains, Cal. of Variations and P.D.E. (to appear). [JT] A. Jaffe and C. Taubes, Vortices and Monopoles, Birkhiiuser, Boston and Basel, (1980). [JS] R. Jerrard and M. Soner, Dynamics of Ginzburg-Landau vortices, preprint. [K] Y. Kuramoto, Chemical Waves, Oscillations, and Turbulence, Springer-Verlag, (1984). [LiL] E. H. Lieb and M. Loss, Symmetry of Ginzburg-Landau minimizer in a disc, Math. Res. Letters 1 (1994), 701-715. [Lin] T. C. Lin, Spectrum of linearized operators at the radial solutions of the GinzburgLandau equation. [LL] T. C. Lin and F. H. Lin, Minimax solutions of Ginzburg-Landau equations, preprint. [L] F. H. Lin, Solutions of Ginzburg-Landau equations and critical points of the renormalized energy, Analyse non Lineaire, IHP. To appear. [L2] __ , Some dynamic properties of Ginzburg-Landau vortices; and A remark on the previous paper, both to appear in CPAM. [L3] __ , Mixed vertex-antivertex solutions of Ginzburg-Landau equations, Arch. Rat. Mech. Analysis (to appear). [LR] C. Lefter and V. D. Radulescu, On the Ginzburg-Landau energy with weight, C. Rend. Acad. Sci. Paris. To appear.

Static and Moving Vortices in Ginzburg-Landau Theories

III

J. Neu, Vortices in complex scale fields, Phys. D 43 (1990), 385-406. P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equations, Jour. Functional Analysis, 130 (1995), 334-344. [PeR] L. Peres and J. Rubinstein, Vortex Dynamics for U(I)-Ginzburg-Landau Models, Phys. D. 64 (1993), 299-309. [PRJ L. Pismen and J. D. Rodriguez, Mobilities of singularities in dissipative GinzburgLandau equations, Phys. Rev. A 42 (1990), 2471-2474. [PZM] Y. Pomeau, S. Zaleski and P. Manneville, Disclinations in liquid crystals, Quart. App!. Math. 50 (1992), 535-545. [RS] J. Rubinstein and P. Sternberg, On the slow motion of vortices in the GinzburgLandau heat flow, preprint. [S] I. Shafrir, Remarks on solutions of Deu = u(1 ~ [U[2) in ffi.2, C.R. Acad. Sci. Paris, t. 318, Serie I, (1994), 327-33l. [SL] L. Simon, Lectures on geometric measure theory, Austr. Nat. Univ. CMA, (1983). [SL2] L. Simon, Asymptotics for a class of non-linear equations with applications to geometric problems, Annals of Math 118 (1983), 525-·572. [St] M. Struwe, On the asymptotic behavior of minimizers of the Ginzburg-Landau model in 2-dimensions, J. Diff. Equations 7 (1994). [TW] Q. Tang and S. Wang, Ginzburg-Landau equations of superconductivity, preprint. [N] [M]

Progress in Nonlinear Differential Equations and Their Applications, Vol. 29 © 1997 Birkhauser Verlag Basel/Switzerland

Wave maps MICHAEL STRUWE

Mathematik, ETH-Zentrum, CH-8092 Zurich

ABSTRACT. In these lectures we outline the known results concerning existence, uniqueness, and regularity for the Cauchy problem for harmonic maps from (1 + m)-dimensional Minkowski space into a Riemannian target manifold, also known as a-models or wave maps. In particular, we mark the limits of the classical theory in high dimensions and trace recent developments in dimension m = 2, substantiating the conjecture that in this "conformal" case the Cauchy problem is well-posed in the energy space.

1.

Local existence. Energy method

1.1. The setting Let (M, 'Y) be an rn-dimensional Riemannian manifold without boundary, the "domain" of our maps, and let (N,g) be a compact, k-dimensional Riemannian manifold, with aN = 0, the "target". For simplicity, in these lectures we only consider the case M = ]Rm; however, many of the results presented below can easily be extended to the case of a compact domain manifold, for instance, to the case M = T m = ]Rmjzm, the flat torus. Moreover, by Nash's embedding theorem, we may assume that N c ]Rd isometrically for some d > k. We denote as TpN C Tp]Rd ~ ]Rd the tangent space of N at a point p, and we denote as Tf N the orthogonal complement of TpN with respect to the inner product (-,.) on ]Rd. TN, T.L N will denote, respectively, the corresponding tangent and normal bundles. Moreover, since N is compact, there exists a tubular neighborhood U28 (N) of width 26 of N in ~d such that the nearest neighbor projection 7rN: U26 (N) -> N is well-defined and smooth. For M and N as above we consider smooth mapsu:]R x 1'vI -> N ~ ]Rd on the space-time cylinder ]R x M. The space-time coordinates will be denoted as z = (t, x) = (xa)O<::;a::;m and we denote &'l a~a U = oau = Uxa the partial derivative of U with respect to x a , O:S: O::S: rn. Also let D = \7) = (a~a )O<::;a::;m. ]R x M will be endowed with the Minkowski metric TJ = (TJa(3) = diag(l, -1, ... , -1) and we raise and lower indeces with the metric. By convention, we tacitly sum over repeated indeces. Thus, for example, oa = TJ a(30(3, where (r)a/3) = (r)af3)-1 (= (TJo:(3) in our setting). Moreover,

eft,

Michael Struwe

114

is the wave operator and

"21 (8a u,8a u) ="21 (2 IUtl -1~ul 2) is the Lagrangean density of u.

1.2. Wave maps Let u: JR x M ---+ N be sufficiently smooth. A (compactly supported) variation of U is a family of maps U,: JR x M ---+ N depending smoothly on a parameter EEl - EO, EO[ for some EO > 0, with Uo == U and such that u, == Uo outside some compact region Q c JR x M for all E. Given a map 'P E Co(JR x M; IR d ), an admissible variation may be obtained, for instance, by letting u, = 7fN(U+E'P) for lEI < 2bll'PIIL~' where 7fN: U26 (N) ---+ N is the smooth nearest neighbor projection defined in Section 1.1. A map U then is a wave map if U is a stationary point for the Lagrangean

=

.c(u; Q)

~

h

(8 a u, 8a u) dz

lEI < EO,

with respect to compactly supported variations u"

dd .c(u,; E

Q)I

,=0

= 0,

where Q strictly contains the support of u, - u. In particular, for the variation u, = 7fN(U equation

° :E =

=-

.c(7fN(U + E'P); Q)

1,=0 =

in the sense that

+ E'P)

above we then obtain the

10 (8 u, 8 (d7fN(U) . 'P)) dz a

a

10 (8 8 u, d7fN(U) . 'PI dz a

a

for all 'P E Co(JR x M; JRd); that is, Du(z) -1 Tu(z)N for all z E JR x M, or

Du -1 TuN for short. To understand and let Vk+1,"" Vd pEN for p near Po k < l :S d, such that

this relation in more explicit terms, fix a point Zo E JR x M be an orthonormal frame for N, smoothly depending on = u(zo). Then we can find scalar functions ).,1: JR x M ---+ JR, near z = Zo there holds

Tl

Du = ).,1(VI

0

u);

in fact, Al = {Du, VI

0

u)

= 8 a (8a u, VI =

0

UI - (8a u, 8 a (VI

-(8a u, dVI(U) . 8 a u)

0

u))

= _AI(u)(8a u, 8 a u)

Wave maps

is given by the second fundamental form Al of N with respect to wave map equation takes the form

115 VI.

Thus, the (1.1 )

where A = Alvl is the second fundamental form of N. We regard the term on the right of (1.1) as a Lagrange multiplier associated with the target constraint u(JR x M) eN. 1.3. Examples In certain cases equation (1.1) takes a particularly simple form. 1.3.1. tion

The sphere.

For N = Sk C JRk+l equation (1.1) translates into the equa-

Du = (lV'uI2 -lutI2)u.

Indeed, since u -.l TuSk it suffices to check that

1.3.2. Geodesics. Suppose 1': JR -+ N is a geodesic parametrized by arc-length and u = I' 0 v for some map v: JR x M -+ R Compute

Note that 1" is parallel along 1'; that is, I'I/(s) 1- T"((s)N for all s E R Thus, u satisfies (1.1) if and only if v solves the linear, homogeneous wave equation Dv = O. 1.4. Basic questions In view of the hyperbolic nature of equation (1.1), in particular, in view of Example 1.3.2, it is natural to study the Cauchy problem for equation (1.1) for (sufficiently) smooth initial data (1.2)

The basic questions we shall ask are the following. Local well-posedness in the smooth category: Does the initial value problem (1.1), (1.2) for smooth data always admit a unique smooth solution for small time It I < T? The smoothness hypothesis on the solution and the data may be weakened. In fact, for a function u E Lroc(JR x M; N) it is possible to interpret equation (1.1) in the sense of distributions provided Du E Lfoc(JR x M). More specifically, for s E No we let HS(M; N) = {v E Hs,2(M; JRd); v(M) c N} denote the Sobolev space of maps v: M -+ N such that v possesses square integrable distributional derivatives of any order up to s. Moreover, we say that u E Lroc(IR x M; N) is a weak solution of (1.1), (1.2) of class H S provided (gt)au(t) E

Michael Struwe

116

H s - a (M) for all (J S s, locally uniformly in t, and if u weakly solves (1.1) and assumes the initial data (1.2) in the sense of traces. Then we can pose the problem of Regularity: What is the minimal regularity of the data to ensure unique local solvability of (1.1), (1.2) in the same regularity class? Global well-posedness: Does there exist a regularity class such that the Cauchy problem (1.1), (1.2) admits a unique solution in this class for all time? We do not consider explicitly the issue of stability, that is, continuous dependence of solutions on the data. However, quite often stability is related to uniqueness. 1.5. Energy estimates Let e(u) = ~IDuI2 be the energy density of a map u:lR x M

E(u(t)) =

lm

----+

N, and let

(e(u))(t) dx

be the total energy of u at time t. Note that, if u solves (1.1), we have the conservation law d 0= (Du, Ut) = dt =

(lutI2 ) -2-

. - dIV(V'U, Ut)

+ (V'u, V'Ut)

!e(u) -div(V'u,ut).

Hence, if Du(t) has compact support, upon integrating over lRm we find that d

dtE(u(t)) = 0; that is, total energy is conserved. A similar energy estimate also holds on light cones. In particular, it follows that the diameter of supp(Du(t)) grows with speed at most 1 and hence Du(t) has compact support for all t whenever supp(Du(O)) is compact. 1.6. L 2-theory The above energy inequality may be generalized to obtain a priori bounds for higher derivatives, as well. Consider the Cauchy problem

Du = j

in lR x M

u It=o= g, Ut It=o= h, where j,g, and h are smooth functions such that supp(Du(O)) = supp(h, V'g) is compact and supp(f(t)) is compact for any t. Then we have

117

Wave maps

and hence by Holder's inequality d d IIDu(t)II£2'dt IIDu(t)ll£2 = -dt E(u(t)) ::::;

1

{t}xIRm

IfllUtl dx

: : ; Ilf(t)II£2II Ut(t)II£2 : : ; Ilf(t)II£2IIDu(t)II£2' It follows that

d

dtIIDu(t)ll£2 ::::;

Ilf(t)II£2·

Similarly, for any multi-index I = (i o, ... ,i m ) E N6+ m with 01 = II",o:; we obtain

III =

I;",i o" letting

(1.3) for all t. Integrating in time, thus we find that for any I E N6+ m there holds

and similarly for T < O. Note that, using the equation Du = f, we can express any derivative DOl u(O) in terms of spatial derivatives of 9 and h of orders III + 1 and IJI, and space-time derivatives of f at t = 0 of order II -11, respectively. For instance, Utt

Letting

= Du + Au = f + Ag.

IlvIIU

X ),2

=

sup

O'S,t'S,T

Ilv(t)II£2,

therefore for any s E No we obtain the estimate

with a constant C = C(8). 1. 7.

Local existence for smooth data The results of the preceding section apply to obtain a-priori bounds for smooth solutions u to (1.1), (1.2) by letting f:= A(u)(o",u,o"'u), etc. The class of admissible data for s E No is given by

Hg+l(M; TN) = {(uo, ud

E

Lfoc(M; TN);

Uo E HS+1(M;]Rd),Ul E HS(M;]Rd),sUpp(Ul, \luo) cc ]Rm}. Note that by Sobolev's embedding HS <:.......+ L oo for 8 > W-. Therefore, and using interpolation, whenever for some constant Co the estimate sup IID s o+lu(t)II£2::::; Co

O'S,t'S,T

(1.4)

Michael Struwe

118

is satisfied for some So

>

~,

then for any sEN we can estimate

uniformly in 0 ::; t ::; T . By (1.3), therefore

and Gronwall's inequality yields bounds for Ds+1u in L'Xo,2, provided (uo, ud E H~+1(M; TN). In particular, the estimate (1.4) will be valid for some T > 0 if we fix a constant Co > IID so+1 u(O)llp, a constant depending only on UO,Ul, and So. Similarly, by the contraction principle, one can show the existence of a unique solution u of class Hs+ 1 on a small time interval 0 ::; t ::; T, provided (uo, ud E H~+l(M;TN) for some s > ~. In this way we obtain

Theorem 1.1 Fix initial data (uo,ud E H~o+l(M;TN), where So > !J}. There exists T > 0 and a unique solution u: [0, T] x M -+ N of (1.1), (1.2) such that sup IIDso+lu(t)llp <

00.

OSotSoT

Moreover, if (uo, ud E H~+1(M; TN) for some s > so, then sup IID s+1 u(t)ll£2

OSotSoT

< 00.

In particular, if Uo, Ul are smooth, also the solution u is smooth on [0, T] x M. For more details and references, see for instance [27] or [31].

1.8. A slight improvement The local existence theorem in the preceding section did not use the special structure of the nonlinear term in (1.1) nor its geometric interpretation. Using the fact that the term on the right of (1.1) is a "null-form" in the derivatives of the components of u = (u 1 , ... ,u d ), that is, the fact that

is a sum of bilinear forms whose symbol

vanishes on the null cone

Klainerman-Machedon [32] obtained the following slight improvement of Theorem 1.1 in low dimensions.

119

Wave maps

Theorem 1.2 Suppose m :S 3. Then for any data (uo,ud E H;(M;TN) there exists a unique local solution u of class H2. If (uo, Ul) E HS, s > 2, then so is u.

The proof in [32] uses special L2-estimates for null forms and is quite involved. A very simple proof, based on energy estimates alone, however, can be given if one uses the geometric interpretation of (1.1). This observation is due to [49]. Proof. (i) First we derive local a-priori estimates for D 2 u for smooth solutions u. Let 8 be any first order derivative. Differentiating equation (1.1), we obtain

D(8u) = 8 (A(u) (8",u, 8"'u)) = dA(u)(8u, 8",u, 8"'u)

+ 2A(u) (8",8u, 8"'u)

with data

(8u It=o, But It=o)

E

H~(M; TJRd).

Note that, by orthogonality (Ut, A(u)(·, .)) = 0,

(But, A( u)(8",8u, 8"'u)) = -(Ut, dA(u)(8u, 8",8u, 8"'u)). Hence we obtain

r

: E(8u(t)) = (D(8u), 8ut)dx t J{t}XM

:S ClldA(u)lluXl . fMIDU(t)13ID2u(t)1 dx. Since N is compact, dA is uniformly bounded on N. Moreover, by Sobolev's embedding, if m = 3 we can estimate

fMIDU(t)13ID2U(t)1 dx :S IIDu(t)lli61ID 2 u(t)ll£2 ::::: ClID 2u(t)lli2

=

CE(Du(t))2.

Similarly, if m = 2 we estimate

fMIDU(tWID2U(t)1 dx :S CIIDu(t)II£2IID2u(t)lli2 ::::: CE(u(t))1/2E(Du(t))3/2. Thus, in both cases we arrive at a Gronwall type inequality

~E(DU(t))

:S CE(Du(t))'

with some 'Y > 1. A local-in-time H2-bound follows. (ii) To show uniqueness in the class H2 let u and v be solutions of (1.1) on [0, T] x M sharing Cauchy data

(u It=o= v It=o,ut It=o= Vt It=o)

E

H~(M;TN).

Michael Struwe

120

Observe that, since (Ut, A(u)(-, .)) = 0, etc. we have

(Ut - vt,Du - Dv) = (Ut - Vt, A(u)(8 u, 8 u) - A(v)(8 v, 8 v)) Q

Q

Q

Q

= (Ut, (A(u) - A(v))(8Q v,8 Q v)) - (Vt, (A(u) - A(v))(8Q u,8Q u))

s Clu -

vlID( u - v) I (IDuI2

+ IDvI2).

Thus, if m = 3, the energy inequality and Sobolev's embedding give

ddt E((u - v)(t)) =

S

cJ

{t}xM

J

{t}xM

(Ut - Vt, Du - Dv) dx

lu - vllD(u - v)I(IDuI2

+ IDvI2) dx

S C(IIDu(t)lIi6 + IIDv(t)lIi6 )1I(u - V)(t)IIL6I1D(u - V)(t)lI£2 S C(IID 2u(t)lIi2 + IID 2v(t)lIi2) IID(u - v)(t)lIi2

S C(IID2Ullioo,2

+ IID2Vllioo,2)E((u -

v)(t)),

and uniqueness follows. For m = 2 the argument is similar. (iii) An H2-solution preserves higher regularity of the data. Indeed, by Theorem 1.1 it suffices to show this for data (uo, Ul) E H~ (M; TN). Let u: [0, T] x M --+ N be a local H2-solution of (1.1), (1.2). We claim that u is of class H3 on [0, T] as well. For this, by Theorem 1.1 it suffices to prove an a-priori estimate for IID3u(t) IIL2. As before let 8 be a first order differential operator. For simplicity, at first we consider only spatial derivatives. Note that

(8 2ut, 82 (A(u) (8 u, 8 u))) = 2(82ut, A(u)(8 82u, 8 u) + A(u) (8 8u, 8 8u)) + 4(82Ut, dA(u)(8u, 8 8u, 8 u)) + (8 2ut, d2A(u) (8u, 8u, 8 u, 8 u) Q

Q

Q

Q

= I

Q

Q

Q

Q

Q

Q

+ II + III.

Clearly, by Sobolev's embedding theorem we can bound

J

{t}xM

IIIII dx s

cJ

{t}xM

IDul 4 1D3ul dx

S CIIDu(t) IILoo IIDu(t)lIi6I1D3u(t)lI£2 S CIID 2u(t)lIi2(1 + IID3u(t)lIi2) S C(1 + E(D 2u(t))),

r

i{t}XM

IIII dx s C

r

IDul 21D 2ullD3ul dx i{t}XM S CIIDu(t)lIi6I1D 2u(t)IIL6I1D3U(t)lI£2 S CIID 2u(t)lIiAD3U(t)lIi2 S CE(D 2u(t)).

121

Wave maps

Before we can estimate the first term in a similar fashion we need to express I in a more convenient way. Using orthogonality (Ut, A(u)(·, .)) again, we have

(8 2Ut,A(u) (8 82u, 8 u)) = 8(8ut, A(u)(8 82u, 8 u)) - (But, A(u)(8 83u, 8 u)) - (But, A(u)(8 82u, 8 8u)) - (But, dA(u) (8u, 8 82u, 8 u)) = 8(8ut, A(u) (8 82u, 8 u)) + (Ut, dA(u)(8u, 8 83u, 8 u)) Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

+ (Ut, dA(u) (8u, 8 82u, 8 8u)) - (But, dA(u) (8u, 8 82u, 8 u)); Q

Q

Q

Q

moreover,

(ut,dA(u) (8u, 8 83u, 8 u)) = 8(ut, dA(u)(8u, 8 82u, 8 u)) - (But, dA(u) (8u, 8 82u, 8 u)) - (Ut, d2A(u) (8u, 8u, 8 82u, 8 u)) - (Ut, dA(u) (82u, 8 82u, 8 u)) - (Ut, dA(u)(8u, 8 82u, 8 8u)). Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Q

Similarly,

(8 2ut,A( u)(8 8u, 8 8u)) = 8(8ut, A(u)(8 8u, 8 8u)) Q

Q

Q

Q

- (But, dA(u)(8u, 8 8u, 8 8u)) - 2(8ut, A(u) (8 82u, 8 8u)), Q

Q

Q

Q

and Thus, as above we conclude that

r

i{t}XM

I dx:::; C

r

i{t}XM

(IDuI2ID2UI

+ IDuI4)ID3ul + IDullD 2ul 3dx

:::; C(l +E(D 2 u(t))).

!

Hence there holds

E(
For 8 = gt we use the fact that 82Ut = (Du)t computations, we can estimate

r

i{t}XM

+ Llut.

Repeating the previous

(Llut, 8 2 A(u) (8 u, 8 u)) dx :::; 0(1 + E(D 2u(t))). Q

Q

Moreover, by Young's inequality

(Dut,82 (A(u)(8 u, 8 u))) = (8(A(u)(8 u, 8 u)), 82 (A( u)(8 u, 8 u))) :::; C(IDuI3 + IDuIID2UI) (IDuI4 + IDul 21D 2ul + ID 2ul 2 + IDullD3ul Q

Q

Q

Q

Q

:::; C(I Du I7 + ID u l5 1D 2ul + IDul 41D3 u l + IDul 31D 2ul 2 + IDul 21D2ullD3ul + IDuIID2uI3) :::; C(IDuI 7

+ IDul 41D3ul + IDul 21D 2ullD3u l + IDuIID2uI3)

Q

Michael Struwe

122

and

r

i{t}XM

r

i{t}XM

IDul 7 dx :S CIIDu(t)llux, IIDu(t)1116 :S CIID 2 u(t)1112 (1 + IID3u(t)II£2) :S C(l + E(D 2u(t)))1/2,

IDullD 2 ul 3 dx :S

CIIDu(t)IIL61ID 2 u(t)II£2IID 2 u(t)lli6

:S ClID 2 u(t)lli21ID3u(t)lli2 :S CE(D 2u(t)). Thus we also find that

and hence that

which yields the desired a-priori bound. (iv) Existence of local H2-so1utions can now be obtained as follows. Approximate the given data (uo, ud E H;'(Mi TN) by smooth data (uQ' ur) E H~(Mi TN) in the topology of H;'(MiTN). The approximate solutions un exist on a uniform time interval 0 :S t :S T and IID 2un (t)IIL2 is uniformly bounded for n E Nand o :S t :S T. Hence as n --; 00 a sub-sequence un~u

weakly in H2 ([0, T] x

M)

and by Rellich's theorem un --; u strongly in HI ([0, T] x

and hence u solves (1.1) in the distribution sense.

M).

In particular,

0

It is conjectured that the initial value problem for (1.1), (1.2) is locally wellposed for data of class Hg+I(Mi TN), where s 2: m;-2. In particular, in the "conformal case" m = 2, we expect the initial value problem to be locally well-posed for finite energy data, and hence, since energy is conserved (by classical solutions), we expect the existence of global unique solutions in this case. In Lecture 3 we will give some partial results in this regard. 1.9. Global existence, the case m = 1 If m = 1, as observed by Shatah [41] we can exchange the roles of x and t in our derivation of the conservation law

123

Wave maps

in Section 1.5 to obtain

Taking the t-derivative of the first and the x-derivative of the second equation and adding, we thus find that e( u) solves the linear homogeneous wave equation

De(u)

=

02 ( ot 2

-

(2) ox 2 e(u)

=

o.

(1.5)

From this fact we easily deduce: Theorem 1.3 Suppose m = 1 and let (uo, ud E H~(M; TN), s 2:: 2. Then (1.1), (1.2) admits a unique global solution u of class HS. Proof. By Sobolev's embedding, Hi H~(M;TN)

'--+

LOO for m

1. Hence, if (uo,ud E

d e(u) It=oE H~, dt e(u) E L2

and the energy inequality applied to (1.5) yields the a-priori bound

E(e(u)(t)) ::; E(e(u)(O)) < 00, uniformly in t E R Hence, by Sobolev's embedding again, e(u) is uniformly apriori bounded on space-time in terms of the data. The assertion of the Theorem then follows by the same reasoning as used in the proof of Theorem 1.1. 0 2.

Blow-up and non-uniqueness

2.1. Overview In Lecture 1 we convinced ourselves that the initial value problem

Du

=

-A(u)(oQu, oQu ) ~ TuN u It=o= uo, Ut It=o=

Ui

on lR x M, on M,

(2.1) (2.2)

is locally well-posed for sufficiently regular initial data (uo,ud E H~+1(M;TN), s > '¥', see Theorem 1.1. In this lecture we now investigate the behaviour of solutions for large time. In particular, depending on the dimension m of the domain and geometric properties of the target, we shall observe a decay of regularity, blow-up of higher derivatives and non-uniqueness of weak solutions beyond such blow-up points. These results indicate the limits of a regularity theory for (2.1) in dimensions m 2:: 3 and raise the question whether there exists a class of weak solutions for which the initial value problem for wave maps is well-posed in a global sense. For better perspective and comparison, in the next section we give a brief survey of the known regularity results for harmonic maps in the elliptic (stationary) case and for the associated parabolic flow.

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Michael Struwe

2.2. Regularity in the elliptic and parabolic cases In the elliptic case we consider weak solutions u E Hl~c(M; N) of the equation

-

~u =

A(u)(V'u, V'u) -1 TuN

(2.3)

with finite static energy

Here, M may be a smooth, compact m-manifold, possibly with boundary, or M = ]Rm. Associated with (2.3) is the heat flow Ut -

~u =

A(u)(V'u, V'u) -1 TuN, u

It=o= uo,

(2.4) (2.5)

which is the L 2-gradient flow for Est in the space Hl(M; N).

2.2.1. Geometric constraints. If the sectional curvature KN of N is non-positive, the Bochner identity for (2.3) implies that

and hence an a-prIOrI Cl-bound for smooth solutions. The same is true for the heat flow (2.4), (2.5). Hence the family u(t,·) of maps generated by (2.4), (2.5) is relatively compact in any Cl-topology and accumulates at a smooth limit U oo : M ----; N. Note, moreover, that (2.4) implies the identity

d IV'u 12 - dlv . (V'u, Ut) + IUtl 2 = o. "21 dt Upon integrating this equation over [S, T] x M, we deduce that for any T there holds

1s

T

Letting S ----; 0, T ----;

fMlutl2 dxdt + Est (u(T)) :s; Est(u(S)).

00,

>S>0 (2.6)

we find the a-priori estimate

in particular, Ut ----; 0 smoothly, as t ----; 00. From this observation, Eells-Sampson [10] derived their fundamental existence result:

Wave maps

125

Theorem 2.1 Suppose KN 'S O. Then for any smooth map Uo: M ----; N there exists a smooth harmonic map Uoo : M ----; N homotopic to Uo.

In fact, for KN 'S 0 every weakly harmonic map u E H1(M; N) is smooth. The curvature constraint on the target can be replaced by the condition that the range of u is contained in a strictly convex geodesic ball on the target N or that the range u(M) is the domain of a strictly convex function; see Hildebrandt [26] or Jost [28], [29] for a survey. Theorem 2.1 may be false if the condition KN 'S 0 is violated. The following result by Lemaire [33] and Wente [50] shows that, for instance, maps Uo: B2 = B1 (0; ]R2) ----; S2 of degree 1= 0 and which are constant on the boundary of B cannot be represented by harmonic maps. Theorem 2.2 If u E H1(B2; S2) is harmonic with u laB:::::: const., then u:::::: const.

Proof. Let u laB:::::: p E S2 and let 1f: S2 \ {p} ----; ]R2 denote stereographic projection from the antipodal point of p. Also let i:]R2 ----; ]R2 denote the involution i(x) = -x. Extend u to a map u E H1(]R2; S2) by letting

Since 1f- 1 i 1f induces an isometry of S2 and since u and 'Vu are (weakly) continuous along aB, u is weakly harmonic and hence smooth by Helein's result, Theorem 2.4 below. From (2.3) it follows by direct computation that the Hopf differential associated with u, "'( X 1 + zx . 2) = 1- . U- x 2 '±' Uxl 12 - 1U x 2 12 - 2·Z Uxl 0

0

is a holomorphic function on ]R2 ~ C. Moreover, by conformal invariance of Dirichlet's integral, E L1 (JR 2 ), and hence, by the mean value property of holomorphic maps, :::::: o. That is, u is conformal. Since u:::::: p on aB, we have 'Vu:::::: 0 on aBo But, by results of HartmanWintner [23], the branch points (where 'Vu = 0) of u are isolated or u : : : const. D

2.2.2. Partial Regularity. If we drop all geometric constraints on the target, for m 2 3 there is no hope of proving regularity for weakly harmonic maps u E H1(M; N). In fact, Riviere [37] constructed examples of weakly harmonic maps with finite energy which are everywhere discontinuous. On the other hand, for maps u that minimize E among maps v E H1(M; N) with the same boundary data, partial regularity results are available. Indeed, by results of Schoen-Uhlenbeck [39], [40] and Giaquinta-Giusti [18], energy-minimizing maps are smooth on the complement of a closed "singular set" of finite (m - 3)dimensional Hausdorff measure. In particular, as was shown earlier by Morrey [36],

126

Michael Struwe

energy-minimizing maps in dimension m = 2 are smooth. The example of the map u(x) = I~I:M = Bm = Bl(O;JRm) ---+ sm-l = N, which is energy minimizing if m 2': 3 [3], [34], shows that the above results cannot be improved. Halfway between weakly harmonic maps and energy-minimizing maps lies the class of stationary maps u E Hl(M; N) which, by definition, are weakly harmonic and, in addition, are critical points of Est also with respect to variations of the independent variables. By results of Evans [11] and Bethuel [2] these latter maps are smooth away from a singular set of dimension :S m - 2. There are analogous global existence and partial regularity results for the evolution problem (2.4), (2.5). In particular in the "conformal" case m = 2 we have the following result of Struwe [47], which was extended to the case 8M "I ¢ by Chang [4]. Theorem 2.3 For any initial map Uo E Hl (M; N) there exists a unique, global, weak solution u: [0, oo[xM ---+ N of (2.4), (2.5), satisfying the energy inequality (2.6) for all S < T, and which is smooth away from finitely many points (ti' Xi), 1 :S i :S I :S CEst(uo). Each singularity (t,x) is related to a smooth, non-constant harmonic map ii: S2 ---+ N in the sense that for suitable sequences Rn ---+ 0, tn / t, Xn ---+ X we have

as n ---+ 00, where u: JR2 ---+ N is smooth, harmonic and extends to a smooth, nonconstant harmonic map ii: S2 ---+ N. (We refer to this "bubbling-off" process as "separation of spheres".)

Originally, uniqueness was only established among partially regular solutions as in the statement of Theorem 2.4. By results of Freire [13], [14] and Riviere [38], in case N = Sk (and probably also for general targets) uniqueness also holds among weak solutions of class Hl satisfying the energy inequality (2.6) for all 0< S < T. By an example due to Chang-Ding-Ye [5], singularities actually may develop in finite time, even if the initial data are smooth. Theorem 2.3 therefore is best possible. If m 2': 3, the existence of global, partially regular solutions to (2.4), (2.5) was derived in [6], based on the monotonicity formula and a-priori estimates for (2.4), (2.5) from [46]. However, there is no uniqueness in the energy class [9]. Also in the time-independent case the situation improves drastically if m = 2. In fact, we have the following beautiful result of Helein [24]. Theorem 2.4 Any weakly harmonic map u E Hl(M; N) is smooth. 2.3. Regularity in the hyperbolic case In short, one can say that all the problems with regularity of weakly harmonic maps and/or well-posedness of the evolution problem (2.4), (2.5) in the class of

127

Wave maps

Hl-solutions are present in the hyperbolic regime, as well. Thus, contrary to the title of this section, in the sequel we will not discuss any regularity properties of wave maps. Instead, we will show the break-down of regularity and loss of uniqueness for the initial value problem (2.1), (2.2) in dimensions m 2 3. The examples we discuss indicate that there is hardly any regularity to be gained (in high dimension) from geometric conditions that we may impose on the target. Moreover, in order for (2.1), (2.2) to be well-posed in a suitable class, one still needs to identify a further "entropy condition" that will ensure uniqueness of weak solutions in this class. The situation in this regard thus is analogous to the situation for the parabolic evolution problem (2.4), (2.5) in dimensions m 2 3. With techniques available at this time we can therefore only hope to prove well-posedness of the initial value problem (2.1), (2.2) in case m = 2, analogous to Theorem 2.3 for the parabolic problem (2.4), (2.5) in this case. Some recent results in this regard will be presented in Lecture 3 . 2.3.1. Blow-up. The simplest way to produce blow-up is to show the existence of self-similar solutions x

u(t,x) =v(m) to (2.1) with non-constant smooth Cauchy data uo

= V,

= X . \7v

Ul

at t = -1. Introduce similarity coordinates x

~=m in the backward light cone from 0 and denote with wE

sm-l.

The Minkowski metric

Ixl = r,

I~I

= p, x = rw,

~

= pw

then can be expressed as 2

2

ds = dT - T

2

( d p2 (1 _ p2)2

+ 1 _p2 p2 dw 2)

.

Hence U is stationary for the standard Lagrangean £ if and only if v (~) = v (p, w) is stationary for the reduced Lagrangean 1

12sim(V) = -2 at

T

= 1. That is, vpp

U

J{

22

(1- P )

1

2

-p 2 Ivpl 2 + -2-lv wl p

}

Pm-l =+1 dpdw (1 - p2)-2

solves (2.1) if and only if v solves the equation

m- 1

+ ( -p- +

(m -

3)P)

1-p

2

Vp

1

+ P2( 1-p2)~WV ~ Tv N .

(2.8)

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Michael Struwe

Remark that equation (2.8) is an elliptic harmonic map equation on the unit m-ball B with the hyperbolic metric

dp2

p2

( 1-p2)2

+ -1-p - 2 dw

2

(2.9)

.

We seek solutions v of equation (2.8) that extend smoothly to the "boundary" p = 1 of B and hence can be continued smoothly to all of ]Rm. Since information propagates with speed:::; 1 the unique solution u of (2.1), (2.2) with initial data Uo

= v,

Ul

=

X

0

at t = -1

\7v

then will coincide with v( I~I) inside the backward light cone Ixl v1=- const. on B, we obtain blow-up at t = O. 2.3.1.1. The case m = 2. If m = 2, equation (2.8) becomes

Multiplying by p~vp and integrating with respect to

and hence

r p2(1 lSl

p2)lv p l2 dw -

r Iv lSl

w l2 dw

w E

< -t and, if

8 1 , we obtain

= Co.

Inspection at p = 0 shows that Co = O. Hence for p = 1 we obtain Vw = 0; that is, v(l,') == const. Recall that by the Riemann mapping theorem the metric (2.9) on B = Bl (0; ]R2) is locally conformal to the standard metric. In fact, define

CJ(p)

= exp

(-11 p

~)

P 1-p2

and observe that the metric

dO"

2222

+0"

dw

= 0"

" ( p2 (1dP_ p2) +dw 2 0 = P 2

)

()2

2

(d P

2

(1- p) (1 _ p2)2

P

+ 1 _ 2p2 dw

2)

is conformal to the metric (2.9) on B. That is, the map (p, w) f---+ (0", w) is a conformal diffeomorphism 'IjJ from B, endowed with the metric (2.9), to B with the standard metric. By conformal invariance of Dirichlet's integral and hence of the harmonic map equation (2.3) in m = 2 dimensions, thus v induces a harmonic map v = v 'IjJ-l E HI n C°(i3; N) on the standard disc with v leJB== const. By Lemaire's result Theorem 2.2, therefore we obtain the following result from [49]. 0

129

Wave maps Theorem 2.5 Ifm

=

= v(m)

2 and ifu(t,x)

solves {2.1} for

Ixl s Itl,

where v

extends to a smooth map on a neighborhood of Bl (0), then v == const. 2.3.1.2 The case m 2: 3. In high dimensions, following Shatah-Tahvildar-Zadeh [45], we can obtain self-similar solutions to (2.1) as follows. We consider as target an m-dimensional manifold N with rotationally symmetric metric

in spherical coordinates h of the special form

> 0, wE

sm-l

and we attempt to find solutions to (2.1)

u(t, rw) = h(t, r)w, where we also express x = rw E lR m in terms of spherical coordinates. Moreover, we make the ansatz u(t,x) = v(I~I)' that is, h(t,r) =
Lsim(V) = -2

J

22

(1- P )

1

2

-p l
p

l)g

2

(
m-l

P m+l dpdw, (1 - p2)-2-

and equation (2.8) takes the form


m- 1

where

P

+

(m - 3)P) 1

-p2

(m - l)f(
o.

(2.10)

f(
For special target metrics g, equation (2.10) admits non-constant solutions

g2(
for some fixed number
> 0 such that 1 <
f(
for 0 <

~(l(

<

where c = J m~l then solves (2.10) for 0 < p < c-1<po = Jm;;l<po = Po, and Po > 1 if m 2: 3. Note that for 9 as above, the radius of convexity of N around 0 is

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Michael Struwe

which is larger than c for m ::: 4 and equals c if m = 3. By changing the metric g( rp) on N suitably for rp > c, and by changing the initial data for h off B1 (0), we may thus construct solutions to (2.10) which blow up in finite time, with initial data having compact support and such that the target manifold is convex, if m ::: 4, and only slightly fails to be convex, if m = 3. A more detailed analysis shows that in 3 space dimensions blow-up may occur also for more general metrics on the target surface:

Theorem 2.6 (Shatah-Tahvildar-Zadeh [45]) Suppose 9 E Coo satisfies g(O) = 0, g'(O) = 1 and suppose g' has a smallest positive zero rp*. Also suppose that gil (rp*) =/= o. Then there is a class of regular initial data such that the corresponding Cauchy problem for equivariant harmonic maps from M3+ 1 into N has a solution that blows up in finite time. 2.3.2. Non-uniqueness of weak solutions. In particular, Theorem 2.6 applies to the sphere, where g( r.p) = sin rp, r.p* = ~. Shatah-Tahvildar-Zadeh construct a solution r.p to (2.9) on [0,00[, satisfying

rp(l) = rp* and having the asymptotic expansion for p ---+ 00:

rp(p) rp' (p) They consider the corresponding function h( t, r)w = rp (f) w as a weak solution of (2.1), that is, 2 sin 2h htt - hrr - -h r + - - 2 - = 0, (2.11) r 2r with singular initial data at t = 0, given by

(?:.) t ht(O,r) = h1(r) = ~ = lim!!:...rp (?:.) r t"O dt t h(O, r)

= ho(r) = a =

lim rp

t"O

(r=/=O) , (r=/=O).

(2.12)

Thereby, h is a weak solution of (2.11) say, on [0, 1] x ~3, if there holds

11 1

00

{-ht'ljJt+hr'ljJr

+ 2~2'ljJsin2h}r2drdt=

1 'ljJ(o,r)~r2dr 00

(2.13)

for any 'ljJECOO([O,l] X~3) such that 'ljJ(t,x) = 'ljJ(t,r) , 'ljJ(l,.) =0, and supp'ljJ(t)c BR(O) for some R > O. Moreover, h assumes the initial data (2.12) in the sense

131

Wave maps that

all H

Ilh(t, r) -

1 1;,2

(1R 3 ) ----+ 0

b

Ilht(t, 7') - ~11£2(1R3)

----+

0

(t

----+

0),

(t

----+

0).

Note that ho E Hl~~' hI E Lroc' On the other hand, also the function

h(t, 7') = {

r >t rst

~ (.T) , ~*.

weakly satisfies (2.11), (2.12) on [0,1] x ]R3, with Dh E L XJ ([0,1]; L2(BR(0))) for any R > 0, showing that weak solutions are in general not unique. To verify that h solves (2.13), for any 7/J we split

11

XJ

I

o

0

I}

{ -h t 7/Jt + h,,7/Jr + - 2 7/J sin 2h r2 dr dt 2r

1

00

0

1jJ(0, r) _r· b '2 dr r

={ Jotjoo {... }r2drdt- JorXJ7/J(0,r)~r2dr}+ t t{ ... }r2drdt=I+II. 7' Jo Jo t

Clearly, since Dh(t, r) == 0 for r S t, the second integral II = O. l'vIoreover, since h == h for r ~ t, and since h satisfies (2.13) the first integral reduces to the boundary term 1 2

t

1= J2 Jo (ht(t, t)

+ hr(t, t)) 7/J(t, t)t

dt

which also vanishes on account of ht

+ hr = =

7' '(7') (7') t + t1, ~ t

t2 ~

~ (1 -

D~' G) =

0

for 7' = t.

Observe that It induces a solution u of (2.1) with E(u(t);BI(O)) > E(u(t);BI(O)) for any t E]O, 1], where u is the solution corresponding to h. Hence there may be a chance of restoring uniqueness by some entropy principle. 3.

The conformal case m

=2

3.1. Overview The results presented in Lecture 2 leave little hope for the development of a satisfactory theory of existence, uniqueness, and stability for wave maps in high dimensions m ~ 3, even under very stringent geometric conditions on the target and/or very restrictive symmetry assumptions on the maps involved. By contrast, as is illustrated by the absence of self-similar solutions, Theorem 2.5, the situation seems to be much better in dimension m = 2, due to conformal invariance of Dirichlet's integral in this dimension. Thus, we are tempted to

132

Michael Struwe

conjecture that a result analogous to Theorem 2.3 for the "heat flow" related to harmonic maps of surfaces M also holds for the Cauchy problem for wave maps u:lR x M -> N. In this last of three lectures we will show that this conjecture is true for equivariant maps to surfaces of revolution and we sketch some recent developments towards a general theorem of well-posedness of the Cauchy problem in dimension 2.

3.2. The equivariant case The results that follow are mostly due to Shatah-Tahvildar-Zadeh [44], [45] and Shatah-Struwe [42], [43]. 3.2.1. Co-rotational maps. As in Lecture 2, Section 2.3, again we consider as target a surface of revolution N with metric ds 2 = dh 2 + g2(h) dw 2 written in terms of polar coordinates h > 0, W E S1. We assume that 9 is smooth with g( -h) = -g(h) and g'(O) = 1. Moreover, we either suppose that

g(h) > 0

l

and

or that there exists q1

if h > 0

h,g (s)1 ds -> 00

(3.1)

as h -> 00,

(3.2)

for 0 < h < ql

(3.3)

> 0 such that

g(qd

= 0,

g(h)

>0

and 9 is odd around ql as well as around qo = 0 (and hence periodic of period 2qI). In the latter case, for ease of exposition only, in the following we will also assume that 9 is even around ~. Case (3.1), (3.2) corresponds to a non-compact target surface N, including the standard plane g(h) == h or metrics of negative curvature like g(h) = sinh(h); condition (3.2) rules out sharp cusps "at infinity". Case (3.3) corresponds to a compact target, including the standard sphere g(h) = sin(h). Remark that (3.3) also implies (3.2). Moreover, we consider maps u: lR x lR 2 -> N such that, expressing a point x E lR 2 in polar coordinates x = rw, the angle w E S1 is preserved by u and h(t,x) = h(t,r). Such maps will be called co-rotational. For such u we have 1 (u) = "21 27r.c

J{I

ht I2

-

Ihr I - g2 (h) } r dr dw dt 2

~

and u is stationary for .c if and only if h: lR x [0,00[-> lR satisfies

htt

-

1 hrr - -h r r

f(h)

+ -r2 - = 0,

(3.4)

133

Wave maps

where

f(h) = g(h)g' (h).

Moreover, for smooth solutions the energy (scaled with a factor 27f) 1 -E(u(t)) 27f

1 2

= -

1

00

0

{

g2 2 (h) } Ih t l2 + Ihr 2 + r l

rdr

=

Eequi(h(t))

is conserved. Lemma 3.1 If u is a co-rotational map with E(u(t)) ~const. for all t E IR, then

°

the associated map h is continuous on IRx ]0, oo[ and h(t,') extends continuously to [0, oo[ for every t E IR, where g(h(t, 0)) = for all t. Proof. Since for any ro

°

> the integral

1

00

ro

~ 2rOl Eequi (h(t))

Ihr(t, rW dr

is uniformly bounded, by Sobolev's embedding H 1 ,2 '---> C 1 / 2 we conclude that h(t, .) is locally Holder continuous on ]0,00[, uniformly in t E R Since h t E Lroc(IRx]O,ooD, moreover, h(',r) is continuous in t for almost every r > 0. Hence, the map t f---7 h(t,') E CO(]O, oo[) is continuous by the theorem of Arzela-Ascoli. In order to prove continuity at r = 0, let

Then, by Holder's inequality, for any t E IR we have

(0

Jo IG(h)rl dr

~

(0

~

((0

(hW

Jo

~ dr

~ Eequi(h(t);Bro(O)) ---YO

(ro ---Y 0).

J o Ig(h)llhrl dr

1

)1/2(rro J Ih o

r

l2 rdr

)1/2 (3.5)

It follows that limr--->o G (h( t, r)) exists for any t and hence, by strict monotonicity of G, that limr--->o h(t, r) = h(t, 0) exists for any t. Finally, since

1

00

g2(h)

--dr < 00, o r

it follows that g(h(t,O)) =

°for any t.

o

In case (3.1) Lemma 3.1 implies the boundary condition

h(t,O)=O

for all t.

(3.6)

In case of assumption (3.3), from Lemma 3.1 we only deduce that h(t, 0) = dql for some d E Z.

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Michael Struwe

Lemma 3.2 There is a constant ro > 0 such that

fO

> 0 with the following property. If there exists

E(u(t); Bro(O)) < fO

(3.7)

for all t,

then h(t,O) is constant in time. Proof. From (3.5) we deduce that IG(h(t,O)) -G(h(t,ro))I:S

fO

for all t. Since h(t, ro) depends continuously on t and since G(h(t, 0)) = d G(qd for some d = d(t) E Z, if Cfo < G(ql) it follows that h(t, 0) is constant in time. D Finiteness of E(u(t)) and finite propagation speed for (3.4) also implies the asymptotic boundary condition for r ~ Ro

+ It I

(3.8)

for some number Ro > 0 and some qo E IR such that g(qo) = O. We may normalize qo = O. Together, (3.5) and (3.8) imply the uniform bound sup G(h(t, r)):S lim G(h(t, ro)) r

ro----+oo

+ 2Eequi (h(t))

= G(O)

+ E equi (h(t)).

From assumption (3.2) we then obtain Lemma 3.3 If h: IR x [O,oo[--t IR corresponds to a co-rotational wave map u with

data (uo, Ul)

E H~,

then h is uniformly bounded.

For later reference we also introduce the set

H~([O,ooD = {(ho,h1);ho E Hl~; ncO([O,ooD,ho(O) = dOql for some do

E

Z,

ho(r), hl(r) == 0 for large r}. The initial value problem for co-rotational wave maps u with data (UO,Ul) E H~ (1R2 ; TN) at t = 0 then corresponds to the initial-boundary value problem (3.4) for data (h It=o,ht It=o) E H~([O,ooD. Due to energy conservation and the semi-linear character of (3.4) it is not hard to show the existence of a global weak solution to the initial-boundary value problem (3.4) of class H~. However, it is not clear whether this solution is unique and whether the solution preserves any additional regularity properties of the data. Fortunately, we can transform equation (3.4) to a form where these questions can be answered.

135

Wave maps

3.2.2. The transformed equation. of h we introduce the map

To eliminate the singularity in (3.4), instead

rp(t, r) = h(t, r) - h(t, 0). r

Recall that, by Lemma 3.2, h(t,O) = dql, where d E Z is independent of t E JR for maps h corresponding to wave maps u which depend continuously on time in the Hl_ topology. Note that formally rp satisfies

3 rptt - rprr - -rpr r

f(rrp)-rrp

+

r

3

= O.

Expanding g'll (0) gill (0) f(rrp) = g(rrp)g'(rrp) = (rrp + -6-(rrp)3 + ... )(1 + -2-(rrp)2 + ... )

= rrp +

~g"'(0)(rrp)3 + ... ,

we can express the nonlinear term as

f(rrp) - rrp _ K( ) 3 r3 - rrp rp , where K is smooth, K(h) = K( -h), and

K(O) =

_~glll(O)

equals 2/3 the curvature of N at O. Also regarding

3 rptt - rprr - -rpr = Drp, r

as the wave operator on JR x JR4, acting on the radially symmetric function rp( t, x) = rp(t, r) for x E JR4 with Ixl = r, thus we arrive at the equation Drp - K(rrp)rp3 = 0

(3.9)

In case (3.1), moreover, (3.8) translates into the boundary condition

rp(t, r) = 0

for r 2 Ro + Itl.

(3.10)

Finally, in case (3.1), (3.2) hold, using boundedness of h (Lemma 3.3) we infer that there exists a constant C, possibly depending on h, such that

h(t,r) ~ Cg(h(t,r)).

136

Michael Struwe

Hence

Eequi(h(t)) 2::

1 (XJ { h2 } 2Jo IhtI2+lhrI2+ C2r2

=~

rdr

1 {1~tI2 + I~r + ;12 + ~:2 } 00

r3 dr

and it follows that (~, ~t) is bounded in the energy space H1(JR. 4 ; TJR). Thus, solutions h of (3.4) of class HI with data (ho, hI) E H1 correspond to solutions ~ of (3.9) of class HI with data in H1, and conversely. For a compact target surface, with 9 satisfying (3.3), condition (3.8) translates into

~(t,r)

= _

d(t)qI

for r 2:: Ro + Itl·

r

(3.11)

Remark that d(O) = do E Z corresponds to the degree of the initial map

uo:JR. 2

;:;'

82

--t

N;:;' 8 2 .

Hence for initial degree do =I- 0, from (3.11) we conclude that

for small t. HI-solutions h of (3.4), (3.6) with do =I- 0 therefore do not correspond to solutions ~ of (3.9) of class HI, but only to solutions which are locally of class HI.

3.2.3. Well-posedness, a model case. consider the model case

To set the stage for the general result first

Equation (3.9) in this case reads D~+~3 =0

with Cauchy data ~

It=o= ~o,

(3.12) ~t

It=o= ~I,

(3.13)

where (~o, ~d E H1(JR.\ TJR). Equation (3.12) is a special case of a class of semi-linear wave equations involving critical growth exponents for which a full theory of existence, uniqueness and regularity has been developed in the past years, starting with the work of Struwe [48] on radial solutions of the equation

in 1988. The symmetry condition in [48] was removed by Grillakis [20], still in 3 space dimensions.

137

Wave maps

The insight how to treat the higher-dimensional cases, in particular the case m = 4 which is relevant here, came from the work of Kapitanskii [30] who pointed out the use of the Strichartz inequalities for the analysis of semi linear wave equations like (3.12). Grillakis [21] then was the first to realize that the Strichartz estimates and the crucial decay estimate from [48] could be combined to prove regularity for the equation (3.14) in dimensions m ~ 5, where 2* = ~~2 is the "Sobolevexponent" in dimension m. More efficient use of the Strichartz estimates was then made by ShatahStruwe [42], [43] who extended the regularity results for (3.14) to dimensions m ~ 7 and, moreover, proved that the initial value problem for (3.14) with finite energy data (rpo, rpd E Hl is well-posed in all dimensions m ~ 3. In particular, the result from [43] applies to the Cauchy problem for (3.12), (3.13). Theorem 3.4 For any (rpo, rpd E Hl there exists a unique solution rp of (3.12), '1/2 . 1/2 (3.13) such that (rp, rpt) E C°(lR; Hl) n Lq('R.; Bq x B;; ), where q = 130 and

Lq('R.; B~/2) is the Besov space of functions rp with "half a spatial derivative" in Lq ('R. x 'R.4). If (rpo, rpd E Coo) then rp E Coo) as well. Remark that by Sobolev's embedding rp E Loo('R.; L4('R.4)) and hence by interpolation rp E L 5('R. X 'R.4).

n L ¥ ('R.; L ¥ ('R. 4))

The proof of uniqueness now follows easily from the Strichartz estimate (3.15)

111PIILli(l~XlR4) ~ CIID1PIIL1,)1(lRxlR4) for 1P: 'R. x ]R4 with vanishing Cauchy data

'lj; It=o= 0 = 'lj;t It=o . For simplicity we assume that the initial data (3.13) have small energy. Then the square of the L 5 -norm of any solution rp of (3.12), (3.13) as in Theorem 3.4 is bounded by the energy of the initial data. Let rp, rp be two solutions of (3.12) as in Theorem 3.4 sharing Cauchy data (rpo, rpd E H~ at t = O. Then 1P = rp - rp satisfies ID'lj;1 = Irp3 - rp 31~ CI'lj;I(rp2 + rp2) and hence by (3.15) and Holder's inequality

11'lj;llc\'l(lRXlR4)

~ CII1Plldf(lRXlR4) (1Irplli s(lRxlR4) + Ilrplli s(IRxlR

4 ))

~ CE(u(O)) 11'lj;II Lli (lRx]R4) , which implies that 'lj; = 0, if E(u(O)) is sufficiently small.

o

138

Michael Struwe

The slightly improved space-time integrability of the solutions obtained in Theorem 3.4 (that is, u E L5 instead of L4 , only) also suffices to propagate further regularity of the data; in particular,

(ho, hI) E H~ corresponding to data (<po, O. In particular, I ~ EO/EO, (Similarly for negative time.) ii) If N is compact, the same result holds; however, the estimates for

(UO,UI) E H~(M;TN)

139

Wave maps

we hope to show the existence of a unique, global, weak solution u: IR x M ---) N of the Cauchy problem

Du = -A(u) (8",u, 8"'u) -.l TuN, u

It=o= Uo,

Ut

(3.16) (3.17)

It=o= Ul·

While we cannot yet solve this problem, we will discuss an approximation method to prove existence and we present some partial results on the relevant convergence problem. 3.4.

Approximate solutions

3.4.1. Penalty method. Suppose as in Section 1 that N is compact, isometrically embedded in IR d , and there is a tubular neighborhood U21i(N) of width 28 of N with smooth nearest neighbor projection 'lrN: U21i(N) ---) N. Also let X E COO(IR) be a function such that X(s) = s for s :S 82 , X(s) == const. for s 2: 28 2 , X' 2: 0, X" :S 0, and let dist(p, N) be the distance from a point p E IRd to its nearest neighbor on N. Note that

then extends to a smooth function on IRd whose gradient is given by

°

X' (dist 2 (p, N)) (p - 'lrN(p)) ,

if p E U21i (N), and otherwise. For LEN consider solutions u L : IR x M ---) IRd of the Cauchy problem

Du L + LXi (dist 2 (u L , N))(u L

- 7T'N(U L )) =

0

with data (UO,Ul) E H;(M;TN) at t = O. Equation (3.18) implies the conservation law

where

1

e(u L ) = -IDu L I2

2 is the energy density for the free wave equation. Let

and let

on 1R. x M

(3.18)

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Michael Struwe

Upon integrating (4.4) over M, thus we find that

d

dtEL(uL(t)) = 0; in particular,

EL(UL(t)) = EL(uL(O)) = E(uL(O)) =

~

r {IUlI

2 JM

2

+ lV'uoI 2 } dx

for all L and all t. It follows that, as L -r 00, a sub-sequence

Du L ~ Du weakly-* in L 00 (JR; L2 (M) ) . Moreover, since uL(O) = Uo for all L, and in view of the compactness of the restriction (trace) operator H 1 ,2 (JR x M) '---+ L2 ({ t} x M) for any t, in L2(M), locally uniformly in time. Hence, by Fatou's lemma, also

r

J{t}XM

r

X(dist 2(u,N)) dx::; liminf

L-too J{t}XM

1

::; LEL(uL(t))

X(dist 2(u L,N)) dx

-r 0

(L

-r

00)

for any t, and it follows that u: JR x M -r N. Finally, since (3.18) has propagation speed::; 1, for data (uo,ud E Hl also DuL(t) has uniformly compact support for any t and thus

(u,Ut)

E LOO(JR;H~(M;TN)).

However, while (3.18) implies that

Du L -.l T"'N(uL)N at all points in space-time, it is not clear that this relation, and hence (3.16), persists in the limit L -r 00. In special cases, the analysis is, in fact, quite simple.

3.4.2. The sphere. We slightly modify the approximation scheme if N = Sk. For LEN, following Shatah [41], we consider solutions uL:JR X M -r JRd, d = k + 1, of the equation on JR x M

(3.20)

with Cauchy data (uo,ud E Hl(M;TN) at t = O. The initial value problem for (3.20) admits global weak solutions u L such that Eh(uL(t))

= E(uL(t)) =

+!:..

E(uL(O)) =

r IIu

4 JM

L

I2 -

112 dx

= Eh(uL(O))

~ 1M {IUl I2 + lV'u oI2 } dx,

141

Wave maps

uniformly in t, for all L. A sub-sequence (u L ) hence converges to a limit u in the sense that, as L ~ 00, in L2(M) for all t, weakly-* in Loo(lR; L2(M)), and for any t, by Fatou's lemma.

Hence u:lR x M ~ Sk. In order to pass to the limit in (3.20), observe that the nonlinear term always points in the direction of u L . Taking the exterior product with u L , thus from (3.20) we obtain the equation (3.21 ) for all L. In the limit L

therefore also the equation

~ 00,

OOi(OOiU /\ u)

= Du /\ u = 0

(3.22)

is valid in the sense of distributions, which implies that u weakly solves (3.16). Moreover, multiplying (3.21), (3.22) with a 2-vector cp E Co(lR x M) and integrating by parts on [0, oo[ xM, we obtain

1 1M 00

((OOiU L /\u L) - (OOiU/\U))OOicpdxdt

r

=

(Ul/\ Uo - Ut /\ uo)cpdx

J{O}XM

In the limit L conclude that

~ 00,

the left hand side vanishes. Since cp(O,·) is arbitrary, thus we

(Ut(O) - Ul) /\ Uo

= 0;

that is, Ut(O) = Ul in the sense of traces. Here we used the fact that both Ut(O) are tangent to Sk along Uo, that is,

Therefore, as t ~ 0, Ut(t,') ---,

limsup~ t--->O

2

Ul

weakly in L2 as t ~ O. On the other hand

JrM IUt(t")12dx+~2 JrM lV'uol2dx

:::;limsupE(u(t)) t--->o

:::; lim sup liminf Eh(uL(t))

t-->O

Ul

L-->oo

:::; E(uL(O)) =

~ 1M (I U lI 2 + lV'uoI 2 ) dx.

and

142

It follows that

Michael Struwe lim sup Ilut(t, .) 11£2 ~ t-...O

IluIII£2·

Together with the fact that Ut (t, .) ~ UI weakly in L2, this implies strong convergence Ut(t,') -+ UI in L2 as t -+ 0. That is, U attains the prescribed initial data continuously in H~(M;TN). Hence we have proved: Theorem 3.6 Suppose N = Sk, and let (UO,UI) E H~(M;TN). Then there exists

a global weak solution U of {3.16}, {3.17} of class HI. Remark that U need not be unique; see Section 2. The above method can be generalized to the orthogonal group N = SO(n), as was observed by Freire [15]. The approximating equation is

Du+LVuF(u) =0,

(3.23)

where

F(u) = fMlutu

_11 2 dx,

and where u t denotes the transposed matrix u and 1·1 is the norm induced by the scalar product (A, B) = trace (At B). Given Cauchy data (uo, UI) E H~(M, TSO(n)) at t = 0, for any L there exists a solution u L : IR x M -+ IR nxn of (4.8) with

Moreover, (u L, uf) is bounded in Loo(lR; H~(M; TlRnxn)) with DuL(t) having uniformly compact support and, as L -+ 00, a sub-sequence in L2(M), locally uniformly in t, weakly-* in L 00 (1R; L 2 (M) ) , where u:1R x M -+ SO(n) and (u,Ut) E Loo(IR;H~(M;TSO(n))). Remark that TuSO(n) = uT1SO(n), where T1SO(n) = so(n) denotes the Lie algebra of SO(n). Recall that so(n) consists precisely of the anti-symmetric matrices A, At = -A. Moreover, the orthogonal complement of TuSO(n) with respect to (', .),

T;;SO(n) = uTlSO(n), where Tl SO( n)

= (so( n)).l , consists precisely ofthe symmetric matrices B = Bt.

Wave maps

143

Indeed, any matrix M may be split 1 M = -(M + M t) 2

+ -1 (M - M)t 2

into its symmetric and anti-symmetric part and we have (A, B)

= trace (At B) = -trace (AB)

= trace (Bt A) = trace (BA) = trace (AB) = 0 whenever A = _At, B = Bt. Similarly, since F(u) = F(u t ), we have

for any u. Thus, from (3.23) we obtain

for every L. Passing to the limit L

----t 00,

then we find

that is, utOu E (so(u))-L. Since u E SO(n), we have uut = 1 and thus, finally, Ou E u(so(u))-L = T;;SO(u), as desired. It is conceivable that Theorem 3.6 extends to any homogeneous space as target. Observe that the condition Ul E TuoN is crucial in showing that the initial data are attained continuously in L2. If Ul rt Tuo N one can show that as t ----t 0 we have convergence Ut(t) ~ d7rN(UO)Ul, the projection of Ul to TuoN, weakly in L2. However, due to the loss of the energy of the normal component, we cannot show strong convergence. Added in proof By recent work of Freire (final version of [15]) and Yi Zhou ([52]), Theorem 3.6 is indeed true for any compact homogeneous space N. 3.5. Convergence For general targets the problem of convergence is more difficult. Let us first consider the stationary case.

144

Michael Struwe

3.5.1. The stationary case. Suppose, for simplicity, that M is a compact surface without boundary; for instance, M = T2 = ~2 I'll}. Moreover, let N be a compact k-dimensional manifold, without boundary, isometrically embedded into some Euclidean ~d, and suppose, for simplicity, that TN admits a smooth orthonormal frame field (el,"" ek). That is, at each point pEN the collection (el (p), ... , ek (p)) is an orthonormal basis for TpN, smoothly varying with p. By a construction due to Helein [25] and Christodoulou-Tahvildar-Zadeh [7] this latter hypothesis can be made without loss of generality in the context of harmonic maps. Indeed, if the original target N does not have a parallelizable tangent bundle we can embed N as a totally geodesic submanifold of a compact manifold N that has this property by taking two copies of a tubular neighborhood of N in ~d, endowed with the product metric of N x ~d-k, and gluing them together along their boundaries. The standard basis of ~d then yields the desired frame field for TN, at least near the range N of our maps. Moreover, since N c N is totally geodesic, for any map u: M ---t N c N C ~d the component orthogonal to TuN of the Laplacian ~u in TuN vanishes. In particular, a harmonic map u: M ---t N will be harmonic, regarded as a map u: M ---t N. Henceforth, therefore we replace N by N and assume N = N. Consider a sequence (u L ) of maps u L E H I ,2(M; N) such that u L ~ u weakly in HI and suppose

(3.24) where strongly in H-l,

Theorem 3.7 Under the above assumptions, u is (weakly) harmonic.

This result is due to Bethuel [1]. A drastically simplified proof was recently given by Freire-Miiller-Struwe [16], which we present below. From now on, moreover, it is convenient to use the language of differential forms. Thus, we let d,8 be the exterior differential and co-differential, respectively. For a I-form 'P = 'POI. dxOl. we have 8'P = a~l 'PI + ~'P2' for a 2-form b = {3 dx l Adx 2 we have 8b = _~{3dxl + a~1{3dx2. Moreover, we define the Hodge Laplacian on forms as ~ = d8 + 8d, acting as the standard Laplacian on the coefficients of the forms. (We always assume M = T2 = ~2 l'l}, so that we can use the standard 1-forms dxl,dx 2 as basis.) Finally, we contract I-forms 'P = 'POI.dxOl.,'l/J = 'l/JOI.dxOl. using the metric on M = T2 by letting 'P . 'l/J = 'PI'l/JI + 'P2'l/J2. Let el, ... , ek denote the orthonormal frame for TN. Then for each L the collection el u L, ... , ek u L is a frame for the pulled back bundle (UL)-ITN; that is, at each x EM, the collection el (u L(x) ) , ... , ek (uL(x)) is an orthonormal base for TuL(x)N. Other such frames may be obtained by rotating this frame; that is, by letting 0

0

where RL = (Rf;) E SO(k).

145

Wave maps

We express du (respectively du L ) in terms of ei (respectively ef) as

du

= Biei,

Bi

= (du, ei).

Also denote the connection I-form of a frame field (ei) as

Note that

(b.u , e"I\ = 8B , -

W "J .

B·J'

Hence u is harmonic if and only if for any i

(3.25)

in the distribution sense. Note that the frames (ed, (ef) are only determined up to rotation, that is, up to a gauge transformation in the bundle of frames. In particular, by choosing ei = Rij (ej u) such that (ei) minimizes 0

~i JMlveil2 dx, we obtain the following result of Helein [25]. Lemma 3.8 For any u E H 1 ,2 (M; N) there exists a frame (e;) for

u-1T

N such

that the associated connection satisfies the Coulomb gauge condition 8Wij

= 0,

1 S i, j S k.

Moreover,

~i J)ve;1 2 dx 'S ~i fM1V(ei

0

u)12 dx S CE(u).

In the following we assume that (ei), (ef) are in Coulomb gauge. Hence, in particular, (ef) is bounded in Hl and, passing to a further sub-sequence, if necessary, we may assume that ef ~ ei weakly in Hl and w{j ~ Wij weakly in L2, as L -+ 00. By Hodge decomposition, for Wij (respectively, wt) we have Wij

= daij + 8bij + H ij ,

where aij and bij are normalized by the condition

{ aij dx = { bij = 0,

1M

1M

and where Hij is a harmonic I-form (a constant linear combination of dXl, dx 2 if M = T2). By mutual orthogonality, II da ij

lIi2 + 118bij lIi2 + IIHij lIi2

=

IIWij

lI}o2.

Michael Struwe

146

In particular (b{j) is bounded in HI, (H6) is bounded in any smooth topology, and we may assume that b{j ~ bij weakly in HI as L ---t 00, while H6 ---t Hij smoothly. Moreover, the Coulomb gauge condition implies

hence aij = 0, and similarly for a{j. Consider the term

8b{j . Of - 8bij . OJ. Let us fix, say, i = 1 and consider only the term involving j = 2 in this sum. For brevity we write 0, OL instead of 01 , of, etc. Let

Then

LL

8b . 0 =

f)Lf)L f)Lf)LL (f)x 1 f3 f)x 2 U - f)x 2 f3 f)x 1 U ,e2)

has the structure of a Jacobian determinant. Due to this particular structure, a special weak compactness property holds. In fact, from [35], Lemma IV. 3 we have the following lemma. Lemma 3.9

8b L . OL ~ 8b . 0 + L,jEJl/j8xj

weakly in the sense of distributions, where J is an at most countable set.

Since J is countable, the capacity of the set X = {x j } j E J vanishes and there exists a sequence of functions 'PI E COO(M), 0 ~ 'PI ~ 1, such that 'PI == 0 in a neighborhood of X for each l and 'PI ---t 1 in H 1 ,2(M) as l ---t 00. Hence for any 'P E Coo we have

and for the proof of (3.25) it suffices to show that (3.26) for any 'P E Coo vanishing near X. Now we use our assumption that

as L

---t 00

in the sense of distributions.

147

Wave maps

Moreover, since ef ~ ei weakly in £2, we also have weak convergence bef ~ be i in the distribution sense. Thus, with error terms 0(1) ---> 0 as L ---> 00, we have

1M (be

i -

Wij' ej)
=

1M (bef - Wij' ej)
= 1M (wt . ef - Wij' ej) 00 by Lemma 3.9, on account of the fact that

The time-dependent case.

We consider the following model situation. Let

(uJ~) be a sequence of wave maps uL: IR x M ---> N with E(uL(t)) :::; E(uL(O)) :::; C,

uniformly in t and L. We may assume that, as L

---> 00,

in L2(M), locally uniformly in time weakly-* in L oo (IR; L2(M)). Then, by a result of Freire-Miiller-Struwe [16] there holds Theorem 3.10 Under the above assumptions, the limit map u: IR x M

--->

N weakly

solves the wave map equation (3.16)'

Below, we indicate the main steps in the proof of Theorem 3.10. Again, as in the stationary case, we may assume that TN is parallelizable. Let e1,"" ek be an orthonormal frame field for TN and for u, respectively u L , consider corresponding rotated frames for the pull-back bundle u-1TN, given by for z where R

=

(Rij): IR x M

--->

= (t, x)

E IR x M,

SO(k).

Note that (3.16) is equivalent to the relation

in the distribution sense. (Recall that we raise indeces with the Minkowski metric.) That is, for the proof of Theorem 3.10 we have to show that for any

0 there holds

Co

(3.27) Fix such

E

> O. The energy inequality for (3.16) implies:

Michael Struwe

148

Lemma 3.11 There is a sub-sequence (u L) such that the f-concentration set of (u L ),

has vanishing Hl,2 -capacity; that is, there exists a sequence of cut-off functions 'Pl E Hl,2 n VXl(JR x M) such that 0 :S 'Pl :S 1, 'Pl == 0 in a neighborhood of SE and 'Pl -+ 1 in Hl,2 as l -+ 00. Since

r

JJRXM

(8 aOi,a - W'ij0j,a)'Pl'P dz

-+

r

iJRXM

(8 aOi,a - W'ijOj,a)'P dz

as l -+ 00, it hence suffices to prove (3.27) for testing functions 'P that vanish in a neighborhood of Se. Scaling suitably, we may assume that the support of 'P is contained in a fundamental domain Q for T3 = JR3 /7.,3. Extending u L , ef, etc. suitably outside Q, we may also regard u L , ef, etc. as functions on T3. (The modified functions u L , of course, only satisfy (3.16) in Q.) On T3 we impose the Coulomb gauge condition (with respect to the Euclidean background metric) by choosing RL: T3 -+ SO(k) such that Ei

r IDefl2dz = minEi r ID(Rij(ej iT3 R iT3

In this gauge, we have

ou L))1 2 dz.

8aWij,a = 8euclWij = 0

and (ef) is bounded in Hl,2(T3) with Ei hiDefl2 dz :S Ei hlD(ei

0

uLW dz :S CE(uL(O)) :S C.

Hence we may assume that ef ~ ei weakly in Hl~; (JR x M) and

of wt

= (du L , ef) = ota dx a ~ 0i = (du, ei), = (def,er) =wt,a dxa ~Wij = (dei,ej)

weakly in L2 as L -+ 00. Moreover, by a simple measure-theoretic argument, the set of concentration points Zo of (ei), satisfying limsupr- 1 r->O

r

i Br(zo)

Eil Deil 2 dz > 0,

has vanishing Hl,2-capacity and we also may assume that 'P vanishes near such points.

149

Wave maps

Finally, since u L solves (4.1) and since

(L

(0)

-+

in the distribution sense, for the proof of Theorem 3.10 it suffices to show: Lemma 3.12 For LEN sufficiently large there holds I

r

JRxM

Proof. By weak convergence liminf L--">oo

r (w.t,aef JT3'

a -

(wt,ne;'a - w'0ej,o,)CP dzl :s;

wt ~

Wij

E.

in L2(Q),

w'0ej,o:)cpdz

= liminf L--">oo

r wt,a(ef.,a JT3

ej,a)cpdz

In the following, again we consider some fixed pair of indices i, j and we omit these indices for brevity. Next, let wL = da L + Deucl bL + HL be the Hodge decomposition of w L(= Wf2)' normalized by the requirement that

ra

JT3

L

dz =

rb

JT3

L

1\ dx a

= 0,

a = 0, 1,2.

By mutual orthogonality

It follows that HL -+ H smoothly as L -+ x. Moreover, the Coulomb gauge condition implies at = 0. Finally,

tJ..b L

= dw L = det

1\ de~

o

exhibits the crucial determinant structure.

In the time-dependent setting we will need the following result on Jacobian determinants, due to Coifman-Lions-Meyer-Semmes [8].

cp, 1/) E Hl,2 then Ildcp 1\ d1jJIIHl :s; Clldcpll£2lld1jJII£2·

Lemma 3.13 If

dcp 1\ d1jJ belongs to the Hardy space Hl and

Decompose

(e L - e)cp = (d((u L - u)cp),e L) + 0(1) = (d(u L - u)cp, eL -

= Af + A~ + 0(1), where 0(1)

-+

0 in L2 as L

-+

00.

e)

+ d((u L

-

u)cp, e)

+ 0(1)

Michael Struwe

150

Denote by af, i = 1,2, the solution to

where D;uel is the adjoint of Deuel with respect to the Minkowski metric. By using a result of Campanato and Giaquinta [17], we show that the functions (af2) are bounded in BMO(T 3 ). In fact, using precise estimates in Morrey spaces a~d our definition of Se, we can show that Lemma 3.14

limsupllafllBMo ~ CE, L-HXJ

limsuplla~IIBMo ~ L-+oo

CVE·

By Hi_BMO duality (Fefferman-Stein [12]) then we have, with error 0(1) as L

----+

0

----+ 00,

r wL . (e L - e)ipdz = iT3r DbL . (Af + A~)dz + 0(1)

iT3

= =

r b ~(af + a~)dz + 0(1) iT3 L .

r def i\ de~ . (af + a~)dz + 0(1) iT3

de~IIHl (1lafllBMo + Ila~IIBMo) eVE + 0(1),

~ Clldef

:s

i\

+ 0(1)

as desired. Concluding remark: Observe that the proof of Theorem 3.10 might be adapted to show that the weak Hi-limit of approximate solutions u L to (3.16) with range on N is a wave map. However, for the sequence (u L ) defined by the penalty method in Section 3.4.1 it is difficult to control the energy of the functions u L in direction normal to N. For this reason we cannot (yet) use Theorem 3.10 to obtain existence of global weak solutions. Added in proof The existence of global weak solutions to the Cauchy problem for wave maps u : JR x JR2 ----+ N with initial data of class Hi was recently obtained by Miiller-Struwe [51], using the convergence result outlined in Section 3.5.2 and the viscosity approximation to (1.1) suggested by Yi Zhou [52].

Wave maps

151

References [1] F. Bethuel: Weak convergence of Palais-Smale sequences for some critical functionals, Preprint (1992) [2] F. Bethuel: On the singular set of stationary harmonic maps, Preprint (1992) [3] H. Brezis, J.-M. Coron, E. Lieb: Harmonic maps with defects, Comm. Math. Phys. 107 (1986) 649-705 [4] K.-C. Chang: Heat flow and boundary value problem for harmonic maps, Ann. lnst. H. Poincare, Analyse Non-Lineaire 6 (1989) 363-395 [5] K.-C. Chang, W.-Y. Ding, R. Ye: Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Diff. Geom. 36 (1992), 507515 [6] Y. Chen, M. Struwe: Existence and partial regularity results for the heat flow for harmonic maps, Math. Z. 201 (1989) 83-103 [7] D. Christodoulou, A. Shadi Tahvildar-Zadeh: On the regularity of spherically symmetric wave maps, Preprint [8] R. Coifman, P.L. Lions, Y. Meyer, S. Semmes: Compensated compactness and Hardy spaces, J. Math. Pures App!. 72 (1993), 247-286 [9] J.-M. Coron: Nonuniqueness for the heat flow of harmonic maps, Ann. lnst. H. Poincare, Analyse Non Lineaire 7 (1990) 335-344 [10] J. Eells, J.H. Sampson: Harmonic mappings of Riemannian manifolds, Am. J. Math. 86 (1964) 109-169 [11] L.C. Evans: Partial regularity for stationaTY harmonic maps into spheTes, Arch. Rat. Mech. Ana!., 116 (1991), 101163 [12] C. Fefferman, E.M. Stein: HP spaces of seveml variables, Acta Math. 129 (1972), 137-193 [13] A. Freire: Uniqueness fOT the harmonic map flow in two dimensions, Calc. Var. 3 (1995), 95-105 [14] A. Freire: Uniqueness fOT the harmonic map flow from sUTfaces to geneml targets, Comm. Math. Helv. 70 (1995), 310-338, correction Comm. Math. Helv. 71 (1996) 330-337 [15] A. Freire: Global weak solutions of the wave map system to compact homogeneous spaces, submitted (1996) [16] A. Freire, S. Muller, M. Struwe: Weak conveTgence of haTmonic maps from (2 + 1)dimensional Minkowski space to Riemannian manifolds, Preprint [17] M. Giaquinta: Introduction to TegulaTity theory fOT nonlineaT elliptic systems, Lectures in Mathematics, Birkhiiuser Verlag, (1993) [18] M. Giaquinta, E. Giusti: The singulaT set of the minima of ceTtain quadmtic functionals, Ann. Scuola Norm. Sup. Pisa (4) 11 (1984) 45-55 [19] J. Ginibre, A. Soffer, G. Velo: The global Cauchy pToblem fOT the cTitical non-lineaT wave-equation, J. Funct. Ana!. 110 (1992), 96-130 [20] M. Grillakis: RegulaTity and asymptotic behaviouT of the wave equation with a critical nonlinear'ity, Anll. of Math. 132 (1990) 485-509 [21] M. Grillakis: Regularity fOT the wave equation with a critical nonlineaTity, Comm. Pure App!. Math. 45 (1992), 749-774 [22] M. Grillakis: Classical solutions fOT the equivaTiant wave map in 1 + 2 dimensions, Preprint [23] P. Hartman, A. Wintner: On the local behavioT of solutions of non-pambolic partial differential equations, Amer. J. Math. 75 (1953), 449-476

152

Michael Struwe

[24] F. Helein: Regularite des applications faiblement harmoniques entre une surface et une varitee Riemannienne, C.R. Acad. Sci. Paris Ser. I Math. 312 (1991) 591-596 [25] F. Helein: Regularite des applications faiblement harmoniques entre une surface et une varitee Riemannienne, C.R. Acad. Sci. Paris Ser. I Math. 312 (1991) 591-596 [26] S. Hildebrandt: Nonlinear elliptic systems and harmonic mappings, Proc. Beijing Sympos. Diff. Geom. Diff. Eq. 1980, Gordon and Breach (1983) 481-615 [27] L. H6rmander: Non-linear hyperbolic differential equations, Lectures 1986-1987, Lund, Sweden [28] J. Jost: Harmonic mappings between Riemannian manifolds, ANU-Press, Canberra, 1984 [29] J. Jost: Nonlinear methods in Riemannian and Kiihlerian geometry, DMV Seminar 10, Birkhiiuser, Basel (1991) [30] L. Kapitanskii: The Cauchy problem for semilinear wave equations, part I: J. Soviet Math. 49 (1990), 1166-1186; part II: J. Soviet Math. 62 (1992), 2746-2777; part III: J. Soviet Math. 62 (1992), 2619-2645 [31] S. Klainerman (lecture notes in perparation) [32] S. Klainerman, M. Machedon: Space-time estimates for null forms and the local existence theorem, Preprint [33] L. Lemaire: Applications harmoniques de surfaces riemanniennes, J. Diff. Geom. 13 (1978) 51-78 [34] F.H. Lin: Une remarque sur l'application x/lxi, C.R. Acad. Sc. Paris 305 (1987) 529-531 [35] P.L. Lions: The concentration-compactness principle, the limit case II [36] C.B. Morrey: Multiple integrals in the calculus of variations, Grundlehren 130, Springer, Berlin 1966 [37] T. Riviere: Applications harmoniques de B3 dans S2 partout disconues, C. R. Acad. Sci. Paris 314 (1992) 719-723 [38] T. Riviere: Regularite partielle des solutions faibles du probleme d'evolution des applications harmoniques en dimension deux, Preprint [39] R.S. Schoen, K. Uhlenbeck: A regularity theory for harmonic maps, J. Diff Geom. 17 (1982) 307-335, 18 (1983) 329 [40] R.S. Schoen, K. Uhlenbeck: Boundary regularity and the Dirichlet problem for harmonic maps, J. Diff. Geom. 18 (1983) 253-268 [41] J. Shatah: Weak solutions and development of singularities in the SU(2) cy-model, Comm. Pure Appl. Math. 41 (1988) 459-469 [42] J. Shatah, M. Struwe: Regularity results for nonlinear wave equations, Annals of Math. 138 (1993) 503-518 [43] J. Shatah, M. Struwe: Well-posedness in the energy space for semilinear wave equations with critical growth, Inter. Math. Res. Notices 7 (1994), 303-309 [44] J. Shatah, A. Tahvildar-Zadeh: Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds, Comm. Pure Appl. Math. 45 (1992), 947-971 [45] J. Shatah, A. Tahvildar-Zadeh: Non uniqueness and development of singularities for harmonic maps of the Minkowski space, Preprint [46] M. Struwe: On the evolution of harmonic maps in higher dimensions, J. Diff. Geom. 28 (1988) 485-502

Wave maps

153

[47] M. Struwe: On the evolution of harmonic maps of Riemannian surfaces, Math. Relv. 60 (1985) 558-581 [48] M. Struwe: Globally regular solutions to the u 5 Klein-Gordon equation, Ann. Scuola Norm. Pisa 15 (1988) 495-513 [49] M. Struwe: Geometric Evolution Problems, Nonlinear Partial Differential Equations in Differential Geometry, lAS/Park City Mathematics Series vo1.2 (AMS, November 1995) [50] R.C. Wente: The differential equation b.x = 2Hxu 1\ Xv with vanishing boundary values. Proc. AMS 50 (1975) 59-77 [51] S. Miiller, M. Struwe: Global existence of wave maps in 1 + 2 dimensions with finite energy data, Top. Meth. Nonlin. Analysis, special volume in honor of L. Nirenberg's 70th birthday (to appear) [52J Yi Zhou: Global weak solutions for the 1+2 dimensional wave maps into homogeneous space, submitted (1996)

Index 11/8 Conjecture, 23 asymptotically fiat, 39 asymptotically free, 46 basic classes, 6 blow-up (wave maps), 127 blowing up, 16 "bubbling" phenomenon, 4 co-rotational maps, 132 Compactness Theorem, 85 connected sum theorem, 11 Coulomb gauge, 34 definite forms (realizability), 20 Donaldson invariant, 4 Donaldson series, 5 dynamical law, 101 Einstein field equations, 35 energy functional, 73 energy inequality, 50, 116 Ginzburg-Landau equations, 74 Ginzburg-Landau parameter, 72 Gor'kov-Eliashberg evolution equation, 74

Noether's Principle, 36 non-uniqueness (wave maps), 130 null forms, 53 optimal regularity exponent, 45 partial regularity, 125 penalty method, 139 Penrose incompleteness theorem, 48 positive energy condition, 32 reducible solution, 10 renormalized energy, 87 scalar wave equation, 40 scaling exponent, 41 Seiberg-Witten equations, 9 Seiberg-Witten moduli space, 9 self-similar solutions, 127 semilinear wave equations, 40, 137 simple type 4-manifold, 5 space-time norms, 61 spine-structure, 8 Strichartz inequality, 51, 57 Structure Theorem, 83 Temporal gauge, 34 Thorn Conjecture, 13

ltl-BMO duality, 150

vortex motion, 100

indefinite forms (realizability), 22 Initial Value Problem, 37 intersection form, 19

wall-crossing formula, 15 wave map, 32, 114 weak compactness, 146 weak solutions (wave maps), 115

Jacobian structure, 146 Kuranishi map, 21 local existence, 51, 117 Lorentz gauge, 34 Maxwell equations, 33 minimax solutions, 88 mobilities, 101 moduli space (Yang-Mills), 3

Yang-Mills equations, 33

BIRKHAUSER Progress in Nonlinear Differential Equations and rheir Applications i.Ilim

Naim IIrezis.

Oepartement de Mathematiques, UnlVersite P, et M. Curie 4, Place Jus$ieu, 75252 Paris Cedex OS, France and Department of Math~mat\~, Rutg~rs Univel'Sity New Brunswick. NJ 08903, US,A.

Progress in Non/in.ear ond Differential Equotions and Their Applications is a book series that lies lit the interface of pure and applieD. mathematics. Mony differential equations lire motivllted b.l' problems llrising in slich diversified fields as Mechanics. Physlcs, Differential Geometry, Engineering, Control Theory, Biology, ond Economics. This series is open to both the theoreticol and applied aspects, hopefully stimulating a fruitful int£raction betweell the two sides. It WiU publi$h monographs, polished notes arising from lectures and seminars, graduote. level texts, and proceedings of focused and refereed conferences. We entouroge prepclration 0/ manuscrlpb i(l some form 0/ reX for delivery i(l comera-reody copy, which leads to rapid publication, or in electronic form for inter/acing with loser prif/t£rs or typesetters. Ploponl$ s~ould be ~nt directly to (he rdito( o( to: Birkhliuw 8ostOfl. 675 Hamthu.setts A~nue. Cambridge. MIl 02139

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Partial Differe(ltial Equations and Functional Analysis. In Memory of Pierre Grisvard J. (ea, D. (henais, G. Geymonat, J.l. Lions (Eds)

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