Fattorini H.O., Kerber A. The Cauchy Problem

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1.6. The Maxwell Equations

details of the computation follow:

ly,

i, E(x)=- 1 Dxdiv9,(x-Y)dy

ae IYI

4,ff

Dy divy p (x - y) dy

47'lyl a E IYI

yl2divyp(x-y)da

- 41r

lyl =e IYI

//' Yldivyp(x

Y) dy

417

Iyl'e lyl

f yl2

div9)(x-y)do

41rlyl=elY1

f 1

+ 4,rJIy1=e

IY1°(YiT I(x-Y)+YIY2'Pz(x-Y)

+YIY3T3(x- y)) do

1 (

+ 41r flvi

IYIS

((IY12 -3Yi

-Y)-3YIY24'2(x -Y)

-3y1Y3CP3(x-Y))dY

where do indicates the area differential on IY I = e. We take now limits as e -> 0. The

first surface integral is easily seen to tend to zero on account of the fact that y, IYI -2

is homogeneous of degree - 1. In the second integral we note that y yj I y 1- 4 is

homogeneous of degree - 2 with

YIY-da-

yl =e IYI

and write 9)j(x - y) = 9)j(x)+((pj(x - y)- p(x)). The final result is

Qj(3)9)j(x-Y)dY

lI(x)=;97,(x)+

lim f

j=I e-'O IYI%e

IYI

ifj=1

ifj=2,3

(1.6.16)

where the SIj are smooth off the origin, homogeneous of degree zero, and satisfy

Iyl

= I

Stj(y)da=0 (j=1,2,3).

Accordingly, the second inequality (1.6.14) for , follows from the Calderbn-

Zygmund theorem for singular integrals (Stein-Weiss [1971: 1, p. 255]); i2 and 43

are similarly handled. It remains to be shown that r1=

a.e. which reduces to

proving that a function f E L2 (8t 3), which satisfies A f = 0 in the sense of distri-

butions, must be identically zero. To do this we note that if p (=- 6D, then 9) * f is an

ordinary harmonic function tending to zero at infinity, thus by Liouville's theorem

(Stein-Weiss [1971: 1, p. 40]) p * f is identically zero; since 9) is arbitrary, f = 0 a.e.

as desired.

The Cauchy Problem for Some Equations of Mathematical Physics

We observe finally that each of the inequalities (1.6.14) implies the other, so

that the proof of Lemma 1.6.2 is finished.

We can now deal with the complete system (1.6.1), (1.6.2), (1.6.3) in

Lp(R3)6 for I <p <oo. Let P be the closed subspace of Lp(Q83)6 consisting of all

vectors u with

dtv(u1,u2,u3)=div(u4,u5,u6)=0.

(1.6.17)

Consider 9SS, the restriction of 91 to D(ii5) = D(91)n L P, the space of all 6-vectors

with components in S satisfying (1.6.17). Assuming that the Cauchy problem for

u'(t) = s?.1Su(t) (1.6.18)

is well posed in Cy- P, we shall obtain a contradiction below.

Let uED(11)=56. Write

u=b+m

where the vectors b, m correspond to the decomposition (1.6.12) in the first and last

three coordinates of u, so that b ELF P and

rot(W1,W2,W3)=rot(W4,W5,W6)=0.

(1.6.19)

By virtue of Lemma 1.6.2, to E L2 (B8 3) 6 and W to = 0, where W is the operator (1.6.4)

as defined in L2 (Q8

3)6

. Accordingly (a i A i + 02A2 + a3 A3) F m = 0 and it follows

that

f_i

expl -it Y_

ajAj)

Jm =m. (1.6.20)

`` i=1

Now, if u(.) is the solution of (1.6.11) with u (0) = u , we have, by virtue of (1.6.6)

and (1.6.20),

u(t) = - i

expl - it j

aiAi)

i=1

ajAj

6j 1) +m

_5 expiti=1

=b(t)+ro.

If SS(-) is the propagator of (1.6.18), an approximation argument shows that

b(t) = SS(t)b, thus for t fixed we must have

IIu(t)IIp<IIb(t)IIp+UmIIp<C'Ilblip+Ilmlip<CIIuIIp (uES)

(where we have used both inequalities (1.6.14) at the end). However, this estimate is

forbidden by Example 1.6.1. This is a contradiction and shows that the Cauchy

problem for (1.6.18) cannot be well posed in e P for any p in the range 1 < p < 00

except for p - 2.

Let es be the subspace of CO(083)6 consisting of all vectors u

satisfying (1.6.17). That the Cauchy problem for (1.6.18) is not well posed in

the space LYS (that is, that the Cauchy problem for Maxwell's equations

(1.6.1), (1.6.2), (1.6.3) is not well posed in the supremum norm) is of course

1.6. The Maxwell Equations

not news for the physicist; in fact, an electromagnetic wave that is initially

small everywhere may become enormously large (even infinite) in certain

regions at a later time through the phenomenon of "focusing of waves."

Examples of these regions are the focuses and caustic surfaces of geometri-

cal optics. (See Kline-Kay [1965: 1], especially p. 326 for a discussion of

caustics; for explicit examples of focusing of waves for the wave equation,

to which the system (1.6.1), (1.6.2), (1.6.3) can be reduced, see Bers-John-

Schechter [1964: 1, p. 13].)

The failure of the present treatment in the LP (p * 2) and the Co

cases for the Schrodinger and Maxwell equations confronts us with the

following problem. Assume, say, that we want to predict the behavior of an

electromagnetic field at some fixed point xo in space. Then it is obvious that

mean square estimates like (1.6.8) are of small comfort and we really need

bounds in the supremum norm. Although existence of this type of estimate

seems to have been ruled out by the previous counterexamples, we can try to

remedy the situation by measuring the initial state of the field in a different

norm. To make the result more widely applicable, we work with the

symmetric hyperbolic system

D,u= Y, AjDju+Bu, (1.6.21)

j=1

where u = (u

1, ... ,

and A1,.. . , A,,,, B are complex constant matrices,

Al,...,Am self-adjoint, and B skew-adjoint. The basic space for the treat-

ment will be HS(R m )v, consisting of all vector functions u = (u 1, ... , with

(I+ IaI2)s/2FU(a) E L2(Rm)v

endowed with the L2-norm of (1 + I a 12 )Si2 q u(a) (see Section 8). It is

exceedingly simple to extend the Fourier analysis carried out for (1.6.5) to

the space L2(Rm)°; the operator li? is

0u= T AjDju+Bu (1.6.22)

j=1

where D(d) is the space of all u E HS(Rm)' such that 6 u (always under-

stood in the sense of distributions) belongs to HS(R m )v;

equivalently,

u E HS belongs to D(() if and only if

(1+Ia12)S/2I -i

aAj+B)u(a)EL2(Rm)°

j=1

(where the factor - i and the summand B may of course be omitted). If

uo E D(l1), the function

u(t)=-' exp(- it ajAj+tB)u (1.6.23)

j=1

The Cauchy Problem for Some Equations of Mathematical Physics

is a solution of

u'(1)=¢u(t) (-oo<t<oo). (1.6.24)

Moreover, if u E D(d), we have

m l

(eu, u)HS = f (1+ I0 12)S

u(a))+(Bu(a), u(a))) do.

j_1

JJJ

(1.6.25)

Since Aj' = Aj, (Aju(o ), 0(a)) is real; on the other hand, B'= - B so that

(Bil(a), u(o)) is imaginary, and

Re(( ',u,u)Hs=O.

(1.6.26)

This implies again that an arbitrary solution of (1.6.24) has constant HS

norm, hence the Cauchy problem for (1.6.24) is properly posed. As we shall

see below, this will yield estimates in the LP and Co norms for the solution

in terms of the HS norm of u(0). Let u be an arbitrary element of HS(IRm)°

Since (1+ Ia12)S/2u E L2 and (1 +

Io12)-S/2 E L2 if

s > m/2, we obtain

from the Schwarz inequality that u(o)= (1+ IoI2)S12u(o)(1+

Io12)-S/2 be-

longs to L' (R m) with norm

Ilull1 < KS/21101 H=,

where KS = f(1 + 1012 )-s d o. It follows from well known properties of the

Fourier transform that u (if necessary modified in a null set) belongs to

Co (68 m) with norm

Ilull,o< (2,g)

m/211u11.'< (2,r)-m/2KS12IIuIIHs.

If s 5 m/2, estimates in the supremum norm are no longer possible; we use

LP norms instead. Let

2<p<m2 Zs<oo

and p' defined by p' -' + p -' =1 so that

1<m+2s<p'<2.

Define next q = 21p'>- 1 and q' by q'-' + q-' =1; we have

,_ 2 m+2s

2-p'> 2s

hence

(1.6.27)

2 p's

p'q's=2-p'>m,

and we obtain from Holder's inequality for the exponents q, q' applied to

1.6. The Maxwell Equations

the product

Iulp'

= I(1 +

IaI2)sI2uIp'(1 + I012)-p'9/2 that

u E LP' with norm

Ilulip'<K(p)IIuIIHs

where K(p) = K", h =1 /p'q' _ (2 - p')/2p'= (p - 2)/2p, r = p'q's/2 =

p's/(2 - p') = ps/(p - 2). We make now use of the Hausdorff-Young theo-

rem for Fourier integrals (Stein-Weiss [1971:

1, p. 178]) to deduce that

u E LP with norm

Ilullp <

(277)_°16

uII p-,

where d = m (p - 2)/2p.

We collect all these observations.

1.6.3

Theorem.

The Cauchy problem for (1.6.24) is well posed in

- oo < t < no in all spaces E = Hs(Rm)

s >, 0, in particular in E = L2(l8my;

if u is a solution in L2 such that u(0) E H', then u(t) E HS for all t and

IIu(t)IIH'=IIu(0)IIH=

(-oo<t<oo).

(1.6.28)

For s > m/2, u(t) (after eventual modification is a null set) belongs to

Co(IRm) and

IIu(t)IIco < CIIu(0)IIH'

(- 00 < t < oo)

(1.6.29)

with C = (2ir)-m/2Ks/2. Ifs < m/2 and p satisfies (1.6.27), then u(t) E LP

and

IIu(t)IILP<C(p)IIu(0)I6 (-x<t<oo)

(1.6.30)

with C(p) = (277)-dK(p). In particular, if s = m/2, u(t) E Lp and the

estimates (1.6.30) hold for any p > 1.

Observe that estimates for the derivatives of u(t) can be obtained in

the same way. For instance, ifs > m /2 + k and a = (a,, ... , am) is a multi-

index of nonnegative integers with I a I < k, the Fourier transform of D"u is

(- ia)au E HS-k; hence estimates of the form

IID°`u(t)Ilc0 ,<Cllu(0)IIHS (-oo<t<oo) (1.6.31)

hold. The same remark applies of course to LP norms; details are left to the

reader.

We note finally that results of precisely the same form hold for the

Schrodinger equation. The proofs are identical: once (1.6.28) is established

by direct Fourier analysis, inequalities (1.6.29) if s > m /2 or (1.6.30) if

s < m12 apply.

1.6.4 Example. Prove in detail the results about the Schrodinger equation

mentioned above (see Lemma 1.4.2).

1.6.5

Example. The theory of the symmetric hyperbolic system (1.6.21)

can be easily modified to embrace the case where B is an arbitrary v X v

52 The Cauchy Problem for Some Equations of Mathematical Physics

complex matrix. Write

B=B1+B2,

where B 1 =

(B + B') is self-adjoint and B2 = '(B - B') is skew-adjoint.

Using (1.6.25) we obtain

Re(d u, u)

.I S f (1+ 1a12)5K(B10(a), u(a))I da

wIl ull Hs

(1.6.32)

instead of the stronger condition (1.6.26), where w is the maximum of the

absolute value of the eigenvalues of B 1. Accordingly, if is an arbitrary

solution of (1.6.24), we must have

ID,IIu(t)1121 < 2w11u(t)112

which is easily seen to imply that

IIu(t)II s ew1'1jju(0)II

(-oo<t <oo).

(1.6.33)

This estimate guarantees continuous dependence on the initial datum.

Existence of solutions is proved in the same way.

1.6.6 Example.

Relativistic quantum mechanics: the Dirac equation. A

relativistic description of the motion of a particle of mass at with spin

(electron, proton, etc.) is provided by the Dirac equation

ihD,=ihc Y, aj

(1.6.34)

J=1

Here h is again Planck's constant, c is the speed of light, a1, a2, a3, 0 are the

4 X 4 matrices

0 0

0 1

0 0 0 -i

0 1 0

0 i

0 0 ' 2

0 -i

0 0 '

0 0 i

0 0 0

0 0 1

0 1

0 0

0 -1 0 1

0 0

a3 __

0 0 0 '

0 0 -1 0 '

0 -1

0 0

0 0 0 -I

and , (x, t) is defined in R 3 for each t and takes values in C 4, that is,

= (t 1, l 2,1'3,1'4). For each t,

t)12 = EI4'k(x, t)12 is interpreted as a

probability density as for the Schrodinger equation. (See Schiff [1955: 1, p.

323] for additional details.) In the case of a free particle (U = 0), the

equation (1.6.34) can be written as a symmetric hyperbolic system of the

1.6. The Maxwell Equations

form (1.6.21), where

cajD' + ca2D2 + ca3D3 + i

/6. (1.6.35)

Thus Theorem 1.6.3 applies; in particular, the Cauchy problem for the

Dirac equation is well posed in E = L2(R3)4 and the conservation of norm

property (1.6.28) holds.

In perturbation results in Chapter 5, a better identification of the

domain of 6 will be necessary. The following result holds for the general

symmetric hyperbolic equation (1.6.22):

1.6.7 Example. thatAssume

jm-i

detE aiAj) *0

(laI=1).

(1.6.36)

Th hen we ave

D(¢)=H'(ff8"')

(1.6.37)

the norm of H' (R') ° equivalent to the graph norm I l u l l + I I d u I l in D(?).

Note first that, independently of Assumption (1.6.36), H' c D(d) and

I I

?uII < C I I uIIHI for u E H'. On the other hand, if (1.6.36) is satisfied, we use

the facts that each Eat Aj is nonsingular for any a,

I a I =1 and that the

function a - EajAj is homogeneous of degree I to conclude the existence of

a constant 0 > 0 such that if u E C',

Eaj Aj u

J=1

%0Ial lul

(aERm)

(1.6.38)

wherefrom the opposite inclusion and inequality among norms follow.

Condition (1.6.36) is satisfied for the Dirac equations (where det(Eajaj) =

- c41014 ), but not for the Maxwell equations since in that case det(Eaa Aj)

= 0. We note (see the next Example) that condition (1.6.36) is necessary for

the coincidence of D(G) with H'.

In the applications of perturbation theory to the Dirac equation it is

important to compute the best constant 0 in (1.6.38). This is immediate

since a,a, + a2a2 + a3a3 is an orthogonal matrix, the length of each row (or

column) being del. Accordingly, I (a ,a,+ a2a 2+ a3a3) u I = c l a l 1U1 so that

0 = c. (1.6.39)

1.6.8

Example.

Assume det(EajAj) = 0 for some a * 0. Then

D(Q) - H'(R-)

1.6.9 Example. The apparent lack of a more precise characterization of the

domain of the Maxwell operator 91 in (1.6.4) is in fact due to the neglect of

The Cauchy Problem for Some Equations of Mathematical Physics

conditions (1.6.3). To see this we consider the restriction Ws of ll to the subspace

= C5 X S C- Lz(Qt3)6 of all vectors u = (6, 9C) satisfying (1.6.17). We have

s(6, `.x)112 = c2(IIrot

&;II2 +IIrot 9CII2). (1.6.40)

Let

2, 3) be a vector in C3 such that

a =

a R 3. Then weyy check easily that

a2J3 - a3

a1J312 +

Iaq

a2J112 = Ia121J12.

This identity, applied pointwise to the Fourier transform of a vector 6 E C25, shows

that rot6 E L2(R3)3 if and only if F E H'(R3)3; applying this and (1.6.40) to both

components

and 9C of an element of D(9ts) we conclude that

D(9ts)=H'(R3)6nCp2

(1.6.40)

1.6.10 Example.

The operators Of (9t the Maxwell operator (1.6.4)) and

ki (d the Dirac operator (1.6.33)) are self-adjoint, the first in L2(R3)6, the

second in L2(R3)4. More generally, id is self-adjoint in L2(Rm)°, where d is

the symmetric hyperbolic operator (1.6.22). It suffices to apply the relevant

portion of Example 1.4.6; the formal adjoint of d is

E AjDj+B'

l=1

1.7. MISCELLANEOUS NOTES

The classical Cauchy problem is that of solving a general partial differential

equation or system P(u) = 0, the Cauchy data of the unknown function u

(its value and the value of its normal derivatives up to order r - 1, r the

order of the equation) prescribed on a hypersurface

of m-dimensional

Euclidean space l8 m. Extending early work of Cauchy, Sophia Kowalewska

succeeded in proving in 1875 that the Cauchy problem for a wide class of

analytic partial differential equations can always be solved (if only locally)

if the surface and the Cauchy data of u are analytic and

is nowhere

characteristic. This result, known as the Cauchy-Kowalewska theorem, is

usually seen nowadays in the simplified version of Goursat (see Bers-John-

Schechter [1964: 1]; a different proof has been given by Hormander [1969:

1]). A misunderstanding seems to have subsequently arisen in relation to the

possibility of solving the Cauchy problem above for nonanalytic Cauchy

data along the lines of the following argument: approximate the Cauchy

data of u by analytic Cauchy data; solve the equation; take limits. The

fallacy involved in overlooking continuous dependence of the solution on its

Cauchy data was pointed out by Hadamard in [1923: 1, p. 331 (although he

was aware of it no less than twenty years before): "I have often maintained,

against different geometers, the importance of this distinction. Some of

them indeed argued that you may always consider any functions as analytic,

1.7. Miscellaneous Notes

as, in the contrary case, they could be approximated with any required

precision by analytic ones. But, in my opinion, this objection would not

apply, the question not being whether such an approximation would alter

the data very little but whether it would alter the solution very little."'

Hadamard presents the classical counterexample to the approximation

argument-namely, the Cauchy problem for the Laplace equation (which

will be found in its essential features in Chapter 6)-and singles out those

Cauchy problems for which continuous dependence holds; correctly set

problems in his terminology. Other names (well set, well posed, properly

posed) are in current use now.

The concept of a properly posed problem in the sense of Hadamard

does not fit into the abstract framework introduced in Section 1.2. To

clarify this we restrict ourselves to the particular case where the equation is

linear and of first order and where the hypersurface is the hyperplane

xm = 0. The requirement that .

be nowhere characteristic implies that the

equation or system can be written in the form

m-1

Dtu= Y_ Aj(x)Dju+B(x)u+f(x),

(1.7.1)

j=1

where u, f are vector functions of (x1,. .. ,

x,,) and the A, B are matrix

functions of appropriate dimensions; we assume that A j, B, f have deriva-

tives of all orders. The Cauchy data of u reduce to its value for xm = 0,

u(xl,...,Xm-1,0)=4,(xi,...,Xm-1). (1.7.2)

The Cauchy problem is properly posed in the sense of Hadamard if solutions

exist for, say, (p infinitely differentiable and depend continuously on q ;

according to Hadamard, continuous dependence is understood in the topol-

ogy of uniform convergence (on compact sets) of derivatives up to a certain

order, not necessarily the same for the data and the solution. It is remarka-

ble that continuous dependence in this sense is a consequence of existence

and uniqueness, as the following result (where we assume that f = 0) shows:

1.7.1 Theorem.

Let Xm > 0. Assume a unique infinitely differentia-

ble solution u of (1.7.1), (1.7.2) exists in

I x

I < Xm for every T E 6 (R m -1).

Then, given a, b > 0 and an integer M >, 0 there exists an integer N >_ 0

such that if (q7n) c 6 (Rm-1) with the support of each 9)n contained in

lim

I D#99n(x)I = 0

n- oo

uniformly in Otm-1 for / =

,$m_1) with I$I < N, then

lint IDaun(x)I =0

n - 00

1 From J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential

Equations, (New Haven: Yale University Press, 1923).

The Cauchy Problem for Some Equations of Mathematical Physics

u n i f o r m l y in X

I x,,, I < X. f o r all I a l < M.

Proof

Let 6Db the subspace of 6 (118 m -1) consisting of all q) with

support in x2 +

+ xm_ 1 < b2. The space 6Db is a Frechet space equipped

with the translation invariant metric

p(w,

) = E 2-kllro - CIk/(1 +IIq) - 0Ilk),

(1.7.3)

k=0

where 11'11k denotes the maximum of all partial derivatives of order < k.

Similarly, the space CX ) of all infinitely differentiable functions of x1,... , xm

defined in

I xm I < X,,, is a Frechet space endowed with the translation

invariant metric

d(u, v) = E E 2-(k+1)IIu - vIIk,1/(1 +IIu - vllk,l),

(1.7.4)

k=01=0

where 11'11k

denotes the maximum of all partial derivatives of order < k in

x +

+ x,2" _ 1

< 12,

I xI < X,,,. The linear operator C5 qq = u (from 6Db into

Cx( )) from Cauchy datum q2 to solution u provided by the assumptions of

Theorem 1.7.1 is easily seen to be closed, thus it must be continuous by

virtue of the closed graph theorem for Frechet spaces (Banach [1932: 1, p.

41]). It follows that there exists t > 0 with

d(u,0) <

if p(gi,0) < S

with I > a. Accordingly, if N is so large that

2-(N+1) +2-(N+2) +

... < 8/2

and T is such that

IIPIIN<8/4

(note that

11-11k

increases with K), we shall have p (qi, 0) < S so that

II ull M, r/(l + II uII M,1) < ,

thus II uII

M,1

must remain bounded. Hence there

exists a constant C (depending on a, b, M) such that

IIUIIM,a<CIIpIIN

(1.7.5)

This concludes the proof.

We note that a rather obvious approximation argument based on

(1.7.5) shows that (1.7.1), (1.7.2) will actually have a solution for every

initial datum with support in x2+

+X21

< c2 < b2 having continu-

ous derivatives of order < N, these solutions having continuous derivatives

of order < M in xi +

x,2"_,-<a2,

Ixml < Xm; it suffices to approxi-

mate the qp in question in the norm 11-11N by a sequence (q)") in 6D with

support in x2 +

+ x"2,_ < b2 and let n - oo; of course, we assume that

M >_ 1 so that limits can be taken in the equation (1.7.1).

There is evidence in [1923:

1] that Hadamard was aware of the

relation between existence of solutions for sufficiently differentiable initial

data and continuous dependence. Contrasting properly and improperly

posed problems, he states on p. 32: "On the contrary, none of the physical