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

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5.6. Symmetric Hyperbolic Systems in Sobolev Spaces

307

Given an operator A E e ,(E) and another Banach space F with

F - E, we say that F is A-admissible if and only if

(exp(tA))F= S(t; A)F c F (t >,0) (5.6.4)

and t -' S(t; A) is strongly continuous in F for t > 0. (Motivation for this

definition is obviously provided by the previous remarks on the equation

(5.6.2), where E = L2, F= H'.)

If A is an arbitrary operator in E with domain D(A), we denote by

the largest restriction of A to F with range in F, that is, the restriction of

A with domain

D(AF) = (u E D(A) n F; Au E F). (5.6.5)

5.6.1 Example.

(a) Let, as in Example 2.2.6, E = L2(- oo,0), and define

Au = u' (with D(A) the set of all u E E such that u' exists in the sense of

distributions, belongs to E and satisfies u(0) = 0), and let F be the Sobolev

space H'(- oo, 0) (see Section 8). Then F is not A-admissible. (b) Let E and

A be as in (a), but take F as the subspace H01(- oo, 0) of Hl(- oo, 0) defined

by the condition u(0) = 0. Then F is A-admissible. (c) Let, as in Example

2.2.7, E = L2(0, oo), Au = u' defined as in (a) but without boundary condi-

tions at 0. If F = H'(0, oo), then F is A-admissible.

In Example (a), AFU = u', where D(AF) consists of all u E E with

u', u" E E. The characterization of AF in (b) and (c) is similar. We note that

the two affirmative examples in (b) and (c) are consequences of the

following result for m = 1: (d) Let A E e ,(E), m >, 1, F = D(Am), IIuli

F =

II(X I - A)muII E, where X E p(A) is fixed. Then F is A-admissible.

5.6 .2

Lemma.

Let F -' E, A E (2+(E ). Then F is A-admissible if

and only if R(X; A) exists for sufficiently large A and (a) R(X; A)F c F with

IIR(X;A)"II(F)<C(X-w)-n

(n=1,2,...).

(5.6.6)

(b) R(X; A)F is dense in F. Conversely, if (a) and (b) hold, the operator AF

belongs to 3+(F) and

S(t;AF)=S(t;A)F (t>l0).

(5.6.7)

Condition (b) is unnecessary if F is reflexive.

Proof. We observe first that, if F is A-admissible, the definition

(5.6.5) means that the operator on the right-hand side of (5.6.7) is none

other than the restriction of S(t; A) to F. Also, S(t; A)F is a strongly

continuous semigroup in F; let A be its infinitesimal generator and C, w two

constants so large that both inequalities IIS(t; A)II(F) <Cew` and

IIS(t; A)II(E) < Cew` hold. If u E F, we have S(t; A)u = S(t; A)u so that, in

view of formula (2.1.10) with n =1 and of the fact that F -' E, R(A; A)u =

R(X; A)u for A > to. It results that R(A; A) is simply the restriction of

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Perturbation and Approximation of Abstract Differential Equations

R(A; A) to F,

R(A; A) = R(A; A)

F-

(5.6.8)

If u E D(A), we have

,4u=F- lim h-'(S(t; A)u - u)

h->0+

=E- lim h-'(S(t;A)u-u)

h-.0+

=E- lim h-'(S(t;A)u-u)=Au,

h-.0+

which shows that A C AF. On the other hand, let u E D(AF); then u =

R(A; A)v = R(A; A)FVfor some v E F, and in view of (5.6.8), u must

belong to D(A). Thus A = AF and

R(A; AF) = R(A; A)F.

(5.6.9)

It is obvious then that (5.6.6) follows from (5.6.9) and the fact that

R(A; AF) satisfies inequalities

(2.1.29); moreover, since R(A; A)F=

R(A; AF)F by (5.6.9), the denseness condition (b) follows.

Conversely, assume that (a) and (b) hold. If u e F and A is large

enough, v = R(A; A)u E F; moreover, Av = AR(A; A)u - u E F so that v E

D(AF) and

(AI-AF)R(A;A)u=u.

This obviously shows that (Al - AF)D(AF) = F; since, on the other hand,

AI - AF is one-to-one because Al - A is, it results that R(A; AF) exists and

(5.6.9) holds. Condition (b) plainly insures denseness of D(AF), and in-

equalities (5.6.6) imply, via Theorem 2.1.1 and Remark 2.1.4, that AF E

e+(F). It only remains to show that

S(t; AF) = S(t; A) F

and this can be done making use of (5.6.9) and expressing each semigroup

by means of the resolvent of its infinitesimal generator; in fact, using

formula (2.1.27),

S(t;AF)u=F- lim (n)nR(n;AF)nu

n +00

t t

=E-nlim0(t)nR(t;A)nu

=S(t; A) u

for u E F as claimed.

It remains to be shown that the denseness condition (b) can be

dispensed with when F is reflexive. Observe first that (5.6.9) has been

established without use of (b). If R(A; A)F= R(A; AF)F= D(AF) is not

dense in F, there exists some u E F, u - 0 that does not belong to the closure

5.6. Symmetric Hyperbolic Systems in Sobolev Spaces

309

D (A F). Consider

un = AnR(An; AF)u = anR(An; A) U,

where An - oo. In view of the first inequality (5.6.6), (u,,) is bounded in F

and, since F is reflexive, we may assume that (u,,) is F*-weakly convergent

in F to some v E F.10 But D(AF) is F*-weakly closed in F, so that v

ED(AF)

. Now,

AnR(A,; AF)R(X,; AF)u = R(A1; AF)AnR(An; AF)u - R(A1; AF)v

(5.6.10)

F*-weakly in F whereas, by the second resolvent equation and the first

inequality (5.6.6),

(AnR(A,; AF)- I)R(A1; AF)u

= (A,

AF)- A,R(A1; AF))u - 0

as n - oo showing, in combination with (5.6.10), that

R(A1; AF)u = R(A1; AF) V,

(5.6.11)

which implies u = v, a contradiction since v belongs to D (A F) and u does

not. This completes the proof of Lemma 5.6.2.

Another useful criterion for A-admissibility of a subspace is

5.6.3

Lemma.

Let F- E, A E (2+(E), K: F- E an operator in

(F; E) such that the inverse K-' exists and belongs to (E; F). In order that F

be A-admissible, it is necessary and sufficient that A, = KAK-' belong to

(2+(E ). In this case, we have

S(t; A,) = KS(t; A)K- (5.6.12)

Proof

By definition, D(A,) = D(AI - A,) is the set of all u E E

such that K-'u E D(A) and (JAI - A)K-'u E F; accordingly,

D((AI - A,)) = KR(A; A)F.

(5.6.13)

Moreover, (Al - A,)

= K(AI - A)-'K- 1; thus if A E p(A), we have

(AI-A,)-'=KR(A;A)K-'.

(5.6.14)

Note, however, that we have not written R(A; A,) on the left-hand side of

(5.6.14) as we cannot state without further analysis that (XI - A,)-' E (E)

(since K may not be continuous as an operator from E into E). Assume now

that F is A-admissible. We have already seen in Lemma 5.6.2(b) that

R(A; A)F is a dense subset of F and that R(A; A)F E (F); this plainly

implies that D(A,) is dense in E (because of (5.6.13)) while it results from

1°It should be remembered that the first inequality (5.6.6) does not guarantee that

AR(X; AF)u - u as A -> oc except if u is known to be in D AF (see Example 2.1.6).

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Perturbation and Approximation of Abstract Differential Equations

(5.6.14) that (XI - A,)-' E (E), thus A E p(A,). Taking powers in both

sides of (5.6.14), we see that

R(A; A,) = KR(A; A)"K-', (5.6.15)

thus R(A; A,) satisfies as well inequalities (2.1.29) and consequently A, E

C'+(E ). Equality (5.6.12) follows from (5.6.15) and the exponential formula

(2.1.27) (see the end of the proof of Lemma 5.6.2(a)).

We prove now the converse. Assume that A, = KA K-' c= C, (E).

Reading (5.6.15) backwards, we see that R(A; A)" = K-'R(A; A,)"K, hence

R(A; A)F C F and (5.6.6) is satisfied for A sufficiently large. On the other

hand, R(A; A)F= R(A; A)K-'E = K-'R(A; A,)E, thus K-'D(A,) must be

dense in F since D(A,) is dense in E. Hence the hypotheses of Lemma 5.6.2

are satisfied.

hold if

In view of Theorem 5.1.2, the assumption that KAK-' E e+(E) will

KAK-' = A + Q (5.6.16)

with Q everywhere defined and bounded. This is the form in which Lemma

5.6.3 will be applied to the equation (5.6.2).

5.6.4 Theorem. Let the matrices A,,...,A,,,, B be continuously dif-

ferentiable in R ', and assume all the functions

IAk(X)I,ID'Ak(x)I,IB(x)I,ID'B(x)I (1<j,k<m)

(5.6.17)

are bounded in Q8 m by a constant M,. Then (a) The operator

(d')* defined

in Section 3.5 belongs to e (1, w; E) with E = L2 (118 m )'for some w. (b) If S(-)

is the propagator of

u'(t)=Cu(t) (-oo<t<oo),

(5.6.18)

we have

S(t)H'cH' (-oo<t<oo),

(5.6.19)

where H' = H'(Rm)v. Finally, if 6 H' is the largest restriction ofd to H' with

range in H', the operator dH, belongs to e+(H'); in particular, any solution

of (5.6.18) with initial value u(O) in H' remains in H' for all t, is

continuous in the norm of H', and satisfies

Ilu(t)IIHI<C,e0-'t1IIu(0)IIHI

(-oc<t<oc),

(5.6.20)

for constants C,, co, independent of u(-).

Proof. The statements regarding the behavior of the operator d in

L2 are immediate consequences of Corollary 3.5.5." The rest of Theorem

5.6.4 will of course result if we show that F = H' is both d-admissible and

"Boundedness of I DkA(x) I and I B(x) I insure inequality (3.5.37).

5.6. Symmetric Hyperbolic Systems in Sobolev Spaces

311

(- i )-admissible. Since - d satisfies assumptions of exactly the same type

as d, we need only consider the latter operator. We use Lemma 5.6.3 as

follows. Consider the operator A = (I -

where 0 is the Laplacian;

that is, if F indicates, as in Section 8, the Fourier-Plancherel transform,

cu = q-'((1 + 1612)'/2Yu(a)).

In view of the Fourier transform characterization of Sobolev spaces over R m

(see again Section 8), it is obvious that R is well defined and bounded as an

operator from H' into L2, and the same can be said of its inverse

g- 'u = 9-'((I+

Ia12)-1126J-u(o))

as an operator from L2 into H'. We note also that both A and A-' map the

space S into itself. We introduce next the operator

91u=

'(IaI 6j-u(0))

(5.6.21)

from H' into L2. It is obvious that if u E H2,

IIRu- JUIIL2 <CIIUIIL2.

(5.6.22)

We note that R (resp. R) is known as the Bessel (resp. Riesz) potential

operator of order - 1; see Stein [1970: 1, p. 117]. We shall also make use

the Riesz transforms R k defined by

9tku= q-'((iak/I('I)Fu(a)) (1<k<m) 12 (5.6.23)

It is plain that each Rk is a bounded operator from L2 into L2 and that

m m

R =

Dklk =

%kDk

(5.6.24)

k=1 k=1

where, as usual, Dk is the operator of differentiation with respect to Xk. The

Riesz transforms can be expressed by means of singular integral operators

as follows:

9'tku(x) = lim

CmJ

Y+1

u(Y) dy

(5.6.25)

EGO

Ix - YI

(Cm = 2m/21T -1/2r((m + 1)/2))) for each u E L2(Rm)v, the limit understood

in the norm of L2; if u is smooth, the limit also exists pointwise (See Stein

[1970: 1, p. 57].)

5.6.5 Lemma.

Let be a continuously differentiable

v X v

matrix-valued function defined in R ', and let

(9't, A)u(x) = Jt(A(x)u(x))-A(x)(Ru(x)). 13 (5.6.26)

12 These definitions coincide with the ones in Stein modulo constants unimportant for

our purposes.

13 Expressions like (ft, A), (9, Ak), and

so on, are defined in a corresponding way.

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Perturbation and Approximation of Abstract Differential Equations

Then, if u E 5°,

(91,A)u(x)= li mCm1

A(x)-A(y)u(y)dy,

(5.6.27)

Ix-yI r Ix-

ylm+1

where Cm is the constant in (5.6.25).

Proof.

In view of (5.6.24), we have

(t,A)u= F, (kDk(Au)-ARkDku)

k=1

(kA-Atk)Dku+ kkDkAu. (5.6.28)

k=1

Now,

( tkA-Ak)Dku(x)

= lim CmJ

Y+1

(A(Y)-A(x))Dku(Y) dy

1x-Y1>F Ix-YI

(the limit understood pointwise). Observe that using the divergence theorem

in the region

I x - y I > e, we obtain

{ Y-

m+1

(A(Y)-A(x))Dku(Y)}

Ix -

YI>E k=1 Ix - YI

mxk

Ix - VI>E k=l Ix- ylm+I

D A(Y)u(Y) dY

+ A(y)-A(x)

u(Y)

Ix-yIr

Ix-Y1`

_ f

A(y)- A(x)

u(Y) da,

- jx-Y1=r

Ix-YIm

(5.6.29)

where da is the hyperarea differential on Ix - yI = e.

Since A(y) =

A(x)+EDkA(x)(yk - xk)+ o(Iy - xl)

as y - x and the integral of

(xk - Yk )/ I x - y I m over I x - y I = e is zero, it is not difficult to

show that

the last integral in (5.6.29) tends to zero with E. Taking advantage of

(5.6.28), (5.6.27) follows.

We make now use of the following powerful result on singular

integrals.

5.6.6 Theorem.

Let A(.) be as in Lemma 5.6.5; assume in addition

that the derivatives DkA(x), 1 < k < m are bounded in jRr by a constant M.

Let 1j be an even, homogeneous function of degree - (m + 1) locally integrable

5.6. Symmetric Hyperbolic Systems in Sobolev Spaces

313

in I x I > 0. Finally, let 1 < p < oo, and define

CEEu(x)=J b(x-y)(A(x)-A(y))u(y)dy. (5.6.30)

YI>E

Then tE maps continuously L"(Rm)P into itself and satisfies

IIGEuiIp<CMllullp,

(5.6.31)

where C depends on p and Tj, but not on u or e. Furthermore, (yEu converges in

LP as e- 0; thus the operator

CSu= limCSEu

E 0

satisfies

Il(yullp < CMII UII,.

(5.6.32)

For the proof (of a more general result) see Calderon [1965: 1], where

additional information is given.

We can combine Lemma 5.6.5 and Theorem 5.6.6 to deduce that if

MI is a bound for DkA in Rm, then

II(R, A)uiI L2

I<CMIIIuiIL2

(u E

Now, (9, A) = (31, A)+ (A - R, A), where in view of (5.6.22), A - 3t is

bounded in L2, and the operator of multiplication by A is also bounded; it

then follows that

II(g, A) U11 L2 < CMIII uII L2,

(5.6.33)

where the constants C in both inequalities do not depend on

(this is

not particularly important now but will be vital in later applications in

Section 7.8).

Let u, as before, be a function in 5 A simple calculation shows that

qdq-'u = du + F, (9, Ak)Dkg-'u+(R, B)c-'u.

(5.6.34)

k=1

Making use of Theorem 5.6.6 and of the easily verified fact that

IIDkq-'uIIL2 s IIUIIL2,

we see that there exists a bounded operator QI in L2 such that

(5.6.35)

(St,Ak)Dk

'u=Q1u (uES").

(5.6.36)

k=1

On the other hand, noting that A _ A-' - ED k a -'D k, we obtain

(9 B)A-u= I

st-

I - k=1) A 'u-Bu (5.6.37)

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Perturbation and Approximation of Abstract Differential Equations

On account of the fact that each D"R-' is bounded (see (5.6.35)) and of the

differentiability properties postulated for B, there exists a second bounded

operator Q2 in L2 such that14

(R,B)R-'u=Q2u (uE Y').

(5.6.38)

To prove (5.6.16) for A = d and K = R we still have to examine the

domains of the unbounded operators involved. Let u e D(C?). Making use

of Lemma 3.5.2, we can choose a sequence (u") in S" such that

u"-4u, du"-du

in L2. Set

Q=Q1+Q2,

write (5.6.34) for u,,, and take limits using (5.6.36) and (5.6.38); we obtain

that RdR-'u" - Ou + Qu, thus CSR-'u" - R - u + Qu), and it follows

from the closedness of d that R -' u E D(d) and 9R-'u = R -' (gu + Qu) E

H'; thus Rd R-'u = d u + Qu, showing that

RdR-'DCT+Q. (5.6.39)

It follows that R(d - XI)R-' D d + Q - Al; a fortiori, if inverses exist,

a (e -,\I) `a-, D (¢ +Q- AI)- (5.6.40)

But if A is sufficiently large, it must belong to both p(C) and p(d + Q)

(Theorem 5.1.2), thus the operator on the right-hand side is everywhere

defined and (5.6.40) is actually an equality. The same must be true of

(5.6.39), that is,

R9 R - '=C + Q.

(5.6.41)

We obtain then from Lemma 5.6.3 that H' is Cadmissible. This ends the

proof of Theorem 5.6.4.

As it can be expected, results of the type of Theorem 5.6.4 can be

obtained in Sobolev spaces Hr, r >, 1, if additional smoothness assumptions

are imposed on the coefficients of L. In fact, we have:

5.6.7

Theorem. Let the matrices Al,...,A,,,, B be r times continu-

ously differentiable in R m and assume all the functions

ID"Ak(x)l ,ID'B(x)I

(0<jal<r, 1<k<m)

(5.6.42)

are bounded in R' by a constant Mr. Then conclusion (a) of Theorem 5.6.4 is

valid and (b) holds replacing H' by Hr. In particular (5.6.19) becomes

S(t)HrcHr (-co<t<oo)

(5.6.43)

14Boundedness of (R, B).'-1 (in fact, of (.R, B) itself) can be of course proved using

Theorem 5.6.6. However, the present argument will be significant in Section 7.6.

5.6. Symmetric Hyperbolic Systems in Sobolev Spaces

315

and every solution u(.) of (5.6.18) with initial value in H' remains in H' for all

t, is continuous in the norm of H' and satisfies

Il u(t)II H' < CreWr1tlllu(0)IIH,

(-00<t<00)

for constants Cr, wr independent of u(.).

We limit ourselves to prove Theorem 5.6.7 in the case r = 2, since no

new ideas are involved in the passage to the range r > 2. The role played

when r =1 by R will now be taken over by R 2, which is a bounded operator

from H2 into L2 with (R2)- I = R-2 : L2 - H2 bounded as well. In view of

(5.6.41), we have

A2CTR-2= R(ReR-')R-' = RCJR-' +RQR- '

thus our only task is to show that S1QR-' is everywhere defined and

bounded; this will obviously hold if we prove that Q is a bounded operator

in H'. We do this next. Let A be a v X v matrix-valued function satisfying

the hypotheses on Ak in Theorem 5.6.7 for r = 2, lj a function as in Theorem

5.6.6. Writing, as we may,

q,u(x)= f

tj(Y)(A(x)-A(x-Y))u(x-y)dy,

lyI>E

we see that if u E S qEu is differentiable and

DkCSu(x)

= f

1i(x-Y)(DkA(x)-DkA(Y))u(Y) dy

Ix - yl>E

+ f Cj(x-y)(A(x)-A(y))Dku(y)dy

Ix -

A>E

(0 < k < m). (5.6.44)

We can then use Theorem 5.6.6 to deduce that each D' (E u converges in L2

as e - 0 to a function v E L2. Since (YEu - (E u in L2, it follows that

D' (E u = Vk (in the sense of distributions) belongs to L2, thus (E u E H. In

view of the estimate (5.6.32), we have

IINIHl < CM2II uII HI

(u E X79).

(5.6.45)

We apply now these remarks to the operators (R, Ak); in view of (5.6.27), it

follows from (5.6.45) that

II{l,Ak)uIIH'<CM21IuIIH'

(U P)'

(5.6.46)

Observing that

II9'u - Null Hi < CIIuIIH.

(u E H')

and arguing as in the comments following Theorem 5.6.6, we see that

(5.6.46) implies

(5.6.47)

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Perturbation and Approximation of Abstract Differential Equations

hence, in view of (5.6.36) and of the fact that 5'(R') is dense in H' (IB' )

Q, is a bounded operator in H'. As for Q2, defined by (5.6.38), it follows

directly from (5.6.37), the hypotheses on B for r = 2 and again the denseness

of S that it is a bounded operator in H'. Hence Q E (H') and the proof of

Theorem 5.6.7 for r = 2 is complete.

5.7. APPROXIMATION OF ABSTRACT

DIFFERENTIAL EQUATIONS

It is often the case that solutions of an initial value problem

u'(t)=Au(t) (t>- 0), u(0)=uo, (5.7.1)

where A E C'+, can be approximated by the solutions of

un(t)=Anun(t) (t>-0),

u(O)=u,,,

(5.7.2)

with u,, - u and An - A in some way or other. Usually, An is an operator in

a different Banach space En; one such instance is that where A is a

differential operator in a space of functions defined in a domain 2 c R m

and An is a finite difference approximation to A, in which case En may be a

space of "discrete" functions defined only at certain grid points; moreover,

instead of solving (5.7.2) (which would correspond to the "method of lines"

in numerical analysis), we may also discretize the time variable, so that the

differential equation in (5.7.2) may be discarded in favor of some finite

difference approximation like

un(t+T)=TAnun(t)+un(t). (5.7.3)

We examine in this section and the next an abstract scheme including these

instances.

In what follows, E and En (n >, 1) are (real or complex) Banach

spaces. The first result is a sort of generalization of (a particular case of)

Theorem 1.2. We shall henceforth denote by 11.11 the norms of E and of

(E; E) and by I I - I I

the norms of En, of (E; En) and of (En; En), except

when more precision is necessary.

5.7.1

Theorem. Let the operators Qn E (E; En) (n >, 1) be such that

(I I Qn u I 1n; n >- 1) is bounded for any u E E. Then there exists M >, 0 such that

IlQnlln<M (nil).

(5.7.4)

Proof. Denote by CY the space consisting of all sequences u =

(ul, u2,...), un E En such that Ilunlln is bounded for n > 1. It is plain that the

norm

(lull = Ilul6 = sup Ilunlln

n>1