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

Подождите немного. Документ загружается.

7.8. Time-Dependent Symmetric Hyperbolic Systems in the Whole Space

437

for 0 < s, t < T. Taking advantage of this improved inequality, we can prove,

making use of A+(t) defined by (7.7.30) and of A- (t) defined by (7.7.34)(see

Figures 7.7.1 and 7.7.2) that if u E F, D1S(t, s)u exists at s = t and equals

A(s)u. We show next that (7.7.31) holds in 0 5 s, t < T and use (7.7.32), now

valid for h of arbitrary sign (as a consequence of (7.7.31) for 0 < s, a, t S T)

to show that DS(t, s)u exists in 0 < t < T and satisfies

D1S(t,s)u=A(t)S(t,s)u

there. The proof of (7.7.15) in 0 < s < T follows the same lines and is

therefore omitted.

It is interesting to note that we have just fallen short of proving that

(t, s) - S(t, s)u is F-strongly continuous for u E F in Theorem 7.7.13 (of

course, this would imply strong E-continuity of (t, s) - A(t)S(t, s)u, thus

continuity of the derivatives of the solutions therein constructed). This

additional bit of information can be established under hypotheses of the

type used in Example 7.7.12.

7.7.14

Example (Kato [1970: 1]).

Let the assumptions of Example 7.7.12

be satisfied both for and for the family A(.) defined in the statement of

Theorem 7.7.13. Then the conclusions of said theorem hold with the

additional result that, if u e F,

(t,s)-->S(t,s)u

is F-continuous in 0 < s, t < T.

7.8. TIME-DEPENDENT SYMMETRIC HYPERBOLIC

SYSTEMS IN THE WHOLE SPACE

The results of last section will be applied here to the abstract differential

equation

u'(0=i(t)u(t) (-oo<t<cc).

(7.8.1)

Here (fi(t) = (d0' (t))*, where do (t), E' (t) are defined as in Sections 3.5 and

5.6 (with domain D(&0) = D(6') = 6D1') with respect to the time-dependent

partial differential operator

L(t)u= E Ak(x,t)Dku+B(x,t)u.

(7.8.2)

k=1

Precise assumptions on the coefficients will be given later; the space E is

438 The Abstract Cauchy Problem for Time-Dependent Equations

once again L2 = L2 (R -) and F= H= H (R `)

Verification of the vari-

ous hypotheses in Theorem 7.7.5 involves different degrees of difficulty.

Stability of the family &(-) in L2 poses no problem, since each d(t) will

belong to C+(1, w; L2) for sufficiently large w under assumptions similar to

those of Section 5.6 for each t (see Remark 7.7.2). In contrast, d (t )-admissi-

bility of H' and stability of are far from trivial to verify, as a glance

at Section 5.6 will show: there we needed a powerful result on singular

integrals merely to show that H' is d -admissible. Fortunately, "dynamic"

versions of the auxiliary results in Section 5.6 will do the trick also in the

time-dependent situation. The first of these extensions is a descendant of

Lemma 5.6.3 and takes care of F-admissibility and F-stability in one stroke.

7.8.1 Lemma. Let F - E, (A(t ); 0 < t < T) a stable family in E.

For each t E [0, T], let K(t) : F - E be a bounded invertible operator (K(t) E

(F, E), K(t) -' E (E, F)) such that the family

(A,(t)=K(t)A(t)K(t)-';0<t<T)

(7.8.3)

is stable in E; assume, moreover, that

IIK(t)II(F;E)<N, IIK(t)

'II(E;F)

N (0<t<T)

(7.8.4)

and that the map t -* K(t) E (F; E) is of bounded variation. Then F is

A(l)-admissible for 0 < t < T and the family

F) is stable in F.

Proof The statement on A(t)-admissibility of F is a direct conse-

quence of Lemma 5.6.3. Combining (5.6.9) with (5.6.15), we obtain

R(X; A(t)F) = R(X; A(t))F

= (K(t)-'R(X; A,(t))K(t))F

= K(t)-'R(X; A,(t))K(t), (7.8.5)

where X belongs both to p(A(t)) and p(A(t,)) (note that this will happen if

A is sufficiently large independently of t, as both A(.) and A&) are stable).

If0<t1< <tn<T,wehave

n n

i R(X;

A(tj)F) =

K(tj)-'R(X; AI(tt))K(tt)

J=t I=t l

= K(tn)

{

H R(x; Aj(ti))(I + Lj)}R(X; A1(t1))K(t,),

l=2

(7.8.6)

where

Li = (K(tj)-K(tj-1))K(tj-1)

= K(tj)K(tj-,)

'- I. (7.8.7)

7.8. Time-Dependent Symmetric Hyperbolic Systems in the Whole Space 439

We estimate now the (F)-norm of (7.8.6). We expand this product and

arrange the resulting summands as a polynomial in the Lj; note that, due to

lack of commutativity of the factors involved in each summand, the differ-

ent Lj will appear sandwiched between consecutive R(X; A(te)), each such

LI corresponding to the choice of Ll rather than I in the parenthesis

(I + Lj). If C1, w, are the stability constants for A

and N is the constant

in (7.8.4), we can then estimate each summand by

(7.8.8)

where LI

, ... , Ll are the Lj appearing in that particular product (note that

each such Lj provokes the breaking up of the stability product (7.7.3) into

two pieces and thus forces us to take powers of C, on the right-hand side,

the exponent being equal to the number of Lj in the product plus one). It

follows then that (7.8.7) can be estimated by

N2CI

(1+CIIIL,II)}(X-w1)-n. (7.8.9)

J=2

We bring now into play the hypothesis that is of bounded variation,

combined with the first inequality (7.8.4), obtaining

Y_ IIL;II <NV,

l=l

V the total variation of K(-); accordingly, the factor between curly brackets

in (7.8.9) is bounded by exp(CI NV) and there exists C > 0 such that

U R(X; A(tj)F) <C(X - w,) n. (7.8.10)

l=l

(F)

This ends the proof of Lemma 7.8.1.

As in the time-independent case, Lemma 7.8.1 will only be used

when A, is a bounded perturbation of A (see (5.6.16)).

7.8.2 Corollary.

Let E, F, A(-), K(.) be as in Lemma 7.8.1, but

assume that

K(t)A(t)K(t)-'=A(t)+B(t),

(7.8.11)

where each B(t) belongs to (E) and

JIB(t)II(E)<M (0<t<T).

(7.8.12)

Then the conclusions of Lemma 7.8.1 hold.

Proof. We only need to show that A(.)+ is stable, and this has

already been done in Lemma 7.7.4.

It is remarkable that a slight stiffening of the assumptions in Corollary

7.8.2 will totally eliminate the restrictions to the differentiability of t -

440 The Abstract Cauchy Problem for Time-Dependent Equations

S(t, s) u in Theorem 7.7.5. This is perhaps of marginal interest to us in

relation with the application we have in mind, since some of these restric-

tions can be eliminated anyway on the basis of Theorem 7.7.13; however, it

is many times the case in applications that the reinforced assumptions are

satisfied without any special effort on our part.

7.8.3 Theorem. Let E, F be Banach spaces with F - E, and let

(A(t); 0 < t <T) be a stable family in E such that F c D(A(t)) and

A(t): F-* E is bounded in 0 < t <T; moreover, if A(t) is the restriction of

A(t) to F, the map

t- A(t)E(F,E)

(7.8.13)

is continuous. For each t E [0, T], let K(t) be a bounded and invertible

operator from F to E (K(t) E (F, E), K(t) -' E (E, F)) such that

t- K(t)E(F,E)

(7.8.14)

is strongly continuously differentiable in 0 < t < T (i.e., t -* K(t)u is continu-

ously differentiable in E for each u E F). Assume, finally, that

K(t)A(t)K(t)-'=A(t)+B(t), (7.8.15)

where B(t) E (E) for each t and t - B(t) is strongly continuous in 0 < t <T.

Then the Cauchy problem for

u'(t) = A(t)u(t) (7.8.16)

is properly posed in 0 < t < T with respect to solutions defined in (D).5

Moreover, if u E F, S(t, s)u is continuous in F for 0 < s < t < T.

Proof.

It follows easily from the uniform boundedness theorem that

the family ((t'- t)(K(t')- K(t)), t'

t) is uniformly bounded in (F; E); in

other words,

IIK(t')-K(t)II(F;E)<DIt'-tI (7.8.17)

for 0 < t, t' < T. We can then use the remarks on inverses6 in Section 1 to

deduce that t -' K(t) -' E (E; F) is as well Lipschitz continuous in 0 < t < T;

in particular, IIK(t)II(F,E) and II K(t)-'II(E,F) are bounded in 0 < t <T.

This shows that the assumptions of Corollary 7.8.2 are amply satis-

fied. It follows that all the hypotheses in Theorem 7.7.5 are themselves

fulfilled, except of course that we have not assumed that F is reflexive.

Combing the proof we soon realize that the reflexitivity assumption was

only used to assure that

S(t, s)F c F

(7.8.18)

(see the comments following (7.7.32)); thus, if (7.8.18) can be established by

5See the statement of Theorem 7.7.13. Strictly speaking, solutions will be only defined

f o r

(D) becomes now s .

t .

6See the comments preceding (1.1).

7.8. Time-Dependent Symmetric Hyperbolic Systems in the Whole Space

441

other means, the conclusions of Theorem 7.7.5 will follow in full. We shall

in fact exceed this by proving that K(t)S(t, s)K(s)-' is everywhere defined

and bounded in E and

(t, s) -* K(t)S(t, s)K(s)-'

(7.8.19)

is strongly continuous in E for 0 < s < t < T. In fact, if this is true, it follows

that

(t, s) -* S(t, s) = K(t) -'(K(t)S(t, s)K(s)-')K(s)

is strongly continuous in F, hence t - A (t)S(t, s)u is strongly continuous in

E. Since, by virtue of the arguments in Remark 7.7.9, equality (7.7.35) holds

everywhere in s < t < T, all the claims in Theorem 7.8.3 follow. The proof of

the strong continuity of (7.8.19) is fairly intricate and will be divided in

several steps for the sake of convenience. We bring back into the light the

operators Sn (t, s) used in Theorem 7.7.5 to construct the propagator S(t, s)

(see (7.7.21) and (7.7.22) and define

Qn(t, s) = K(t)S,, (t, s)K(s)

(7.8.20)

7.8.4

Lemma.

(a) Under the assumptions of Theorem 7.8.3, we have

IIQn(t,S)II<C(1+DNT)

e(w+CM+CDN)(t-s)+CDNT

(0 <s<t<T)

(7.8.21)

where C, w are the stability constants of A( ), M is a bound for II B(t )II in

0 < t < T as in (7.8.12), D is the constant in (7.8.17), and N is a bound for both

IIK(t)II(F,E) and K(t) -'11

(E, F)

in 0 < t < T as in (7.8.4). (b) Qn is strongly

convergent as n - oo, uniformly in 0 < s, t < T.

Proof. It has been established in Lemma 7.7.2 that if the family

A(.) is stable, then

{A(t)+B(t); 0 <t <T)

is stable if each B(t) is bounded and (7.7.12) holds; in the present situation,

this inequality is a consequence of the strong continuity of B(.) and of the

uniform boundedness theorem (see Section 1). (Recall that if C, w are the

stability constants of A(-), A(-)+

possesses stability constants C, W +

CM; since, on the other hand, the hypotheses on in Lemma 7.8.1 are

satisfied, it follows that the conclusions there hold.) We obtain from (7.8.11)

that

R(X; A(t))"u = K(t)-'R(X; A(t)+B(t))"K(t)u (u E F) (7.8.22)

for 0 < t < T, n > 1, and A sufficiently large; making use of the exponential

formula (2.1.27), we deduce that

S(s; A(t)) = K(t) -'S(s; A(t)+B(t))K(t)

442

The Abstract Cauchy Problem for Time-Dependent Equations

for s > 0, 0 < t < T. Let now 0 < s < t < T and let k, 1 be two integers that

stand in the relation to s and t described in the definition of S,,, that is,

k-1

T < s <

1-1 T<t<

T (7.8.23)

n n n

(see the comments around (7.7.21)). Then we have

Q.(t,s)

=K(t)K(1

n 1 T)

I S(t

1 n

I T A(1

n1T)+B(1 nIT))K(1 n1 T)

K T

T A

T +B T K T

t-i

(,nl

) In

(jn

j=k+,

l 1

\,nl

/1 l,nl /

K(k-1

T)-IS(kT-s;A(k__T)+ BJ

T))K(k-1 T)K(s)-i

n n n n n

(7.8.24)

if k < I (the product in the middle is taken equal to I if k = I - 1), and

Qn(s, t)

=K(t)K(kn 1 T)

I S(t-s;A(kn 1 T)+B(kn 1 T))K(kn 1 T)K(s)-1

(7.8.25)

when k = 1. We define now

L(t, s) = K(t)K(s)-t- I

=(K(t)-K(s))K(s)-' (0<s,t<T)

(7.8.26)

(note that the operator Lj in (7.8.7) is none other than L(tj, tj- I)) and

denote by C25

the operator constructed from A(-)+ in the same way Sn

was constructed from A(.); clearly

II(25n(t, s)II

<Ce("+cM)('-s)

We note that

(7.8.27)

=I+L(jn1T,jn2T

and replace this expression in the interstices of the product (7.8.24). We

develop then in products of the L(( j -1)T/n,(j -2)T/n) much in the

same way as (7.8.6), but ordering the sum this time by ascending powers.

7.8. Time-Dependent Symmetric Hyperbolic Systems in the Whole Space

443

After doing this we obtain

Qn(t,s)= {I+L(t,

with

R,, (t, s)

1-1

j=k+1

T,s)}

(7.8.28)

T)}Rn(t,s){I+L(k

n J

I T)L(> n 1

T's)

+ E

(25

nt'j-1T)L(j-IT j-2T

j=k+1

n '

jY-1

i-17)L(i-17 i-2T n(i-IT

i=k+1

n n n

(7.8.29)

where all the terms after C

disappear if k=1 (the total number of

summands in (7.8.29) is 1- k + 1).We can make the notation more orderly

as follows. Define inductively operators R n o (t, s ), R.n 1(t, s ), ...

in the

triangle 0 < s < t < T by Rn0 O =

Rn m(t,S)°

Snt,j-1 T)L(j

j=k+1

n n

It is then easy to see that we can write

)Rn,m-1(j

n I T,s

(7.8.30)

1-k

Rn(t,s)= E Rn,m(t,s).

(7.8.31)

M=0

(Note that if (t, s) is a fixed point in the triangle 0 < s < t < T and n - 00,

the number of summands in (7.8.31) will also tend to infinity.)

Observe next that, by virtue of (7.8.17),

hence

L(j

lT'j2T)I

<TDN,

(7.8.32)

n n

JIRn,1(t1 S)Il <

TC2DNe("+cM)(t-s)

C2DN(t - s +

e("+cM)(t-S)

(7.8.33)

Accordingly, each term in the sum (7.8.30) corresponding to m = 2 is

444

The Abstract Cauchy Problem for Time-Dependent Equations

bounded in norm by

C3(DN)2n ((7.8.34)

Hence

IIRn.2(t, s)II < C3(DN)2n

(nT-S)g(ca+CM)(t-s)

j=k+1

C3(

- s +

T)2e(W+CM)(t-s)

(7.8.35)

noting that

T j

t+2T/n

n+1(-T-s)<

(r-s)dr.

(See Figure 7.8.1.) Replacing the estimate into the expression for Rn3 3 and

proceeding inductively in a similar fashion, we obtain

IIRn,m(t, s)II<

Cm+1(DN)

- s+ m T)m

e+CM)(t-s) (7.8.36)

in0<s<t5T,(m=0,1,2,.... We deduce from (7.8.31) that

IIRn,m(t, s)II,

IIRn(t, s)II <

Ce("+CM+CDN)(t-s)+CDNT

(7.8.37)

The second inequality implies (7.8.21).

7.8. Time-Dependent Symmetric Hyperbolic Systems in the Whole Space 445

We prove now (b). As a first step, swe shall construct the "evolution

operator" C (t, s) of the equation u'(t) = (A(t)+ B(t))u(t) by a perturba-

tion series like the one used in Example 7.1.12 and prove that C5 - C5. It

should be pointed out that convergence of the C5 cannot be proved by

applying to A(.)+

the machinery of Theorem 7.7.5, since this family

does not fulfill the hypotheses therein (in particular, F may not be A(t)+

B(t)-admissible and, even if it were, t -* A(t)+B(t) may not be (F; E)-

continuous). Define to (t, s) = S(t, s),

C5m(t,s)u= ftS(t,a)B((,)CSm_1(a,s)uds (m>l).

It is rather easy to show inductively (see again Example 7.1.12) that each

is strongly continuous in 0 < s < t < T and that an estimate of the form

Cmm1!

(m,1)

(7.8.38)

holds there, the constants C, w, M with the same meaning as before. It

follows that the series

E (m(t,s)

M=1

(7.8.39)

is uniformly convergent in 0 < s < t < T in the norm of (E), hence its sum

C (t, s) is strongly continuous there and satisfies

(t, s)II

<Ce("+MC)(t-s).

(7.8.40)

The next step is to show that

(25,, (t, s) - C (t, s) (7.8.41)

strongly, uniformly in the triangle 0 < s < t < T. This is done as follows. We

observe first that the operators C5 constructed from the "discretized"

family A,,(.)+ B

can be obtained from the S -similarly constructed

from

means of a perturbation series in the same way C was

obtained from S; precisely, if we set CS,,,0(t, s) = Sn(t, s),

CS,,,m(t,s)u= (m, l),

(7.8.42)

then

Y_ (25n,m(t, s) = CSn(t, s) (7.8.43)

M=0

strongly, uniformly in 0 < s < t < T. (The proof of this relation is a some-

(7.8.43) can be heuristically justified as follows: S, (t, s) and the function

on the left hand side satisfy the same differential equation in s < t <T and the same initial

condition at t = s.

446

The Abstract Cauchy Problem for Time-Dependent Equations

what tedious exercise involving the explicit expression (7.7.21) for S,,, the

analogous formula for C

and the perturbation argument used in Theorem

5.1.2 to construct S(t; A + B) from S(t; A) in the particular case where B is

bounded; we omit the details.) We compare now the two series (7.8.39) and

(7.8.43) term by term, beginning with the observation that the approxima-

tions to C25

furnished by (7.8.42) satisfy the same inequality (7.8.38) obeyed

by the t m; given then e > 0, we can choose an integer mo so large that

I1tm(t,s)-(25n,m(t,s)II <e/2

(7.8.44)

m=m0+I

in 0 < s < t < T, this estimate being independent of n. We note next that,

since t o = S, (tI, t

2, ...

are all strongly continuous, given u E E fixed,

each of the sets

{ `G' m(t,s)u; 0 <s <t <T}

is compact in E. We bring into play the obvious estimate

IISS(t,s)BB(s)II <CMe"(`s) (0<s<t<T)

valid for all n, and the fact that SS(t, s)BB(s) - S(t, s)B(s) strongly,

uniformly in the same region: an elementary compactness argument then

shows that, given S > 0, we can pick n o so large that

II(S(t,a)B((Y)-Sn(t,a)Bn(a))tm(a,s)ull <S

in 0 < s < a < t < T for n > no and 0 < m < m

and we can even include the

inequality

IIS(a,S)u-Sn(a,s)uII <S

in the same range of s, a, n, by eventual augmentation of no in view of the

fact (proved in Theorem 7.7.5) that Sn - S strongly, uniformly in 0 < s <

a < T. We have now

(t,s)u-C25nj(t,s)u=

tSn(t, (Y)Bn(a)(S(a, s)u

- S(a, s)u) da,

(7.8.45)

so that, in view of all the previous precautions,

IItJ t,S)u-G'n ](t,S)uII <T(1+CMe"T)S.

Replacing this inequality into the inductive successor of (7.8.45) and repeat-

ing the trick, we obtain

Iltm(t,s)u-' n,m(t,s)ull <e/2m0

(7.8.46)

in 0 < s < t < T for n >, n o, 1 < m < m 0 if 8 is chosen adequately small.