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

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7.8. Time-Dependent Symmetric Hyperbolic Systems in the Whole Space 447

Putting this inequality together with (7.8.44) and taking advantage of the

arbitrariness of t-, (7.8.41) follows.

With that relation in our bag, we go back to (7.8.30) and (7.8.31).

The next step in the proof of part (b) of Lemma 7.8.4 will be to show that

each R,,,,n has a (strong) limit as n -* oo; precisely,

R,,,,n(t, s)u

s)u (7.8.47)

for each u E E, uniformly in 0 < s < t < T, the operators Rm defined induc-

tively by the formulas A. (t, s) = S (t, S),

R,n(t,s)u= fStS(t,o)K'(o)K(a)-'Rm_j(o,s)udo (m>,1).

(7.8.48)

This is quite obvious when m = 0, since Rn,0 = C5,,. Assume then that

(7.8.47) holds for all indices < m - 1; let u E E and consider the following

function in the interval [0, T]:

8n(a)

25t,

(L:± T)K'(a)K(j-

n n

Rn,m-1

J-2T<a<--1

for j=k+1,...,1,

n n

g(a) = 0 elsewhere. Taking into account that

f(;-

)T/n

K(a)K(J-2)

uda=K(JT)K

l(y-2

(j-2)T/n

n \ n n

L(i-1 T j-2T)u,

n n

it is obvious that

u- u

gn(a)do.

(7.8.49)

We use now the convergence relation (7.8.41), (7.8.47) (for indices < m -1)

and the uniform estimates (7.8.27) and (7.8.37) to deduce that 9n( o ) -*

C5 (t,a)K'(a)K(a) Rm_t(a,s)u in 0<a<T with

Ilgn(a)II

uniformly

bounded there, thus we can take limits in (7.8.49) using (a vector-valued

analogue) of the dominated convergence theorem and obtain (7.8.47).

Making use of (7.8.40) and of the recursive formula (7.8.48), we

easily obtain the estimate

IIRm(t,s)II <

Cm+t(HN)m

(t -s)me(w+MC)(r-s)

(m 31)

(7.8.50)

in 0 < s < t < T, where H (resp. N) is a bound for I I K'(o )I I

(F; E) (resp.

for

448

The Abstract Cauchy Problem for Time-Dependent Equations

II K(a)-'II(E F) in 0 < a < T. Combining this with (7.8.37) and with the

convergence relation (7.8.47) obtained a moment ago, we obtain that

t-k

R,, (t, s) = F, Rn,m(t,s)-*R(t,s)

M=0

Rm(t, s)

(n -oo)

(7.8.51)

m=0

strongly, uniformly in 0 < s < t < T. The final step takes advantage of

(7.8.28) and of the estimate (7.8.32): the result is

Qn(t,s)- R(t,s)

(n- oo)

(7.8.52)

in the same sense as in (7.8.5 1).

We summarize the different steps in their logical order; arrows

indicate strong convergence, uniform in 0 < s < t < T, as n - oo.

(1) Sn(t,S)

("evolution operator"

(7.8.41)

and

of An(t)+Bn(t) de- comments

fined before (7.8.27))

(2) Rn.m(t,S)

(defined in (7.8.30))

(3) Rn(t, S)

I-k

Y_ R,,,m(t,s)

M=O

(t,s)

following

("evolution operator" of

(7.8.47)

and

comments

following

(7.8.51) and preceding

comments

A(t)+B(t) defined in

(7.8.39))

Rm(t, S)

(defined in (7.8.48))

R(t,s)= E km(t, s)

M=0

(4) Qn(t,s)

R(t,s)

(3) above and (7.8.28)

End of proof of Theorem 7.8.3.

As pointed out at the beginning of

the proof

(see

(7.8.19)),

we only

have

to show that Q(t, s) =

K(t)S(t, s)K(s)-' is everywhere defined and (E)-strongly continuous in

0 < s < t < T. This is done as follows. Pick u e E. Since Qn (t, s) =

K(t)S,,(t, s)K(s)-'u converges in E by Lemma 7.8.4(b), we obtain pre-mul-

tiplying by K(t)-' that Sn(t, s)K(s)-'u is convergent in F, a fortiori in E;

since

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

in E, it follows that

S,,(t, s)K(s)-'u - S(t, s)K(s)-'u in F, thus S(t, s)K(s)-'u E F and

K(t)SS(t, s)K(s) -'u - K(t)S(t, s)K(s)-'u

in E, the convergence being uniform in 0 < s < t < T by virtue of Lemma

7.8.4(b). Since each K(t)Sn(t, s)K(s)-'u is continuous in 0 < s < t < T, our

7.8. Time-Dependent Symmetric Hyperbolic Systems in the Whole Space 449

claims on K(t)S(t, s)K(s)-' follow and the proof of Theorem 7.8.3 is

complete.

The manipulations in the proof of Theorem 7.8.3 are perhaps too

intricate to let the basic idea of the argument shine through. An heuristic

justification of why we should expect K(t)S(t, s)K(s)-' to be strongly

continuous is this. Differentiating formally and using (7.8.15), we obtain

D,(K(t)S(t, s)K(s) ') = K(t)A(t)S(t, s) K(s) '

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

_ (A(t)+B(t)+K'(t)K(t)-')(K(t)S(t,s)K(s)-'),

whereas K(t)S(t, t)K(t)-' = I, thus K(t)S(t, s)K(s)-' should be the

propagator of

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

and must then be everywhere defined and strongly continuous. Naturally, it

would be hard to try to establish this along the lines of Theorem 7.7.5, since

the hypotheses there are far from being satisfied

for B( )+

We close this section with the announced application of Theorem

7.8.3 to the time-dependent symmetric hyperbolic operator (7.8.2) (and to

the associated differential equation (7.8.1)).

7.8.5 Theorem.

Let the matrices A,(x, t),...,Am(x, t), B(x, t) be

defined and continuously differentiable with respect to x1,.

.. , x,,, in 118

m X [O, T].

Assume all the functions

IAk(x,t)I,ID'Ak(x,t)I,IB(x,t)I,ID'B(x,t)I

(1Sj,kSm)

(7.8.53)

are bounded in R' X [O, TI. Assume, moreover, that Ak (x, t), DiAk (x, t), and

B(x, t) are continuous in t for 0 < t < T, uniformly with respect to x E 118'.

Then the Cauchy problem for (7.8.1) is properly posed in both senses of time in

0 < t <T (with respect to solutions defined by (D)), with F= H'(Rm)P.

Proof. Fortunately, most of the work needed to verify the assump-

tions of Theorem 7.8.3 in this case was already done in Section 5.6 in for the

stationary equation. Continuity of the map (7.8.13) is immediate (we only

use here the hypotheses on Ak(x, t), B(x, t), not those on the partial

derivatives). The role of the operator function K(t) is played by the

constant function

Using the results of Section 5.6, we see that

Rd (t)A-' = Ca(t)+Q(t)

(0 <t <T),

(7.8.54)

450

The Abstract Cauchy Problem for Time-Dependent Equations

where Q(t)=Q1(t)+Q2(t), both operators being bounded for each t (see

their definition in (5.6.36) and (5.6.38). Our only task is then to show that

QI (' ), Q2(') are strongly continuous. If u E 5°,

(Q2(t')-Q2(t))U- I

a-1 - C-' Dkq-IDk)(B(

kL=1

(7.8.55)

The second term on the right-hand side obviously tends to zero as t' - t

uniformly with respect to u bounded in the L2 norm. As for the first, taking

into account that

tends to zero pointwise in x E R m while remaining uniformly bounded

there, we see that it tends to zero also (although of course not uniformly

with respect to u). On the other hand, it follows from boundedness of B and

DjB that IIQ2(')II is uniformly bounded in 0 < t <T, thus

is strongly

continuous. Now,

(Q1(t')-Q1(t))u= E

k=1

Since I DJAk(x, t')- DJAk(x, t)I -0 as t'- t -* 0, uniformly for x E Rm, it

follows from Theorem 5.6.6 that QI is actually continuous as a (L2)-valued

function, hence strongly continuous. It follows then that Q 1(. )+

is strongly continuous and all the hypotheses of Theorem 7.8.3 are

fulfilled. Since these hypotheses are also satisfied

for

the

family

(- Q (T - t); 0 < t < T) (which obeys the same assumptions as 9 (-)), the

claims of Theorem 7.8.5 follow.

A time-dependent counterpart of Theorem 5.6.7 can be easily proved.

7.8.6

Theorem.

Let the matrices AI(x, t),...,Am(x, t), B(x, t) be r

times continuously differentiable in R' X [0, T j with respect to x1,. .. , xm and

assume all the functions

ID"Ak(x,t)I, ID"B(x,t)I

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

(7.8.56)

are bounded in R' X [0, T ]. Suppose, moreover, that the D "A(x, t) (I a I <r)

and the DOB(x, t) (I# I < r - 1) are continuous in t for 0 < t < T, uniformly

with respect to x E R m. Then the conclusion of Theorem 7.8.5 holds; moreover,

S(t,s)HrcHr (0<s,t<T),

each S(t, r) is a bounded operator in Hr and (t, s) - S(t, r) is continuous

from 0 <s, t <T into Hr. In particular, a solution u(t) that starts in Hr

(u(s) E Hr) remains in Hr and satisfies

IIU(t)IIHr<CrIIU(S)IIH' (0<s,t<T)

for a suitable constant Cr independent of u.

7.9. The Case Where D(A(t)) is Independent of t. The Inhomogeneous Equation

451

The proof can be carried out modifying that of Theorem 7.8.6 along

the lines of Theorem 5.6.7 and is therefore omitted.

7.9. THE CASE WHERE D(A(t)) IS INDEPENDENT OF t.

THE INHOMOGENEOUS EQUATION

The following result is an important particular case of Theorem 7.8.3.

7.9.1 Theorem.

Let (A(t); 0 < t <T) be a stable family of densely

defined operators in E with stability constants C, w. Assume that

F= D(A(t))

(7.9.1)

does not depend on t, and that, for some X > w, the function

t - A(t)R(X; A(0))

(7.9.2)

is strongly continuously differentiable in 0 < t < T. Then the conclusions of

Theorem 7.8.3 hold for the equation

u'(t) = A(t)u(t).

(7.9.3)

Proof.

Let F= D(A(t)), and let 11.11

be the graph norm of A(t) in F

or, even better, the equivalent norm

Ilullr=ll(XI-A(t))ull.

(7.9.4)

Plainly F is a Banach space endowed with any of the IIWe show that all

these norms are equivalent (even if no assumptions are made on the map

(7.9.2)). Let s, t E [0, TJ and

a sequence in E with u -> u,

A(t)R(A; A(s))u - v. Since R(X; A(s))u - R(X; A(s))u and A(t) is

closed, it follows that A(t)R(A; A(s))u = v, thus A(t)R(X; A(s)) is itself

closed; since it is everywhere defined, it results from the closed graph

theorem that A(t) R (X; A(s )) is bounded. Then, if u E F and w =

(XI - A(s))u,

(lull, =

li(XI- A(t))ull

= Il(XI -

A(t))R(X; A(s))wll

<Cll wll = Cll(XI - A(s))ull = Cllulls,

which justifies our claim.

We check the other hypotheses. It is obvious that F -> E. Using the

fact that (7.9.2) is strongly continuously differentiable, we apply the uni-

form boundedness theorem to the family

((t'- t){A(t')R(X; A(0))-A(t)R(X; A(O))); t * t')

obtaining

IIA(t')R(X;A(O))-A(t)R(X;A(0))II<Cit'-tI

452

The Abstract Cauchy Problem for Time-Dependent Equations

and this shows that t -* A(t) E (F; E) is continuous (in fact, Lipschitz

continuous). Define

K(t)=JAI-A(t).

(7.9.5)

Then IIK(t)ull=llulltfor uEF,IIK(t)-'vllt=llvllfor vEE,so that K(t)E

(F, E) and K(t) -' E (E, F). Finally, it is obvious that (7.8.15) holds with

B(t) = 0 and that the hypothesis on (7.9.2) implies that K(t) u is continu-

ously differentiable if u E F = R(X; A(0))E. The assumptions of Theorem

7.8.3 are thus verified in full. This completes the proof.

Going back to the general set-up of Sections 7.7 and 7.8 we consider

the inhomogeneous equation

u'(t) = A(t)u(t)+ f(t). (7.9.6)

If F, satisfy the hypotheses of Theorem 7.7.5 and f is a E-valued

continuous function in 0 < t < T, every solution of (7.9.6) in the sense of

Definition (D) must perforce be of the form

u(t)=S(t,s)u(s)+jtS(t,s)f(s)ds (s<t<T).

To see this, take v(a) = S(t, a) u(o), s < a < t and operate in a way similar

to that used to show uniqueness for the homogeneous equation (see the end

of the proof of Theorem 7.7.5 and Remark 7.7.8); the result is the equality

D+v(a)=S(t,a)f(a) (s<a<t).

It is then an easy consequence of Lemma 7.7.7 that

u(t)=S(t,s)u(s)+ jtS(t,o)f(a)do;

(7.9.7)

(it should be noted that this argument depends only on s-differentiability of

S(t, s) and thus holds even if F is not reflexive.) Conversely, if the

assumptions on f are strengthened, (7.9.7) will be a solution of (7.9.6). We

limit ourselves to proving two results, which can be considered to be

"dynamic" counterparts of parts (a) and (b) of Lemma 2.4.2.

7.9.2 Lemma.

Let the assumptions of Theorem 7.8.3 hold, and let

f (t) be F-valued and F-continuous in 0 < t < T. Then

u(t) = ftS(t,o)f(a)do (7.9.8)

is a genuine solution of (7.9.6) ins < t < T (in the sense of Definition (D).8

Proof. Under the present assumptions S(t, s) is F-strongly continu-

ous in 0 < s < t < T, thus

(t, s)-A(t)S(t,s)f(s,'

8See footnote 5, p. 440.

7.9. The Case Where D(A(t)) is Independent of t. The Inhomogeneous Equation

453

is E-continuous there. This justifies the following computation:

f A(r)u(r) dr = f t{ f rA(r)S(r, o) f(o) dal dr

s ` s

=f'( f'A(r)S(r, o) f(a) dr } do

= ft{ f'DrS(r,o)f(a)dr} do

f'S(t,a)f(a)do- ftf(a)do

(7.9.9)

and shows that is a solution of (7.9.6) in the sense claimed above. This

ends the proof.

Other versions of Lemma 7.9.2 can be proved under different hy-

potheses. We limit ourselves to the following result:

7.9.3 Example.

Let the assumptions of Theorem 7.7.7 be satisfied, and let

f be strongly measurable and integrable (as a F-valued function) in 0 < t < T.

Then u(t) is continuous in E, u(t) E F almost everywhere, t - A(t)u(t) is

strongly measurable and integrable in E and

u(t) = f A(r)u(r) dr+ f tf(r) dr

(s <t <T).

(7.9.10)

The proof follows that of Lemma 7.9.2 almost verbatim; here we use again

the fact that weak continuity implies strong measurability to show that S is

F-strongly measurable in 0 < s < t < T; therefore, A(t) S(t, a) f (a) is E-

strongly measurable there (as well as strongly measurable separately in each

variable) justifying the calculations.

7.9.4

Lemma.

Let the assumptions of Theorem 7.9.1 be satisfied,

and let f be an E-valued continuously differentiable function in 0 < t < T. Then

u(.), given by (7.9.8) is a genuine solution of (7.9.6) in s < t < T.

Proof.

Replacing if necessary

by A(.)- aI for a sufficiently

large, we may assume the stability constant w is negative, and thus set

K(t) = A(t). We have

(A(t)-'f(t))'= -

A(t)-'A'(t)A(t) (t)+A(t)-'f'(t)

= A(t)-'g(t), (7.9.11)

where g(t)

A'(t)A(t)-'f(t)+ f'(t) is E-continuous. Hence we have

DQ(S(t,a)A(a)-'f(a))=-S(t,a)f((,)+S(t,a)A(a)-'g(a),

(7.9.12)

where we have used the properties on a-differentiability of S(t, a) proved in

454

The Abstract Cauchy Problem for Time-Dependent Equations

Theorem 7.7.5 under much weaker hypotheses. We integrate now (7.9.12) in

s < a < t, obtaining

u(t) = - A(t)-'f(t)+S(t, s)A(s)-'f(s)+jtS(t, a)A(a)-'g(a) da

=-A(t)-'f(t)+v(t). (7.9.13)

Since A(s)- 'f (s) E F and g(.) is (E)-continuous,

is F-continu-

ous, and by virtue of Lemma 7.9.2, v(.) is a solution of

v'(t) = A(t)v(t)+A(t)_'g(t)

in s S t 5 T. Then

u'(t)=-(A(t)-'f(t))'+v'(t)

=A(t)v(t)

=A(t)u(t)+f(t)

as claimed.

(7.9.14)

We close this section with a few results on the homogeneous equation

(7.7.1) (the proofs are omitted). In the first one (Example 7.9.4 below) the

conclusions of the theorems in Sections 7.7 and 7.8 are essentially un-

changed, but the assumptions are considerably weaker. The first task is to

modify in a convenient way the notion of stability.

A family (A(t); 0 < t < T) of densely defined operators in E is called

quasi-stable if there exists C > 0 and a function w(.), upper Lebesgue

integrable in 0 5 t < T, such that R(A; A(t)) exists for A > w(t) and

fl R(A1; A(te))

<Cfl (x1-w(ti))

(7.9.15)

i=1

forall(tj),(Aj)such that 01< tl1< tz1< <tn<T,and X,>w(tl),...Xn>

w(tn). The product on the left-hand side of (7.9.15) is ordered as in the

definition of stability.

*7.9.5 Example (Kato [1973: 1]). Let E, F be Banach spaces, F -*

E,(A(t); 0 S t < T) a quasi-stable family of operators in E such that

F c D(A(t )) and A(t) : F - E is bounded; moreover, if A(t) is the restric-

tion of A(t) to F, the map

t -* A(t) E (F; E)

(7.9.16)

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

invertible operator from F onto E such that

t-*K(t)E(F,E) (7.9.17)

is strongly continuous, and there exists another strongly measurable func-

tion t - K(t) E (F; E) such that

IIK(t)Ilsa(t) (OSt<T)

7.10. Miscellaneous Comments

455

(f3 integrable in 0 < t < T),

K(t)u= ftk(s)uds+K(0)u (0<t<T,u(=-F)

(7.9.18)

and

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

(7.9.19)

where B(.) is a strongly measurable (E)-valued function such that

IIB(t)II<y(t)

(0<t<T),

(7.9.20)

y integrable in 0 < t < T. Then the conclusions of Theorem 7.8.3 hold in full

for the equation

u'(t) = A(t)u(t). (7.9.21)

In other words, the Cauchy problem for (7.9.19) is properly posed in

0 < t < T with respect to solutions defined in (D).

The proof runs along the same lines as that of Theorem 7.8.3, the

main difference being that the sequence of partitions (tk = (k -1)T/n) used

in the construction of the approximating operator S must now be replaced

by a suitable sequence of uneven partitions.

Even more general results can be obtained if one is willing to accept

less stringent definitions of solution; in fact, we have

7.9.6

Example (Kato [1973: 1]).

Let the assumptions in Example 7.9.6

hold, except in that the map (7.9.16) is only assumed to be strongly

measurable (in the topology of (F; E)) and

1 II A (t )I I ( F; E)dt < 00'

(7.9.22)

Then the Cauchy problem for (7.9.22) is properly posed with respect to the

following definition of solution:

(E) asolution of (7.9.21)in s<t<Tif and only if u(t)EFa.e,

A(t) u(t) is strongly measurable and integrable in s < t < T, and

u(t) =

f`A(T)u(T)dT+u(s)

(s<t<T).

7.10. MISCELLANEOUS COMMENTS

The notion of properly posed abstract Cauchy problem was introduced by

Lax (see Section 1.7) including the time-dependent setting treated here.

Operators S(t, s) satisfying (7.1.7) and (7.1.8) were called propagators by

Segal [1963:

1]; other names in use are evolution operators or solution

operators, when arising from a differential equation like (7.1.1). This equa-

tion is usually called an evolution equation; curiously, the name seems not to

be used for its time-invariant counterpart (2.1.1).

456

The Abstract Cauchy Problem for Time-Dependent Equations

The facts on the equation (7.1.1) pertaining to the case A(t) bounded

are all well known (and rather natural generalizations of even better known

results for finite-dimensional vector differential equations). For deep in-

vestigations of the equation (7.1.1) in this setting see the treatises of

Massera-Schaffer [1966: 1] and Daleckii-Krein [1970: 1]. Of special interest

is the result in Example 7.1.8. Some manipulations with formula (7.1.31)

and the definition of the exponential function show that the propagator

S(t, s) can be obtained in the form

m(n)

S(t, s) = lim [1 (I+ (tn,j - tn,j_1)A(t (7.10.1)

j=1

In the finite-dimensional case, this limit was taken by Volterra [1887: 1],

[1902: 1] as the definition of the product integral of the matrix function A(-)

in the interval s < r S t (Volterra's name for (7.10.1) is different). The

product integral is denoted in various ways, for instance

r r

`(I+ A(r) dr),

11(I+ A(r) dr), He a(r) it

(7.10.2)

s s

and coincides with

exp(1 IA (r) dr)

when the different values of A(.) commute. A calculus of matrix-valued

functions based on this integral and on a corresponding product derivative

was developed by Volterra, Schlesinger, Rasch, and others. (We note that

(7.10.2) is the left integral of Volterra; the right integral is defined inverting

the order of the factors and is used to construct the propagator or the

equation S' = SA.) For a detailed exposition of the calculus see Volterra-

Hostinsky [1938:

11. During the fifties, a revival of interest took place,

motivated on the one hand by quantum mechanical applications (see

Feynman [1951: 1]) and on the other hand by the search for generalization

of representation results for matrix-valued analytic functions of the type of

Nevanlinna's theorem; see, for instance, Ginzburg [1967: 1]. The definition

of product integral extends in an obvious way to continuous operator-val-

ued functions in infinite dimensional spaces. See Daleckii-Krein [1970: 1,

Ch. III] for a brief account. A thorough exposition with many applications

can be found in the recent monograph of Dollard and Friedman [1979: 11,

where several generalizations are presented as well.

The material in Sections 7.2, 7.3, and 7.5 is entirely due to Kato and

Tanabe [1962: 11; we have followed closely the original exposition incorpo-

rating some simplifications in the estimates in Corollary 7.2.4 due to Krein

[1967: 1]. We note, however, that the original estimates of Kato and Tanabe

are more precise. Section 7.4 is due to Kato-Tanabe [1967: 1]; some of the