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

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1.7. Miscellaneous Notes 57

problems connected with V 2 u = 0 is formulated analytically in Cauchy's

way. All of them lead to statements such as Dirichlet's, i.e. with only one

numerical datum at every point of the boundary. Such is the case with the

equation of heat. All this agrees with the fact that Cauchy's data, if not

analytic, do not determine any solution of any one of these two equations.

"This remarkable agreement between the two points of view appears

to me as an evidence that the attitude which we adopted above-that is,

making a rule not to assume analyticity of data-agrees better with the true

and inner nature of things that Cauchy's and his successors' previous

conception."2

The first result of the type of Theorem 1.7.1 was given by Banach

[1932: 1 p. 44] as an application of his inverse function-closed graph

theorem; the result there is not restricted to Cauchy problems, however.

Many variants have been proved using the same basic idea: we note the

following result, due to Lax [1957: 2], where the phenomenon of finite

domains of dependence is deduced from mere existence and uniqueness of

solutions of (1.7.1), (1.7.2) for q) having arbitrary growth at infinity.

1.7.2

Theorem. Let Xm > 0. Assume a unique infinitely differentia-

ble solution u of (1.7.1), (1.7.2) exists in IxmI < X,,, for every qv infinitely

differentiable in fm-'. Then, given a > 0 and an integer M > 0, there exists

b, C > 0 and an integer N > 0 (all three depending on a, M) such that

IIuIIM,a <CIITIIN,b.

(1.7.6)

The proof is essentially the same as that of Theorem 1.7.1; this time

the space of Cauchy data is C(m)(Rm-1) consisting of all T infinitely dif-

ferentiable in Rm-' endowed with a metric of the type of (1.7.4). Inequality

(1.7.6) implies that the solution u in x

+ x,

, _ 1 < a 2,

IxmI <

depends only on the restriction of the initial datum to xi +

+X2._

1 <

b2.

An abstract version of

the Cauchy problem incorporating

Hadamard's basic ideas could be formulated along the lines of Section 1.2

by looking at the differential operator on the right-hand side of (1.7.1) as a

x,,; dependent linear operator in C(°°)(Rm-1) or 6l. (IRm-') with values in

C(°°)(Rm-'); setting xm = t, F= C(°°)(Pm-'), and E = 6l. (K'

C(O0)(Rm-'), depending on whether the setting is that of Theorems 1.7.1 or

1.7.2, we are led to the following abstract Cauchy problem in the sense of

Hadamard: given two linear topological spaces E, F with E C F (but where

the topology of E is not necessarily that inherited from F) and a family of

operators (A(t ); 0 < t < T) with domains D(A(t)), the Cauchy problem for

u'(t)=A(t)u(t) (0<t<T) (1.7.7)

is properly posed in the sense of Hadamard if solutions (defined in an

2From 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

obvious way) exist for initial data u(0) in a dense set D C E and depend

continuously on them, in the sense that lim u., (t) = 0 in F, uniformly in

0 < t < T whenever (the generalized sequence) (u., (0)) converges to zero in E

(note that in the examples considered above D(A(t)) = E and A(t) : E -* F

is continuous). Different abstract schemes of this type, corresponding to

different definitions of solution could be devised. A setting where E and F

are Banach spaces is the following: E is the Banach space BC(N)(Rm-') of

functions with bounded and continuous partial derivatives of order < N

(equipped with the norm 1 1.1 1 N, the supremum of all these derivatives in

Rm-') and F = BC(M)(R'-'). This last version, however, is probably not

very faithful to Hadamard's formulation since it incorporates a priori

restrictions on the behavior of the solutions at infinity (implicit in their

membership in BC(M)(R`) for all t), which are wholly absent from his

original definition of correctly set problem; however, these restrictions at

infinity allow us to treat as well characteristic Cauchy problems (such as the

usual initial value problem for the heat equation) for which the local

schemes in Theorems (1.7.1) and (1.7.2) are inadequate, since solutions may

not exist or uniqueness may fail (see Hormander [1961: 1, p. 121]).

The abstract Cauchy problem in the sense described above does not

seem to have been studied in full generality; for the case where E and F are

Banach spaces, see Krein-Laptev-Cvetkova [ 1970: 11, where T = oo and the

problem is understood in a somewhat different sense (see also Sova [1977:

1]). On the other hand, there exists a vast literature on properly posed

"concrete" Cauchy problems for partial differential equations, undoubtedly

one of the central questions in the theory: see, for instance, Petrovskii [1938:

1], A. Lax [1956: 11, Lax [1957: 2], Hormander [1955: 1] and [1969: 11

(additional references can be found there), Strang [1966: 1], [1967: 1], [1969:

1],

Flaschka-Strang [1971:

1],

Gelfand-Silov [1958: 3], and Friedman

[1963: 1].

The abstract Cauchy problem for differential equations in Banach

spaces was introduced by Hille [ 1952: 3] (see also [ 1953: 2], [195 3: 3], [1954:

1], [1954: 2], [1957: 1], Phillips [1954: 2]). It should be pointed out that

Hille's prescription for a well-set Cauchy problem is somewhat different

from that in Section 1.2 not only in the minor technical matter of definition

of solution but in that continuous dependence is not assumed; the matter of

interest is that solutions (that satisfy a certain growth condition at infinity)

should be unique. That continuous dependence follows from existence and

uniqueness in certain cases can be seen much in the same way as in

Theorems 1.7.1 and 1.7.2, as the following result (which is due to Phillips

[ 1954: 2] in a slightly more general formulation) shows.

1.7.3 Theorem.

Let A be a densely defined operator in the Banach

space E such that p(A) * 0. Assume that for every u E D(A) there exists a

unique solution of

u'(t) = Au(t) (t >_ 0) (1.7.8)

1.7. Miscellaneous Notes

in the sense of Section 1.2 with u(O) = u. Then the Cauchy problem for (1.7.8)

is properly posed in the sense of Section 1.2.

Proof.

Let T> 0, C(') = 0)(0, T; E) the Banach space of all E-val-

ued continuously differentiable defined in 0 < t < T endowed with the

norm I I u(. )I I i = supl I u(t )I I + supl I u'(t )I I (the suprema taken in 0 < t < T ).

Consider D(A) equipped with (the norm equivalent to) its graph norm

II(X I - A) ull for some A E p(A) and the operator CS : D(A) -* C(') defined

by CS u = u(-), u(.) the solution of (1.7.8) with u(0) = u. If u -> u in D(A)

and e v(.) in CO), it is easy to see taking limits in (1.7.8), and using

the fact that A is closed that v(-) is a (then the only) solution of (1.7.8) with

v(0) = u; it follows that CS is closed, hence bounded by the closed graph

theorem. Given u E D(A), it is plain that v(.) = R (A) C5 u is a (then the

only) solution of v'(t) = Av(t) with v(O)=R(X)u, so that SR(X)u=

R(X)CSu. Hence we have, for u(.)=CSu,

supllu(t)II = supll(AI- A)CSR(A)uII

<C'Ilv(-)II<

<CII(AI- A)R(X)uII = CIIull.

Since T is arbitrary, the result follows.

The definition of properly posed abstract Cauchy problem given in

Section 1.2 was introduced by Lax in a 1954 New York University seminar

(where A was allowed to depend on time). The time-invariant version was

published in Lax-Richtmyer [1956:

1]. Although also inspired by Hada-

mard's basic idea, this formulation is less general since solutions and initial

data are measured in the same norm. Yet, it seems to be ample enough

(together with its time-dependent counterpart, to be examined in Chapter 7)

to model linear physical systems where a "measure of the state" such as

energy, amount of diffusing matter, or temperature, remains bounded or at

least does not become infinite in finite time; ideally, the norm of the space

E should be this measure (although the choice of norm is many times

dictated by mere mathematical convenience, as is the L2 norm in the

treatment of the heat equation). However, not all physical systems fit into

this scheme; see Birkhoff [1964: 1, p. 312]. The term propagator was used to

denote the operator S(t) by Segal [1963: 1].

An interesting problem in this connection is: given a system (or class

of systems) of partial differential equations, we seek functional norms that

allow the Cauchy problem to fit in the scheme of Section 1.2, or, alterna-

tively, we look for conditions on the system that permit the use of a given

functional norm like the L2 or the C norms. For results in both directions

see Birkhoff-Mullikin [1958: 1], Birkhoff [1964: 1], Lax-Richtmyer [1956: 1]

Richtmyer-Morton [1967:

1], Kreiss [1959:

1], [1962: 11, [1963:

11, and

Strang [1969: 1], [1969: 2], [1969: 3], [1969: 4]. See also L. Schwartz [1950:

1], where the "state spaces" are spaces of distributions. We note that the

surge of interest in the theory of the abstract Cauchy problem during and

60 The Cauchy Problem for Some Equations of Mathematical Physics

after the fifties was motivated in part by the rapid development of finite-

difference methods (to approximate solutions of differential equations)

made practical by the availability of high-speed computers.

The notion of properly posed problem in Section 1.2 may of course

be modified in many ways: solutions may be defined in suitably weak

senses, the function in the a priori estimate (1.2.3) can be assumed only

finite, and so on. Different properties of the propagator result. Some of

these variants will be examined in Section 2.5.

In relation to the examples in Sections 1.4 and 1.6, we note that the

Schrodinger, Maxwell, and Dirac operators are skew-adjoint (that is, i times

a self-adjoint operator). This was implicitly known when the corresponding

equations were introduced in physics, since the conservation of norm

property for the solutions (required on physical grounds) is a consequence

of skew-adjointness, as we have seen in the corresponding examples. An

abstract formulation of this argument is:

1.7.4 Example.

Let A be a self-adjoint operator in the Hilbert space H.

Then the Cauchy problem for

u'(t) = iAu(t) (1.7.9)

is well posed in - oo < t < oo and, for any solution u(.) we have

Ilu(t)II = Ilu(0)II

(1.7.10)

The proof is essentially the same as that for, say, the Schrodinger

operator, the role of the Fourier transform being taken over by the spectral

integral and the functional calculus.

We should mention, however, that the rigorous proof of skew-

adjointness for the Maxwell, Schrodinger, and Dirac operators is rather

recent: for the last two see Kato [1951: 1], [1951: 21.

In the matter of Example 1.7.4, we point out that the converse is as

well true (see Theorem 3.6.7) with a partial converse holding in the case

where the Cauchy problem for (1.7.9) is well posed in t >, 0 and (1.7.10)

holds there (Theorem 3.6.6). Equation (1.7.9) is sometimes called the

abstract Schrodinger equation (see J. L. B. Cooper [1948: 1]).

Example 1.6.1 generalizes an earlier result of Littman [1963:

11,

where the nonexistence of LP estimates for the wave equation (p $ 2) is

deduced from direct estimation of the explicit solution. For LP estimates for

symmetric hyperbolic systems, not necessarily with constant coefficients (as

well as for estimates in other functional norms), see Brenner [1972:

1],

[1973:

1], [1975:

1], [1977:

1], and Brenner-Thomee-Wahlbin [1975:

11;

estimates for the Schrodinger and other equations are in Da Prato-Giusti

[ 1967: 1], [1967: 2], Lanconelli [ 1968: 1], [1968: 2], [1970: 11, and Marshall-

Strauss-Wainger [1980: 1]. Most lie deeper than Theorem 1.6.3.

Example 1.2.2 deserves some comment. As pointed out there, it does

not apply to any nontrivial differential operator A in the usual Banach

1.7. Miscellaneous Notes 61

spaces of classical analysis. However, many differential operators are con-

tinuous in certain linear topological spaces (such as the spaces of Schwartz

test functions 6D, S or the distribution spaces 6D', S') hence we may ask

whether (1.2.10) makes sense in these conditions. The answer is in the

negative, as the series is in general divergent.

We note finally that it is possible in some cases to deduce that a

Cauchy problem is properly posed in the sense of Hadamard from the fact

that it is properly posed in the sense of Section 1.2; this is usually done

using "comparison-of-norms" results like the Sobolev embedding theorems.

An example of this sort of reasoning is that following Theorem 1.6.3,

although estimates in the HS norms involve of course growth conditions at

infinity. Estimates in local norms could be obtained, however, by using

domain of dependence arguments.

Chapter 2

Properly Posed Cauchy Problems:

General Theory

We continue in this chapter our examination of the Cauchy problem

for the equation u'(t) = Au(t) initiated in Sections 1.2 and 1.5. The

main result is Theorem 2.1.1, where necessary and sufficient condi-

tions are given in order that the Cauchy problem be well posed in

t > 0. These conditions involve restrictions on the location of the

spectrum of A and inequalities for the norm of the powers of the

resolvent of A. The proof presented here, perhaps not the shortest or

the simplest, puts in evidence nicely the fundamental Laplace trans-

form relation between the propagator S(t) and the resolvent of A; in

fact, S(t) is obtained from the resolvent using (a slight variant of) the

classical contour integral for computation of inverse Laplace trans-

forms. Section 2.2 covers the corresponding result for the Cauchy

problem in the whole real line, and the adjoint equation. The adjoint

theory is especially interesting in nonreflexive spaces and will find

diverse applications in Chapters 3 and 4.

One of the first results in this chapter is the proof of the

semigroup or exponential equations S(0) = I, S(s + t) = S(s)S(t) for

the propagator S(t). We show in Section 2.3 that any strongly

continuous operator-valued function satisfying the exponential equa-

tions must be the propagator of an equation u'(t) = Au(t). Finally,

Section 2.4 deals with several results for the inhomogeneous equation

u'(t) = Au(t)+ f(t).

2.1. The Cauchy Problem in t > 0

2.1. THE CAUCHY PROBLEM IN t , 0

We examine in this section the question of finding necessary and sufficient

conditions on the densely defined operator A in order that the Cauchy

problem for

u'(t) = Au(t) (t > 0) (2.1.1)

be well posed in t > 0. An answer is given in Theorem 2.1.1; the conditions

on A bear on its resolvent set p (A) and on the behavior of the resolvent

R(A) = (XI - A)-' in p(A).

We assume that the Cauchy problem for (2.1.1) is well posed in the

sense of Chapter 1. Let

be the propagator of (2.1.1). Then

S(0)=1, S(s+t)=S(s)S(t), s,t,O. (2.1.2)

The first equality (2.1.2) is evident from the definition of S. To prove the

other one, let u e D and t, 0 fixed, and consider the function u(s) =

S(s + t)u, s, 0. Due to the fact that A does not depend on t, is a

solution of (2.1.1); hence it follows from (1.2.7) that u(s) = S(s)u(0) =

S(s)S(t) u. The second equality in (2.1.2) results then from denseness of D.

An important consequence of (2.1.2) is the following. Let t > 0, n the

largest integer not surpassing t. Then S(t) = S(t - n)S(1)"; accordingly,

IIS(t)II < II S(t - n)II IIS(1)II" < C. C" = Cexp(nlogC) < Cexp(tlogC), where

C = sup(C(t ); 0 < t < 1) (here is the function in our definition of

well-posed Cauchy problem in Section 1.2). It follows that there exist

constants C, w such that

IIS(t)II <Cewt (t>0).

(2.1.3)

This implies that every solution (and also every generalized solution) of

(2.1.1) grows at most exponentially at infinity.

We can deduce from (2.1.2) more precise information on the behavior of

at infinity. In fact, let w(t)=1ogflS(t)Ij (t> 0). Clearly (which may take

the value - oo) is subadditive-that is, w(s + t) < w(s)+ w(t) for all s, t > 0. It

follows then essentially from a result in Pblya and Szego [1954: 1, p. 17, Nr. 98] that

too= inf

(0(t)

= lim

w(t)

<oo,

t>O t t

where we allow the value - oo for wo. Clearly (2.1.3) holds for any w > wo (but for

no w < wo); it may not necessarily hold for wo itself.

Let A be a complex number such that Re A > w (w the constant in

(2.1.2)). Define

Q(X)u= f oo e-"tS(t)udt (uEE). (2.1.4)

The norm of the integrand is bounded by C I I u I I exp(w - Re A) t, thus by

Lemma 1.3, (2.1.4) defines a bounded operator in E. We have already seen

Properly Posed Cauchy Problems: General Theory

in Example 1.2.2 that in the case where A is bounded, Q(A) = R(A), which

suggests that the same equality may hold in general. This is in fact so, at

least if we assume A closed. In fact, let u E D, T > 0. Making use of Lemma

3.4, we have

A f e-AtS(t)udt= f e-X`AS(t)udt

= f e-At(S(t)u)'dt

e-TS(T)u-u+A f e_XIS(t)udt.

Letting T -* oo and making use again of the fact that A is closed, we see that

Q(A)u E D(A) and AQ(A)u = AQ(A)u - u. Let now u E E,

a se-

quence in D with u - u. Then AQ(A)u = AQ(A)u - u - AQ(A)u - u

while evidently Q(A)u -* Q(A)u. Accordingly Q(A)u E D(A) and AQ(A)u

= AQ(A)u - u; in other words, Q(A)E C D(A) and

(AI - A)Q(A) = I. (2.1.5)

It is plain that (2.1.5) shows that (XI - A) : D(A) - E is onto. We show

next that (Al - A) is one-to-one. In fact, assume that there exists u E D(A)

with Au = Au. Then u(t) = extu is a solution of (2.1.1); since 11u(t)II =

e(Re')tIIuII, this contradicts (2.1.3) unless

u = 0. We see then that (Al - A) is

invertible and that (Al - A)-1 = Q(A) is bounded. (Compare with (1.2.13).)

The fact that p(A) contains the half plane Re A > w (in particular,

that it is nonempty) has some interesting consequences. In fact, let u e D,

A E p(A). Clearly u(t) = R(A)S(t)u is a solution of (2.1.1), so that in view

of (1.2.7),

R(A)S(t)u = S(t)R(A)u (t>0).

(2.1.6)

Since D is dense in E, (2.1.6) must hold for all u e E. Noting that

D(A) = R(A)E, we see that

S(t)D(A)9;D(A) (t>0)

and, making use of (2.1.6) for (Al - A)u instead of u and then applying

(Al - A) to both sides, we obtain

AS(t)u = S(t)Au

(u E D(A), t > 0).

(2.1.7)

Observe next that if we integrate the differential equation (2.1.1) and make

use of (2.1.7), it follows that

S(t)u-u= f`(S(s)u)'ds= fIS(s)Auds

for u E D. Applying R(A) to both sides of this equality and using (2.1.6), we

obtain

R(A)(S(t)u - u) = R(A) f 'S(s)Auds

=f tS(s)AR(A)uds.

2.1. The Cauchy Problem in t > 0

Both sides of this equality are bounded operators of u, thus it must hold for

all u E D(A). Making use of the fact that R(X) is one-to-one, we conclude

that

S(t)u-u= f tS(s)Auds= f tAS(s)uds

(2.1.8)

0 0

for all u E D(A). But then must be a solution of (2.1.1) for any

u E D(A); that is, the assumption that the Cauchy problem for (2.1.1) is

well posed implies that we may take

D = D(A). (2.1.9)

All elements of D(A) are then initial data of solutions of (2.1.1). We try

now to obtain more information about R(X) = Q(X). Estimates for the

norm of arbitrary powers of R(A) can be obtained from (2.1.4) by differen-

tiating both sides with respect to X. On the left-hand side we use formula

(3.5), whereas on the right-hand side we differentiate under the integral sign.

(That this is permissible, say, for the first differentiation can be justified as

follows on the basis of the dominated convergence theorem: the integrand

corresponding to the quotient of increments is h- 1(e-0"')'- e-Xt)S(t)u,

which is bounded in norm, uniformly with respect to small h by

I 1I

")t The higher differentiations

can be handled in the same

way.) We obtain the formula

R(1\)"u= 1 f OOtn-le-XtS(t)udt (ReA>

w, n > 1).

(2.1.10)

(n -1)! 0

Noting that the integrand is bounded in norm by Ct n - l e-(Re X -")`, we

deduce from (2.1.10) that

IIR(X)nII<C(ReX

-w)-'

(ReX>w,n>,l). (2.1.11)

It is remarkable (but not unexpected, in view of a result in Widder [1931; 1]

that identifies Laplace transforms of bounded functions) that inequalities

(2.1.11) are as well sufficient for the Cauchy problem to be well posed. In

fact, we have

2.1.1 Theorem.

Let A be closed. The Cauchy problem for (2.1.1) is

well posed and its propagator satisfies (2.1.3) if and only if a(A) is contained in

the half-plane Re X < w and R(A) satisfies inequalities (2.1.11) (C and w the

same constants in (2.1.3)).

We have already proved half of Theorem 2.1.1. To show the other

half we begin by constructing certain solutions of (2.1.1). Let u E D(A3),

w'> w, 0. Define

u(t)=u+tAu+2t2A2u

JW,+iooe3tR(X)A3udX

(t>0).

(2.1.12)

21T1

t-t,,o A

Properly Posed Cauchy Problems: General Theory

A simple deformation-of-contour argument shows that the integral on the

right-hand side vanishes for t < 0, so that

u(0) = U. (2.1.13)

It is also plain that

Ilu(t)II <C(u)ew` (t>-0). (2.1.14)

Differentiating under the integral sign (justification of this runs along the

same lines as that of (2.1.10)) and making use of another deformation-of-

contour argument and of Lemma 2.1, we obtain that u(.) is continuously

differentiable, u(t) E D(A) for all t and

u'(t)-Au(t)=-it2A3u

/'w'+fooe`(AI-A)R(A)A3udA=O

(t>0).

2771 Jw'-ioo

We see then that u(.) is a solution of (2.1.1) satisfying the initial condition

(2.1.13). We try now to improve the estimate (2.1.14) for

To this end

we observe that, making use of the equality u'(t)- Au(t) = 0, of an argu-

ment similar to the one leading to (2.1.5), and of the fact that (Al - A) is

one-to-one, we obtain

f00 e-kni(t)dt=R(A)u (Rea>,w').

(2.1.15)

(Note that the integral in (2.1.15) makes sense in view of (2.1.14); a more

direct, but less tidy proof of (2.1.15) can be achieved computing its left-hand

side by replacing u(-), given by (2.1.12), into it.) Differentiating now

(2.1.15) repeatedly with respect to A and arguing once again as in the

justification of (2.1.10), we obtain

R(A)nu = (n

1 0! /'osn-le-asu(s)

ds (ReX > (,o').

Hence, if t > 0,

)nRI

t )nu= f 'yn(t,s)u(s)dd (n>w't), (2.1.16)

where

yn(t s)=

(n) Sn- le -nslt

(s>0,t>0).

(n-1)1.

It is immediate that each yn is positive and infinitely differentiable. It

follows from (2.1.16) applied to A = 0, or directly, that

f'Oyn(t,s)ds=1

(t>O,n>_l). (2.1.17)

Clearly yn (t, 0) = y,, (t, oo) = 0. Moreover, yn, as a function of s, increases in