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

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2.4. The Inhomogeneous Equation 87

Let be a solution of (2.4.1), (2.4.2), and let S be the propagator

of the homogeneous equation. It is easy to show (see the last lines of the

proof of Theorem 2.3:1) that if t > 0, the function s - S(t - s) u(s) is

continuously differentiable in 0 < s < t with derivative - S'(t - s)u(s)+

S(t - s)u'(s) = S(t - s)(u'(s)- Au(s)) = S(t - s) f (s). Therefore, we ob-

tain after integration the familiar formula of Lagrange:

u(t)=S(t)uo+ f`S(t-s)f(s)ds.

(2.4.3)

We turn now to the question of which conditions on uo, f will guarantee

that u(-), defined by (2.4.3), is indeed a solution of (2.4.1). Clearly we need

U0 E D(A), as can be seen taking f = 0. On the other hand, if A is bounded,

(2.4.3) is a solution of (2.4.1) for any f continuous in t >, 0 (the proof is

immediate). This is not true in general when A is unbounded. (See, however,

Example 2.4.7.)

2.4.1 Example. Prove the statement above about bounded operators.

Give an example of unbounded A E C'+ and a continuous f such that (2.4.3)

is not a solution of (2.4.2).

Necessary and sufficient conditions on f in order that (2.4.3) be a

solution of (2.4.1) are not known. Two simple sufficient conditions are given

in the following result.

2.4.2

Lemma. Assume that, either (a) f(t) E D(A) and f, Af are

continuous in 0 < t < T, or (b) f is continuously differentiable. Moreover, let

U0 E D(A). Then (2.4.3) is a solution of the initial value problem (2.4.1),

(2.4.2).

Proof. Obviously we may assume that uo = 0. To prove that (2.4.3)

is a solution of (2.4.1), it is sufficient to prove that u(t) E D(A), and that

Au(-) is continuous and

u(t)= ft(Au(s)+f(s))ds (0<t<T). (2.4.4)

We work at first under assumption (a). In this case, the fact that Au(t) E

D(A) follows directly from Lemma 3.4 and from (2.1.7); moreover,

Au(s) = J0SS(s-r)Af(r)dr (0<s<T),

(2.4.5)

from which continuity can be readily verified. (See the following Lemma

2.4.5.) To obtain (2.4.4), it suffices to integrate (2.4.5) in 0 < s < t, reverse

the order of integration, and make use of the fact that S(s)Au = (S(s)u)',

which holds for any u E D(A) (see again (2.1.7) and following comments).

In case assumption (b) is satisfied instead, we observe, arguing as in the

Properly Posed Cauchy Problems: General Theory

beginning of this section, that the function

v(s)= f t-SS(r)f(s)dr (0<s<t<T)

is continuously differentiable in 0 < s < t with derivative

v'(s) S(t - S) f (s)+ f- SS(r) f'(s) dr.

Integrating in 0 < s < t, we obtain

u(t)= f/S(s)f(0)ds

+ ff( ft-SS(r)

f'(s) dr) ds.

Using (2.3.7) we deduce that

Au(t) = S(t) f (0)- f (t)+ f0 tS(t - s) f'(s) ds

=S(t)f(0)-f(t)+ 0S(s)f'(t-s)ds. (2.4.6)

Integrating (2.4.6) and reversing the order of integration, we obtain (2.4.4).

This ends the proof.

It is natural to expect that if additional conditions are imposed on

the operator A (or, equivalently, on the propagator of the homogeneous

equation (2.4.1)), it should be possible to weaken the assumptions on f.

Some instances of this heuristic principle will be examined in Section 2.5(j).

Even in the general case, the assumptions can be relaxed if one is willing to

accept weaker definitions of solutions of (2.4.1). This was discussed in Section 1.2 in

relation to the homogeneous equation, and it is possible to extend these ideas to the

nonhomogeneous equation (2.4.1).

We call a function

a mild solution of (2.4.1), (2.4.2) in 0 < t <T if

u(t) E D(A) for almost all t, Au(.) is integrable in the sense of Lebesgue-Bochner in

0<t<T, and

u(t)= ft(Au(s)+f(s))ds+u0 (0<t<T). (2.4.7)

Naturally, in order for (2.4.7) to make sense, we have to assume that f itself is

integrable. Lemma 2.4.2 admits an immediate generalization in this context.

2.4.3 Example. Let assumptions (a) and (b) in Lemma 2.4.2 be replaced by (a')

f (t) E D (A) for almost all t and f(-), Af are integrable in 0 < t < T, or (b')

f (t) = fog(s) ds, where g is integrable in 0 < t <T. Then u(t), defined by (2.4.3), is

a mild solution of (2.4.1) if uo r= D(A). (Hint: the proof of Lemma 2.4.2 can be

imitated word by word if we substitute the Riemann by the Lebesgue-Bochner

integral.)

However, the assumptions in Example 2.4.3 are still excessive for

some applications. Copying then the definition made in Section 1.2 for the

2.4. The Inhomogeneous Equation

homogeneous equation, we call an integrable function

a weak solution

of (2.4.1), (2.4.2) in 0 < t < T with f itself integrable if for every u* in the

domain of the adjoint operator A*, and for every test function 9) with

support in (- no, T] we have

f T((A*u*, u(t))+(u*, f(t)))gp(t) dt

JT(u*,

u(t))q)'(t)

dt - (u*,

uo)q'(0).

(2.4.8)

It is not difficult to see that weak solutions are represented as well by

the Lagrange formula. To prove this we need the following two results.

2.4.4

Example.

Let CT(A*) be the space of all D(A*)-valued functions

u*(.) defined in - no < t < oo, with compact support contained in (- no, T],

continuously differentiable (in the norm of E*) and such that

continuous. Let 6 T®D(A*) be the space of all functions of the form

EpPk(t)uk, where the sum is finite and T,, T2,... are test functions with

support in (- no, T], uf, u*2,... are elements of D(A*). Then '6 T®D(A*) is

dense in CT(A*) in the following norm:

=supllu*(t)Il+supllu*'(t)ll+supllA*u*(t)ll,

the suprema taken in - no < t < T.

2.4.5 Example. Let f be integrable in 0 < t < T. Then

u(t)= f`S(t-s)f(s)ds

(2.4.9)

is continuous in t >, 0.

We only have to note that, if t < t',

IIu(t')-u(t)II < f0

tII(S(t'-s)-S(t-s))ff(s)II ds

+ f t IIS(t'-s)f(s)II ds.

The two integrals can be handled by means of the dominated convergence

theorem. In the first we observe that the integrand tends to zero as t' - t -* 0

while being bounded by a constant times l I f (- )I I; the same estimate applies

to the second integral while the shrinking of the domain of integration takes

care of the convergence to zero.

2.4.6

Theorem.

Let f be integrable in 0 < t < T, uo an arbitrary

element of E. Then (2.4.3) is a weak solution of (2.4.1), (2.4.2), that is, (2.4.8)

holds for every u* E D(A*) and every 99 E 6D with support in - no < t < T.

Conversely, let

be a weak solution of (2.4.1), (2.4.2) with f integrable and

uo E E. Then

obeys the Lagrange formula (2.4.3) almost everywhere (and

can be modified in a null set in such a way that it becomes continuous and

satisfies (2.4.3) everywhere).

Properly Posed Cauchy Problems: General Theory

Proof.

Let f be integrable in 0 < t < T. Then there exists a sequence

(fn) of smooth (say, infinitely differentiable) E-valued functions such that

f lf(t)-f(t)11

oo.

(This can be proved with the aid of mollifiers; see Section 8.) Let (un) be a

sequence in D(A) such that u - uo. It follows easily that if

is the

function obtained from uo and fn(.) in formula (2.4.3), then Aun(t)-u(t)II

- 0 uniformly in 0 < t < T. Since each un is a genuine solution of (2.4.1),

(2.4.2) (as was proved in Lemma 2.4.2), equality (2.4.8) holds for u(.).

Taking limits, we obtain the same equality for u(.), which proves that u(.)

is in fact a weak solution as claimed.

Conversely, let be an arbitrary weak solution. Making use of

Example 2.4.4, we observe that (2.4.8) can be generalized to

f T((A*u*(t), u(t)) + (u*(t), f(t))) dt

f T(u*'(t), u(t)) dt

- (u*(0), uo)

(2.4.10)

for all u*(.) in CT (A*) (in fact, (2.4.8) obviously implies (2.4.10) for u*(.) in

61,T(&D(A*) and the density result takes care of the rest). Let now q' be a

fixed positive test function with integral 1 and define Tn(s) = ngp(ns); let

A# be the Phillips adjoint of A, and

the Phillips adjoint of

(see

Section 2.3). Choose to in the interval (0, T) and u* in D(A*) and define

v,*(t)=- f S*(s-t)g7n(s-to)u*ds (t<T).

Slight variations on the argument used in Lemma 2.4.2 (a) easily show that

v,*(t) is a solution of v,*'(t)+A*v,*(t) = q)n(t - to)u* in t < T with "final

condition" v,*(T) = 0. Let finally

be a test function that equals 1 in the

interval 0 < t < T. Define u* (t) = X(t)v,*(t). Clearly u* E CT(A*). Writing

(2.4.10) for u* and reversing the order of integration in the double integral,

we obtain

T(u*

u(t))9pn(t - to) dt = f

T(u*, f `S(t

- s).f(s) ds)(pn(t - to) dt

+ f

T(u*, S(t)uo)Tn(t

- to) dt (2.4.11)

if n is large enough.

We invoke now a well-known result in approximation theory (Hille-

Phillips [1957: 1, p. 91]) according to which if g is an integrable scalar

function, then Jg(t)ggn(t - to) dt - g(to) whenever to is a Lebesgue point of

g. But the set e of Lebesgue points of has full measure in 0 < t < T

(Hille-Phillips [1957: 1, p. 88]), whereas every Lebesgue point of u(.) is a

Lebesgue point of (u*, u(.)). Making use of this and of the fact (proved in

2.5. Miscellaneous Comments

Example 2.4.5) that t - fo S(t - s) f (s) ds is continuous, we let n - oo in

(2.4.9), and conclude that

(u*, u(t)) _ (u*, f `s(t - s) f(s) ds + s(t)uo)

for t E e. Since u* E D(A#) is arbitrary and D(A#) is weakly dense in E*

(see the remarks after Theorem 2.2.11), we see that the Lagrange formula

holds in e: then in view of Example 2.4.5,

can be modified outside of e

in such a way that it becomes continuous in 0 < t < T and (2.4.3) holds

everywhere.

2.4.7 Example.'

Let co be the space of all sequences

(n >_ 0)

satisfying lim

0, endowed with the norm I I

I I = sup I

A the operator

defined by

A _ (- n6n), (2.4.12)

the domain of A consisting of all

E co such that the right-hand side of

(2.4.12) belongs to co. Then A E e+(1,0) (the propagator of u'= Au is

S(t) d = (exp(- nt) n)) and (2.4.3) is a solution of (2.4.1), (2.4.2) in 0 < t < T

for any u0 E D(A) and any co valued f continuous in 0 < t < T.

2.5.

MISCELLANEOUS COMMENTS

Strongly continuous groups of unitary operators in Hilbert space were

characterized by Stone [1930: 1], [1932: 1]. His celebrated result belongs to

the subject matter of Chapter 3 and will be commented on there. Uniformly

bounded groups of bounded linear operators in Banach spaces were

studied by Gelfand [1939: 1], who defined the infinitesimal generator A and

proved that U(t) u = exp(tA)u, where exp(tA) u is defined by the exponen-

tial series, convergent in a certain dense subset of E. Related results are due

to Fukamiya [1940: 1]. Some special semigroups of linear transformations in

function spaces were considered by Hille [1936: 1]. Two years later Hille

[1938: 1] and Sz.-Nagy [1938: 1] discovered independently a representation

theorem for semigroups of bounded self-adjoint operators in Hilbert space:

this result also belongs to the class spawned by Stone's theorem and will be

referred to in Chapter, 3. Other early results in semigroup theory can be

found in Hille [ 1938: 2], [1939: 1], [1942: 1], [1942: 2], and [ 1947: 11; there is

also related material in Romanoff [1947: 1]. The ancestors of Theorem 2.3.2

due to Cauchy and P51ya were already pointed out in Section 2.3. This

theorem (now called the Hille- Yosida theorem) was discovered indepen-

dently by Hille [1948: 1] and Yosida [1948: 11 in the particular case where

'Personal communication of A. Pazy (who credits the result to T. Kato).

92 Properly Posed Cauchy Problems: General Theory

C = 1. Hille's proof culminates in the construction of S by means of formula

(2.1.27), the Hille approximation of S(.); in contrast, Yosida uses formula

(2.3.20), the Yosida approximation of S(.). The proof for the general case

(C >_ 1) was discovered, also independently, by Feller [ 1953: 2], Miyadera

[1952: 1], and Phillips [1953: 1]; all three papers use the Yosida approxima-

tion. Groups S(.), which are continuous in the (E )-topology, were shown to

have the form exp(tA) for some A E (E) by Nathan [1935: 1] (his method is

different than those in Lemma 2.3.5 and Example 2.3.10). A generalization

of this result (where

takes values in a Banach algebra) was proved

independently by Nagumo [1936: 11 and Yosida [1936: 11.

The proof of Theorem 2.3.2 presented here is due to Hille [1952: 2].

Its basis is the following formula to recover a function f from its Laplace

transform f,

f(t)= lim

\t/

f(n)ltl'

goo

(2.5.1)

due to Post [ 1930: 1 ] in the scalar case; see also Widder [ 1946: 1, Chapter 7].

The obvious relation between the characterization of strongly continuous

semigroups by inequalities (2.1.11) and Widder's identification in [1931: 1]

of Laplace transforms of bounded functions was already pointed out in

Section 2.1 (see also Widder [ 1971: 11).

The first result on "measurability implies continuity" is due to

von Neumann [1932: 1], who considered groups of unitary operators. The

particular case of Example 2.3.15, where S is strongly measurable in the

norm of (E), is due to Hille [1948: 11, while the full result was proved by

Miyadera [1951:

1] and Phillips [1951:

11, which showed that a crucial

hypothesis in earlier proofs of Dunford [1938: 1] and Hille [1948: 1] was

actually unnecessary.

Example 2.3.11 is due to Hille [ 1942: 1 ] (see also [ 1948: 1 ]). Formula

(2.3.19) is one of the exponential formulas that justify writing S(t) = exp(tA);

other such formulas are the Hille and Yosida approximations of S. An

ample collection of these can be found in Hille-Phillips [1957: 1, Sec. 10.4]

together with some general principles on their obtention. Example 2.3.12 is

due to Dunford and Segal [1946: 1] where it can also be found a short proof

of (2.3.19) for semigroups with S(t )I I < 1. Example 2.3.8 is a particular case

of a result on smooth semigroups (see the bibliography in Section 4.11).

Lemma 2.3.9 is due to Krein and Sobolevskii [1958: 11.

It is worth pointing out that, although the relation between semi-

groups and abstract differential equations was in the forefront from the very

beginning (see Yosida [1949: 11, [1951: 1], [1951: 2], [1952: 11, and Hille

[1952: 1]) and was one of the main incentives for development of the theory,

most of the results in this chapter have appeared for the first time in the

"semigroup-generator" formulation rather than in "equation-propagator"

language.

2.5. Miscellaneous Comments

In a more general form, Lemma 2.2.3 was proved by Phillips [1955:

1]. Lemma 2.4.2 on sufficient conditions for the solvability of the nonhomo-

geneous equation (2.4.1) is also due to Phillips [1953: 1] as well as the entire

# -adjoint theory ([1955: 1], [1955: 2]).

In the rest of this section we comment on several extensions and

generalizations of the results in this chapter.

(a) Discontinuous Semigroups.

The Cauchy problem for an abstract

differential equation u' = Au has been studied under assumptions weaker

than the ones in Section 1.2 by Krein ([1967: 1] and references there). On

the one hand, solutions are only assumed to be continuously differentiable

and to belong to the domain of A for t > 0 (although continuity in t > 0 is

retained). On the other hand, the function in (1.2.3) is merely assumed

to be finite for all t > 0; this means that a sequence of solutions whose

initial values tend to zero will only tend to zero pointwise (compare with (b')

in Section 1.2). A Cauchy problem for which these weaker assumptions hold

is called correct or well posed by Krein; in his terminology, the setting used

in the present work is that of uniformly correct or uniformly well posed

problems.

Assume the Cauchy problem for u' = Au is correct in the sense of

Krein. Then we can define the propagator S(t) in the same way used in

Section 1.2 and show that it satisfies the semigroup equations (2.1.2). If u is

an arbitrary element of E, we can approximate it by a sequence

in the

subspace D of possible initial data. Since S(t)u,, -+ S(t) u pointwise in t > 0,

it follows that

is strongly measurable there, thus strongly continuous in

t > 0 by Example 2.3.15. However, it is not in general true that S(.) is

strongly continuous at t = 0 or even bounded in norm near the origin; all we

know is that solutions starting at D assume their initial data-that is,

S(t) u - u as t -' 0 + for u e D. (We note, incidentally that, by virtue of

Example 2.3.16, a Cauchy problem that is well posed in the sense of Krein

in (- oo, oo) must be uniformly well posed there, at least if solutions are

defined as in Section 1.5.)

This, and many other applications, motivate the study of semigroups

that are strongly continuous for t > 0 but not at t = 0, or even semigroups of

a more general nature. An extremely ample theory has been developed by

Feller [1952: 2], [1953: 1], and [1953: 2] with probabilistic applications in

view. In its more general version, even less than weak measurability is

assumed. A generation theorem for semigroups strongly continuous in t > 0

is given by Feller [1953: 2]; the notion of infinitesimal generator is under-

stood in a sense somewhat different from that in Section 2.3.

Different classes of semigroups strongly continuous for t > 0 have

been staked out by Hille, Phillips, Miyadera, and others on the basis of their

properties of convergence to the identity as t - 0. Ignoring certain addi-

tional assumptions, some of these classes can be described as follows.

94 Properly Posed Cauchy Problems: General Theory

Assume that S satisfies

0IIIS(t)ulldt<oo

(2.5.2)

for all u e E. In the terminology of Hille, S is of class Co if

lim S(t)u -emu

(uEE), (2.5.3)

t-.0+

of class (1, C1) if (2.5.3) holds in Cesaro mean, that is, if

lim

1 fS(s)uds=u (uEE) (2.5.4)

t-0+ t

and of class (1, A) if (2.5.3) holds in Abel mean; in other words,

lim X f"* e-'S(t)udt=u (uEE). (2.5.5)

X-.o0

(We note that strong continuity of S for t > 0 can be seen to imply that

I I S(t )I I = 0(e"t) as t - no, hence the integral in (2.5.5) makes sense for

sufficiently large X.) Obviously, the class Co coincides with e. Other classes

are (0, C,) and (0, A), which are obtained respectively from the classes

(1, C1) and (1, A) by reinforcing condition (2.5.2) to

f 'IIS(t)II dt <oo

(2.5.6)

(where the integral makes sense due to the fact that

IIS(t)II = sup{IIS(t)ull; (lull <1)

is lower semicontinuous, hence measurable in t > 0). For general references

on these and other types of semigroups see Hille-Phillips [1957: 1]; other

papers on the subject are Abdulkerimov [1966: 1], Mamedov-Sobolevskil

[1966: 1], Miyadera [1954: 1],[1954: 2], [1954: 3],[1954: 4], [1955: 1], [1972:

1], Miyadera-Oharu-Okazawa [1972/73: 11, Oharu [1971/72: 11, Okazawa,

[ 1974: 1 ], Orlov [ 1973: 1 ], and Phillips [ 1954: 1 ]. Other classes of semigroups

with discontinuities at t = 0 are the "semigroups of growth n" of Da Prato

[ 1966: 11, [1966: 2], characterized by polinomial growth at zero (t" I I S(t )l I < C

for 0 < t < 1). For other results on these semigroups and related classes see

Okazawa [1973: 1], [1974: 1], Ponomarev [1972: 2], Sobolevskii [1971: 2],

Wild [ 1975: 11, [1976: 1], [1976: 2], [1977: 11, Zabreiko-Zafievskii [ 1969: 1 ],

and Zafievskii [ 1970: 11.

(b) Systems, Huygens' Principle, Ergodic Theory, Markov Processes.

The state at time t of many physical systems can be described (or modelled)

by a point x(t) in a state space X. Some of these systems obey (in the

terminology of Hadamard [ 1923: 11, [1924: 1], [1924: 2]) the minor premise

of Huygens' principle. This can be formulated as follows; given two times

s < t, the state of the system at time t is totally determined by the state of the

2.5. Miscellaneous Comments 95

system at time s. In other words, there exists a flow or evolution function

I (t, s) : X -* X such that

x(t) = 4(t, s; x(s)). (2.5.7)

We note that other systems, such as those described in Chapter 8 of this

work, do not obey the major premise of Huygens' principle since the state

x(t) may depend on all (or several) of the past states, that is, on the history

(x(s); s < t), or subsets thereof. This can be remedied by taking the

histories as new states. We restrict ourselves, however, to systems satisfying

(2.5.7).

It follows immediately from the definition of (D (t, s) that

4) (t, t) = i = identity map,

(2.5.8)

t(t,s)o11(s,r)= I(t,r)

(r<s<t).

(2.5.9)

The system is reversible if 1 is defined for all s, t (not just for s < t) and

(2.5.9) holds for r, s, t in arbitrary position; here not only the future is

determined by the present but the past itself can be known from the present.

A system (reversible or not) is time-invariant if I only depends on t - s; we

can then write 1(t, s) ='Y(t - s) and equations (2.5.8) and (2.5.9). become

*(0) = i,

(2.5.10)

$(r) o 4'(s) _ 4'(r + s). (2.5.11)

It should be pointed out that the above framework is not limited to

deterministic systems; for instance, x(t) may be a probability distribution

(as in diffusion processes or in quantum mechanics) or some space average

in statistical mechanics. In all these cases a "particle-by-particle" descrip-

tion of the system is either impracticable or senseless.

In many cases of interest, the state space X can be given a structure

of normed linear space in which the functions D (t, s) are continuous

operators depending on s, t smoothly in one way or another; if the system is

time-invariant, the operators fi(t) linear, and the time dependence strongly

continuous, the systems will be amenable to the theory in Section 2.3; in

particular, we deduce that the evolution of the system will be governed by

an abstract differential equation

x'(t) = Ax(t). (2.5.12)

An example which fits into this kind of framework is that of a Markov

process. Here the states x(t) of the system are countably additive proba-

bility measures x(t; di) defined in a set 9 (for instance, x(t; e) may be the

probability of finding in the subset e c 0 at time t a particle performing

Brownian motion), which we assume defined in a common a-algebra C of

subsets of Q. The law (2.5.7) governing the evolution of the system takes the

Properly Posed Cauchy Problems: General Theory

form

x(t; e) = D(t, s; x(s))

= fnP(t,

s, ; e)x(s;

(e E C5 ),

(2.5.13)

where P(t, s, ; e) is called the transition probability of the process. (In the

Brownian motion example, P(t, s, J; e) is the probability that the particle

falls in e at time t starting at at time s.) To give sense to (2.5.13) in general,

we make the following assumptions on the transition probabilities:

(i) If 0 < s < t and E 0, then P(t, s, ; dry) is a probability measure

in Sl defined in C5; moreover,

P(t, t, ; dii) = St(dri)

(t > 0), (2.5.14)

where S£ is the Dirac measure centered at .

(ii) For each e E C5, the function P(t, s,

; e) is C5-measurable in Q.

(iii)

(2.5.15)

Condition (i) guarantees that the function 1 maps probability mea-

sures into probability measures and that (2.5.8) holds. Condition (ii) is an

obvious technical

requirement while

(2.5.15), called the Chapman-

Kolmogorov equation of the process, implies (2.5.9). The Markov process in

question is temporally homogeneous if P depends on t - s, in which case we

can use transition probabilities of the form P(t, ; e) and the Chapman-

Kolmogorov equation becomes

P(s + t, ; e) = faP(s, ; dri)P(t, ri; e). (2.5.16)

We go back for a moment to the general description of a system

sketched above. It is often the case that the states of a system are unob-

servable in practice, the only information available being that given by a

measuring instrument. What one can read off the instrument will in general

be of the form

f (X (0)

(2.5.17)

with f a real-valued function (for instance, the probability densities x(t,

of a Markov process may only be known through space averages

f,(x(t)) = (2.5.18)

It follows that, instead of taking the states themselves as elements of a linear

space, we may obtain a mathematical model better adapted to reality by