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

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5.9. Miscellaneous Comments 337

a2(l+e2)IIAul12+b2(1+e-z)IIu112 for any a>0 so that (5.9.1) lies in be-

tween the versions of (5.3.14) for a < 1 and for a = 1. The proof of Corollary

5.3.5 in full generality is due independently to Wiist [1971: 1].

Theorem 5.3.6 is due to Hille [1948: 1] (see also Hille-Phillips [1957:

1, p. 417]), although instead of (5.3.15) it is postulated there that IIPR(A)II < 1

for some sufficiently large A; in the present version, this is a consequence of

the assumptions.

Theorem 5.4.1 and Examples 5.4.2 and 5.4.3 are due to Kato [1951:

1]; the characterization of the domain of A in Theorem 5.4.5 we have taken

from Kato [1976: 1]. Theorem 5.4.7 is (essentially) the result of Stone [1932:

11 and Friedrichs [1934: 1] to the effect that any semi-bounded symmetric

operator possesses a self-adjoint extension. The method used here is that

due to Friedrichs, and can be considered as a forerunner of the Lax-

Milgram lemma (Lemma 4.6.1) that covers a somewhat similar situation.

We refer the reader to the treatise of Kato [1976: 1] for many additional

results on the Schrodinger, Dirac, and related equations. The theory of

"perturbation by sesquilinear forms," of which Theorem 5.4.8 offers a

glimpse, has been greatly extended, in particular to nonsymmetric forms: a

thorough exposition can be found in Kato [1976: 1]. Other works on

perturbation of differential operators are Schechter [1971: 11 (where other

references are given), Jorgens-Weidmann [1973: 1], Goodall-Michael [1973:

1], Gustafson-Rejto [1973: 11, Kato [1972: 1], [1974: 1], Agmon [1975: 11,

Semenov [1977: 11, Bezverhnii [1975: 1], and Powers-Radin [1976: 11.

The material in Section 5.6 is due to Kato [1970: 1] and is basic in

his treatment of time-dependent symmetric hyperbolic systems (to be found

in Sections 7.7 and 7.8).

Approximation of the abstract differential equation (5.7.1) by dif-

ference equations like (5.7.3) was discussed for the first time by Lax (see

Lax-Richtmyer [1956: 1] and references there). In this version, approxima-

tion proceeds in the space E and the operator An is required to "have the

limit A" in a suitable subspace of E. Its most celebrated result is the Lax

equivalence theorem (Lax-Richtmyer [ 1956: 1 ], Richtmyer-Morton [ 1967: 11)

of which Theorem 5.8.4 is a generalized version. Another treatment of the

approximation problem is that of Trotter [1958: 1], which we have followed

here. In this work the approximating sequences (E,,, are introduced and

convergence of the semigroups S (and of their discrete counterparts e5,,) is

related with convergence of the resolvents (Theorems 5.7.5, 5.7.6, 5.8.2, and

5.8.3). Theorem 5.7.6 is known as the Trotter-Kato theorem (see Trotter

[1958: 1], Kato [1959: 1]). The notion of extended limit of an operator and

the results on them proved here are due to Kurtz [1969: 11, [1970: 1], in

particular those linking extended limits of the An and convergence of the

semigroups S and the discrete semigroups C5 (Theorems 5.7.11 and 5.8.4).

For additional literature on discrete semigroups, see Gibson [1972: 1] and

Packel [1972: 1].

338

Perturbation and Approximation of Abstract Differential Equations

The treatment of continuous and discrete approximations we have

followed, although amply sufficient for our purposes, obscures the fact that

what is at play here is the convergence of a sequence of vector-valued

distributions given convergence of their inverses or their Laplace trans-

forms. This point of view leads to an extension of the theory to equations

much more general than (5.7.1) (such as integrodifferential or difference-dif-

ferential equations); see Section 8.6 for additional information.

We note finally that the results on numerical treatment of the

equation ut = Au + cu (Examples 5.7.12 and 5.8.5) are not understood as

"real" applications, especially in that the way we deal with the boundary

condition u = 0 would make any numerical analyst shudder. Actual finite-

difference schemes taking boundary conditions into account are usually far

more sophisticated in that the grids at which the function is computed

"follow the boundary" suitably. For examples, see Forsythe-Wasow [1960:

11 or Richtmyer-Morton [1967: 11.

We comment below on some developments related with the material

in this chapter.

(a) Addition and Perturbation of Infinitesimal Generators. Let A, B

be two square matrices of the same dimension. A classical formula (that can

be traced back to Lie in a more general form) states that

nlim0(exp(nA)exp(nB)) =exp(t(A+B))

(5.9.2)

(see Cohn [1961: 1, p. 112]). It is natural to inquire whether an infinite-

dimensional extension is possible. The first question is, of course, whether

A + B E (2+ whenever A and B belong to C+. In this form the question is too

restrictive to admit any reasonable answer; in fact, if A is an unbounded

operator in e and B = - A, then A + B is not closed, hence does not belong

to C'+. The following formulation avoids trivialities of this sort.

Addition Problem:

Let A, B belong to e+ (or to a subclass thereof).

Does A + B, with domain D(A) n D(B), possess an extension (A + B), in

C+ (or in a subclass thereof)? If the answer is in the affirmative, does

(A + B), coincide with the closure A + B?

Representation Problem:

Assuming the answer to the addition prob-

lem is in the affirmative, can exp(t(A+B)e) be obtained by means of

formula (5.9.2)?

Perturbation Problem:

Let A be an operator in C'+ (or in a subclass

thereof). Find conditions on the operator P in order that A + P (with

domain D(A) n D(P)) possess an extension (A + P), in C'+ (or a subclass

thereof). Can exp(t(A + P)e) be obtained directly from exp(tA) (say, as in

(5.1.19))?

Obviously, the perturbation problem is more general than the addi-

tion problem in that P itself may not belong to E.

5.9. Miscellaneous Comments

339

The first systematic consideration of the addition problem is that of

Trotter [1959:

1], whose work in relation to Theorem 5.3.1 was already

noted above. Among other results in [1959:

1], the following negative

answer to the addition problem is given.

5.9.1

Example.

Let (p be a function defined in - oo < x < oo, absolutely

continuous and with derivative q)' satisfying

0 < c < V(x) < C

a.e. (5.9.3)

It follows from the first inequality that T- I exists and is likewise absolutely

continuous. Define

SS(t)u(x) = u(p(T-'(x)+t))

for u E E = L2(- oo, oo) and - oo < t < oo. Some simple arguments involv-

ing no more than change of variables show that SS,(.) is a strongly

continuous group with

IISq'(t)II<C/c (-oo<t<oo)

and infinitesimal generator

(ATu)(x)=4)'((P-'(x))u'(x),

where D(A,,) consists of all absolutely continuous u e E such that u' c= E

(hence D(Aq,) does not depend on T). Let A = Aq, B = - A. with qp(x) = x

in oo<x<oo, fi(x) = x for x < 0, 4'(x) = 2x for x >, 0. Then

(A + B)u(x) = - h(x)u'(x),

(5.9.4)

where h(x) =1 for x >, 0, h(x) = 0 for x < 0. It is easy to see that A + B is

again defined by (5.9.4), this time with domain consisting of all u E E such

that u is absolutely continuous in x >, 0 and u' E=- L2(0, oo). It results then

that for any A > 0 the function u(x) = 0 for x < 0, u(x) = e-" for x >, 0

belongs to D(A + B) and satisfies

Xu-(A+B)u=0.

(5.9.5)

If A + B possesses an extension (A + B)

e C

(2+, we must have A + BC

(A + B)e, which shows, in view of (5.9.5), that every A > 0 belongs to

a((A + B)e), a contradiction.

We note incidentally that, when E is a Hilbert space (as in the

example just expounded) the addition problem always has a solution when

both A and B belong to (2+(1,0) and D(A)n D(B) is dense in E; in fact,

since A + B is dissipative, by Lemma 3.6.3 there always exists an m-dissipa-

tive extension (A + B)e. In a general Banach space this extension may not

exist (Example 3.6.5), but a partial result can be derived from Remark

3.1.12: A + BE (2+(1,0) if an only if (XI -(A + B))D(A + B) is dense in E.

Under these hypotheses, it was also shown by Trotter [1959: 1] that the

340

Perturbation and Approximation of Abstract Differential Equations

representation problem has an affirmative answer, that is, (5.9.2) holds. The

limit is here understood in the strong sense, uniformly on compact subsets

of t > 0. We refer the reader to Chernoff [1974: 11 for a deep study of the

addition and representation problems and additional references. Other

works on the subject and on the perturbation problem are Angelescu-

Nenciu-Bundaru [ 1975: 1 ], Uhlenbrock [ 1971: 11, Belyi-Semenov [ 1975: 1 ],

Chernoff [ 1976: 11, Dorroh [ 1966: 11, Da Prato [ 1968: 21, [1968: 3], [1968: 4],

[1968: 5], [1968: 6], [1968: 7], [1969:

1], [1970:

11, [1974:

11, [1975: 1],

Goldstein [1972:

1], Gustafson [1966: 1], [1968: 1], [1968: 2], [1968: 3],

[1969: 11, Gustafson-Lumer [1972: 1], Gustafson-Sato [1969:

11, Kurtz

[1972:

1],

[1973: 11, [1975: 11, [1976:

1], [1977:

11, Lenard [1971: 1],

Lovelady [ 1975: 1 ], Lumer [ 1974: 1], [1975: 1], [1975: 3], Miyadera [ 1966: 1 ],

[1966: 2], Mlak [1961: 11, Okazawa [1973: 2], Rao [1970: 11, Showalter

[1973: 11, Suzuki [1976: 11, Sunouchi [1970: 1], Vainikko-Slapikiene [1971:

11, Webb [1972: 11, Semenov [1977: 2], and Yosida [1965: 2].

(b) Approximation of Abstract Differential Equations. In relation

with the material in Sections 5.7 and 5.8, we mention the scheme developed

by Jakut [1963: 11, [1963: 2], wherein the spaces E in Trotter's theory are

quotient spaces of E by suitable subspaces and time-dependent operators

are allowed. For an exposition of this theory see the treatise of Krein [1967:

1]. See also Gudovic [1966: 1] for another abstract scheme. References to

the earlier bibliography can be found in the book of Richtmyer-Morton

[1967: 1]. We list below some of the more recent literature, comprising both

continuous and discrete approximation; some of this material refers to

specific types of partial differential equations (treated as abstract differen-

tial equations in suitable function spaces). See Belleni-Morante-Vitocolonna

[1974: 11, Crouzeix-Raviart [1976: 1], Gagimov [1975: 1], Cannon [1975: 11,

Gegeckori-Demidov [1973: 1]. Geymonat [1972: 1], Grabmuller [1972: 1],

[1975: 11, Gradinaru [1973: 11, Groger [1976: 1], Gudovic-Gudovic [1970:

11, Ibragimov [1969: 1], [1972: 11, Ibragimov-Ismailov [1970: 1], Ismailov-

Mamedov [1975: 1], Net as [1974: 11, Nowak [1973: 1], Oja [1974: 1], Pavel

[1974: 11, Ponomarev [1972: 1], Rastrenin [1976:

1], Sapatava [1972: 1],

Seidman [1970: 11, Showalter [1976: 11, Topoljanskii-Zaprudskii [1974: 1],

Vainikko-Oj a [1975:

11, Veliev [1972:

11, [1973:

11, [1973: 2], [1975: 11,

Veliev-Mamedov [1973: 1], [1974: 11, Zarubin [1970: 1], [1970: 2], Zarubin-

Tiuntik [ 1973: 11, Takahashi-Oharu [ 1972: 1 ], Ujishima [ 1975/76: 1 ], Raviart

[ 1967: 1], [1967: 2], Goldstein [ 1974: 2], Oharu-Sunouchi [ 1970: 1 ], Piskarev

[1979: 1], [1979: 3], the author [1983: 1].

Roughly speaking, a singular perturba-

tion of an abstract (not necessarily differential) equation is one that changes

the character of the equation in a radical way. For instance, the algebraic

time-dependent equation Au(t)+ f(t) = 0 becomes the differential equation

eu'(t) = Au5(t)+ f(t) by addition of the "small term" eu'(t) to the right-

5.9. Miscellaneous Comments

341

hand side; similarly, the first order differential equation u'(t) = Au(t (tf(t)

may be singularly perturbed into the second order equation euf (t)+ u'(t) _

Aue(t)+ f(t). In these and other similar cases, the subject of study is the

behavior of the solution

as e - 0 (hopefully, the limit will exist and

equal the solution of the unperturbed equation). Another instance of

singular perturbation is that considered in Example 5.9.4, where a Cauchy

problem that is not properly posed is approximated by properly posed

problems in a suitable sense. We limit ourselves to a few examples (which

exhibit some of the typical features of the theory) and some references.

5.9.2

Example.

Let A be an operator in C'+(C, - w) for some w > 0 (so

that, in particular, 0 E p (A)), and let f( - ) be a E-valued function continuous

in t > 0. The solution ue of the initial-value problem

eu'(t)=AuE(t)+f(t)

(t>0), u(0)=uEE (5.9.6)

with e > 0 is given by

ue(t)=S(t/e)u+ s J'S(s/e)f(t-s)ds

=S(t/e)u+ f S(s/e)(f(t-s)-f(t))ds

+ S(t/e)A-f (t) - A- f (t) (5.9.7)

(strictly speaking, ue will be a genuine solution of (5.9.6) only if u E D(A)

and f satisfies, say, one of the two sets of assumptions in Lemma 2.4.2:

under the present hypotheses ue is just a generalized solution in the sense of

Section 2.4). To estimate the integral on the right-hand side of (5.9.7) in an

interval 0 < t 5 T, we take r > 0, a sufficiently small and split the interval of

integration (0, t) in the two subintervals (0, re) and (er, t); after the change

of variable s/e = a, we see that the norm of the integral does not surpass

C fore-"°Il f(t-ea)-f(t)II da+C'omaxTII1(s)IIe-W'

(5.9.8)

uniformly in 0 < t < T. Clearly, a judicious choice of r and e (in that order)

will make (5.9.8) arbitrarily small. It follows from (5.9.7) that for each

8 > 0,

converges uniformly in 8 < t < T to uo(t) = - A- 'f(t). (The

lack of uniform convergence in 0 < t S T is of course unavoidable since

uE(0) = u while u0(0) = A-'f(0).) More generally, it can be shown that

convergence is uniform on compacts of t > t(e) if t(e)/e - oo (see definition

below). For additional information on this problem see Krein [1967: 1, Ch.

4].

5.9.3 Example.

Let A be the infinitesimal generator of a cosine function

in a Banach space E (see 2.5(c)). The solution of the initial-value

problem

e2ue(t)+ue(t)=AuE(t)+.ff(t),

ue(0)=ue,

u'(0)=ve (5.9.9)

342

Perturbation and Approximation of Abstract Differential Equations

can be written in the following form:

r/2E2C(t/E)uE+A(t;e)(IUE)+C5(t;E)(IUE+e2VJ

uE(t)=e

2 2

+ jtCS(t-s;E)f,(s)ds,

(5.9.10)

where the operator-valued functions A' (t; E) and C (t; e) are defined as

follows:

te-r/zee

/t/EI,(((t/e)2-s2)t/2/2e)

SG (t;

J l ((t/E)2_s2)1/2

C(s)uds,

/zee

C' (t; E)u=

e-r

Ot/eio(((t/e)2-s2)t/2/2E)C(s)uds

(5.9.11)

(5.9.12)

(see the author [1983: 4]); these formulas, which are examples of transmuta-

tion formulas (see Section 5(f)), are due to Kisynski [1970: 1]). On the other

hand, the solution of the initial value problem

uo(t) = Auo(t)+ f(t),

u0(0)=u,

(5.9.13)

is given by

uo(t)=S(t)u+ ftS(t-s)f(s)ds (5.9.14)

with

z/4t00

S(t)u=f e C(s)uds (5.9.15)

67t

(see 2.5(f)). As in the previous example, "solution" is understood here in the

generalized sense. Under suitable assumptions on uE, vE, it can be proved

that

converges to in various ways. We state a few results below.

Given a function e -* t(E) > 0 defined for e > 0, we say that a family

of functions

converges uniformly on compacts of t >_ t(E) (as E - 0) to

if and only if

lim

sup

IIge(t)-g(011=0

E-+0t(e)<r<a

for every a > 0.

(i) (The author [1983: 4].) Consider the homogeneous equation

(fE = 0), and assume that

U,-W, e2VE , u - w

(e -* 0),

(5.9.16)

where w is arbitrary and u is the initial condition in (5.9.13). Then

uE(t) -* uo(t)

(e- 0)

(5.9.17)

5.9. Miscellaneous Comments

343

uniformly on compacts of t >_ t(e), uniformly with respect to u bounded, if

t(E)/E2 -4 oo

(e-0).

(5.9.18)

Condition (5.9.18) is necessary in order that the convergence result above be

valid.

Obviously, uniform convergence on compact subsets of t >, 0 will not

hold in general since uE - u. If this condition is assumed and if u E D(A),

Kisynski [1970: 1] has obtained a very precise estimate, reproduced in (ii)

below in a slightly weaker form. It implies uniform convergence on com-

pacts of t >, 0 if uE -- u and E2vE -* 0. To expedite the formulation denote by

C, co two constants such that

IIC(t)II<CeW1'1

(-oo<t<oo).

(ii) Let, as before,

(resp. u0( )) be the solution of (5.9.9) (resp.

(5.9.13)). Then, if u E D(A), we have

II uE(t)-uo(t)II <CE2(1+(02t)ew2`IIAuII

+ Ce"2`IIu, - uII+

CE2e02`IIvtII

(t >, 0). (5.9.19)

Uniform convergence on compacts of t > 0 can also be obtained under

"crossover" of initial conditions (see (5.9.16)) if correction terms are added;

these terms neutralize the contribution of vE to u, the initial datum in

(5.9.13). A typical result is:

(iii) (The author, [1983: 4].) Let

be the solution of (5.9.9) with

uE = u + O(E), vE = E-2v + 0(e`), and let be the solution of the initial

value problem

E2v"(t)+vv(t)=0, ve(0)=E-2v (5.9.20)

that dies down as t - oo (so that vE(t)

e`/£Zv). Then, if u, v E D(A),

we have

UJO = u0(0+VJ0+0(E) (5.9.21)

uniformly on compacts of t > 0.

This is the prototype of a family of results where asymptotic develop-

ment in powers of e are obtained for u,(-); in general, we assume that

UE = UO + EUI + ... + ENUN + O(EN+I) and

VE =

E_2VO

+ E-IV, + ... +

EN-2VN-2 + O(EN-') and obtain an asymptotic development of the form

UE(t)=UO(t)+EUI(t)+

+ENUN(t)

+ vO(t/E2)+EVI(t/E2)+ ... + ENVN(t/E2)+O(EN+I ),

(5.9.22)

where vo = vE is the correction term in (5.9.21) and v I, ... , V N are higher

order correctors.

344 Perturbation and Approximation of Abstract Differential Equations

Among the convergence results for the nonhomogeneous equation we

have (iv) below, which involves the derivative u'(-); here u,(-) is the

solution of (5.9.9) with ue(0) = u'(0) = 0.

(iv) Let E be a Hilbert space, 1 < p < oo, f£, f E LP((0, T); E), T > 0.

Assume that

in LP as a-*0. Then ue(t)-+uo(t) uniformly in

0 < t <T,

in LP as e- 0, where

is the solution of

(5.9.13) with u0(0) = 0. Here the derivative of uo is understood as in

Example 2.5.8. (See the author [1983: 4]).

For a rather complete survey of the results available in this perturba-

tion problem as well as for additional references, see the author [1983: 4],

where some applications to partial differential equations are also covered.

Other works on the subject are Dettman [1973: 1], Friedman [1969: 2],

Griego-Hersh [1971: 1], Kisynski [1963: 1], Latil [1968: 1], Nur [1971: 11,

Schoene [1970: 1], Smoller [1965: 11, [1965: 2], and Sova [1970: 2], [1972: 11.

5.9.4 Example. Consider the heat equation

ut = KAu (t > 0) (5.9.23)

in R', with K > 0: we look at (5.9.23) in the space E = L2(Rm), the domain

D(A) consisting of all u E E such that Du E E. Working as in Section 1.4,

we can show that 0 belongs to (2+ (in fact, to d), the propagator of (5.9.23)

given by

S(t)u =

g-'(e-"10

12,,F

u), (5.9.24)

where 9 indicates Fourier transform. Let T > 0, rl > 0, and v E E be given.

Consider the (control) problem of finding u E E such that

IIS(T)u - v11 < n.

(5.9.25)

The "obvious" solution (taking u = S(- T) v) is meaningless since

S(-T)v= 6-'(e"I"I'Tfv)

(5.9.26)

may not belong to E (or even be defined in any reasonable sense) for an

arbitrary v E E. However, S(- T )v E E if v E

- IGD(Rm) thus we may

select w E -' 61(R ') with 11w - v I1 <

,l and define u = S(- T) w. Unfor-

tunately, this solution is far from satisfactory from the computational point

of view, since very small changes in w (in the L2 norm) may produce

enormous variations in S(- T )w.

A more advantageous approach is the following. Consider the equa-

tion

Ut = KAU + eA2u, (5.9.27)

where e > 0. Although the operator KA + e& does not belong to C+,

- (K h + e&) does. If we denote by S,(-) v the (in general weak) solution of

5.9. Miscellaneous Comments 345

(5.9.27) in t < 0 satisfying Sf(0)v = v, we have

Se(-t)v=F-I(e-(p1.14_II.11)t6v).

(5.9.28)

It follows from (5.9.24) and (5.9.28) that

S(T)S,(-T)v- vasE-40

for any v E E, hence an element u E E satisfying (5.9.25) will be obtained

by setting u = S£(- T) v for E sufficiently small; since Sf(- T) is a bounded

operator in E, small variations of v (with E fixed) will only produce small

variations of u and its numerical computation will pose no problem.

The argument above is a typical instance of a quasi-reversibility

method. These methods were introduced and systematically examined by

Lattes and Lions [1967: 11, although particular examples were used previ-

ously by Tikhonov and other workers in the field of ill-posed problems (see

the forthcoming Section 6.6)). Other references on quasi-reversibility meth-

ods are V. K. Ivanov [1974: 11, Kononova [1974: 1], Lagnese [1976:

1],

Melnikova [1975: 11, and Showalter [1974: 11. See also Payne [1975: 11 for

additional information and references.

The works below are on diverse singular perturbation problems for

abstract differential equations. See Belov [ 1976: 1], [1976: 2], Bobisud [ 1975:

11, Bobisud-Boron-Calvert [1972: 1], Bobisud-Calvert [1970: 11, [1974: 1],

Groza [1970: 11, [1973: 11, Janusz [1973: 11, Kato [1975: 2], Mika [1977: 1],

Mustafaev [1975:

1], Osipov [1970:

1], Veliev [1975:

1], and Yoshikawa

[1972: 11. For an exposition of singular perturbation theory and associated

asymptotic methods, see the treatise of Krein [1967: 1], where additional

references can also be found.

Chapter 6

Some Improperly Posed Cauchy Problems

Certain physical phenomena lend themselves to modeling by non-

standard Cauchy problems for abstract differential equations, such as

the reversed Cauchy problem for abstract parabolic equations (Section

6.2) or the second order Cauchy problem in Section 6.5, where one of

the initial conditions is replaced by a growth restriction at infinity. If

these problems are correctly formulated, it is shown that they behave

not unlike properly posed Cauchy problems: solutions exist and

depend continuously on their data.

6.1.

IMPROPERLY POSED PROBLEMS

Many physical phenomena are described by models that are not properly

posed in any reasonable sense; certain initial conditions may fail to produce

a solution in the model (although the phenomenon itself certainly has an

outcome) and/or solutions may not depend continuously on their initial

data. One such phenomenon is, say, the tossing of a coin on a table.

Assuming we could measure with great precision the initial position and

velocity of the coin, we could integrate the differential equations describing

the motion and predict the outcome (head or tails); however, it is obvious

that for certain trajectories, arbitrarily small errors in the measurement of

initial position and velocity will produce radical changes in the predicted

outcome. In studying phenomena like these we are then forced to abandon

346