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

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I.1. The Heat Equation in a Square

LP spaces, p * 2 or in spaces of continuous functions), the L2 norm of

arbitrary solutions remains constant and, finally, solutions do not

become smoother than their initial data. Conclusions of the same type

are obtained for the Maxwell and Dirac equations in Section 1.6;

these are the prototype of results on symmetric hyperbolic systems to

be found in Chapters 3 and 5.

1.1. THE HEAT EQUATION IN A SQUARE

Consider the equation

2 z

=KAu=K

ax2+ d

(t>_ 0)

(1.1.1)

(K a positive constant) in the square 2 = ((x, y); 0 < x, y < Ir) with initial

condition

u(x,y,0)=uo(x,y) (0<x,y<IT)

(1.1.2)

and boundary condition

u(x,y,t)=0 ((x,y)EF,t>, 0), (1.1.3)

where r is the boundary of 2. It is customary to call u = u(x, y, t) a

classical solution of (1.1.1), (1.1.2), and (1.1.3) if u is continuously differen-

tiable up to the degree required by the equation and the initial and

boundary conditions and satisfies identically all of them. We make precise

these somewhat vague requirements by assuming that: u is continuous in

Sl x [0, oo), u1 exists and is continuous in the same region, uX, uy, uxx, u,, uy

exist in 9 x [0, oo) and are continuous in Sl x [0, oo), the equation (1.1.1)

holds in 2 x [0, oo), and the initial and boundary conditions are satisfied.

We can write the initial-boundary value problem (1.1.1), (1.1.2),

(1.1.3) as a pure initial value problem for an ordinary differential equation

in a Banach space as follows. Using the notation in Section 7 we take

E = Cr(Sl), the space of all continuous functions in Sl that vanish at the

boundary t, endowed with the supremum norm, and define an operator A

in E as follows:

Au=K(uXX+uyy)=KAU

(1.1.4)

where D(A) consists of all u E E such that u, uy, uXX, uXy, uyy exist in 52

and are continuous in S2, and Au E E (that is, Au = 0 on r). Let u be a

classical solution of (1.1.1), (1.1.2), and (1.1.3) in the sense made precise

above. Define a function t - u(t) E E in t >, 0 as follows:

u(t)(x, y) = u(x, y, t)

(t >'0, (x, y) E S2).

(1.1.5)

28 The Cauchy Problem for Some Equations of Mathematical Physics

Clearly u(t) E D(A) for all t >_ 0. On the other hand, if we define v(t) E E

v(t)(x, y)=ur(x, y,t)

(t>'0, (x,y)(=-S2)

(note that u1= KAU E E by hypothesis), we see, on account of the uniform

continuity of ut on compact subsets of S2 X [0, oo), that v is continuous in

t >, 0. Moreover, if t > 0, 1 h I < t, we have

IIh -'(u(t + h) - u(t))- v(')IIE

= sup Ih-'(u(x, y,t+h)-u(x, y, t))-ut(x, y,t)I

(X, Y)Ea

-0 as h-*0

by virtue of an elementary application of the mean value theorem and of the

continuity of ur in S2 x [0, oo); the same can be proved when t = 0, but we

have to take h > 0. It turns out that

is continuously differentiable in the

sense of the norm of E in t > 0 and

u'(t)=Au(t) (t>_0). (1.1.6)

On the other hand,

u(0) = uo, (1.1.7)

where uo is the initial function in (1.1.2). Conversely, let t -* u(t) E E be a

function defined and continuously differentiable in t >_ 0, and such that

u(t) E D(A) and (1.1.6) is satisfied for t > 0. Then it is rather easy to show

that

U(X, y, t) = u(t)(x, y)

(t ? 0, (x, y) E 9)

(1.1.8)

is a classical solution of (1.1.1), (1.1.2) (with uo(x, y) = u(0)(x, y)), and

(1.1.3). This shows the complete equivalence of our original initial-boundary

value problem and the initial value or Cauchy problem (1.1.6), (1.1.7).

Making use of this equivalence we try to obtain additional information on

the latter, taking advantage of the fact that (1.1.1) can be explicitly solved

by separation of variables. Let

uo(x, y) _ E E amnsin mxsin ny (1.1.9)

m=1 n=1

with

E E (m2+n2)lamnl <oo.

(1.1.10)

m=1 n=1

Then we obtain a classical solution of (1.1.1), (1.1.2), and (1.1.3) setting

X00

u(x, y, t) =

e-rc(n2+m2)tamnsin mxsin ny. (1.1.11)

m=1 n=1

1.2. The Abstract Cauchy Problem

(this follows from elementary theorems on differentiation of multiple

Fourier series, which can be found, for instance, in Zygmund [1959: 1]) and

a fortiori a solution of (1.1.6) and (1.1.7). On the other hand, let

be an

arbitrary solution of (1.1.6) and (1.1.7), and let u(x, y, t) be the function

defined by (1.1.8). Then it follows from the maximum principle for the heat

equation (Bers-John-Schechter [1964: 1, p. 96]) that

lu(x,Y,t)I<

sup

((x,y)(-A,t%0)

(

,n) Et2

IIu(t)IIE < IIkOIIE

(t>10)'

(1.1.12)

We have then proved that solutions of the Cauchy problem (1.1.6), (1.1.7)

exist for uo satisfying (1.1.10) (note that the subspace of all such uo is dense

in E) and that arbitrary solutions are continuously dependent on their

initial value (as inequality (1.1.12) and linearity of the equation (1.1.6)

show). Clearly this last property implies in particular that solutions are

unique; that is, if two solutions coincide for t = 0, they coincide forever.

1.2. THE ABSTRACT CAUCHY PROBLEM

Taking as motivation the observations in the previous section we examine

the equation

u'(t) = Au(t) (t >, 0) (1.2.1)

where A is a densely defined operator in an arbitrary (real or complex)

Banach space E. A solution of (1.2.1) is a function t - u(t) continuously

differentiable for t > 0 (i.e., such that u'(t) = lim h -'(u(t + h)- u(t)) exists

and is continuous in the norm of E in t > 0, the limit being one-sided for

t = 0) and such that u(t) E D(A) and (1.2.1) is satisfied for t > 0. We may

of course define solutions in intervals other than [0, oo), say (- oo, oo) or

[0, T I.

We say that the Cauchy problem (or initial value problem) for (1.2.1)

is well posed (or properly posed) in t > 0) if the following two assumptions

are satisfied:

(a) Existence of solutions for sufficiently many initial data. There

exists a dense subspace D of E such that, for any uo E D, there

exists a solution of (1.2.1) in t > 0 with

u(0) = uo (1.2.2)

(b) Continuous dependence of solutions on their initial data. There

exists a nondecreasing, nonnegative function C(t) defined in t 3 0

such that

IIu(t)II<C(t)IIu(0)II

(t,0) (1.2.3)

for any solution of (1.2.1).

The Cauchy Problem for Some Equations of Mathematical Physics

Note that (b) refers to any solution of (1.2.1), that is, not only to the

solutions in (a), where u(0) E D. Note also that, since our definition of

solution requires that u(t) e D(A) in t, 0, we must have

D c D(A). (1.2.4)

Hypothesis (b) can be given the following equivalent but slightly

more natural form:

(b') Let {u(.)) be a sequence of solutions of (1.2.1) with un (0) -* 0.

Then un(t) -* 0 uniformly on compacts of t, 0.

We examine some immediate implications of assumptions (a) and

(b). Let u E D. Define, for t , 0,

S(t) u = u(t), (1.2.5)

where u(.) is the (only) solution of (1.2.1) with u(0)= u. In view of (1.2.3),

S(t) is a bounded operator in D (with norm < Qt)). Since D is dense in E

we can extend S(t) to a bounded operator in E, which we denote by the

same symbol, and

IIS(t)II <C(t)

(t, 0). (1.2.6)

The operator-valued function S(.) is called the propagator (or solution

operator or evolution operator) of the equation (1.2.1). If

is an arbitrary

solution of (1.2.1), then

u(t)=S(t)u(O) (t, 0), (1.2.7)

where uo = u(0). This is just the definition of S(t) when u(0) E D; if u(0)

does not belong to D, let (un) be a sequence of elements of D such that

un -* u(0). Then it results from (1.2.3) that S(t)un - u(t), whereas, by

definition of S outside of D, S(t)u(0) = lim S(t)u,,, thus proving (1.2.7) in

general.

Let u be an arbitrary element of E, and let (un) be a sequence in D

with un -* u. It follows from (1.2.6) that the (continuously differentiable)

functions un converge uniformly to S(.) u on compacts of t , 0, thus

S(.) u is continuous in t , 0. The function u(t) = S(t) u will be called a

generalized solution of (1.2.1). Clearly, need not be a genuine solution of

(1.2.1) if u does not belong to D.

Generalized solutions of (1.2.1) can be defined in a way more akin to the

general distribution-theoretical idea of solution of a differential equation. Let A be

closed and let u(t) be a continuous (or just locally integrable) function in t > 0.

Then we say that is a weak solution of (1.2.1) and (1.2.2) if and only if, for every

u* E D(A*) and every test function 9) E 6D we have

f oo(A*u*,u(t))gp(t)dt=-

f'(u*,u(t))q,'(t)dt-(u*,uo)gq(0).

(1.2.8)

As the following result shows, both definitions are equivalent.

1.2. The Abstract Cauchy Problem

1.2.1

Lemma. Assume the Cauchy problem for (1.2.1) is properly posed, and

let be a continuous (or only locally integrable) weak solution of (1.2.1) and (1.2.2).

Then

u(t)=S(t)u0 (t>O) (1.2.9)

(almost everywhere if

is only assumed to be locally integrable). Conversely, every

function of the form (1.2.9) is a weak solution of (1.2.1) and (1.2.2).

Lemma 1.2.1 shows that generalized and weak solutions of (1.2.1) and (1.2.2)

coincide. We postpone the proof until Section 2.4, where a more general result will

be shown (Theorem 2.4.6).

The case where the operator A in (1.2.1) is everywhere defined and bounded

is of no special interest in applications since nontrivial differential operators are

never bounded in the spaces considered in this volume or in other usual Banach

spaces of functions. However, the theory becomes utterly simple here and provides

some heuristic suggestions for the general case.

1.2.2 Example.

Let A E (E). Then the Cauchy problem for (1.2.1) is properly

posed in t > 0 with D = E. To see this we define S(f) for any complex by

S(3") =

A" (t>0), (1.2.10)

n0 n.

the series being convergent in the norm of (E) uniformly on compacts of C since

113"nA"II

< ICI"IIAII"; moreover,

IISU)II < F

i_1,"

IIAIIn =

e1SI11A11

n=0

(1.2.11)

It follows from the elementary theory of vector-valued analytic functions (see

Section 2, especially Lemma 2.5) that S is analytic in C and that

S'(3') = AS(3') = S(3')A.

Hence u0 is a solution of the initial value problem (1.2.1), (1.2.2) for any

UO E E thus proving (a). In view of (1.2.11) we only have to show that solutions of

(1.2.1) with the same initial data coincide or, equivalently, that if is a solution

with u(0) = 0, then u(t) = 0 (t > 0). To see this, we integrate u'(s) = Au(s) in

0 < s < t, apply A on both sides, use (1.2.1), integrate again, and so on. We obtain

u(t) (n 11)! f`(t-s)"-'A"u(s)ds, (1.2.12)

whence IIu(t)II < Ca"II All n/n! in 0 < t < a, where C is a bound for IIu(t)II there.

Letting n - oo, it follows that u vanishes identically in t > 0.

There exists an interesting Laplace transform relation among the propagator

and the resolvent operator R (A) = (A I - A) -1 corresponding to the scalar

relation L(e°`) =1/(k - a). (L indicates the Laplace transform.) Noting that the

partial sums of (1.2.10) are bounded in the norm of (E) by exp(13'I IIAII) we take

A E C such that Re X > IIAII and integrate exp(- X t)S(t) in t > 0; term-by-term

integration is easily justified and yields

f00e-A"S(t)dt=

k-(n+1)An=R(k).

(1.2.13)

0 n-0

The Cauchy Problem for Some Equations of Mathematical Physics

(See (1.3).) This formula, generalized to unbounded A in Chapter 2, will be the basis

of our treatment of the abstract Cauchy problem. Of some importance also is the

representation of the propagator as the inverse Laplace transform of R (X ); in our

case, we can verify by means of another term-by-term integration that

SM=217if e"'R(X)dX

(1.2.14)

where F is a simple closed contour, oriented counterclockwise and enclosing a (A) in

its interior. Formula (1.2.14) has counterparts for unbounded A (see Example 2.1.9).

Note also that (1.2.14) is nothing but the prescription to compute efA according to

the functional calculus sketched in Example 3.12.

1.2.3 Example.

(a)

Using (1.2.14) show that if rM = sup(Re X; i E a(A))

and if w > rM, then

IIS(t)JI <Ce" (t>_ 0) (1.2.15)

for a suitable constant C. (b) Show that (1.2.15) does not necessarily hold if w = rM.

1.3. THE DIFFUSION EQUATION IN A SQUARE

Equation (1.1.1) describes the evolution in time of the temperature of an

homogeneous plate occupying the square 0 < x, y < it (tc is the ratio of the

conductivity to the specific heat; see Bergman-Schiffer [1953: 1, Ch. 1]),

the boundary condition (1.1.3) expressing the fact that the temperature at

the boundary is kept equal to zero. In this case, the choice of the space E in

Section 1.1 is natural enough. However, the equation is also a model for

diffusion processes; in that case u(x, y, t) is the concentration at (x, y) at

time t of the diffusing substance while the boundary condition (1.1.3)

expresses that particles reaching the boundary F are absorbed. (See

Bharucha-Reid [1960: 1, Ch. 3], where the interpretation of K is also

discussed.) In the diffusion case the supremum norm has no obvious

physical meaning; on the other hand, Ju(x, y; t) dx dy is the total amount of

matter present at time t and this suggests that the L' norm is the natural

choice here. (Observe, incidentally, that only nonnegative solutions should

be admitted since densities cannot be negative; the same is true for heat

processes if we measure temperatures in the absolute scale. We shall

comment on this later in this section). Since no additional complication is

involved, we take E = LP(S2), where 1 < p < oo. The main difference with

the case E = Cr(S2) considered in Section 1 is that now functions in E are

only defined modulo a null set, and thus it makes no sense trying to impose

the boundary condition on every function in E. This difficulty, however,

may be readily circumvented by including the boundary condition in the

definition of the domain of A. In fact, we define here D(A) to be the set of

all u such that ux, uy,, uXx, uX3,, uyy exist in 9, are continuous in 12, and such

that u = 0 on F. Condition (a) of the definition of well-posed Cauchy

1.3. The Diffusion Equation in a Square 33

problem with E = LP(2) is a consequence of the same condition in E =

Cr() in fact, since Cr() c LP(S2) and convergence in C_r(S2) implies

convergence in LP(0), a solution of (1.1.6) and (1.1.7) in Cr(S2) is as well a

solution in LP(Q). (We take D again to be the set of all uo whose Fourier

development (1.1.9) satisfies (1.1.10).)

We check now condition (b). Let be an arbitrary solution (in

E = LP(S2)) of

u'(t) = Au(t) (t >, 0), (1.3.1)

U(O) = uo. (1.3.2)

For each t >, 0 we develop u(x, y, t) = u(t)(x, y) in Fourier series,

u(x, y, t) - F, Y- a,,,,, (t)sinmxsinny. (1.3.3)

We have

n=l m=1

a' Jt)= lim

amn(t+h)-amnlt)

h-0

4 u(x, y, t + h)- u(x, y, t)

= lim -f

sin mx sin ny dx dy.

0712 a

(1.3.4)

The quotient of increments in the integrand converges in the norm of LP(S2)

to Au(t)(x, Y) = KDu(x, y, t). An application of Holder's inequality shows

that we can take limits under the integral sign, obtaining

a;,,n(t)

=-f iu(x,y,t)sinmxsinnydxdy

4K (M2 + n2)

_ -

fu(x,

y, t)sinmxsinnydxdy

_ -K(m2+n2)amn(t) (t0).

Since a,,,,, (0) = amn, where (amn) are the Fourier coefficients of u0, it follows

that amn(t) = a nnexp(- K(m2 + n2)t) and thus that u(x, y, t) equals the

sum of its Fourier series in t > 0:

u(x, y, t)

00 00

Y, Y.

e-,,(mz+"Z)`an,nsinmxsinny

m=1 n=l

°°

-f2

m=l n=1

(t>0,(x,y)En),

where the interchange of summation and integration is easily justified and G

is well defined and infinitely differentiable for t > 0. We deduce next some

The Cauchy Problem for Some Equations of Mathematical Physics

immediate properties of G. Observe first that, as pointed out in Section 1.1,

if uo E D, then the function u(x, y, t) defined by (1.3.5) is a classical

solution of (1.1.1), (1.1.2), and (1.1.3) to which we can apply the maximum

principle; it follows in particular that if uo is nonnegative, the same is true

of u(x, y, t). Then, if {(p) is a mollifying sequence in 61(R2) (see Section 8),

(1.3.6)

for (x, y), (x', y') E 9, and n large enough. But the integral (1.3.6) ap-

proaches G(x, y, x', y', t) when n --> oo, so that

G(x, y, , rl, t) >_0

(t>0, (x, y),

,q) E U).

(1.3.7)

On the other hand, if n is a positive integer, we can choose q)n E 6 (R 2) with

support in SZ such that 0 < rl) < 1 and

(1.3.8)

Again by the maximum principle,

n) d dn < 1

(1.3.9)

for (x, y) E 2, whereas, by virtue of (1.3.8), the integral on the left-hand

side of (1.3.9) converges to JG(x, y, , rl, t) did q as n - oo. We deduce then

that

(t>0,(x,y)En). (1.3.10)

We go back now to (1.3.5). Let 1 < p < oo, p' -' =1- p -'. Applying Holder's

inequality we obtain

u(x, Y, ')I

f G(x,y,

t t t

f G(x,

'n,

uo(S,'n)I G(x, y, S,'n, t)'/P'dS dal

I/P

(

I/P'

(fG

t)Iuo(n)IPdEdii)

(f G(x, y, ,rl,t)d

,d71)

(1.3.11)

where we have used (1.3.7). It follows from (1.3.10) and from the immediate

fact that G is symmetric in (x, C) and in (y, n) that

'The argument is modified in an obvious way when p = I.

1.3. The Diffusion Equation in a Square 35

for

r1) E U. Taking then the p-th power of (1.3.11) and integrating in SZ

we obtain

Ilu(t)Ilp < Iluollp

(t % 0) (1.3.12)

for

1 < p < oo, where II Ilp indicates the norm of LP(Q) and is the

LP-valued function u(t)(x, y) = u (x, y, t). Since u(.) was an arbitrary

solution of (1.3.1) and (1.3.2), inequality (1.3.12) completes the proof that

the Cauchy problem for (1.3.1) is well posed in LP(S2).

We note that (1.3.7) together with the representation (1.3.5) (which

holds for an arbitrary solution of (1.3.1)) establish the important fact that a

solution whose initial value is nonnegative remains nonnegative forever, a

fact that is essential if the diffusion interpretation of (1.3.1) is to make

sense. The same property holds of course for the solutions in Cr(Sl) treated

in Section 1.2 since, as we have pointed out before, a solution of (1.1.6) in

Cr(Sl) is automatically a solution of (1.3.1) in LP(f ).

We have seen in Section 1.1 that in the Cr(Sl) setting there was a

complete correspondence between classical solutions of the original equa-

tion (1.1.1) and solutions of the abstract differential equation (1.1.6). In the

LP(U) case, however, one wonders what kind of solution of (1.1.1) is

provided by solutions of (1.3.1). Differentiability of u in the space variables

of course depends on how the domain of the operator A is defined. It turns

out that differentiability with respect to t in the LP sense involves no more

than existence of the partial derivative in the sense of distributions and

"mean continuity" conditions on it. We make this more precise below.

1.3.1 Lemma. Let U be an arbitrary domain in Euclidean space R"',

and let t - u(t) be a function with values in LP(Q) (1 < p < oo) defined and

continuously differentiable in 0 < t <T. Then the function u(x, t) = u(t)(x)

satisfies Dtu = v in the sense of distributions, where v(x, t) = u'(t)(x). Con-

versely, let u(x, t), v(x, t) be functions in LP(Sl x (0, T)) such that Du = v in

the sense of distributions and assume that v(t)(x) = v(x, t) is continuous in

0 < t <T as a L'(S2)-valued function. Then we can modify u(x, t) in a null set

in Sl x (0, T) in such a way that u(t)(x) = u(x, t) is continuously differentiable

as a Lp(1)-valued function and u'(t) = v(t) (0 < t <T ).

Proof.

Let T E 6D with support in Sl x (0, T). Then we have

S2 x(0, T)

u(x, t)D,cp(x, t) dxdt

= lim f u(x,t)h(p(x,t+h)-g)(x))dxdt

h-.0 Stx(0,T)

nim

fSt(oT>h(u(x,t-h)-u(x,t))gp(x,t)dxdt

v(x,t)p(x,t)dxdt.

x(O,T)

The Cauchy Problem for Some Equations of Mathematical Physics

(Here we take I h < d, where d is the distance from the support of T to the

top and bottom of the cylinder 2 x (0, T).) This proves the first half of

Lemma 1.3.1. To prove the converse, we consider the function

w(x,t)= frv(x,s)ds (xESZ,0<t<T).

This function is defined almost everywhere in SZ x (0, T), and an integration

by parts shows that, always in the sense of distributions,

Dtw=v

in 9 x (0, T) (note that w E LP(0 x (0, T)) if v does). It then follows that

Dt(u - w) = 0, thus u - w must equal almost everywhere a function of x

alone; this implies that

h(u(x,t+h)-u(x,t))=

ft+hv(x,s)ds

(1.3.13)

almost everywhere (of course we take h sufficiently small; if t = 0, then

h > 0, if t = T, then h < 0). We obtain making use of Holder's inequality

that

h (u(x, t+h)-u(x, t)) - v(x, t))

P I/P

= (fal I Ir+h(v(x,s)-v(x,t))ds

dx)

S I

I f Iv(x,s)-v(x,t)IPdxds

I/P

-*0 as h-*0

`h flX(r,t+h)

(with the adequate modifications if h < 0). This ends the proof.

Lemma 1.3.1 shows, roughly speaking, that solutions of an equation like

(1.1.1) considered as an equation in a LP space will only be weak solutions in the

sense of distribution theory, at least as regards t-dependence. This need not concern

us unduly since classical solutions are in no way "required by nature" (see

Truesdell-Toupin [1960: 1, p. 232]) and weak solutions of one kind or another are

perfectly acceptable to model physical phenomena. On the other hand, weak

solutions are, from a mathematical point of view, usually conducive to simpler and

more powerful theories, as made evident by the work of Sobolev, Friedrichs, and

Bochner in the thirties and forties and by the widespread use of distribution theory

in the modern treatment of partial differential equations.

It should be pointed out, however, that Lemma 1.3.1 is somewhat

irrelevant for the heat-diffusion equation and will only come fully into its

own in the next two sections. The reason for this observation is the

following result, where St is again the square ((x, y); 0 < x, y < 7r).

1.3.2 Lemma.

Let

be an arbitrary solution of (1.3.1) in LP(2),

15 p < oo or in Cr(SZ). Then u(x, y, t) = u(t)(x, y) = (S(t)u)(x, y) can be