Mathematical Analysis and Numerical Methods for Science and Technology || Evolution Problems: Variational Methods

Chapter XVIII. Evolution Problems: Variational Methods

Introduction. Orientation

In the preceding chapters, we have considered some methods of resolution oflinear evolution problems leading to an explicit formulation of the solutions (method of diagonalisation, Laplace transform, semigroups). As we have mentioned, the field of application of these methods is limited and, in particular, does not cover the case of equations whose coefficients depend on x and t which occur when we consider non-homogeneous media whose properties evolve in the course of time. Further, the explicit formulation of the solution obtained makes use of elements (eigenvectors in the case of the method of diagonalisation, the evolution operator G(t) in that of semi groups ... ) which are not, in general, numerically attainable. Variational methods have already been considered in Chap. VII and it has been seen that the origin of these methods are principally in the mode of expression of physical laws (like the principle of virtual work in mechanics) on the one hand, and on the other hand in the research of solutions in finite dimensional spaces for stationary problems of the form Au = f, with A an elliptic operator. These motivations are transposable into the framework of evolution phenomena. Firstly, the intervention of time leads to replacing stationary systems (described in a Hilbert space H) by 'dynamical' systems (that is to say dependent on time). We shall make the mathematical formulation of such systems precise by limiting ourselves to finite time intervals. The dynamic states will be, at present, functions of time t E ]0, T[ with values in H. The most natural Hilbert space framework is then the space .Yt' = L 2 (0, T; H) (if H = L2(Q) with Q an open set of [Rn, then .Yt' = L2(QT ) = L2(Q x (0, T)))(l). In fact, the space of dynamic states which has a physical sense must be more restrictive: on the one hand the continuity of t seems physically necessary, (but CC([O, T]; H) does not have Hilbert space structure); on the other hand, we impose in general that each of these states has finite energy at each instant t, which is expressed by: u(t) E V for t E [0, T] with Va Hilbert space contained in H, where u represents a dynamic state. We shall define, in this chapter (§ 1.2), a space W which fulfills all the requisite conditions (Hilbert space, of continuous functions and finite energy for t E ]0, T[).

(1) Note that the space L 2(0, + 00, H) is, in general, 'too narrow' for a space of dynamic states, which forces us to consider time intervals ]0, T[ with T finite.

R. Dautray et al., Mathematical Analysis and Numerical Methods for Science and Technology© Springer-Verlag Berlin Heidelberg 2000

468 Chapter XVIII. Variational Methods

We have established that numerous evolution problems considered in the preceding chapters lead naturally to weak variational problems of the type: find t ...... u(t), u(t) E V, satisfying in the sense of.@' (]O, T[)

(1)

i) -~ (u( . ), v) + a(u(.), v) = (f(.), v) for all v E V dt

in the case of an equation of first order in t

ii) ~(u(.), v) + h(u'(.), v) + a(u(.), v) = (f(.), v) for all v E V (2)

dt

in the case of an equation of second order in t ,

(with given initial conditions), where we denote by a and b some bilinear (or sesquilinear) forms over V with (at least) a V-coercive, or V-coercive with respect to H; f models a source assumed known. The goal of this chapter is to adapt the variational methods seen in Chap. VII, for stationary problems, to the case of (mixed) evolution problems; this allows us, amongst other things, to give a constructive approximation to some solutions, likewise in the case of equations with coefficients depending on x and t. The interest of these methods resides therefore: on the one hand in the fact that they give the possibility of treating the majority of examples considered in the preceding chapters, (with the notable exception of hyperbolic systems of first order; see the chapters Transport and Numerical Methods); on the other hand that they prepare the ground for different approximation methods (see later in Chap. XX) and that they are particularly well adapted to nonlinear problems. These variational methods are also often the most natural for physical problems. However, they probably demand the greatest effort from the reader in order to comprehend them, the proofs need numerous stages and the formulations may sometimes appear 'abstract'. We shall consider, on the one hand the first order problem (1)i), with the initial condition u(O) = uo and a(u, v) replaced by a(t; u, v)(3); on the other hand the following problem of second order in t, more general than (l)ii): Find t ...... u(t), with u(t) E V, satisfying, in the sense of.@' (]O, T[)

{

c( . ; u"( . ), v) + b(.; u'( . ), v) + a( . ; u( . ), v) = (f(.), v)

(2) with the initial conditions

u(O) = uo, u'(O) = u 1 , uo, u1,f suitably given,

where a(t; u, v), b(t; u, v), c(t; u, v) are continuous bilinear (resp. sesquilinear) forms suitably given. We shall make the functional framework precise in § 1 and recall in § 1.6 certain weak compactness results. The Galerkin formulation (with variable bases) already considered in Chap. XII is recalled in §2. This constitutes the guiding

(2) Where u' denotes the derivative of u with respect to time. (3) Again with the hypothesis that a(t; u, v) is V-coercive (or V-coercive with respect to H).

§l. Some Elements of Functional Analysis 469

thread of this chapter. Also it is essential, in order to comprehend the chapter, to have present, at each moment, the spirit of the Galerkin scheme indicated in §2.3. The rest of the chapter examines the case of equations of first and of second order in t as well as some examples.

§ 1. Some Elements of Functional Analysis

We shall first of all recall here, in full, the ideas introduced in Chap. XVI concerning vector-valued distributions.

1. Review of Vector-valued Distributions

Let X be a Banach space, ] a, b [ an open set of lit The definitions which we shall recall are valid for an arbitrary open set Q of IRn; dt denotes the Lebesgue measure over ]a, bE.

Definition 1 a) We denote by U(a, b; X) (1 :!( p < + 00) the space of (classes of) functions: t -. f(t): ] a, b [ -. X such that

(1.1) {i) f is measurable for dt ,

ii) IlfIIU(a.b;X) = (f IIF(t)ll~dty/P < + 00 .

b) We denote by LOO(a, b; X) the space of (classes of)functions f from ]a, b[ into X satisfying i) and

(1.2) { ii)' f is bounded almos~ everywhere over] a, b [ and we set

IlfIIC(a.b;X) = mf(M) Ilf(t)ll x ';; M a.e.

This definition being given, we prove (ref. Bourbaki [3] for example),

Proposition 1. For 1 :!( p:!( + 00, U(a, b; X) is a Banach space. Let X and Y be two Banach spaces; Y(X, Y) denotes the space of continuous linear mappings of X -. Y.

Proposition 2. Then

(1.3)

Let u E L 1 (a, b; X) and dE Y(X, Y).

{

i) du E L 1 (a, b; Y)

ii) f du(t)dt = d(f U(t)dt).

Let X' be the dual of X and let <, > the duality coupling between X' and X.


Corollary 1. Ifu E Ll(]a, bE; X) and fE X' we have:

(1.4) \J. I: U(t)dt) = I: <J. u(t»dt.

On the other hand, we know that if [a, b] c IR (therefore a and b finite), then

U([a, b], IR) c+ Ll(a, b; IR), p ~ 1 ,

therefore:

Corollary 2. For p ~ 1, and a and bjinite, equality (1.4) is validfor u E U(a, b; X), fEX'.

We now recall

Definition 2. We call every continuous linear mapping of EtJ(]a, b[) into X a vectorial distribution over ]a, b[ with values in a Banach space X, and we note:

(1.5) EtJ'(]a, bE; X) = ~(EtJ(]a, b[); X) .

Following from this definition (and the results in Appendix "Distributions") we have

Proposition 3. Let A be a linear mapping from EtJ(]a, b[) into X. The following propositions are equivalent:

(1.6) { (i) A is a distribution; (ii) the restriction of A each EtJK(]a, b[) is continuous.

(EtJK(] a, b [) the space of functions of class C{f 00 with support contained in the compact set K, K c ]a, b[). Now let Lfoc(a, b; X), be the space of (classes of) functions u such that;

(1.7) for all compact K c ]a, b[, XKU E Ll(a, b; X),

where XK denotes the characteristic function of the compact set K. Then if qJ E EtJ (] a, b [), we have

(1.8) qJu E £1 (a, b; X) ,

therefore I: qJ(t)u(t) dt has a sense.

Further if qJ E EtJK(]a, b[), then:

(1.9) III: qJ(t)U(t)dtllx ::;;; s~PlqJ(t)I·(LIIU(t)lIxdt). Therefore by using Proposition 3, we obtain

Proposition 4. Let u E Ltoc(a, b; X); the mapping

qJ -+ I: qJ(t)u(t) dt

is a distribution over ]a, b[ with values in X.


We identify the function u with the distribution with which it is associated. We have in effect (ref. L. Schwartz [1])

Proposition 5. The functions u, v E Ll~C (a, b; X) define the same distribution if and only if u and v are equal (in a scalar sense) almost everywhere.

(This means that for all fE X' the functions: t -+ <f, u(t» and t -+ <f, v(t» are equal almost everywhere). If X is a separable space(4l, this proposition implies that u = v a.e. Recall now the definition of the derivative of a distribution f E ~'(] a, b [; X).

Definition 3. LetfE ~(]a, be; X) and m be an integer? O.

Then the mapping cp -+ (- l)m f(d mcp ), cp E ~ (] a, b [), is a distribution that we dt m

dmf denote by -d .

t m

We therefore have

(1.10)

Notation. Ifu E Lloc(a, b; X) and if X is a space of functions of the variable x, for example X = U(Q) then u is identified with a function u(x, t); u(t) denotes the function 'x -+ u(x, t)' for almost all t. The distributional derivative du/dt is identified with the derivative au/at of u in ~'(Q x ]a, b[).

We use thefoliowing notationfor the derivative ofu with respect to t interchangeably

du , au dt or u or at· o

The following situation (which we have met in some examples of Chap. XVI) which we shall consider in greater detail later on, is particularly interesting. Let (X, Y) be a pair of Banach spaces with

(1.11) X c; Y (c; continuous injection) .

Then, we have

(1.12) ~'(]a, be; X)c;~'(]a, be; Y)

and

(1.13) U(Ja, be; X) c; U(]a, be; Y), 1:( p :( + 00

This set, consider u E L2(a, b; X). Then for all cp E ~(]a, b[)

(1.14) ~~ (cp) = - f u(t)cp'(t)dt.

(4) We are in this situation very often. Note however that we must use the space L oo(Q) which is not separable (see for ex. Brezis [1]).


We say that u' = du/dt E L2(a, b; Y) if there exists v E L2(a, b; Y) such that:

{for all cp E .@(]a, b[), v(cp) = - u(cp') ,

(1.15) fb fb i.e.: a v(t)cp(t)dt = - a u(t)cp'(t)dt .

Remark 1. Let f E X' and 9 E .@'(]a, bE; X); we may define a scalar distribution [g, f] by:

(1.16) Vcp E .@(]a, b[), [g,f](cp) = (J, g(cp»

therefore [g,f] E .@'(]a, b[). In particular, suppose that 9 is defined by u E L2(a, b; X): formula (1.16) may be written, taking into account (1.4)

(1.17) [g,f](cp) = f (J,u(t)cp(t»dt, Vcp E .@(]a,b[). o

2. The Space W(a, b; V, V')

The space which we shall define at present is of fundamental importance in what follows. We shall explain the reasons later, after having given the definition and principal properties of this space. We consider two real, separable Hilbert spaces V, H. We denote by «,)) the scalar product and II II the norm in V and (,), I I the corresponding quantities in H. Moreover, we suppose that Vis dense in H so that, by identifying H and its dual H', we have (denoting by V' the dual of V, with norm II II.):

(1.18) V c:+ H c:+ V' ,

each space being dense in the following.

Definition 4. Let a, b E IR = IR u { - 00, + oo}. We denote by W(a, b; V, V') the space:

(1.19) W(a, b; V, V') ~ {u; U E L2(a, b; V), u' E L2(a, b; V')} (5)

Remark 2 i) Definition 4 has a sense from Sect. 1. ii) We may also consider the case where the Hilbert spaces which occur are complex; it is then appropriate to replace the notion of the dual with that of the antidual. 0

(5) When there is no risk of confusion, we use the simplified notation W(a, b) or even W.

§ 1. Some Elements of Functional Analysis 473

We have

Proposition 6. The space W(a, b; V, V') equipped with norm

(1.20)

lIull w = (li u lli2(a,b;V) + Il u/ lli2(a,b;V,))1/2 = (f [lIu(t)1I 2 + IIU I (t)II!]dtY/2

is a Hilbert space.

Proof Let {un}nE 1'\1 be a Cauchy sequence in W(a, b; V, V'), then

i) Un -+ U in L2(a, b; V) ,

ii) u~ -+ u 1 in L2(a, b; V'),

we must show that u' = u1 .

But from i), we have Un -+ U in .@/(]a, b[; V) and also in .@/(]a, b[; V') since .@/(]a, b[; V) c+ .@/(]a, b[; V'). We deduce that u~ -+ u' in .@/(]a, b[; V'). From ii) u~ -+ u 1 in .@/(]a, b[; V') and therefore u' = u 1 (uniqueness of the limit), from which we have the proposition. 0

Remark 3. We have met, in Chap. XV, Method of Diagonalisation, some example of spaces W(a, b; V, V') when V is a Sobolev space. 0

We are now interested in the regularity properties of elements of the space W(a, b; V, V').

Theorem 1. For a, b E IR, every u E W(a, b; V, V') is almost everywhere equal to a continuous function of [a, b] in H. Further, we have:

(1.21) W(a, b; V, V') C+ rcO([a, b]; H) ,

the space rcO([a, b]; H) being equipped with the norm of uniform convergence.

Proof a) We shall start by stating two lemmas which we shall prove later.

Lemma 1. For a, b finite or not, let .@([a, b]; V) be the space of restrictions to [a, b] of functions of.@ (IR; V)(6)

(1.22) .@([a, b]; V) is dense in W(a, b; V, V')

Lemma 2. For a, b finite or not, there exists a continuous prolongation operator P: W(a, b; V, V') -+ W( - 00, 00; V, V').

b) Let u E W(a, b; V, V'), and P be a continuous prolongation operator of W(a, b; V, V') into W( - 00, 00; V, V'). From Lemma 1, there exists a sequence {t/ln}nEI\,j' t/ln E .@(IR, V) satisfying

(1.23) Pu = lim t/ln in W( - 00, 00; V, V') .

(6) Thus !0([ - 00, + 00]; V) = !0(IR, V).


Furthermore, <,) denoting the duality pairing between V' and V, we have:

It d It d Il/In(tW = -00 d(T Il/In((T)IZd(T = -00 d(T (I/In((T), I/In((T))d(T

= 2 foo (I/In((T),I/I~((T))d(T = 2 foo <1/I.((T),I/I~((T)d(T

~ 2 f 00 III/I.((T) IIIII/I~((T) II * d(T ;

by applying the inequality 2ab ~ (a Z + bZ ), it becomes:

(1.24) Il/In(t)IZ ~ foo II I/In((T)IIZd(T + foo 1I1/1~((T)II;d(T from which

(1.25) sup I I/In(t) I ~ 111/1. Ilw· t

Then replacing I/In by (I/In - I/Im) in (1.25) and by using the fact that I/In is a Cauchy sequence in W( - 00, + 00; V, V'), we deduce that I/In is a Cauchy sequence in BO(IR; V), the space of continuous bounded functions of IR in V equipped with the topology of uniform convergence. We then have:

(1.26) 1/1. -> Pu in W( - 00, + 00 ; V, V')

which implies

(1.27)

from which

(1.28) Pu = va.e. and u = v a.e. over [a, b] .

Further, we have

(1.29) Il u ll'60 ([a,b]; V) ~ CIIull w (C constant) .

Thus Theorem 1 is proved subject to verifying lemmas 1 and 2.

c) Verification of Lemma 1. This is done in three stages. 1st stage: We restrict ourselves to the case where a or b is infinite.

o

If [a, b] c 1R(7), we introduce (}i(i = 1,2) E g([a, b]) the space of restrictions of functions of g(J(IR) to [a, b] with

(1.30) {(}l (t) + (}z(t) = 1 for t E [a, b]

(}l(resp. (}z) null in a neighbourhood of b (resp. of a).

(7) a and b are finite.


Then, for all U E W(a, b; V, V'), we have

(1.31)

Introduce:

( 1.32)

We easily verify that

for t E [a,b]

for t > b

for t E [a,b]

for t < a.

(1.33) U1 E W(a, + 00; V, V'), Uz E W( - 00, b; V, V') .

2nd stage: We restrict ourselves to the case where a = - eX), b = + 00.

Let U E W(a, + 00; V, V') and h > 0; set

(1.34)

from which

and

(1.35)

Uh(t) = u(t + h) a.e. t ~ a ,

u~(t) = u'(t + h) a.e. t ~ a

Uh E W(a, + 00; V, V') .

Further, due to the continuity of translations in L Z (see Chap. XVII)

(1.36) {Uh ~ U in L Z(a, + 00; V) as h ~ 0 ,

U;. ~ u' in LZ(A, + 00; V') as h ~ 0,

therefore

(1.37) Uh ~ U III W(a, + 00; V, V') as h ~ 0 .

Let'" E CCOO(IR) be such that 0 ~ "'(t) ~ 1,

( 1.38) { . h

"'(t) = 1 If t ~ a - -2

"'(t) = 0 if t ~ a - h

f ~ = 1

1 ~ = 0

a - h h a a- Z

. h If t ~ a - Z

if t";;;a-h

475


We therefore have by setting

(1.39) () _ {"'(t)Uh(t) Vh t - 0

Vh = Uh a.e. t ~ a, and further

ift~a-h

ift~a-h

(1.40) Vh E W( - 00, + 00; V, V') .

Therefore, it remains to show 3rd stage: 9C(IR; V) is dense in W( - 00, + 00; V, V'). Let U E W( - 00, + 00; V, V'); we shall start by reguiarising u, that is to say approximate it by u. E <iff 00 (IR, V). For this, let j E 9C (IR) such that

(1.41) j ~ 0, tj(t)dt = 1 ;

we set:

(1.42)

then u. is defined by

U.(t) = j. * u(t) = t j.(t - O")u(O") dO" .

Then (see Chap. IV) u. E <iff OO(IR; V) and for e tending towards zero:

(1.43) { u. -+ u in L 2(1R; V) ,

u~ = u'*j. -+ U in L2(1R; V').

It is therefore sufficient to approximate Us by elements of 9C(IR; V); for this, we use a truncation procedure. Let P E 9C (IR) such that

(1.44) { P(t) = 1, for It I ~ 1,

p(t) = 0, for It I ~ 2;

we set Pn(t) = p(t/n) and it is easy to verify that

(1.45) Pnu. -+ u. when n -+ 00 in W( - 00, + 00; V, V') .

Verification of Lemma 2. We proceed in two stages.

o

1st stage: We restrict ourselves to the case where a or b is infinite when [a, b] c IR; for this we use the same method as in the 1st stage of Lemma 1. 2nd stage: Suppose,for example that b = + 00. By translation over the variable h, we can reduce to the space W(O, + 00; V, V') Then let u E 9C([0, + 00]; V). Set

(1.46) ( ) {U(t) if t ~ 0 ,

Pu t = u( - t) if t < 0 .


Therefore Pu E L 2 (0, + 00; V) and further, we have

(1.47) (Pu(t}), = {u/(t} if t > 0, - u/( - t} if t < 0,

and since Pu is continuous at t = 0 (the Dirac rlleasure does not occur in the derivative), we deduce that Pu E W( - 00, + 00; V, V'}. Further, we have:

(1.48) IIPullw(-oo. +(0) ~ 21Iullw(o. +(0) (8).

Since from Lemma 1, ~([O, + 00[; V) is dense in W(O, + 00; V, V'), P can be prolonged into a continuous linear mapping of W(O, + 00; V, V'} -+ W( - 00, + 00; V, V'}.

As Pu is again given by (1.46) a.e., it is therefore such that Pu = u a.e. for t E ]0, + oo[ from which we have the result of the lemma. Thus the proof of Theorem 1 is completed. 0

Remark 4. Consequence of Theorem 4. Trace result For u E W(a, b; V, V') with [a, b] c IR, we may speak of the traces u(a}, u(b} E H.

Remark 5. We can prove that the mapping u -+ u(a} from

W(a, b; V, V') is surjective.

o

This result will be obtained as a consequence of a forthcoming theorem. (Remark 8 cl§n 0

Theorem 2 (Green's formula or, more simply, integration by parts). We assume [a, b] c IR. Let u, v E W(a, b; V, V') then

(1.49) f <u/(t), v(t) dt + f <v'(t}, u(t) dt = (u(b), v(b)) - (u(a), v(a)) .

Proof From the trace result, formula (1.49) has a sense. As it is true for u, v E ~([a, b]; V), it remains so for u, v E W(a, b; V, V') by virtue of Lemma 1 and of Theorem 1. 0

A very useful property for what follows is given by

Proposition 7. For u E W(a, b; V, V'), u E V, we have:

(1.50) <u'(.), v) = :t (u(.),v) in ~/(]a,b[).

Proof a) Let cp E ~(]a, b[), we choose v E W(a, b; V, V') of the form cp ® v with, for the moment, v E V (i.e. such that v(t) = cp(t)v) in (1.49). Then since cp(a) = cp(b) = 0, we obtain:

f <u'(t), cp(t)v) dt + f <cp'(t)V, u(t) dt = 0 .

(8) Omitting V and V' for conciseness.


Since for all t E ]a, b[, <p(t) is a scalar, we have again

f <u'(t), v)<p(t)dt = - f <v, u(t)<p'(t)dt .

Now, observe that u(t), v EVe H and that the duality < , ) is compatible with the identification of H with its dual [which implies that if f E H, v E V, we have

<f, v) = (f,v) = (v,!)],

we obtain, for all <p E E0(]a, b[)

(1.51) f <u'(t), v)<p(t)dt = - f (u(t), v)<p'(t)dt ,

from which we have the result. b) Another proof of (1.50): Let <u'(t), v) is in U(a, b). Then from Corollary 1

f <u' (t)<p(t), v) dt = \ f u' (t)<p(t) dt, v)

= - \ f u(t)<p'(t)dt, v) = - f <u(t)<p'(t), v) dt ,

and since u(t) E Va.e. t E ]a, b[, we have:

f <u'(t), v)<p(t)dt = - f (u(t), v)<p'(t)dt

from which we have the result.

We shall explain the reasons for the introduction of the space W(O, T; V, V'). We wish to resolve the equation:

du dt + Au = 0

in ] 0, T[ where A is a 'variational' operator, therefore

A E Sf(V; V'), V c H c V'.

For Au to have a meaning, it is reasonable(9) that u takes values in V, that is

u E U(O, T; V), 1:;:; p :;:; 00 .

D

(9) But not indispensable. There is, in linear problems a very large latitude for the choice of functional framework.


Then au/at = - Au E U(O, T; V'). We are therefore led to the space of u E U(O, T; V) such that au/at E U(O, T; V'). If, moreover, we want to remain in a Hilbert space framework(lO), we must take p = 2 and we arrive at W(O, T; V, V'). We then have the structure of a Hilbert space and we have given a meaning to the expression 'u takes the value Uo at t = 0' (with a continuous mapping u -+ u(O) E H).

3. The Spaces W(a, b; X, Y)

The situation considered in Sect. 2 is, in fact, a particular case of a more general situation. Let X and Y be two complex separable Hilbert spaces with

(1.52) X ~ Y continuous injection, X being dense in Y.

Example 1. The space X may be defined as the domain of an unbounded operator A in Y, which is positive and self-adjoint, X having a norm equivalent to the graph norm:

(1.53) (1Iull~ + IIAull~)1/2, U E D(A) = X.

A procedure for constructing such an operator A starting from X and Y is the following (which is due to the variational formulation of elliptic operators, see Chap. VII). We denote by D(A) the set of u of X such that the antilinear form (linear if the spaces are real)

v -+ (u, v)x

which is continuous over X for the topology induced by Y. Then

(1.54) (u, vh = (Au, vh , which defines A as an unbounded operator in Y with domain D(A). We then verify (see Chap. VII) that D(A) is dense in Y; A is self-adjoint (i.e. A = A * and D(A) = D(A *) and positive, since

(1.55) (Au, u)y = luli ~ Clul~ (C = constant) Vu E D(A) .

We can then define A 1/2 with domain D(A 1/2) = X and we set

(1.56) A = A1/2 •

Therefore A is positive self-adjoint in Y with domain D(A) = X and we deduce from (1.55)-(1.56) that

(1.57) (u, vh = (Au, Av)y for all u, v EX.

Then by using the spectral decomposition of self-adjoint operators (see Chap. VIII, Chap. XV and Remark 6 later), we recall the following d~finition.

(10) This is, in general, impossible for nonlinear problems.


Under hypothesis (1.52) and A being defined by (1.56), we set

(1.58) def

(X, Y)o = D(A 1-0), (J E [0, 1] (11)

equipped with the graph norm of Al -0. o We now define the space W(a, b; X, Y) by

def (1.59) W(a, b; X, Y) = {u; U E U(a, b; X), u' E U(a, b; Y)}

which, equipped with the norm

( 1.60) II u II w = (II u II ~2(X) + II u' II ~2(Y))1/2 ,

is a Hilbert space. Then, as in the case of the spaces W(a, b; V, V'), we show that E0([a, b]; X) is dense in W(a, b; X, Y). Further (see Remark 6 below), we see that:

{

i) X is dense in (X, Y)o ,

ii) lub, flo :::; C.luli-o.lul~,

iii) W(a, b; X, Y) ~ ~O([a, b]; (X, Y)1/2) .

(1.61)

Then under the hypotheses of Sect. 2, we verify that if X = V, Y = V', we have (X, Y)1/2 = H and the results of Sect. 2 are a particular case of the situation considered in this section. We shall give, in Sect. 4, a proof of a trace theorem in the framework of Banach spaces and in the case where X is the domain of the infinitesimal generator of a semi-group of class ~o. Result (1.61)iii) then comes from (see Remark 6) the fact that - A is such an infinitesimal generator. A direct proof with Hilbert space methods is obviously possible (see Lions-Magenes [1], Huet [1]).

Example 2. An example of the preceding situation We shall show that the properties of the trace of functions of H 1 (Q) (see Chap. IV, §4) are a particular case of (1.61). By using local maps, we return to the case of a regular open set or a half-plane of [Rn.

We therefore consider the space HI (Q) where Q = [R"r. In a precise manner, if x E [Rnwe'separatethevariables'bysettingx = (x', xn)with x' = (Xl"" ,xn - 1 );

then:

(1.62) [Rn = {x E [Rn. x > O} + , n

and r = [R~,- 1 = {x E [Rn; Xn = o} .

(II) With (X, y), = Y and (X, Y)o = D(A) = X, the space (X, Y)o is called (see Chap. VIII, §3 and Lions-Magenes [1]) the 'holomorphic' interpolant of the spaces X and Y.


Let u E HI (1R"t-). We may consider u as a function:

Xn --+ u(.,xn )

where u(., xn) is itself the function: x' --+ u(x'. xn) all these functions being defined a.e. in 1R"t-. It is then easy to verify that we have:

Proposition 8. The following two assertions are equivalent: 1) u E HI(IR"t-); 2)

(1.63)

It follows from this proposition that the trace of u E HI (1R"t-) over the boundary r = IR~;-I (see Chap. IV) occurs as the trace of the function u(., xn) at the origin of the X n •

By setting Xn = t, we have

(1.64) U E HI(IR"t.) ¢> u E W(O, + 00; X, Y)

with

(1.65)

To make the operator A with domain X precise, it is convenient to use the definition of Sobolev spaces over IR~,~ I with the help of the Fourier transform. Then (see Chap. IV, § 1), if (' denotes the dual variable of x', we have:

(1.66)

so that (1.54) becomes:

(u, v)H'(IW-') = r (1 J~"-,

= r Au. vdx' = (Au, V)LZ(W-') . J~"-l

Under the Fourier transform, the operator A therefore corresponds to multiplication by 1 + 1('1 2 (if Lln~1 = Laplacian in IRn~l, A = I - Lln~I)' The operator A = A 1/2( = (l - Lln~ d 1/2 ) then corresponds to multiplication by (1 + 1('12)1/2 and its domain is D(A) = Hl(lRn~l).

Generally, the operator A 1 ~8, e E [0, 1] corresponds, under the Fourier transform, to multiplication by

so that:

(1.67) D(Al~8) = (X,Y)8 = Hl~8(1R~;-1), eE(O,l),


and in particular:

(1.68) D(A 1/2) = (X, Y)1/2 = H 1/2(~n-l) = H 1/2(r) .

As u E W(O, + 00; X, Y)(X, Y given by (1.65)), for all T > 0, the restriction of u to [0, T] is in ~o([O, T]; HI/2(~~;-I)) from (1.6t)iii). We deduce, in particular, that we have:

(1.69)

which again gives the trace result obtained (see Chap. IV) by a direct method. 0

4. Extension to Banach Space Framework

There are no particular reasons to limit ourselves to the Hilbert space framework and it is useful to place the problem of traces in a more general context. For this, let X and Y be two Banach spaces with X c+ Y, the injection of X into Y being continuous with dense image. We introduce for p ~ t:

def du (1.70) Uj,(0, + 00; X, Y) = {u; u E U(O, + 00; X), dt E U(O, + 00; Y)} .

The trace problem is then the following: 1) can we define u(O)? (u(O) will be the trace of u at the origin), 2) can we characterise the space to which u(O) belongs, when u is in Uj,(0, + 00; X, Y)? As an example of this situation, we have particularly:

au X = W1.P(Q) = {u; u E U(Q), -;- E U(Q), i = t, ... , n} (12)

uXi

(a Sobolev space constructed over U(Q)) and Y = U(Q) with p ~ 1. In order to consider, in particular, the case p = t, it is useful (and this does not complicate the following proofs) to introduce a problem a little more general. Let p and ex be two real numbers with:

(1.71) t ~p~ +00;

we denote by W(p, ex; X, Y) the space of (classes of)functions u such that:

(1.72)

(1. 73)

tau E U(O, + 00; X) ,

t a :~ E U(O, + 00; Y) .

The sense of (1.73) is the following: there exists g such that tag E U(O, + 00; Y) and

(12) See Chap. IV.

§1. Some Elements of Functional Analysis 483

that for all q; E ~(]o, + oo[), we shall have:

(1.74) (+00 (+00

- Jo u(t)q;'(t)dt = Jo g(t)q;(t)dt (integrals taken in Y) ;

we equip the space W(p, 0(; X, Y) with the norm:

(1.75) II u II w'.P = max (II tau II UfO, + CJ'J; Xl' II ta ddu II ) , t UfO, + 00; y)

We leave the reader the task of verifying that, equipped with this norm, the space W(p, 0(; X, Y) is a Banach space, We shall now show the following proposition:

Proposition 9 1) Every function u E W(p, 0(; X, Y) is (a,e. equal to) a continuous function of t>O~y'

2) Let U E W(p, 0(; X, Y) (denoted also by ~, p). Then if:

1 (1.76) - + 0( < 1 ,

p

u(t) converges in Yas t ~ 0.

Proof 1) 1st point. Let Y' be the dual of y(13).

For g E U(O, + 00; y), for all y' E Y' we may define the scalar function:

<g, y'): (t ~ <g(t), y'») E U(O + 00)

where <,) denotes the duality pairing between Yand Y'. du

Set Cit = g and introduce for almost all t:

(1.77) v(t) = u(t) - J: g(a)da ,

then for y' E Y', <v, y) E U(O, + 00).

We calculate its derivative in the sense of distributions over ]0, + 00[. For q; E ~(]O, + oo[), we have:

(:t <v,y'),q;) = - «v,y'),q;') = - fo+ oo <v(t),y')q;'(t)dt

= -(I+oo U(t)q;'(t)dt,y,) + I+OO(J: <g(a)'Y')da)q;'(t)dt,

(13) For this part 1), we may also take 9 E L~oc(]O, + 00[; Y); we shall thus find: U E L~oc(]O, + 00[; X) du

and - E L P) (]O, + 00 [; Y) = U E ~O(]O, + 00 [; Y). dt oc


but

I+ ro (J: <g(o"), y' > dO}P' (t) dt = - ( I+ ro g(t)<p(t) dt, y') ,

therefore:

(:t <v, y' >, is independent of t therefore v(t) = y a.e. (y = constant E Y) from which we have from (1.77):

(1.78) u(t) = y + J: g(a)da a.e. in t,

which proves point 1). 2) 2nd point. We deduce from (1.78):

u(t) - u(s) = f g(a)da = fa. :; a-·da ,

from which by using the Holder inequality, we have:

(1.79)

lu(t) - u(s)lr ~ (fla.:; I: da )l 1P (f a-.p'daYIP',

1 1 - + ---;- = 1, s, t > 0 ; P P

we may take s = 0 in the right hand side of (1.79) if I a-·P' da < + 00, which

holds when (1.76) is satisfied. From which we have the result. D

Definition 5. We denote by T(p, a: X, Y) (also noted 1'., p) the space (trace space) to which u(O) belongs when u belongs to W(p, a; X, y), a and p satisfying (1.76).

We leave the reader to verify that, equipped with the norm:

(1.80)

Ta, P is a Banach space.

II a II Ta,p = inf II u II W a• p , UEW a.p u(O) =a

We shall now characterise T •. p when X = D(A) where A is the infinitesimal generator of a bounded semigroup of class Cfjo in Y (see Chap. XVII A). Therefore let {G(t)} be a semigroup of class Cfj 0 in Y satisfying:

(1.81 ) II G(t) II ~ M, t? 0 ,

and let A be its infinitesimal generator.

§ I. Some Elements of Functional Analysis 485

We set X = D(A) and we suppose that D(A) is equipped with the graph norm:

(1.82) laID(A) = laly + IAal y ;

X = D(A) is then a Banach space dense in Y. We shall show

Theorem 3.

( 1.83)

hold.

We suppose that (1.81) and

1 O<-+a<

p

Then the space T(p, a; D(A), Y) coincides with the space of a E Y such that:

(1.84) I+oo t(·-I)PIG(t)a - al~dt < + 00.

The norms 1/ a 1/ T •. p and:

(1.85) (f +oo )I/P Illalll = laly + 0 t(·-I)PIG(t)a - al~dt

are equivalent.

Proof of Theorem 3. (In 3 stages) 1st stage: the Hardy-Littlewood-Polya inequality. We shall show

f + 00

Lemma 3. Let f be a numerical function such that 0 t·Plf(t)jP dt <

suppose that (1.76) holds. Then if we set:

(1.86) 1 ft g(t) = - f(O') dO' , t 0

we have:

+ 00; we

(1.87) ( f+ 00 )I/P 1 (f+ 00 )llP o t·Plg(t)jP dt ~ 1 _ G + a) 0 t·Plf(t)jP dt

Proof of Lemma 3. We set t = e" f(e f) = ](r), and we note that:

g(r) = e-ffoof(e<1)e<1dO',

so that if we introduce X by:

X(t) __ {O if 0 < t < 1 {O if r < 0 therefore x(r) = x(e f

) = e- f 1ft if t > 1 , if r > 0 ,


we have:

(1.88)

Set:

(1.89)

g = X *] (where * denotes the convolution) .

1 8=-+ex,

p

and note that the condition t'f E U(O, + (0) is equivalent to:

now from (1.88), we have: eOt 9 = (eOt X) * (eOt])

so that (the convolution properties to an L 1 function with an LP function):

II eOtg II LP ~ II eOtXII L' ·11 eo111 LP .

Since II eOt X II L' = _1_, we obtain (1.87). Besides, as the function eOt X is ~ 0, this 1 - 8

estimate is the best possible. o 2nd stage. Let U E W(p, ex; D(A), y), we shall show that u(O) = a (which has a sense in Y from Proposition 9) satisfies (1.84). For this, we shall use the following density lemma.

Lemma 4. Let W., p(X) be the space off unctions of W(p, ex; x, Y) (X and Y in the general situation of the beginning of this Sect. 4) which are infinitely differentiable from t > 0 -> X; w" p(X) is dense in W(p, IX; X, Y).

Assume, for the moment, this Lemma 4. Suppose firstly u E w" p (with here X = D(A» Set:

(1.90) - Au + u' = f;

we know (see Chap. XVIIB, § 1) that we have for t ~ /: > 0

(1.91 ) u(t) = G(t - /:)u(/:) + f G(t - o")f((J) d(J ,

from which:

(1.92) G(t - /:)u(/:) - u(/:) = f u'((J)d(J - f G(t - (J)f((J)d(J .

As /: -> 0 we have (with (1.76»:

f u'((J)d(J -> I u'((J)d(J

f G(t - (J)f((J)d(J -> I G(t - (J)f((J)d(J

and u(/:) -> u(O) (by the definition of u, see Proposition 9) in Y.


From (1.92) we therefore deduce:

{ G(t)u(O) - u(O) = rot u'(a)da - rot G(t - a)f(a)da,

(1.93) JI JI u E JYa, p(X) .

Now if u E W(p, IX; x, Y) we may find, from Lemma 4, a sequence {Un}nEN with Un E JYa, p(X) such that Un -+ U in W(p, IX; X, Y) and (1.93) is valid for each Un' We may therefore pass to the limit over Un in (1.93) and we obtain the following result: Let U E W(p, IX; X, Y), u(O) = a; then:

GWa-a lit lit (1.94) = - u'(a)da - - G(t - a)f(a)da , t tot 0

with: - Au + u' = f;

we deduce (by using (1.81)):

IG(t)a - a I 1 it Mit :::; - lu'(a)lyda + - If(a)lyda, t y tot 0

from which, using Lemma 3:

(1.95) ( r+ 00 )l/P Jo t(a-l)PIG(t)a - al~dt

1 (r+oo )l/P :::; 1 - () Jo taPlu'(t)l~dt

+ 1 ~ () (I+oo tapi - Au + U'IPdtYIP ,

which proves (1.84) and further establishes that:

(1.96) III a III :::; C. II a II T •.•.

3rd stage. Now let a E Y satisfy: Illalll < + 00.

We shall show that a E T(p, IX; D(A), Y). For this we introudce:

(1.97) 1 it v(t) = - G(a)a da , t 0

and

(1.98)

{

u(t) = q(t)v(t) ,

q i~ a s~alar function of class C(j 1, with q = 1 in a neighbourhood of 0 ,

q=Olft~l.


If a E D(A) then 1 it 1 it d G(t)a - a

(1.99) Av(t) = - AG(a)ada = - ~d G(a)ada = . to toa t

Since D(A) is dense in Y, by passage to the limit, (1.99) becomes true for all t > ° and for all a E Y. It follows that

(1.100) G(t)a - a

Au(t) = q(t)· . t

We shall show that u E W(p, a; D(A), Y) (we say that u is a lifting of a in the space W(p, a; X, Y)). First of all, due to (1.84) and (1.100), we have:

(1.101) t~Au E U(O + 00; Y).

Since moreover Iv(t)ly ~ Mlal y we see that t~u E U(O, + 00; Y) if t~q E U(O, + 00), which holds if and only if - ap < 1 therefore if:

1 -+a>O. p

Then u' = q'v + v' q and since t~q' v E LP(O, + 00; Y) (a arbitrary) we shall have t~u' E U(O, + 00; Y) if we verify

(1.102)

Note that: 1 1 it v'(t) = - G(t)a - 2 G(a)a da t t 0

1 1 it = - (G(t)a - a) - ~ (G(a)a - a)da , t t 2 0

now t~ [~(G(t)a - a)] E U(O, + 00; Y) by hypothesis and ta[:2 I (G(a)a - a)da]

E U(O, + 00; Y) by using the Hardy-Littlewood-Polya inequality (Lemma 3), a replaced by a-I. From which we have (1.102) and u E W(p, a; D(A), y). As u(O) = a, then a E T(p, a; D(A), Y). Moreover II u IIw ~ CIII a III, from which:

".P

(1.103) II a II T ~ Clil a III ; ".p

(1.102) and (1.103) therefore shows the equivalence of norms and completes the proof of the theorem, subject to verifying Lemma 4. 0

Proof of Lemma 4. Let u E W(p, a; X, Y). We set t = e" u(t) = u(r); then if 1

() = - + a, u E W(p, a; X, Y) is equivalent to: p

1 e9tu E U(IR, X)

(1.104) d-e(9-1)! d~ E U(IR, Y) .


We now introduce j, satisfying (1.41), (1.42) and we set:

we have, for e tending towards zero:

(1.105) {

e9tfi -+ e 9tu in U(~; X)

(9 - 'l)t du, (9 -l)t du . LP(rTll Y) e --+e -In tN.;. dr dr

Thus u, defined by u,(t) = u,(log t) converges towards U in W(p, ex; X, Y) for e -+ 0 and u,(t) is infinitely differentiable from t > 0 -+ X. From which we have the result. o

Generalisation of Theorem 3. For applications (see Example 3, later) it is indispensable to generalise the situation of Theorem 3 a little. Let AI' ... , Aq be a family of infinitesimal generators of semigroups {Gdt)}, ... , {Gq(t)}, of class C(j0, in Y such that:

{II Gj(t) II ::::;; M j , j = 1, ... , q

(1.106) Gj(s)Gi(t) = Gi(t)Gj(t) for all i, j; s, t ;;::: 0

(the semigroups commute) .

We set: q

(1.107) D(A) = n D(A). j=I

It is easy to verify that X = D(A) is a Banach space for the norm: q

(1.108) lIaIID(A) = lair + L IAjalr · j= I

Theorem 4. We assume that (1.83) holds. Then T(p, ex; D(A), Y) coincides with the space of a E Y such that:

q (1+00 )I/P N(a) = .L t(a-I)PIG;(t)a - al~dt < + 00.

,= I ° The norms II a II T and lair + N(a) are equivalent. o.p

The result is valid if p = 00 by replacing, in N(a), the U norms by L 00 norms.

Proof The proof only differs from that of Theorem 3 by the choice of function v for the construction of a lifting of a in the space W(p, ex; D(A), Y). Here we take:

(1.109) { v(t) = hI (t)h2 (t) ... hq(t)a

1 it hi(t) = t J ° Gi(u) du, i = 1, ... , q . o


Example 3. Application to the traces offunctions of W I.P(Q)(14)

Let Q be a regular open set of IRn; we are reduced, by local maps, to the case where Q = 1R"t, = {x E IRn, Xn > O} and we note that:

(1.110) {au

u E wI. P(IR"t,) = u; u E U(IR"t,), aXi

E U(IR"t,) ,

is equivalent to:

(1.111)

We take:

(1.112)

{

U E U(O, + 00; WI.p(IR~;-I))

au - E LP(O + 00' LP(lRn-I)) ::l "x" uXn

.. ) a 1 11 Aj = ~, j = 1, ... , n - ,

i=1, ... ,n}

Ii) Y = U(lRn-I),

uX·

iii) Gj(t)f(x:) = f(x l , .•• , Xj-l' Xj + t, Xj+l" •. , xn-d.

Then D(A) given by (1.107) is:

(1.113)

Under these conditions, Theorem 4 gives:

Theorem 5. The following two conditions are equivalent(15)

i) f E T(p, rx; W1,p(lRn-I), U(lRn-l)

ii) fE U(lRn-l) and for j = 1, ... , n - 1,

(+00 t(<<-I)P( ( If(xl , .•. , Xj + t, ... , xn-d - f(X'WdX')dt < + 00(16). Jo JW- l

We then set by definition:

(1.114)

with

(1.115)

(14) See Chap. IV, §8.

1 (15) Again with 0 < - + C( < 1.

P (16) With X' = (XI' ••. , Xn- d.

1 O=-+rx.

p


We may take as the norm:

(1.116) Ilfllwl-6"(IR.-I) = Ilfll~p(IR,-1) + ~t: I+oo t(IX-1)p L._llf(',Xj + t, ••• )

- f(x')IP dx' dt .

Now consider, for p > 1, the case a. = O. Then from (1.110), (1.111) with Xn = t, if u E W 1,P(IR"t-), the trace of an element u E Wl.P(IR"t-) is such that:

1

(1.117) u(.,O) = u(.,xn)lxn=o = UlrE W 1 -p,p(lRn- 1 ) ,

the mapping u --+ u(., 0) = Yo u being surjective and continuous. If we take p = 2,

1 W 1,2(1R"t-) = H1(1R"t-) and W l -2· 2(lRn-1) = Hl/2(lRn-1);

we recover the result in Hilbert space. o

Remark 6. It is interesting to place the result obtained in Theorem 3 when X and Yare two complex Hilbert spaces into the situation of (1.52). In this case, we have noted that X = D(A) where A = A 1/2 (A is a self-adjoint operator> 0(17) given by (1.54)), the norm I Ix being equivalent to the graph norm of A. The operator - A with domain D( - A) = D(A) = X is therefore the infinitesimal generator of a semigroup of class ~o, {G(t)},~o given (formally) by G(t) = e- A1•

We shall now verify that we have:

(1.118) 1

for 0 ~ a. < 2' T(2, a.; D( -A), Y) = (X, Y)t+IX = D(At- IX ) .

(so that we have

T(2, 0; D(A), Y) = (X, Y)t = D(At) ,

which again gives (1.61)iii)). Firstly, we must make the meaning of D(A 1-0) (see (1.58)) precise. For this, note that the following properties, which we shall recall, result from the Stone-von Neumann-Dixmier theorem(18). Let X and Ybe two complex separable Hilbert spaces satisfying (1.52). There exist: i) a positive measure p., with support in [).o, + 00[, ,1.0 > 0,

ii) a Hilbert integral

(17) That is to say positive with a bounded inverse. (18) See Chap. VIII.


iii) an isometry I1IJ of Y over Jt' (that is to say a unitary operator from Yover Jt') with

{ U E Y is equivalent to v = l1IJ(u) E Jt' ,

f+OO

lul~ = IIv(A)II';"(.lo)dJL(A) , AO

such that we shall have

(1.119) if u E X = D(A), I1IJ(Au) = AI1IJ(U) = AV E Jt'

(that is to say U E X is equivalent to f + 00 A 211 V(A) II ';"(A) dJL(A) < + 00) . AO

Under these conditions, for () E [0, 1], we consider

(1.120) Jt'1-6 = {v E Jt'; AI - 6V E Jt'},

that is to say the set of vector fields of Jt' such that

f+OO A2(1-6)lIv(A)II';"(A)dJL(A) < + 00, AO

and we set (by dejinition)(19)

(1.121)

The space Jt'1-6 equipped with the norm

(1.122) IlvllJt'l_. = (I:oo A2(1-6) II V(A)II';"(A)dJL(A)Y/2

is a Hilbert space dense in Jt'. We equip D(A 1-6) with the norm induced by I1IJ -1 (equivalent to the graph norm(20)).

(1.123)

so that I1IJ is an isometry of (X, Y)6 over Jt'1 -6. From the Holder inequality

f+OO

A 2(1 -6) II V(A) II ~~A)6) II V (A) 11;(A)dJL(A) AO

~ (I:oo A2I1V(A)II';"(A)dJL(A)Y-6(I:00 IIV(A)II';"(A)dJL (A)Y,

(19) See (1.58). (20) The graph norm will be the norm defined by


so that

lul(x, Y). ~ Clull-Olult which is (1.61)ii) ,

Finally, if f is a continuous bounded function of A, we may define the bounded operator in Y, denoted f(A), by

(1.124) f(A)u = IJIt - 1 (), 1--+ f(A)V(A)), v = IJIt (u) ,

Thus, for all t ~ 0 the semigroup {G(t)} associated with the operator - A is defined for all u E Y, by

(1.125) {

G(t)u = 1JIt-1{A 1--+ e-A'v(A), v = lJIt(u)}, with

IG(t)ul~ = L:oo e-2).'llv(A)II~(),)dJl()'). This set, we now verify (1.118).

Note that, from (1.84) a E T(2, 0:; D( - A), Y) with - ~ < 0: < ~ is equivalent, by

setting b = lJIt(a), to: a E Yand

f +OO IG(t)a al 2 f+oo [f+oo (1 e- A')2 ] o t2a t- ydt= 0 t2a AO -t2 Ilb(A)II~(},)dJl(A) dt

f +OO [f+oo [1 -AtJ2 ] = AO II b(A) II ~(A) 0 t 2a - / dt dJl(A)

where

C(o:) = L+ 00 t2a [g(t)]2 dt < + 00 (21) with g(t) = ~ I e- U du,

which shows that a E Yf i-a'

5. An Intermediate Derivatives Theorem

Let H be a complex separable Hilbert space; it is then possible to extend the Fourier transform in the sense of vector-valued distributions (with values in H) (see Chap. XVI) to (classes of) functions of the space L2(1R; H) = L2(H); we again denote (formally) the Fourier transform of u by

u(r) = t u(t)e-ir'dt

The mapping u -+ U is again an isometry (up to a given constant factor) of L 2(H)

(21) We may verify it by the Hardy-Littlewood-Polya inequality (1.87).


over itself and we have the Plancherel formula

(1.126)

Now let IX E ]0, 1 [ . We say that the derivative of order IX of u, denoted Dau is in L 2 (H) if the function r --+ Irlau(r) is in L 2 (H). ----[This definition is motivated by the fact that when IX is an integer Dau = (ir)au and that Dau E U(H) is equivalent to Irla u E U(H) by the Planche reI formula (1.126).] At present, let X and Y be two complex separable Hilbert spaces in the framework (1.52); we denote by W(X, Y), the space W( - 00, + 00; X, Y) defined by (1.59) (with a = - 00, b = + (0). By the Plancherel formula (1.126) and the fact that the Fourier transform exchanges the differentiation D = d/dt and multiplication by ir, to say that u E W(X, Y) is equivalent to

(1.127) L [lu(r)li + IrI2Iu(r)I~]dr < + 00 .

Note again, that by using the spectral decomposition of the self-adjoint operator A associated with the couple (X, Y) by (1.56), we have

(1.128)

f u E W(X, Y) is equivalent to :

lL dr1:oo(lrI2 + ).2)llu(r,).)II~(i.)dli()') < + 00.

This set, we shall show

Theorem 6 (theorem of intermediate derivatives). Let u E W(X, Y). Then,for all IX E ]0,1 [, we have Dau E L2([X, Y]a), the mapping u --+ Dau being continuousfrom W(X, Y) into L2([X, Y]a), with

(1.129) II Dau II L'([X. YJ,) ~ C. II u Ill~;) II Du II ~2(Y) •

Proof By using the spectral decomposition of the operator A with

v(r, ).) = OZ/(u(r))().) , we may write:

I r 12a A 2(1-a) II v(r, A) II ~(i.) = [I rill v(r, ).) II Jf(i.)Fa PII v(r, ).) II x().)] 2(1 -a)

so that by applying the Holder inequality twice in succession with p = -, IX

1 . p' = --, we obtam: first of all

1 - IX

II r lau(r)l[x, Y], = f+ 00 I r 12a ). 2(1 -a) II v(r, ),) II ~(i.) dli(),) "0

~ [1: 00 Irl211 V 11~(i)dli J -[ 1: 00 A211 V 11~().)dli J-a

~ C .llrlu(r)l~alu(rm(1-a) ,


and then

thus

(1.130)

from which we have Theorem 6.

For mEN

(1.131)

{O}, we introduce, more generally, the space

w(m)(x, Y) = {u E U(X); Dmu E L2(y)} .

It is easy to verify that, equipped with the norm

(1.132)

495

D

w(m)(x, Y) is a Hilbert space in which f0(IR; X) is dense. Then we easily deduce from Theorem 6

Corollary 3.

( 1.133)

Remark 7

Let u E w(ml(x, Y). Then,for all j = 1,2, ... , m - 1, we have

f i) Dju E U([X, Y]j/m)

1 ii) II Dju II L2([X. Y]Fm) ::::;; II u III ~~i II Dmu II i~ y) (22)

1) The preceding proof which is essentially built on the Fourier transform and the Plancherel theorem (due to 1.-L. Lions [1]) cannot be applied when X and Yare two arbitrary Banach spaces in the setting of (1.52). 2) In the case of two arbitrary Banach spaces, we may obtain a result, by a different method, which is a little less precise about the intermediate spaces obtained which are trace spaces 'larger' than [X, Y]e when X and Yare Hilbert spaces. We may however obtain a result (which again gives Theorem 6 and Corollary 3 in the Hilbert space framework) when X and Yare two Banach spaces satisfying supplementary hypotheses which are satisfied by the majority of Banach spaces used in applications: U(Q) spaces, W1.P(Q) Sobolev spaces, spaces obtained by interpolation from the preceding ones, Besov spaces etc ... For this, we refer to Artola [3]. D

Remark 8. We denote by

( 1.134) def

w(m)(o, + 00; X, Y) = {u; u E L2(0, + oo;X),Dmu E L2(0, + 00; Y)}.

We equip this space with the norm:

{Ihl lu(t)lidt + Ihl lu(m)(t)l~dt f/2 which is in fact a Hilbert space.

(22) With the notation I - jim = I - (jIm).


We shall show(23)

Theorem 7. Let u E w(m)(o, + 00; X, Y) 1) We have

(1.135)

2) We may define yiu (also denoted u(j)(O)) the 'trace oforderjofu'fo rj = 0,1, ... , m - 1, with

(1.136)

and the mapping

u -+ (u(O), U(l)(O), ... , u(m-l)(O))

m-l is surjective from w(m)(o, 00; X, Y) over n [X, YJ8j"

J=O

Proof of Theorem 7 1 ) We first of all prove

Lemma 5. Every u E w(m)(o, + 00; X, Y) is a restriction to [0, + 00] of a (nonunique) element U E w(m)(x, Y).

Proof of Lemma 5. It is done in two stages. 1st stage. Let ~(IR+, X) be the space of restrictions to IR+ [0, + oo[ of func-tions of ~(IR; X). Then:

(1.137) ~(IR+; X) is dense in Wm(O, + 00; X, Y) .

In effect, if u E w(m)(o, + 00; X, Y), we consider Uh defined by

Uh(t) = u(t + h), t > - h ,

and we denote by Vh the restriction to IR+ of Uh• It is easy to verify:

{i) Vh E w(m)(o, + 00; X, Y)

(1.138) ii) Vh -+ U in this space as h -+ ° .

It is therefore sufficient to approximate Vh, h fixed, by elements of ~(IR+, X) therefore finally to approximate an element u which is the restriction to [0, + 00 [

of a fixed element U 1 E wm( - h, + 00; X, Y). By truncation of U 1 we suppose that u is the restriction to [0, + oo[ of U E w(m)(x, Y). Now by proceeding as for the space W(X, Y) by truncation and regularisation, ~(IR, X) is dense in wm(x, Y) we therefore approximate U by Un E ~(IR; X) and the restrictions Un of Unto IRn approximate U in a required sense. From which we have (1.137).

(23) In fact, this result has already been proved in Chap. VIII, §4, Theorems 5 and 7.


2nd stage. For u E 2& (IR +, X), we use the 'Babitch extension' P defined by

(1.139)

with

(1.140)

{ u(t)

Pu(t) = m - 1 _ . L cju( }t), j = °

if t > 0

t < 0,

L ( - I)k/cj = I, k = 0, I, ... , m - I .

497

The mapping u -+ Pu is continuous from '@(IR+; X) equipped with the topology induced by wm(o, + 00; X, Y) in w(m,(x, Y), which thanks to (1.137), extends into a mapping, again denoted u -+ Pu, which is linear and continuous from w(m'(o, + 00; X, Y) into w(m,(x, Y).

Obviously Pu = u a.e. in [0, + 00 [ ,

from which we have the lemma

2) Point I) of Theorem 7 then follows from Theorem 6 and from Lemma 5.

o

3) Point I) of Theorem 7 being established, the trace result (1.136) follows immediately from Theorem 3 (a = 0) and from the definition of spaces [X, Y]o (Remark 6). Finally the surjectivity of the mapping u -+ {u(O), ... , u(m - 1 '(O)} follows from

j + 1/2 . Lemma 6. Let aj E [X, Y]Oj' OJ = m for} = 0, 1,2, ... , m - 1.

There exists u E wm(o, + 00; X, Y) (non unique) such that

(1.141) yjv = vj(O) = aj , j = 0, 1,2, ... , m - 1.

Such a function v is called a 'continuous lifting' of the element a = {ao,a 1 , ••. ,am-d in the space w(m,(o, + 00; X, Y).

Proof of Lemma 6. To construct such a lifting, we shall use the spectral decomposition of the operator A. For aj E [X, Y]Oj' we set bj = ~(aJ (~ being the isometry of Y over

yt' = fEB yt' (A) dJ1(')')) and we define for q> E 2& (1R)(q>(t) = lover a neighbourhood

of zero) the function:

(1.142) 1. 1

WiA, t) = -:--, bj(A)tJq>(APt) where p = -}. 2m

We then define a lifting Rj of aj in wm(o, + 00; X, Y) by

(1.143) Rj(aJ(t) = ~ -I (Wf)(t)

and we verify that

(1.144) m-I

V = L Rj(a) j=O


is a lifting of Ii = {ao, ... , am - d in the space

w(m)(o, + 00; X, Y) .

Remark 9 1) Numerous questions occur about spaces analogous to w(m)(x, Y) but with weights. For example, this is the case of Theorem 3. For results of this type in weighted Sobolev spaces, we may consult Grisvard [1]. For results about intermediate derivatives in the spaces

W:~)cJX, Y) = {cou E L2(X), cmDmu E L2(Y)}

where cj ( j = 0, m) is a suitable weight, we refer to Artola [2] and for weights Co = t a, Cm = tP to Lacomblez [1], [2]. 2) We similarly consider the following situation which we have met in Chap. XVI, §4. Let ¢ E IR be given. Set:

(1.145)

Equipped with the norm

{L+oo e-2~r[lu(t)li + ID2U(t)I~]dtr/2 ,

~(O, + 00; X, Y) is a Hilbert space. Then, we have the analogue of Theorem 7:

(1.146)

(1.147)

e-~rDu E L2(1R+; [X, Y]1/2)

{i) u(O) E [X, Y] 1/4 ,

ii) u'(O) E [X, YJ3/4 .

In the case of the example treated in Chap. XVI (loc. cit.), we have

X = D(A), Y = H, so that

u(O) E D(A3/4), u'(O) E D(Al/4) and e-~rDu E U(IR+; D(A 1/2).

To prove (1.146), we proceed as for Theorem 7; we note that u E ~(O, + 00; X, Y) is the restriction to [0, + oo[ of a (non unique) element V E ~(IR; X, Y) and that the space .@(IR; X) is dense in ~(IR; X, Y); then for V E '@(IR; X) we have (§' denoting the Fourier transform):

(1.148)

So that: e - ~r Dj u E L 2 ([X, YJj/2)j = 0, 1, 2, is equivalent to

(1.149) , --+ (, - i¢)j§,(U)(, - i¢) E L2(lRt; [X, YJj/2) (24) ;

(1.146) is then immediate (by using Cauchy-Schwarz).

(24) With the usual notation of Chap. XVI, we have:

Y'(e-<'U)(r) = SfU('; + ir) = SfU(i(r - i';)) = Y'(U)(r - i';)

and (1.149) therefore denotes the functions r = Imp f-+ pj Sf U(p) E U ihl p [X, Y]jj2), j = 0, 1,2, (with p = .; + ir).

§ I. Some Elements of Functional Analysis 499

To verify (1.147) we set v = e~~tu and we note that due to (1.146), Dv E LZ([R+; [X, Y]1/Z); from which:

v(O) = u(O) E [X, Y]1/4, V'(O) E [X, Y]3/4

that is to say (1.147). o

6. Bidual. Reflexivity. Weak Convergence and Weak* Convergence(25)

From the definition and the study of spaces W in which we shall work in the following, we shall recall several well known ideas(Z6) which will be useful in the third stage of the method which will be developed later (§2.3). Returning to evolution problems: to find a solution u of this problem, we shall construct a bounded sequence(Z7) {un}. If, in the space in which we are working (containing the un), we have the property P: P: every closed ball is compact (this property is equivalent to: the closed unit ball is compact), then by the definition of compactness, we may extract from the sequence {un} a subsequence converging towards a certain u (a candidate for the solution of the problem). But for a Banach space (or a Hilbert space), the topology due to the norm only has this property P if the space is finite-dimensional. We are thus led to change the topology. We shall introduce in this Sect. 6 the elements necessary to change the topology. We return to this at the end of this § 1.

6.1. Bidual

Recall that the dual X ' (Z8) of a normed space X is a Banach space; X' therefore has a dual, which is also a Banach space called the bidual of X and denoted X". It is interesting to make the relation between X and X" precise. We denote by II II the norm in X, II II * the norm of X' and II II ** the norm of X". <, > denotes the duality between X' and X or between X" and X'.

Theorem 8. Let X be a normed vector space. There exists J E !t' (X, X ") with the following properties:

(1.150) {i) J is injective,

ii) IIj(x) II ** = II x II for all x EX.

Proof (29). Let x E X be fixed.

(25) Called 'weak star'. (26) See Chap. VI, § I. (27) That is to say lunl < K for all n E N. (28) That is to say, to repeat ourselves, the set of continuous linear forms over X with the norm

, (x',x) Ilx II = sup --.

XEX,X;'O Ilxll (29) This proof is a reminder of Chap, VI, § I.


For all x' E X' we have: l<x',x)1 ,,;; Ilx'lI* Ilxll . Thus, the mapping

(1.151) J(x): x' -+ J(x)(x') = <x', x)

is a continuous linear form over X'. Therefore

(1.152) J(x) E X", \:Ix EX.

Now, note that:

(1.153) IJ(x)(x')1 l<x',x)1

l/1(x)II** = sup II 'II = sup II 'II ,,;; IIxll . X'EX' x * X'EX' X *

On the other hand, as a consequence of the Hahn-Banach theorem (see Chap. VI, §1 and Dunford-Schwarz [1]), it is possible to find y' E X' with

(1.154)

(1.155)

{ <y:,X) = IIxll lIy II * = 1 ,

IJ(x)(x')1 2 J(x)(y') therefore II J (x)// ** = sup II x' II * r II y' II *

From (1.153) and (1.155) we deduce

(1.156) IIJ(x)lI** = IIxll .

IIxll '

Thus, the mapping J: x -+ J(x) (which is linear by definition) is injective and continuous from X -+ X" ( J is likewise an isometry of X over J (X) which is thus a closed vector subspace of X"). From which we have the theorem, 0

The fundamental consequence of Theorem 8 is:

{it is possible to identify X with the vector subspace

(1.157) J(X) c X" .

6.2. Reflexivity

Definition 6. We say that X is a reflexive Banach space if:

(1.158) J(X) = X" .

With this definition, we have

Proposition 10. Let X be a reflexive Banach space. Then i) X' is also a reflexive Banach space, ii) if Y is a closed subspace of X, then Y is also reflexive.

For the proof, see Chap. VI, §1.

Examples 4 (see Brezis [1]) 1) Every Hilbert space is reflexive. 2) Let Q be an open set of Ill"; for 1 < p < + 00, U(Q) is a reflexive Banach space.

§ I. Some Elements of Functional Analysis SOl

3) The dual of L 1 (Q) is L 00 (Q). But L 1 (Q) is not reflexive. 4) If V is a separable reflexive Banach space, V' its dual, then if p is such that 1 < p < + 00, U(O, T; V) is a reflexive Banach space, the dual of U(O, T; V) being

1 1 U' (0, T; V'), p' such that - + --; = 1 .

p P

5) Ll(O, T; V) has LOO(O, T; V') for its dual, but is not reflexive. 6) Similarly L 00 (Q) and L 00 (0, T; V) are not reflexive.

6.3. Weak Convergence and Weak* Convergence

o

Recall that {xn}n E 1'.1*, Xn E X for all n, tends to x E X weakly(30) (i.e. for the weak topology of X) as n -+ 00 if

(1.159) (x', xn ) -+ (x', x), 'rIx' E X' .

An important result(31) for the following is then

Proposition 11. Let X be a reflexive Banach space {xn}n E 1'.1* a bounded sequence in X. Then, it is possible to extract from {xn } n E 1'.1* a subsequence {x~} which converges weakly in X.

This result is a result of weak compactness which may be restated

(1.160) 'The unit ball in a reflexive Banach space is weakly compact' .

Definition 7. (Weah convergence). Let X be a normed space and X' its dual. The weak star topology (weak*) is defined in the following manner: we say that {x~}, x~ EX' converges towards x' EX' in X' equipped with the weak * topology, as n-+oo,if

(1.161) lim (x~, x) = (x', x), 'rIx EX.

Examples 5 1) Let Q be an open set of IRn; recall that (Ll(Q))' = LOO(Q). Let {./,;}nel'.l* be a sequence of elements f,. E L 00 (Q). To say that J. -+ fin L OO(Q) weakly * means that

(1.162) { tf,.(X)9(X)dX -+ tf(X)9(X)dX,

'rig E Ll (Q), when n -+ 00 .

2) Let V be a separable, reflexive Banach space, and V' its dual. Let {J.} n E 1'.1* be a sequence of elements J. E L 00 (0, T; V').

(30) We often use the notation (see Chap. VI, §I) x. x or x. -+ x weakly in X. (31) See Chap. VI, §l.


To say that f" -+ fin L 00 (0, T; V') weakly * means that

(1.163) f: <f,,(t), g(t) dt -+ f: <f(t), g(t) dt, '<Ig E U(O, T; V) .

Remark 10. We say that x~ -+ x' in X' is equipped with the weak topology if

(1.164) <x", x~) -+ <x", x') for all x" E X" .

Weak* convergence therefore only uses elements of X == J(X) eX". This idea is obviously of interest only in the cases where the spaces X are not reflexive. D

There exists a property of weak * compactness of the unit ball of the dual of a separable normed space(32).

Proposition 11*. Let X be a separable normed vector space. Let {X~}nEI\I" x~ EX', be a bounded sequence in X' (i.e. sup II x~ II * < + 00).

Then it is possible to extract from {x~} a subsequence {X;"} such that x;" -+ x' in X' equipped with the weak * topology.

6.4. A Common Situation in Applications

In applications (see later §3, §4, §5, ... ), we frequently encounter the following situation. Let {Xn}nEI\I*' (resp. {X~}nEI\I') be a bounded sequence in X (resp. in X'). Then by Proposition 11 (resp. 11 *), it is possible to extract from {x n } (resp. from {x~}) a subsequence {xm } (resp. {x;"}) which converges to x (resp. x') weakly in X (resp. in X' weakly *). Suppose that it is possible to show that x (resp. x') is independent of the subsequence {xm } (resp. {x;"}) extracted (for example because we know that x (resp. x') is the unique solution of our problem). The following two propositions are then useful.

Proposition 12. Let X be a reflexive Banach space. Let {Xn}nEI\I* ?e a sequence of elements of X; we assume

i) Ilxnll ~ C < + 00, '<In E N*, C constant,

ii) the set of cluster points of {xn } for the weak topology is reduced to {x}. Then the sequence {xn } converges towards x, weakly in X.

Proof(33). If Xn does not converge towards x, for the weak topology, then it is possible to find e > 0, x' E X' and a subsequence {xm } extracted from {xn } such that

(1.165) I < x', Xm - x) I ~ e > 0, '<1m.

(32) See Chap. VI, § 1. Theorem 6. (33) See also Chap. VI, § I.

§2. Galerkin Approximation 503

From i), II Xn II ~ c < + 00 and X is a reflexive Banach space. Therefore, we may extract from {xm} a subsequence {x/l} which converges towards an element of X which is none other than x from ii). We therefore have

(1.166) lim I (x', x/l - x>1 = 0.

This contradicts (1.165). 0

In an analogous manner, for weak * convergence, we have

Proposition 12*(33). Let X be a separable (non-reflexive) Banach space. Let {X~}nEN' be a sequence of elements of x'. We assume

i) Ilx~lI* ~ c < + 00, Vn EN *, C constant,

ii) the set of cluster points of {x~} for the weak * topology is reduced to {x'}. Then {x~} converges to x', in X', weakly *. Remark 11. Let X be a Hilbert space. Note again that if Un -+ U weakly in X and Vn -+ v weakly in X, we are not assured of having (un' vn) = (u, v) as shown by the example Vn = Un' Un -+ ° weakly in X, I Un Ix = 1. But if Un -+ U strongly in X and Vn -+ v weakly in X then lim (un' vnh = (u, vh·

o We therefore have the fundamental property, a cornerstone of the variational method, used in the 3rd stage of this method (see §2.3) (and which corresponds to our preoccupation at the beginning of this Sect. 6 of changing the topology): 'For the weak topology, the unit ball of a Hilbert space is compact'. This therefore allows us to work in the remainder of this chapter with the Hilbert spaces L2(0, T; V), L2(0, T; V'), W(O, T; V, V'). However the Banach spaces L <Xl (0, T; V) and L <Xl (0, T; H), which we shall use, are not reflexive; therefore for them, the weak topology does not satisfy the property P of the beginning of Sect. 6. Conversely this property P, for these L <Xl spaces, is satisfied for the weak * topology (Example 5) we therefore only use it with these spaces. Further, for our evolution problem, we have seen (Propositions 12 and 12*) that if the solution of the problem is unique, then the whole sequence {un} envisaged at the beginning of this Sect. 6 will converge(34) to the solution candidate u.

§2. Galerkin Approximation of a Hilbert Space

We firstly give the definition of an approximation of a Hilbert space and some examples.

(34) For the topology having the property P.


1. Definition

We recall the following definition encountered in Chap. XII. Let V be a separable Hilbert space and { Vm} me I'll' a family of finite dimensional vector spaces satisfying the axioms:

(2.1) { i) Vm C V (dim Vm < + (0)

ii) Vm -+ V when m -+ 00 in the following sense:

there exists "1/ a dense subspace of V, such that, for all v E "1/, we can find a sequence(35) {Vm}mEI'II' satisfying: for all m, Vm E Vm and Vm -+ v in Vas m -+ 00.

The space Vm is called the Galerkin approximation of order m (m # dim Vm ) of V(36).

2. Examples

Example 1. Let Wi' w2 , ... , Wm , ••• be an orthogonal basis of V comprised for example of eigenfunctions of a (positive) self-adjoint operator A in H with V = D(A 1/2) c H (with compact injection). We take

(2.2)

{ Vm}mel'll+ forms an increasing sequence of subs paces of Vand by the definition of a hilbertian basis "1/ = U Vm is dense in V. D

m

Example 2. The family {Vm}mEI'II*' Vm = {Wi"'" Wm} is an increasing family of finite dimensional subspaces of V, corresponding to a fixed basis Wi' W 2 , .•. , W m , ••• of V, whose elements are not eigenfunctions of an operator A occurring naturally in the problem. The corresponding approximation of V is the usual Galerkin method with fixed basis. D

Example 3. Consider Q = ]0, 1[ and let V = HMQ), i.e.

(2.3) V= {V;VEL 2 (Q),:: EL2 (Q),V(0)=V(1)=0}.

(35) The phrase 'We can find a sequence' is in the sense of: 'We know effectively how to construct a sequence ... '. (36) This property also implies: for all U E V, there exists a sequence {V .. }melll· with v .. E V .. and v .. -+ v in V for m -+ 00 (for the proof, we shall take a family of orthogonal projectors POI over V .. in V; for m -+ 00, Pm tends to the identity mapping over '"Y which is dense, therefore in V). We shall use this in the remainder of § 3.3.1.

§2. Galerkin Approximation

Let m E N* and h = ..!.... We define (see Fig. I) WI' W 2 , .• ·, W m - I satisfying m

{

Wj is piecewise linear,

(2.4) wj(jh) = I, wj«(j ± I)h) = 0

Wj(x) = 0 if x¢. [(j - I)h, (j + I)h], j = I, ... , m - I .

x

Fig. 1

We take

(2.5)

which is a space of dimension m - I.

505

o Remark 1. In this example, the family of functions {w j } changes completely with the parameter h = 11m. It is therefore a different situation from that of Examples I and 2. This situation corresponds to the general situation of Definition I, the Galerkin approximation with 'variable bases'. However since here (in the case of dimension n = I) the functions of H bCQ) are continuous we may take

(2.6) j/ = V.

Then for v E V, define

m-I (2.7) Vm = L v(jh)wj;

j= I

with the choice

(2.8) dVm I. . ~ = h [v(jh) - v«(j - I)h] over «(j - I)h, jh) ,

that is to say

dv 1 fjh (2.9) -d m = -h v' (0") dO" , over «(j - I)h, jh), I ~ j ~ m - I .

X (j-I)h

We note that ~; is a 'stepwise' approximation to v' = ::. We easily show that

(2.10) vm -+ v in V as m -+ + co. o


Example 4. The general framework of Definition 1 corresponds to approximation by finite elements, the approximation of Example 3 belongs to this framework. We content ourselves here with an example of a particular case. For the general theory and the verification of the details below, we refer to Chaps. XII and XX.

Fig. 2

Let Q c jR2; we consider the space H b(Q) = V; since the space dimension is n = 2, the functions of Hb(Q) are not continuous. Let m c N*; we set h = 11m and we introduce a family of triangles ~h having the following properties:

(2.11)

(2.12)

(2.13)

(2.14)

the family ~ h is finite

{

if T1 , T2 E ~h' Tl "# T2, then

either Tl (\ T2 = 0, or Tl and T2 have a common edge,

or Tl and T2 have a common vertex.

T c Q, VTE ~h.

{diameter T ~ qJ(h) , VT E ~h where

qJ(h) -+ 0 as m -+ 00 .

Finally, if we set

we shall say that

if,

(2.15)

Q h -+ Q as m -+ + 00

{ for all compact K c Q, there exists hK > 0 such that

K c Qh for h = 11m < hk .

This set, let M be an 'interior' vertex of triangles of ~h; we define one function W Mh satisfying:

{i) WMh is affine over each triangle with vertex M

(2.16) ii) WMh(M) = 1 and WMh(P;) = 0, i = 1, 2, ... , k (see Fig. 3).

§2. Galerkin Approximation 507

Fig. 3

Then, we show (see Chap. XII) that

(2.17)

and we denote by Vrn the finite dimensional space generated by the WMh when M describes all the 'interior' vertices of the triangulation. Thus

(2.18) dim Vm = cardinal of the set of points M .

Since the functions of H 1 (Q) are not continuous here(37), we introduce

(2.19) -y = E&(Q)

which is dense in Hb(Q). Then for v E E&(Q), we introduce

(2.20) 11m) M

and we have (see Chap. XII).

(2.21 ) Vrn --+ v in V as m --+ 00. o Remark 2. Note that if we consider m' #- m, as in Example 3, the basis functions WM'h' are completely different from the functions W Mh . From which we have the justification ofthe term sometimes used: the Galerkin approximation with 'variable basis'. 0

3. The Outline of a Galerkin Method

Let P be the exact problem for which we intend to look for the existence of a solution (or to construct the solution if we know of its existence) in a space of functions constructed over a separable Hilbert space V. We suppose that we know in advance that

(2.22) the solution u of problem P is unique.

After having made the choice of a Galerkin approximation Vm of V (see remarks at the end of §3 about the motivation of the choice), it is suitable to define

(37) Since n = 2.


an approximate problem Pm in the finite dimensIOnal space Vm having a unique solution Um'

The procedure of study is the following. i) Stage 1: We define the solution Um of problem Pm. ii) Stage 2: We establish some estimates on Um (called 'a priori' estimates on u) which show that {um} belongs to fixed balls (i.e. independent ofm) of certain normed spaces. iii) Stage 3: By using the results of weak compactness of the unit ball in a Hilbert space (resp. a Banach space) (resp. the dual of a normed space), it is possible to extract from {Um}mEN* a subsequence {u~, }m'EN* which has a limit in the weak (or weak *) topology of spaces which occur in the estimates of stage 2. Then let U be the limit obtained. iv) Stage 4: We show that U is the solution of problem P, therefore the solution sought from uniqueness,

Remark 3. The method is only of limited practical interest for numerical calculation if it only allows us to obtain the solution U by the (non-constructive) extraction of a subsequence. Due to the uniqueness of problem .0, we see that every sequence urn converges weakly towards u. 0

One last stage, which is optional but may be important is: v) Stage 5: strong convergence results. To summarise, the plan of study will be the following:

I. Variational formulation of problem P. II. Uniqueness of the solution.

III. Existence of the solution.

{

i) Stage 1. Formulation of the approximate problem Pm (therefore um) ii) Stage 2. a priori estimates.

iii) Stage 3. Use of the estimates m ---+ 00 (from which we have u) iv) Stage 4. u is the solution of problem P. v) Stage 5. Strong convergence results.

IV. Properties of stability with respect to the given data or continuity of the solution with respect to the data.

Remark 4. In numerical applications, the essential problem is the choice of Vm •

Except in very particular cases, an approximation with fixed basis (see Example 2) leads, for problem Pm' to 'full' matrices and the methods of numerical approximation of systems (linear in our case) limits the size of m. Another limitation of the method may also result from the difficulty of calculating the elements of the matrices occurring in problem Pm. A 'finite element' method, therefore an approximation with variable bases allows a 'local' representation and the elements of the matrices are mostly zero; because of this, the matrices occurring are 'sparse' with m large enough, the functions Wrn which appear have support as small as m is large(38). 0

(38) See Chap. XII.

§3. Evolution Problems of First Order in t 509

Remark 5. The Galerkin method of approximation is not limited only to Hilbert spaces and is interesting particularly for nonlinear problems(39). 0

§3. Evolution Problems of First Order in t

1. Formulation of Problem (P)

1.1. Function Spaces

We are given a pair of real, separable Hilbert spaces V, H; we denote by:

{ (( , )) the scalar product, II II the norm in V

( , ) the scalar product, I I the norm in H .

We suppose that V is dense in H and we identify H with its dual H'. If V' denotes the dual of V (with norm II II *) we have

(3.1) V c+ H C+ V' ,

each space being dense in the following (C+ denotes continuous injection). We denote by W( V) the space W(O, T; V, V'), defined in § 1 with ° < T < 00.

We recall that (3.2) W(V) C+ 'C([O, T]; H) ,

'C([O, T]; V) being the space of continuous functions over [0, T] with values in V equipped with the topology of uniform convergence. We also denote the duality between V' and V by ( , ).

1.2. The Bilinear Form a(t; u, v), t E [0, T]

For each t E [0, T] we are given a continuous bilinear form over Vx V and we make the hypothesis:

(3.3)

for every u, v E V, the function t --+ a(t; u, v)

is measurable and there exists

a constant M = M(T) > ° (independent of

t E ] 0, T [, u, V E V)

such that

la(t; u, v)1 ~ M Ilull'llvll, for all u, v E v. It follows that, for each t E [0, T] the bilinear form a(t; u, v) defines a continuous linear operator A(t) from V --+ V' with

(3.4) sup II A (t) 11..'f'(v, v ') ~ M. tE(O, T)

(39) See Lions [2].


We make the following hypothesis (of coercivity(40) over V with respect to H)

{ there exists A, tX constants, tX > 0 such that (3.5) a(t; u, u) + Alul 2 ~ tXllul1 2 , Vt E [0, T] (41), Vu E V.

Finally (see Chap. VI, §3), we denote by D(A(t)) the space

(3.6) D(A(t)) = {u: U E V; V -+ a(t; u, v)

is continuous over V for the topology of H} .

This is the domain of A(t) in H .

1.3. Examples of Bilinear Forms a(t; u, v)

Let Q be an open bounded set of IRft with boundary r assumed sufficiently regular.

Example 1. We take V = H5(Q); H = L2(Q).

(3.7) a(t; u, v) = a(u, v) = (Vu, Vv), Vu, V E V.

Then (3.1), ... , (3.5) holds (with A = 0 in (3.5)); u E D(A(t)) is equivalent to

(3.8) u E H 5(Q) , L1u E L 2(Q)

which (for sufficiently regular r(42») is also equivalent to

(3.9) D(A(t)) = D(A) = H5(Q) n H2(Q) .

We again denote by ( - ,1) the operator A(t) = A E .P(V, V') .

Example 2. We take V = H1(Q), H = U(Q), and:

(3.10) a(t; u, v) defined by (3.7) .

Then (3.1), ... , (3.5) holds (with A > 0 for (3.5)); u E D(A(t)) = D(A) is equivalent to:

(3.11 ) {i) uEH1(Q), L1uEL2(Q),

ii) (- L1u, v) = (grad u, grad v), Vv E H 1 (Q)

o

condition (3.11)ii) expresses in a general sense that the normal derivative ~: of u

is zero over r (the Neumann condition)(43); we denote here by A the operator A(t) = A E .P(V, V') . 0

Example 3. We take for Va closed subspace of Hl(Q) with

H5(Q) ~ V ~ Hl(Q) and H = L2(Q) .

(40) See Chaps. VI and VII. (41) Or even only a.e. t E [0, T]. (42) For example r of class Cfj2 (with Q locally on one side of n. (43) See Chap. VII.


We set QT = Qx]O, T[ and

(3.12) n i au ov i a(t; u, v) = i, ~! Q aij(x, t) oXi OXj dx + Q ao(x, t) uv dx

with the following hypothesis:

(3.13) aij , ao E LOO(QT) ' i, j = 1 to n,

(3.14) f i, ~! aij(x, tK~j ~ IX. it! I ~;l2 ,

llX > 0, ~i E IR, a,e. III QT .

Then, for A large enough and a.e. t E ]0, T[ we have for all u E H!(Q):

(3.15) a(t; u, v) + Alul 2 ~ IX Ilu11 2 , '<Iu E H!(Q) (44)(45).

Setting a.e. t E ] 0, T [

(3.16) A(t)u =

U E D(A(t)) is equivalent to

1 i) u E V, A(t)u E H = L2(Q)

(3.17) I ii) JQ A(t)uv dx = a(t, u, v), '<Iv E V.

511

To interpret (3.17)ii), we can formally write the Green's formula (see Chap. II)

(3.18) IQ A(t)u. v dx f ;, au . v dr + a(t; u, v) Ji r uV A(t)

where au au

(3.19) = }:>ij ~ cos (n, x;) OV A(t) uXj

with cos(n, x;) = ith direction cosine of the 'outward normal' to r, n;dr 'an element of area' of r, the boundary of Q(46).

(44) Note also that there exists M > ° constant, such that t E [0, T] a.e.

la(t; u, vll ,;; M lIull Ilvll, Vu, v E V.

(45) We remark that (3.15) is only satisfied a.e. t E ]0, T[ and not necessarily Vt E [0, T] (see (3.5)).

(46) .. der " au 2 . Note that u E V ImplIes ~i(t) = L... aij(x, t) - E L (Q), I = 1 to n, t E ]0, T[ a.e.; u E D(A(t)) } ax}

implies A(t)u = div W) = L a~i(t) E U(Q), t E ]0, T[ a.e., and therefore (see Chap. IXA), i ox;

au W) = (~i(t))i~1 too E H(div,Q) has a 'normal' trace ~(t)n = L~i(t)ni = -- in H- I/2(r), t E]O, T[

i aVA(I)

a.e .. Thus in the case where V = H I(Q) (i.e. for Neumann boundary conditions), the domain D(A(t)) of the operator A(t) is given t E ]0, T[a.e. by:

D(A(t)) = {u E HI(Q), A(t)u E L2(Q), ~I = ° (in H-I/2(r))}. aVA(I) r


We again denote by A(t) the operator A(t) E 2'(V, V'). o

1.4. The Exact Problem

If X is a Banach space, we denote by U(X) the space U(O, T; X), 1 ~ p ~ + 00.

We are given, in the framework of (3.1):

(3.20)

and we set:

Problem (P). Find u satisfying

(3.21) u E W(V),

(3.22)

(3.23)

Remark 1

{ :t (u( . ), v) + a(.; u( . ), v) = (f(.), v)

in the sense of .@'(JO, T[ ) for all v E V

u(O) = Uo .

i) From condition (3.2), condition (3.23) has a sense. ii) From (1.50), we note that

(3.24) :t (u(.),v) = (:tU(.),v), 'riVE V.

Remark 2. Preliminary reduction. If we set u = wekt, k E IR, w satisfies

( dW ). _ -k'f() ) (3.22)' dt ( . ), v + a(., w( . ), v) + k(w(.), v) - (e . ,v

(3.23)' w(O) = Uo

o

by changing u to ue kt and choosing k, we can assume that (3.5) holds with A. = 0 (this has no consequences since T is finite). In the following, we shall therefore make the hypothesis:

(3.25) a(t;u,u) ~ IXllull~, 'rItE[0,T](47),UE V.

We remark that it is interesting to write equation (3.22) in vectorial form:

(3.22) :t u(.) + A(. )u(.) = f(·) in the sense of L 2 (V') .

This follows from formula (1.50). o

2. Uniqueness of the Solution of Problem (P)

Theorem 1. We suppose V, H are given and satisfy (3.1) and a(t; u, v) satisfies (3.3), (3.25), uo, f are given and satisfy (3.20).

(47) Or likewise t E [0, T] a.e ..


Then the solution of problem (P), if it exists, is unique.

Proof Let U 1 and U2 be two distinct solutions of problem (P), then w = U 1 - U2 satisfies WE W(V) and

(3.26) { (~; (.), v) + a(.; w( . ), v) = 0, \Iv E V,

w(O) = O.

Then by replacing v by vet) in (3.26) and integrating from 0 to t:

(3.27) 1 rl

21 w(tW + J 0 a(a; w(a), w(a)) da = 0 ,

and since (3.25) holds, we have

(3.28) 1 2Iw(t)12 < 0 => wet) = 0 for all t E [0, T] .

From which we have uniqueness in problem (P).

3. Existence of a Solution of Problem (P)

D

Theorem 2. Under the hypothesis of Theorem 1, there exists a solution of problem (P) and

U E W(O, T; V, V') .

3.1. Stage 1. Approximate Problem (P ... )

Let {Vm}mEN* be a family of finite dimensional vector subspaces satisfying (2.1); V being dense in H and from (2.1)ii)(48), for Uo E H, there exists a sequence {UOm}mEN* such that

(3.29) \1m, UOm E Vm and UOm ---+ Uo in H .

We denote by

(3.30) dm = dim Vm, {~m} j 1, ... , dm, a basis of Vm .

The problem is then:

Problem (Pm). Find

(3.31 )

(48) Since (Vm ) is also a Galerkin approximation of H.


satisfying:

(3.32) { ( dUm(t) ) ~' ~m + a(t; um(t), Wjm ) = (f(t), ~m)'

um(O) - UOm ·

System (3.32) is equivalent to a differential system of order 1 in IRdm of the type

(3.33)

where Bm and dm(t) are matrices (dm, dm) whose elements are {Jjm, a;jm(t) respectively and defined by

(3.34) {Jijm = (W;m, ~m)' a;jm(t) = a(t; W;m, ~m)' i, j = 1, . .. , dm .

We note, that since the ~m are linearly independent, we have

(3.35) det Bm "# O.

Thus, we obtain (resolution of a system of differential equations):

Lemma 1. There exists a unique solution Um to problem (Pm) satisfying:

(3.36) Um E <c°([O, T]; Vm ), u~ E L2([0, T]; Vm ).

3.2. Stage 2. A priori estimates

We multiply equation (3.32) by gjm(t) and we sum from 1 to dm; it becomes

(3.36)a) 1 d 2 . . 2 dt lum(t)1 + a(t, um(t), um(t)) = (f(t), um(t)),

from which, by integration over ]0, t[:

1 ft ft 1 (3.37) 21 um(t) 12 + ° a(O", Um' um) dO" = 0 (f(0"), um(O")) dO" + 2 I UOm 12 ;

using (3.25), we obtain:

1 ft ft 1 (3.38) 2Ium(t)12 + a ° II Um(O") 112 dO" ~ 0 IIf(O") 11* II Um(O") II dO" + 2luom l2;

then by noting that

(3.39) {I UOm I ~ c I Uo I c = constant independent of m

it ait 1 it ° II f(O") II * II Um(O") II dO" ~ 2 ° II Um(O") 112 dO" + 2a ° II f(O") II;' dO" ,


we deduce from (3.38)-(3.39)

(3.40) {~lum(tW + ~1"Um(U)12dU ~ C[luol2 + IT IIf(U)II!dU]

t E [0, T], C a suitable constant, independent of t, m ,

from which we have

Lemma 2. Thefunctions Um' solutions of (Pm) belong to a bounded set of L OO(H) and of L2(V).

3.3. Stage 3. Passage to the Limit for m -+ 00

From Lemma 2 and from (3.4), we deduce that

(3.41 ) A(. )um E a bounded set of L2( V') .

From which, by using the properties of weak (or weak star) compactness of unit balls of the spaces L 2 (V), LOO(H), L 2 (V') we deduce

Lemma 3. We can extract from the sequence {Um}mEN* a subsequence {u;"} having the following properties:

{

i) Um' -+ U weakly in L2(V)

ii) um' -+ u weakly * in L OO(H) ,

iii) A(. )um -+ A(.)u weakly in L 2 (V') .

(Property iii) follows from the fact that A ( . ) is linear and continuous from L 2 (V) into L 2 (V') in particular for the weak topologies and from i).) Consider then ep E .@(]O, T[) and v E "Y. From (2.l)ii), there exists {Vm}mEN*, Vm E Vm, such that Vm -+ v strongly in V. Therefore, if we introduce

(3.42) { I/Im = ep ® Vm (i.e. I/Im(t) = ep(t)vm)

1/1 = ep®v,

we have particularly

Ii) I/Im -+ 1/1 in L2(0, T; V) strongly, m' -+ 00

(3.43) dl/l ii) 1/1;" = dtm' -+ 1/1' in L2(0, T; H) strongly, m' -+ + 00 .

From (3.32), we deduce

(3.44)

{ - fT ( .... (t, ~~. (tl l dt + r art; u •. (tl, ~ •. (tl) dt ~ r (f(t, ~., (t)) dt

I/Im - ep ® Vm , Yep E '@(]O, T[) .


From (3.43)i):

(3.45) f~ (f(t), t/J m(t» dt -+ f~ (f(t), v) <P (t) dt as m' -+ 00 .

From point ii) of Lemma 3, and (3.43)ii):

(3.46) LT (um,(t), t/J;",(t» dt -+ LT (u(t), t/J'(t» dt as m' -+ 00 .

Finally, from point iii) of Lemma 3, and (3.43)i)

f~ aCt; um,(t), t/Jm,(t» dt

(3.47) = LT (A(t) um,(t), t/Jm,(t» dt -+ LT a(t; u(t), t/J(t» dt

as m' -+ 00 .

Thus we can pass to the limit in (3.44) and we obtain:

(3.48) { - LT (u(t), v) <p'(t) dt + f: aCt; u(t), v) <pet) dt = LT (f(t), v) <p (t) dt

Vv E Y and V<p E g&(]0, T[) .

Since Y is dense in V, (3.48) remains true for all v E V if we have shown that u satisfies (3.22).

3.4. Stage 4. u is the Solution of (P)

To show that u is the solution of problem (P), it remains to show that (3.21) and (3.22) are satisfied.

Verification of (3,21). From (3.48), we deduce:

(3.49)

- LT (u(t), v) <p'(t) dt = LT (f(t), v) <p (t) dt -

= f: (f(t) - A(t)u(t), v) <p (t) dt .

Since f E L2(V') and A(. )u(.) E L2(V'), we have

(3.50)

LT aCt; u(t), v) <p (t) dt

{g =f- A(.)UEL2(V')

- LT(U(t),V)<P'(t)dt= LT(g(t), V) <p(t)dt , VVEV, V<pEg&(]O,T[),


which, taking into account (1.15)-(1.17) implies

(3.51 )

and shows (3.21).

u' = du E L2(V') dt

517

Thus from (3.2), u E W(V) is (the class of) a continuous function from [0, T] -+ H. Verification of the initial condition (3.23). Let q> be a function of class rc oo over [0, T], zero in a neighbourhood of T, with q>(0) -:f- 0, with values in IR. Then IjJ = q> Q9 v, V E V is in W( V) and by the integration by parts formula (1.49), we have:

(3.52) LT (u' (t), q>(t)v) dt = - f: (u(t), v) q>' (t) dt - (u(O), v) q> (0) .

From equation (3.22) and (3.51), we also have

(3.53) LT (u' (t), q>(t)v) dt = f: (f(t), v) q> (t) dt - LT a(t; u(t), v) q> (t) dt .

Besides, from equation (3.33) (or (3.32», we deduce

(3.54) LT (u~,(t), vm') q>(t) dt = f: (f(t), vm') q> (t) dt - f: a(t; um,(t), vm') q> (t) dt

and also

(3.55) LT (u~,(t), vm') q>(t) dt = - LT (um,(t), vm') q>'(t) dt - (uom', vm') q> (0) .

If we pass to the limit in (3.54)-(3.55) as m' -+ 00 we obtain:

(3.56) lim iT (u~" vm') q>(t) dt = fT (f(t), v) q> (t) dt - iT a(t; u(t), v) q> (t) dt m'~oo ° 0 °

= LT (u'(t), q>(t)v) dt, from (3.53), and

(3.57) lim iT (U~" vm,)q>dt = - foT (u(t), V)q>'(t)dt - (uo,v)q>(O). m'-+oo 0

From which, by comparing (3.52) and (3.57), taking into account (3.56), we obtain

(3.58) (u(O), v) = (Uo, v), "Iv E V.

From the density of V in H, (3.58) remains true for all v E H and therefore:

u(O) = Uo (which is (3.23» .

Thus

Lemma 4. The function u is the solution of problem (P).

Remark 3. We now give the vector form of(3.32) in the approximate problem (Pm) relative to the space U(O, T; V').


Let: V~ be the set of U E V' such that (u, v) = 0 for all v E Vm ;

P~' the projection in V' over Vm , following V~ :

if {Wj,mL= 1 lodm is an orthonormal basis in H of Vm, P~' is given by:

We can show that (3.32) can be put in the form:

(3.32)y

This formula allows us in particular to see how the preceding method can be simplified if the Galerkin approximation is such that we have: (C) the family (P~')mEI\l is bounded in 2(V'). This condition is always satisfied if the Galerkin approximation is constructed starting from an orthonormal basis in Yf of elements in V (as in Example 1, §2 or Chap. XV); (C) implies that such a basis is also a basis in Vand in V' (see Chap. XV, § 1). We can always place ourselves in this case if we only wish to show the existence and uniqueness of the solution, and not make an explicit calculation of this solution. In effect (3.41) immediately implies:

(3.41 )' P~'A(. )um E a bounded set of U(V') .

As the family P~'f(.) is then also in a bounded set of L 2 (V'), we deduce from (3.32)y that the family dum/dt is in a bounded set of L 2 (V'). Lemma 2 becomes:

Lemma 5. The solution Um of(Pm) remains in a bounded set of L 00 (H) and of W(V).

As a consequence of the weak compactness of the unit ball of W(V), we can extract from the preceding sequence, a weakly convergent (to u) subsequence in W(V) and in Loo(H) weakly*. Since the mapping u E W(V) --+ u(O) E H is continuous from Theorem 1, §1, we deduce that um(O) tends towards u(O) weakly in H, therefore that the initial condition u(O) = Uo is satisfied. 0

This will be used later in problems with delay (see §7).

3.5. Stage 5. Strong Convergence

Remark 4. Thanks to the uniqueness of the solution of problem (P), it IS

unnecessary to extract a subsequence of Um and we have:

(3.59) um --+ U in U(V) weakly and in Loo(H) weakly * . o We now introduce


From (3.40), um(T) remains bounded in H and we can extract {um.} in Lemma 3 with

(3.61) um.(T) -+ Xl weakly in H .

Besides,ifwetake<p E .s&([0, T])null in a neighbourhood ofO(49), with <p(T) #- 0, then by reasoning analogous to that made to show u(O) = Uo we obtain

(u(T), v) = (Xl' v), \:Iv E V

from which we deduce

(3.62) u(T) = Xl .

Taking into account Remark 4 of this §3 and Proposition 12 of § 1, it follows that

(3.63) um(T) -+ u(T) weakly in H .

This set, Xm(T) can be written:

Thanks to Lemma 3 and to (3.63), we deduce

1 fT lim Ym(T) = - -2Iu(TW - a(t; u(t), u(t)) dt . n-oo 0

(3.64)

Besides, from (3.36)a), we deduce by integration from ° to T:

~IUm(TW + f: a(t; um(t), um(t))dt = ~IUom12 + f: (f(t), um(t))dt,

from which

But from equation (3.22)(50)

(3.66) ~luol2 + f: (f(t), u(t))dt = ~IU{T)12 + f: a(t; u(t), u(t))dt.

Thus (3.64), (3.65), (3.66) imply

(3.67) lim Xm(T) = 0.

(49) Instead of rp E ~([O, T]), zero in a neighbourhood of T, in the verification of the initial condition (3.23) (see (3.52) to (3.58)).

(50) In effect, by taking v = u(t) E V in the equation (due to (3.22) and (3.24»: (:~ ( . ), v) + a(., u(.),

. 1 d v) = (f(.), v) we obtam: --lu(tW + a(t,u(t), u(t» = (f(t), u(t)), from which we have (3.66) by

2 dt integration from 0 to T.


Since from (3.25), we have

(3.68) ° ~ at f: II um(t) - U(t) 112 dt ~ X m(T) ,

we deduce from (3.67), (3.68)

Proposition 1. When m --+ 00, we have Um --+ U strongly in L2(V)

Remark 5. (3.67) also implies that um(T) --+ u(T) strongly in H. More generally

(3.69) 'it E [0, T], um(t) -+ u(t) strongly in H .

For this, it is sufficient to remark that for to E ]0, T[ fixed, L2(0, to; V) identifies with a subspace of L2(V). Thus all VEL 2( V) define, by restriction to ]0, to [, an element of L 2(0, to; V). (3.69) then results from Proposition 1 and from consideration of:

4. Continuity with Respect to the Data

4.1. Equality of Energy

If u is the solution of problem (P), then (3.22) (and by operating as for (3.65», we obtain "It E ]0, T[,

(3.70) 1 ft 1 ft 2lu(tW + 0 a(O'; u(o'), u(o'» dO' = 21uol2 + 0 (f(O'), u(O'» dO'

called the 'energy equality', as the quantity

1 ft X(t) = 21 u(tW + 0 a(O'; U(O'), u(O'» dO'

represents the 'energy of the system'.

4.2. Continuity Theorem

We always suppose that (3.25) holds.

Theorem 3. Let (uo, f) and (u6, f*) E H X L2(V') and let u and u* be the corresponding solutions of problem (P), then

(3.71) lIu - u*IIL'(H) ~ [Iuo - u612+ ~llf-f*llhv·T/2

(3.72) II u - u* 11L2(v) ~ fi [I Uo - u612 + ~ II f - f* Ili2(V')J/2

Proof Set w = u - U*, w(O) = Uo - u6, g = f - f*.


Then W satisfies

WE W(V),

(3.73) (~~, v) + a(.; w(.), v) = (g(.), v) in !0'(]0, T[),

w(O) = Uo - u6 .

From (3.70), with u replaced by w

521

(3.74) 1 fl 1 fl 21 w(t) 12 + 0 a(O"; w(O"), w(O")) dO" = 21 w(OW + 0 (g(t), w(t)) dt .

As for the a priori estimates (see Sect. 3.3.2), we obtain:

(3.75) ~lw(tW + ~ t II w(O") liZ dO" ~ ~IW(OW + 21a f: Ilg(t)ll!dt

from which we have (3.71) and (3.72). D

5. Appendix: Various Extensions-Liftings

Remark 6. Consider afresh problem (P); we have assumed that f E L2(V'); suppose at present that

(3.76)

We show that problem (P) again has a unique solution in the space:

W*(V) = {u; U E U(V), U' E LZ(V') + Ll(H)} .

To see this, it is sufficient to show that in this case we have some a priori estimates analogous to (3.40) and which are established in the same manner as Theorem 4 which follows. Note that the space W*(V) satisfies the following property: (proof analogous to that made for W(V)).

W*(V) ~ ~O([O, T]; H) ,

W*(V) is equipped with the following norm:

(u E W*(V)=:. U' = U 1 + Uz , U 1 E L2(V'), U 2 E Ll(H))

Ilullw*(V) = Ilullu(v) + inf {llu 1 1I L 2(V') + Ilu 2 1Iu(H)} ' Ul+U2=U'

which makes W * (V) a Banach space. We can also note that in this case with T < + 00:

def W*(V) ~ W*l(V) = {u, U E L2(V), U' E Ll(V')}

and that W*l(V) ~ ~O([O, T]; H).

(51) Note also that with notation analogous to (1.70)

W*I(V) <=+ WI (0, T; V, V') <=+ ~O(]O, T]; H)

(51)


Similarly for the case f E L 2(V') we establish the continuity of the solution with respect to the data.

Theorem 4. Let (uo, f), (u~, f) E H x Ll(H) and let u, u* be the corresponding solutions of problem (P), then

(3.77) Ilu - u*IIL'(H) ~ j2[luo - U~12 + 21If-f*lli'(H)]1/2

(3.78) lIu - u* IIL2(V) ~ ja[IUO - U~12 + 211f - f* lIi'(H)J1/2 .

Proof Again with w = u - u*, w(O) = Uo - u~, g = f - f*, w satisfies (3.73) and (3.74). Since

(3.79) I f' (g(o'), w(O')) dO' I ~ sup 1 w(t)1 IT Ig(t)1 dt o IEIO, T] 0

~ -41 sup Iw(tW + [IT Ig(t)1 dtJ2 , IEIO, T] 0

from (3.74), taking into account (3.79), we deduce first of all

(3.80) 1 II 21 w(tW + 0 a(O'; W(O'), w(O')) dO'

1 1 [IT J2 ~ -2Iw(OW + -4 sup Iw(tW + Ig(t)1 dt , IEIO, T] 0

from which

(3.81) 1 1 [IT J2 4: sup Iw(tW ~ 2Iw(OW + 0 Ig(t)1 dt

This gives (3.77). From (3.80) and (3.25), we obtain, taking into account (3.81)

(3.82) rT 1 [IT J2 1 ex Jo II w(O') 112 dO' ~ 2Iw(OW + olg(t)ldt + 4:llwlli"(H)

~ Iw(OW + 2[I: Ig(t)1 dtT

This gives (3.78). o

Remark 7. By superposition of the results obtained, which is permissible as the problems considered are linear, we can take f given in problem P with

(3.83)

and there exists a unique solution to this problem, the hypotheses being those of Sect. 1 of §3. 0

§4. Problems of First Order in t. Examples 523

Remark 8. Consider the bilinear form over V x V

a(u, v) = «u, v)) ,

and let A be the isomorphism from V --+ V' defined by

(3.84) (Au, v) = «u, v)), Vu, V E V.

From Theorems 1 and 2, there exists a unique u E W( V), the solution of

{ .) du 0

(3.85) I dt + Au =

ii) u(O) = Uo , Uo given in H .

Thus the mapping u --+ u(O) from W(V, V') = W(V) into H ('trace' mapping) is surjective. The solution u of (3.85) is a 'lifting' in W (V) of Uo E H. It is here a 'continuous lifting' since the mapping Uo --+ u is continuous from H --+ W(V). In effect, from Theorem 3, Uo --+ U is continuous from H --+ L2(V) and from equation i) of (3.85) Uo --+ du/dt = - Au is continuous from H --+ e(V'). It therefore follows that Uo --+ u is continuous from H --+ W( V). D

Remark 9. It is possible to consider two complex Hilbert spaces V and H in the setting of Sect. 1 of §3. Then, this time we identify H and its anti-dual H', and V' denotes the antidual of V, then (3.1) holds. We can then take a sesquilinear form a(t; u, v) continuous over V x V, the coercivity hypothesis are then satisfied by Re a(t; u, v) (real part of a(t; u, v)). By taking the real parts of the equalities which occur for the a priori estimates, all of the preceding results remain valid. Notably, formula (3.70) 'equality of energy' becomes: Vt E ]0, T[,

(3.86) 1 fl 1 fl '2lu(tW + Re 0 a(O'; u(O'), u(O')) dO' = '2luol2 + Re 0 (f(0'), u(O')) dO'

the quantity X (t) = ~ I u(t W + Ret a(O'; u(O'), u(O')) dO' represents the 'energy of

the system' in this case. D

§4. Problems of First Order in t (Examples)

We shall first of all treat, with the methods of §3, three simple examples already studied in the preceding chapters. These allow us to:

- make precise the function spaces in which we work. - prepare the elements of the method, which in Chap. XX lead to numerical results.


1. Mathematical Example 1. Dirichlet Boundary Conditions

Let Q be an open bounded set(52) of IW with boundary r. We take V = HA(Q), H = L2(Q), V' = H~!(Q):

(4.1 ) { a((t; u, v) = a(u, v) = (grad u, grad v) = it! L ::; :;i dx

for all t E [0, T] .

Then (see Example 1 of §3), the bilinear form a(t; u, v) satisfies the hypotheses of Theorems 1 and 2 of §3; thus by setting:

Q T = Q x ] 0, T [, f T = r x ] 0, T [ ,

we may apply Theorems 1 and 2 to Cauchy-Dirichlet boundary value problems of the following type:

(4.2)

(4.3)

(4.4)

where we take

(4.5)

au f . at - L1u = In Q T

ul rT = ° u(. , 0) = uo(.) in Q ,

We deduce that there exists a unique u in:

L2(0, T; HMQ» (l CCO([O, T]; U(Q»

satisfying (4.2) in the sense of distributions in QT, (4.4) holds in the sense of L 2(Q). Boundary condition (4.3) on r T is contained in the fact that u E L 2(0, T; H A(Q».

o

2. Mathematical Example 2. Neumann Boundary Conditions

We now take V = H!(Q), H = U(Q). Then V' = (H!(Q»' is not a space of distributions over Q and consequently L 2(0, T; V') is not a space of distributions over QT' We consider:

(4.6) {

a(t; u, v) = (grad u, grad v)

= L r ~~dx. i JQ oX i oX i

(52) This hypothesis is made here in order for condition (3.25) to hold. If Q is not bounded then (3.5) is satisfied with .Ie > 0, and the conclusions of Theorems 1 and 2 again apply (for all finite T).


The hypotheses of Theorems 1 and 2 of §3 are satisfied(53). 'Formally', these theorems give the solution of the following Cauchy-Neumann problem:

(4.7)

(4.8)

(4.9)

au f . ra at - Au = m ~~T

au = 0 over r T' n 'normal' to r an

u(. ,0) = Uo in D .

In fact, if we take Uo E L 2 (D), f E L 2(0, T; L 2 (D)), Theorems 1 and 2 of §3 give, more precisely, a unique solution u which satisfies:

(4.10)

and:

iT [a(u(t), v) qJ (t) - (u(t), v) qJ"(t)] dt

(4.11) = (uo, v) qJ (0) + iT (f(t), v) qJ (t) dt

for all qJ E ~([O, T[) and for all v E Hl(D) (55).

By choosing v E ~(D) and qJ E ~(]O, T [) in (4.11), we deduce that (4.7) is satisfied in the sense of distributions in DT .

The boundary condition (4.8) over r T is then formally contained in (4.11) (see Chap. VII). In effect, if we integrate by parts in x in (4.11), we obtain, thanks to the fact that v is arbitrary in H 1 (D),

au o. an = m rT •

But this remains formal if we do not make the regularity hypotheses on r precise. Suppose that the boundary of r is Lipschitzian. We may give a meaning to (4.8) in the following manner: Set ~ = - gradxu, ~o = u, [ = (~o, ~). Conditions (4.10) imply that ~ E L2(Dd+ 1 , and (4.7) implies that

divx.t~ = O:tO + divx~ = fE U(DT) ,

therefore, with the notation of Chap. IXA) [ E H (div, DT ); since the cylinder DT

then also has a Lipschitzian boundary, there exists (see Chap. IXA) a trace

(53) In fact, whether or not 0 is bounded (see also Example 2, §3), it is condition (3.5) (with.le > 0) which is realised, and not (3.25). (54) Note that the evolution equation (4.7) can be considered as realised in the space L 2(0, T; H -\ (0)) or even (following the abstract formulation) in L2(0, T; V'). (55) With £0([0, T[) = {v = ullo. TI' U E£0(] - 00, T[}.


Yn[ = rnlO!hE H- 1/2 (aQT) with aQT = r X ]0, T[ u Q X {a} u Q X {T}. It is then permissible to apply Green's formula (see Chap. IXA) for all t/J E H 1 (Q T )

(therefore with trace Yot/J = t/JIO!hE H1/2(aQ T)):

(~, grad t/J) + (div [, t/J) = <Yn[, Yot/J) ,

the bracket in the right hand side denotes the duality H 1/2(QT)' H -1/2(aQ T)' Now we write this again with u

r [uaat/J -gradxugradxt/J + (~~ - L1U)t/J]dXdt = <Yn[,YOt/J) (56).

JnxjO,T[ t at

By difference with (4.11) (written for all t/J = v ® t/J E H1(Q) ® .@([O, T])(57) and more generally for all t/J E H1(QT)) we obtain by using (4.7) and the element e E L 2(aQT)' therefore e E H-1/2(aQT)' defined by

elnxtTl = u(T), elnx{OI = - Uo, elilnxjO,T[ = 0:

<Yn~ - e,Yot/J) = 0, '<It/JEH 1(QT)'

and since the mapping t/J E Hl(QT) ...... Yot/J E H I /2(aQT) is surjective, we obtain:

Yn[ - e = OinH- 1/ 2 (aQ T)'

If we wish to express the Neumann condition on the boundary rT = r x ]0, TL we are led to introduce the space H 6~2(r T) (see Chap. VII, §2 and Lions-Magenes [1]) and its dual (H6~2(rT))'; we then obtain:

h l' au I - 0' (112 ))' t erelore -a - III H 00 (rT . n rT

For other details about the sense of such boundary conditions of (weak) solutions of the Cauchy-Neumann problem (4.7), (4.9), and also (4.8) (eventually taken nonhomogeneous at the boundary), we refer to Lions-Magenes [1], Chap. 3, p. 268 and Chap. 4, p. 87. In the case of strong solutions (see Chap. XVII B, § 1) of such problems (for a given I sufficiently regular, for example IE C6'1([0, T]; U(Q)), then u(t) E D(L1N) (the domain of the Laplacian with Neumann conditions) for all t E ] 0, T [; thus

~~ (t)lr = ° in H- 1/ 2 (r), '<It E ]0, T[ (if r is Lipschitzian).

Remark 1. If I is such that, '<Iv E H I(Q):

(f(t), v) = 1/0 v dx + tIl v dr ,

(56) We are tempted to write

<Yn[, YoofJ) = <u(., T), ofJ(·, T» - <u(., 0), ofJ(·, 0) - (0;1. ' ofJlr'lo.TI) , un I '10. T(

but this is a priori formal, as the space Hl/2(oQT ) is not the sum of the spaces Hl/2(Qx{T}), Hl/2(QX{0}), Hl/2(rx]0, T[). (57) This gives the supplementary term (u(T), v) q>(T) in (4.11).

§4. Problems of First Order in t. Examples

where

then l(f(t), v)1 ~ C [I!o(t)1 + 1ft (t)lw '12(nJli v II

and therefore

The interpretation of the corresponding problem is

1 au fr.' at - Au = 0, In QT

au ov = ft, on r T ,

the initial condition being unchanged.

3. Mathematical Example 3. Mixed Dirichlet-Neumann Boundary Conditions

527

D

We assume r = r 1 U r2, r2 an open set of r, r 1 II r2 = 0 and we denote by V the space of u E Hl(Q) such that ulr, = O. We assume r 1 has non-zero measure. If we equip V with the topology induced by H l(Q), then V is a closed subspace of Hl(Q). Note that from the Poincare inequality, see Chap. IV, §7, Remark 4, or Deny-Lions [1], we may equip V with the norm (equivalent to that of H 1 (Q) in this case(58»):

(4.12) u -+ I grad u IL2(U) •

Then, for the choice (4.1) of a(t, u, v) and the given data:

(4.13) uo E L2(Q), fE L2(0, T; L2(Q)),

we obtain the existence and uniqueness of:

u E L2(0, T; V) II ~O([O, T]; L2(Q)),

also satisfying the initial condition u(O) = uo:

(4.14) ~~ - Au = f in the sense of distributions in QT ,

and the boundary conditions:

(4.15) ulr, = 0 (contained in the fact that u belongs to L2(0, T; V)) ,

(4.16) oul = 0, an r 2

(58) By supposing Q bounded (or bounded in one direction) and connected.


this condition contained in:

(4.17)

f: [a(t, u(t), v) qJ (t) - (u(t), v) qJ'(t)] dt

= (uo, v) qJ (0) + f: (f(t), v) qJ (t) dt

for all qJ E '@([O, T [) and for all v E V . o Contrary to Examples 1 to 3 which we will be able to solve by the method of semigroups, since the bilinear form a(t, u, v) is independent of time, we shall now study a mathematical example for which the methods of the preceding chapters will be less suitable, particularly because of the dependence on time of the coefficients of the equation.

4. Mathematical Example 4(59). Bilinear Form Depending on Time t

All the spaces considered are real. Let Q be an open bounded set of ~n which is sufficiently regular. We consider a closed subspace of H 1 (Q) denoted V and satisfying

(4.18) H MQ) eVe H 1 (Q) (inclusions in the broad sense) .

Then we take H = L2(Q) and we assume a(t; u, v) given by

1 a(t; u, v) = ± r aij(x, t) ~u (x) ~v (x) dx i. j; 1 J U ()Xi ()xj

(4.19) n ou r

+ if:l ai(x, t) oXi (x)v(x) dx + J u ao(x, t)u(x)v(x) dx ,

where

(4.20)

with

{ ± aij(x, tKe j ~ 0( ± I ed 2 , 0( > 0, ei E ~ ,

(4.21) ~:~~iln Q T (0( constant ;:~ependent of x, t in QT) .

We verify that we are in the conditions of application of Theorems 1 and 2 of §3(60).

(59) The problems considered in this example have been treated by localisation and the Laplace transform,in the ref. Agranovich and Visik [I]. When the coefficients do not depend on t, we recover the methods of the Laplace transform, or in the symmetric case (Examples I, 2, 3) of diagonalisation (spectral decomposition). (60) With, again, (3.5) instead of (3.25), satisfied a.e. t E [0, T].


For this, note that from hypothesis (4.20), we have:

(4.22) 1 s~p (sup ess I a;(x, t)l) ~ C 1 I x.teUT

sup ess I ao(x, t)1 ~ C2 (x,t)e!h

so that a.e. t E [0, T] and v E V

(4.23)

(4.24)

Thus (a.e. t E [0, T] and 'rIv E V)

2 ex 2 ( Ci) 2 112 a(t; v, v) + Alvl ~ "2 I grad vb(D) + A - C2 - 2ex Ivl ~ ex1llv

C2 by choosing A - C 2 - _1 > O.

2ex

529

Then, for Uo given in L2(.o) and IE L 2(.oT)' Theorems 1 and 2 of §3 lead to existence and uniqueness of u satisfying

(4.25)

(4.26)

U E L2(0, T; V) (\ CCO([O, T]; L2(.o)) , u(O) = Uo ,

f: [a(t; u(t), v) lp (t) - (u(t), v) lp'(t))] dt

= (uo, v) lp (0) + f: (f(t), v) lp (t) dt

for alllp E EC([O, T [) and for all v E V.

Now introduce the differential operator:

A (x, t, :x) = A(t) = - . t -:- (aij(x, t)-:-) U l,j=1UX; uXj


Then (4.26) implies in particular

(4.27) A (x, t, :x) u + ~~ = f in !J)'(.QT)

(take qJ E ~(]O, T[) and v E ~(.Q)).

Note that (4.25) implies: u, ~u E L2(.QT where i = 1, ... , n and moreover that UXi

u(t): x -+ u(x, t) is almost everywhere in V. From (4.27), we deduce the equality in ~'(]O, T[):

(4.28) {( A (X, ., :x)u + ~~,v) = Lf(X, .)v(x)dx

for all v E V, <, > denotes the duality between V' and V,

and the first member must be equal to

(4.29) lou ) d a(.; u(.), v) + \ot' v = a(.; u(.), v) + dt (u(.), v)

because of (4.26). To summarise,

(4.30)

1) u is the solution of (4.27) in

2) u satisfies the initial condition u(x, 0) = uo(x) a.e. in .0

3) u satisfies boundary conditions which are of two sorts

(if V is distinct from HI (.0)):

i) u(t) E Va.e. tin ]0, T[

ii) ( A (x, t, :x) u, v) = a(t; u(t), v) a.e. t for all v E V.

If we assume r regular, n denotes the outward normal to r, we get a n a

-::1 - = I aij(x, t) cos (n, Xi) -;uV A(/) i. j= 1 uXj

(cos(n, xJ being the ith direction cosine of the normal to n, then by using the Green's formula (see Chap. II, §8), condition (4.30)-3)-ii) is often formally written

(4.31) i au -::1 - v dr = 0 for all v E V,

r uVA(/)

dr denotes the element of area of r. We may again make precise in what sense the boundary condition corresponding to the formal statement (4.31) is satisfied in the case where .0 has Lipschitzian boundary r (with .0 locally on one side of n by posing, in an analogous fashion to the case of the Neumann problem for the Laplacian

au ~o = U'~i = - Iaij-;-, i = 1 ton,[ = (~O'~l""'~n)'

j uXj


Then by (4.25), [E L2(QTr+l, and by (4.27) divx,1 [E L2(QT). Thus (see Chap. IXA), [E H (div, QT) and therefore admits a normal trace yJ = tnIO!hEH-l/2(oQT)' It is permissible to apply the Green's formula:

({, gradljl) + (div [, 1jI) = <Yn[, yoljl> , VIjI E Hl(Q T )

(the bracket denotes the duality H 1/2(oQT)' H- 1/2(oQT))' By making this formula explicit, and by difference with (4.26), we obtain, as in the case of Mathematical Example 2, by introducing the function eEL 2(oQT) (therefore e E H- 1/2(oQT)) defined by the equality:

<yJ - e, yoljl) = 0, VIjI E W(V, H) (61) ,

which expresses both the initial conditions and boundary conditions at the same time. Neumann boundary conditions may be expressed, by introducing the space

vlrT = w,lT = oQ x ]0, T[}

(therefore v c H W(l T)), by:

/ ~'W) = 0, Vw E v. \ oVA(I)

We make precise the choice of V.

Mathematical Example 4.1. V = H6(Q). Then (4.30) always holds.

D

The fact that u(t) belongs to H6(Q) (a.e.) contains the Dirichlet boundary condition ulrr = O. D

Mathematical Example 4.2. Condition (4.30)-3-i) is no longer a boundary condi

tion; (4.31) gives us ;) ou I = O. uV A(I) rT

This condition is called the Neumann condition. D

Mathematical Example 4.3. 1 being assumed regular with 1 = 11 U 1 2 ,

11 n 12 ¥- 0 (11 of capacity > 0), we set liT = 1i X ]0, T[, i = 1,2 and we take:

V = {v E Hl(Q); vlrl = O} .

The boundary conditions are then of two sorts (mixed problem):

(4.32)

(4.33)

(4.30)i) gives Ulr lT = 0 (Dirichlet condition)

(4.30)ii) gives /u I = 0 (Neumann condition) . v A(I) r 2T

D

(61) That is to say IjJ E L2(O, T; V), 1jJ' E L2(O, T; H) with H = U(Q), H~(Q) eVe HI(Q) and therefore W(V,H) c HI(QT ).


Remark 2. A regularity result We place ourselves in the mathematical framework of Mathematical example 4.1. but supposing a(t; u, v) = a(u, v) (independent of t). If Um denotes the Galerkin approximation of order m of the problem corresponding to (4.26), we have, a.e. t E [0, T], or in ~/(]O, T[):

(4.34) {(U~(t), v) + a(um(t), v) = (f(t), v) for all v E Vm C H~(Q).

As an approximation Vm of H ~(Q) consider

(4.35)

where the »j are elements of the special basis defined by

(4.36) { - LI »j = Aj »j

»jlr = ° (i.e. the eigenfunctions of the Laplacian in the Dirichlet problem). By taking v = - Llum in (4.34)(63), we obtain:

(4.37)

from which we deduce by integration in t:

(4.38) ~I Vum(tW + L (Aum, - Llum)ds = L (f, - Llum)ds + ~I VUom 12 •

If now, we assume the coefficients of the operator A (x, :x)' regular enough,

we may apply the Ladyzhenskaya-Sobolevski inequality (Sobolevski [1], Ladyzhenskaya [2], P. L. Lions [1]) (which as A = - LI is none other than a coercivity condition of Ll2 over H2(Q))

(4.39) {there exists ci constant > ° (i = 1, 2) with

(A v, - Llv) ~ clllvll~2(Q) - c21vl 2

lor all v E H2(Q) n H&(Q) .

From (4.38), (4.39) we deduce:

(4 40) 1 2 (t 2 1 2 (t 2 . 2 lVum(t) I + C1 Jo IlumII H2(Q) ds ::::; 2IVuomi + C2 Jo Iuml ds

+ f~ 1/1.ILluml ds

if I E L2(0, T; H). Now, we already know that Um is bounded in £2(0, T; H), consequently, if Uo is given in H&(Q) then lVuoml ::::; clVuol, and since ILluml ::::; II Um IIH2(Q),from (4.40), we

(62) Where { WI' w2 , ••• , Wm } denotes the vectorial space generated by WI' ... , Wm •

(63) Note that with the choice (4.36), v = - Llum E Vm •


deduce:

(4.41)

from which:

(4.42) {Urn remains in a bounded set of L 00 (H b (Q»

Urn remains in a bounded set of L2(H2(Q» .

We then have the following regularity result:

Theorem 1. For

(4.43) {i) Uo given in H b(Q) ,

ii) f given in U(O, T; U(Q» ,

the solution u of (4.26)(65) is such that

(4.44)

provided that (4.39) holds.

5. Evolution, Positivity and 'Maximum' of Solutions of Diffusion Equations in £P (U), 1 ~ p ~ 00

5.1. Positivity of the Data and Positivity of the Solution of a Parabolic Problem

o

We assume that all the spaces which occur are real. Let Q be a bounded regular set of ~n. We are given aij , i, j = 1,2, ... , n with

(4.45) au E L oo(Q T )

n n

1 i, ~ 1 aij(x, _tK~j ~ a. Jl ~;, (a.. constant > 0 independent of x, t) (4.46)

foraB (-((l""'(n)E~, anda.e.(x,t)EQT·

For u, v E Hl(Q), we set

(4.47) · f au av a(t; u, v) = L aij(x, t) -a -a dx. i,j=l g Xj Xi

Then for uo, f satisfying

(4.48)

(64) We therefore denote by Ci, i = 3 and 4 some constants> 0, with the notation C4[UO' f) to denote the dependence of C4 on Uo and f (65) With, recall, a(t; u, v) independent of t. For more general regularity results see Lions-Magenes [1], Vol. 2.


we know that there exists a (unique) u satisfying

i) u E <C°([O, T]; L2(Q)) (\ L2(0, T; Hb(Q))

(4.49) ii) :t (u(.), v) + a(.; u(.), v) = (f(.), v) in the sense of £&'(]O, T[)

for all v E Hb(Q).

iii) u(O) = Uo .

We shall establish the following result:

Theorem 2. We assume that hypotheses (4.45) to (4.48) hold, with moreover:

(4.50) uo(x) ~ 0 a.e. in Q

(4.51) f(t) ~ 0 in the sense of £&'(Q)

(i.e. (f(t), q» ~ for all q> E £&(Q) with q> ~ 0). Then we have,for all t E [0, T]:

(4.52) u(x, t) ~ 0 a.e. in Q .

Proof Let v E H b(Q). We may write v = v+ - v- where v+ and v- denote respectively the positive and negative parts of v. We know (see Chap. IV, §7, Proposition 6) that if v E Hb(Q) then v+ and v- E Hb(Q). We may therefore take v = - u- in (4.49). We note that generally:

(4.53)

so that we obtain:

(4.54) 1 d "2 dt lu-(tW + a(t; u-(t), u-(t)) = - (f(t), u-(t));

from (4.51) the right hand side of (4.54) is ~ 0 so that (4.54) and (4.46) imply

(4.55) d dt 1 u - (t) 12 ~ 0 .

Since, from (4.50), u - (0) = 0, we deduce from (4.55) that u - = 0 from which we have the conclusion of Theorem 2. 0

Remark 3. This property (4.52) is valid for all boundary conditions independent of time and such that the space V is stable under the operation v --+ v +. 0

5.2. A Maximum Principle

The given data are those of Sect. 5.1. We consider

(4.56) 1 du - + A(t)u = 0 dt

u(O) = Uo .


We shall show

Theorem 3. We assume

(4.57) Uo E L 00(.0), M = IUoIL<Xl(U).

Then, u, the solution of (4.56), satisfies

(4.58) uELoo(.oT)' VT>O

and

(4.59) lulL'"(Dr ) ::::;; luoIL~(D) (66) •

Proof We use (4.49) with f = 0 and we take

v=(u-M)+.

This is permissible since v = 0 on F. It becomes (a.e. t E [0, T])

(4.60) ( au +) + at' (u - M) + a(t; u, (u - M) ) = O.

But a(t; u, (u - M)+) = a(t; u - M, (u - M)+)

=a(t;(u - M)+,(u - M)+) ~ 0,

therefore

:tl(U - M)+12::::;; O.

Since(u - M)+(x, 0) = (uo - M)+ = Owededucethat(u - M)+ = o therefore u ::::;; M. Changing u to - u, or taking v = (u + M) - we have the result. D

Remark 4. The preceding proof (and the result) is valid for all boundary conditions independent of time, with a space V stable under v -+ (v - M) +. D

5.3. A Property of 'Diffusion Type' Semigroups

In this Sect. 5.3, we are given aij with i,j = 1, ... , n, independent of t satisfying

n

(4.61) aij E Loo(.o),Laij(xKej ~ (X L a a.e. in .0, (X > 0,

and (for simplicity)

(4.62)

(66) We may also note that:

ij i=l

aij = aji , i,j = 1,2, ... , n .

u(., t) e L<Xl(Q) , Vt e (0, T)

Ilu(.,t)IIL<.;;;luoIL<. Vte(O,T).


We set as usual

(4.63) i au av I a(u, v) =~. aij -;- -;- dx, u, V E H (Q).

'.J Q uX i uXj

Then let {G( t)} I", 0 be the semigroup associated with (4.63) and with the Dirichlet conditions(67) (see Chap. XVII A, §3). We know that this semi group operates in U(Q). We shall show

Theorem 4. The semigroup {G(t) L '" Q operates in U(Q) for p such that 1 ~ p ~ + 00.

Proof 1st stage: G(t) operates in LP(Q), 2 ~ p ~ + 00.

We already know that G(t) operates in U(Q). From Theorem 3, G(t) operates in L OO(Q)(68). We then apply the Riesz-Thorin interpolation theorem (see Zygmund [IJ, Sadovsky [lJ): if qJ is a continuous linear mapping of L qO(Q) into UO(Q) and of V'(Q) into Lq'(Q), 1 ~ qQ, ql ~ + 00 then qJ is a continuous linear mapping ofLq(Q)intoV(Q)forallqwithqo ~ q ~ ql;wetakehereqo = 2,ql = + 00,

qJ = G( t). 2nd stage: G(t) operates in LP'(Q), 1 < p' ~ 2. By transposition of the result of the 1st stage G*(t) ( = G(t) thanks to (4.62)) operates in (U(Q))' = U'(Q), lip + lip' = 1, 2 ~ p < + 00. Theorem 4 is therefore proved for p > 1 (but not for p = 1 since the dual of L OO(Q) is not LI(Q) !). 3rd stage: G(t) operates in LI(Q). Denote by sign(A) (sign of A) the function of A equal to 1 (resp. 0, resp. - 1) for A > ° (resp. A = 0, resp. A < 0). The formal idea is to take v = sign(u) in (4.49) (where f = 0). But this is not possible since, in general, sign( u) is not in H b (Q). We then introduce the following approximation of sign(A):

(4.64) /3,(A) = { ~/e -1

irA ~ e

iflAI ~ e

if A ~ - e.

It is permissible to take v = /3,( u) in (4.49) (where f = 0); it becomes

(4.65) (~:, /3,(U)) + a(u, /3'(u)) = 0 .

But

(67) We can also consider other types of boundary conditions independent of t. (68) Recall that the semigroup {G(t)}", 0 is not ofcJass 'Co in LOO(Q).


therefore (4.65) implies

(4.66)

If we introduce

(4.67)

then (4.66) may be written

(4.68) :t L y.(u)dx ::::; 0

therefore

(4.69) L y.(u(x, t))dx ::::; L y,(uo(x))dx .

We now let e tend towards 0 in (4.69) (we may not pass to the limit beforehand) and we obtain in the limit(69)(70)

(4.70) L lu(x, t)1 dx ::::; L luo(x)1 dx, "It > O. o

Remark 5. We have in fact shown

(4.71) o Remark 6. In the case of Neumann conditions, the 3rd stage may be simplified as we may then directly take v = 1, so that

(4.72) :t L u(x, t)dx = 0,

therefore

(4.73) L u(x, t)dx = L uodx .

But if we decompose Uo into u; - Uo and we denote by Wi (resp. w2 ) the solution relative to the given initial data u; (resp. uo) we have:

u = Wi -- W2, Wi ~ 0, fa wi(x,t)dx = fa u;dx,

L w2 (x, t)dx = L Uo dx ,

from which we easily have (4.70).

(69) It should be noted that

r y,(v)dx = r (v _ ~)dX + r (v + ~)dX + r ~dx. J" Jv:;., 2 Jv<; -, 2 J1vl<;,2€

(70) Some proofs of this type have been introduced by H. Brezis and T. Kato.


Note nevertheless that the proof of the 3rd stage is valid for all the boundary conditions. D

Remark 7. Inequality (4.70) is valid, with the same proof, for the case of non symmetric coefficients and also for coefficients aij(x, t) E L OO(Q x [0, T]) depending ~t D

Remark 8. As we have seen in Remark 6, we may always return (no matter what the boundary conditions) to the case Uo ~ O. Then u ~ 0 and consequently, if u is regular, we have: if u = 0 over r

r (Au)dx = _ r ~u dr ~ 0, Jo JrUVA

so that (compared with (4.72))

(4.74) :t In u(x, t)dx ::::; O.

To justify this, we must assume Uo regular (which does not cause any difficulty for approximation in L 1 (Q)) - and assume that the coefficients are regular, which is possible by approximation - but not in L OO(Q) - and therefore arrive at some technical difficulties much greater than those in the method above. D

Application Example 1. Heat flow equation We have seen in Chap. lA, § 1, that the evolution of the temperature u(x, t) at a point x E Q of a continuous homogeneous medium, which is a heat conductor, is subject to the heat flow equation(71) (72)

{ ~u _ LI u = 0, X E Q, t > 0

U;X,O) = uo(x) , X E Q

where uo(x) is the temperature at the moment t = o. In the case where the temperature is zero on the boundary, we moreover have the boundary condition

u(x, t) = 0, X E aQ, t > O.

This problem is a particular case of problem (4.56) with A(t) independent of t and equal to the negative of the Laplacian with Dirichlet boundary conditions. In addition to existence and uniqueness (already given by the Laplace method and method of semigroups), the results of §3 and 4 show the decrease in L 00 of the temperature u(t) at the moment t: this agrees with physical intuition which suggests that we cannot have creation of heat without a heat source. On the other

(71) With respect to equation (1.75)' of Chap. lA, we have scaled the problem to make the conduction coefficient equal to one. Recall that some restrictive hypotheses are necessary in Chap. lA, § 1 to obtain this model. (72) We further assume that Q is bounded.


hand the decrease of u(t) in (which also follows from the accretiveness of the operator ( - A), see Chap. XVII A, § 3) illustrates the decrease of internal energy in the case considered. 0

6. Mathematical Example 5. A Problem of Oblique Derivatives

We look for a solution to the following problem (with Q bounded and regular):

.) au f' Q 1 at - Au = III T,

(4.75) .. ) () au au 0 F 11 P t at + ov = on T,

iii) U(X, 0) = uo(x) in Q .

We assume that p(t) is a given positive function, with p E Loo(O, T). We take H = L2(Q), V = Hl(Q). If Vm denotes a Galerkin approximation of V, the approximate variational problem associated with (4.75) is: find Um E Vm with a.e. t E [0, T]

(4.76) ( OUm ) (OU m ) (73) - J, at'V + p(t) at' v r + a(um,v) - ( ,v), "tv E Vm

and Um(O) = UOm '

By taking v = Um in (4.76), we obtain:

1 d 2 I OUm J, (4.77) '2 dt lum(t)1 + Jr p at umdF + a(um, um) = ( ,um)·

By taking v = u~ in (4.76), we also obtain:

I 2 I 1 OU m 12 1 d J, ') (4.78) lum(t)1 + Jr p at dF + '2 dt a(um, Um) = ( ,Um .

If Uo is given in V, andfin L 2(0, T; H) then from (4.78), we deduce, taking account of p(t) ~ 0:

(4.79) { Um E a bounded set of L 00 ( V)

u~ E a bounded set of L2(H) ,

estimates which are not sufficient, it would seem, to pass to the limit in the term

f oUm pat vdF.

(73) arum, v) = fa grad um.grad vdx.


If we assume

(4.80) p(t) ~ Po > 0, t E [0, T] ,

then we deduce from (4.78), the supplementary estimate

(4.81) f ~o denoting the trace of order 0 on r

1 ot YOU m E a bounded set of L 2(r T) .

Thus we obtain the following result relating to problem (4.75).


(4.82) ii) f given in L 2(0, T; H) = L2(QT ) {i) Uo given in V = Hl(Q)

iii) (4.80) holds .

Then, there exists a unique u satisfying:

1) u E ~O([O, T]; V), U' E L2(QT ) ,

o 2 2) otYouEL (rT ) ,

(4.83) d r a 3) dt (u(.),v) + a(u(.),v) + Jr P at (You)(Yov)dr = (f(.),v)

in the sense of~'(]O, T[)for all v E Hl(Q)

4) u(O) = Uo (74)

Remark 9 1) The preceding theorem again holds if we replace the Laplacian in (4.75)i) by a second order operator A such that A = A * (in fact here tA, as the spaces being real), given by the bilinear form

(4.84) a(u, v) = f r aij ~u ~v ax + rn aou. vdx i.j~ 1 JQ UX i uXj Jl<

aij , ao satisfy (4.20), (4.21) and, moreover,

(4.85)

(74) Note that as a consequence of equation (4.75)i) and the conditions f and u E L 2 (QT)' we have

au I Llu E UfO, T; L2(Q)), which. with (4.83)1). implies - E L2(0, T; H -1/2(r)), and therefore (4.75)ii) is av r

satisfied in L2(O, T;H-I/2(r)), or even in H- 1/2(r). a.e. t.


2) If A f:. A *, a(u, v) satisfies (4.19), (4.20), (4.21), then Theorem 5 subsists by using some supplementary hypotheses of the type

(4.86)

In effect, from (4.77), we then deduce

1 d [ 2 r 2 ] 1 r I 2 - (f, (4.87) 2 dt Iuml + Jr pumdr + a(um, Um) - 2 Jr p umdr - ,um)

and (by reducing to the half-space by localisation and by using(75) inequality (4.14), Chap. IV) since p' is bounded, we have

(4.88) 1- ~Lplu2drl ~ cllull.lul·

It follows that if Uo is given in H ~(Q) andf E L 2(0, T; H) then, from (4.87), (4.88), we easily deduce:

{ um E a bounded set of L2(0, T; Hl(Q)) n L 00(0, T; L2(Q))

(4.89) p. Youm E a bounded set of L 00(0, T; L 2(r)) .

Then, by passing to the limit and by the methods used for the proof ofthe principal theorem of this chapter, we obtain the following result for probem (4.75):

Theorem 6. We assume:

i) V = Hl(Q), H = L2(Q).

ii) a(u, v) satisfies (4.19) to (4.21)(76).

iii) f given in L2(0, T; H), Uo E Hl(Q).

iv) p satisfying (4.86).

(7S) Or even with the trace inequality:

f'U(XI,OWdXI";; c[f['U(XI,XnW + I::n (X',Xn)12}X'dxn]. valid, with the same c, for u. with u.(x) = u(x', Axn ),). > 0, from which:

f'U,(x',OWdX' ,,;; C[~fU2dX + ). flau/aXnl2dX J. By taking). such that ~ f u2 dx = ). f,au/aXn,2 dx, we obtain (4.88) by:

f'U(X',OWdX' ,,;; 2c[fU2 dX J'2[flau/axnI2dX J'2 (76) With a priori au, a;, ao independent of time, but we may generalise. The interpretation of the boundary conditions (4.75)ii) may again be made in a fashion analogous to that of Example 2.

542

Then, there exists a unique u satisfying:

10 u E L2(0, T; V) n ~O([O, T]; H),

2 0 the equation

Chapter XVIII. Variational Methods

(4.90)

-f: (u(t), v)qJ'(t)dt - f: (u(t),(PqJ)'v)rdt

+ f: a(u(t), V)qJ(t)dt (77)

= f: (f, v)qJ(t)dt + (uo, v)qJ(O) + (uo, v)r(PqJ)(O)

for all v E H 1 (Q) and for all qJ E ~ 1 ([0, T]) zero in a neighbourhood of T. 0

Review of Mathematical Examples. We have seen, by treating Examples 4, 4.1, 4.2, 4.3 and 5, the great adaptability of variational methods of different situations (particularly with coefficients depending on x and t): We have been able, as we stated at the beginning, to adapt suitable function spaces to each problem, to make precise in what sense the equalities are satisfied and the regularity of the solution in the framework of a constructive method, suitable for numerical calculation (see Chap. XX). However, this great adaptability of the variational method, due to the fact that we have worked by projection in finite dimensional vectorial spaces to approximate the solution and that we also search naturally for weak solutions, has its drawbacks: the qualitative aspects of the solution is sometimes less evident than in the methods of preceding chapters. Note that these variational methods are susceptible to extension to nonlinear cases.

7. Example of Application. The Neutron Diffusion Equation

7.1. Equation and Data of Problem

We have seen in Chap. lA, §5, the equations of neutron diffusion; in the present example, we treat the evolution problem relating to the diffusion equation by taking into account explicitly the kinetic energy E(E E 8) of neutrons; the problem to be resolved comprises: - the neutron diffusion equation IA (5.12) where we have denoted the unknown by u (instead of <P), with (given) coefficients dependent on t

(77) (u, v)r = f uvdr = f YouYovdr.


1 au . Iv(E)1 at (x, E, t) - dlV D(x, E, t) Vu(x, E, t) + kt(X, E, t)u(x, E, t)

(4.91) -r ks(X, E', E, t)u(x, E', t)dE'

- vx(E) r kf(X,E')u(x, E', t)dE' =f(x,E,t) (78),

- the initial condition

u(x, E, 0) = uo(x, E) a.e. (x, E) E Q x rff = Q x ]oe, jJ[ ;

- the boundary conditions IA (5.18)

u(x, E, 0) = 0, a.e. x E aQ, E E rff, t > O.

By setting U = u/I v(E)I, we obtain an equation which is formally of type (4.91), with the coefficient of the term au/at equal to 1; we shall now start (79) from these assumptions denoting the unknown coefficient by u.

7.2. Variational Formulation

We introduce the Hilbert spaces:

H = L2(rff; L2(Q)) = U(Q x rff)

with the scalar product (u, V)H = r u(x, E)v(x, E)dxdE JQxcf

and

v = L2(rff; Hb(Q)), with norm: Iluli v = (1." U(E)llhMQ)dE yl2

Then we have

(4.92)

with density. We make the hypotheses:

{i) D and k t are given functions in L OO(Q x rff x ]0, TD ,

(4.93) ii) {D(X, E, t) ~ )' > 0, )' is a constant, kt(X, E, t) ~ )' > 0 .

1 (78) We assume that the set tf of kinetic energies E = -mlv(EW of neutrons (where m is the

2 mass of the neutron) is the interval tf = [IX, p] with 0 < IX < p < + 00, it follows that: 0= (2IX/m)1/2 < Iv(E)1 < (2P/m)I/2. We denote by dE a Radon measure over tf (which may particularly be the Lebesgue measure, or a finite sum of Dirac measures in the case of a multi-group). (79) This does not change the hypotheses made (see (4.93)) over D and Er which become v(E)D and

veE) v(E)E" nor the hypothesis made (see (4.96)) on E, which becomes -- E,(x, E', E, t).

veE')


We then define:

(4.94) {d(t,u,V) = LLD(X,E,t)VxU. VxvdxdE

+ L L l:"r(x, E, t)uvdxdE

and we note that: there exists C > 0, constant (with respect to (t, u, v» such that:

(4.95) Id(t;u,v)l::::;; Cllullv.llvllv a.e. tE[O,T].

Set i(x, E, E', t) = Ls(X, E, E', t) + v. X(E)L f(x, E'). We equally assume that:

(4.96) i is a given function in L 00(.0 x C x C x ]0, T[) ;

we introduce the bilinear form

K(t; u, v) = r f i(x, E, E', t)u(x, E')dE'v(x, E)dxdE . Jox.6' .6'

We verify, by applying the Cauchy-Schwarz inequality twice, that:

{ there exist Co and C t, constants independent of t, such that (4.97)

IK(t;u,v)l::::;; ColulHlvlH ::::;; Ctllullvllvll v . It follows from the hypotheses made that t -+ [d(t; u, v) - K(t; u, v)] is measurable in ]0, T[ for fixed u, v and that for almost all t, the bilinear from [d(t; u, v) - K(t; u, v)] is continuous over V x V. Besides, from (4.93) and (4.97), there exists ex > ° such that a.e. t E [0, T]

(4.98) d(t;u,u) - K(t;u,u) ~ exllull~ - Colul 2

or again

(4.99) d(t;u,u) - K(t;u,u) + Colul 2 ~ exllullL VUE V.

We are in the setting of application of Theorems 1 and 2 of §3 with condition (3.5). If we are given Uo E Hand f E L 2(0, T; V'), we therefore have the existence of a unique function u which satisfies:

(4.100)

i) u E C6°([0, T]; H) n L2(0, T; V) ,

ii) :t(u(.),v) + d(.;u(.),v) - K(.;u(.),v) = <f(·),v)

in the sense of 2.&'(]0, T[), "Iv E V,

iii) u(O) = Uo .

We compare this with the results expressed in Chap. XV, §3.1 where a problem consisting of a simpler diffusion equation has been studied. Equality (4.100)ii) may, with suitable regularity conditions on the given data (see Remark 20 of Chap. XV,


§1), be interpreted, by substituting v = Ix formally(80) in (4.100)ii), where X c: Q,

as the balance of neutrons in the domain X x iff. Relation (4.1 00) i) indicates, again with analogy with Chap. XV, that the unique solution u is a continuous function from [0, T] -+ H. The continuity of u with respect to the given data uo and f has been established here by Theorems 3 and 4 of §3. The continuity of u with respect to the given data D, L" f is examined in Sect. 8: A stability result. In the case of the multi group formulation(81) ofthe diffusion equation, over each of the intervals (lXi' lXi+d the given functions E -+ D, E -+ L" E' x E -+ f are constant in E, E', at the interior of an energy group ('averages' of functions with rapid variation). These supplementary hypotheses concerning these three given functions are compatible with (4.93) and (4.96)i). It follows that all the preceding results are applicable in the multigroup formulation.

Discontinuous variation of given data D, L, and f as functions of x. In practical applications, when two different materials occupy the domains Q 1 and Q 2 side by side, as their given data Dl (x, E, t), D2(x, E, t), LI1 (x, E, t), L'2(X, E, t) and f 1(x, E, t), f 1(x, E, t) are distinct, the problem of equation (4.91), posed in Q = Q 1 U Q2 U S is comprised of given functions D, L" f which are discontinuous (82) at the surface S of transmission between Q 1 and Q 2 (these given functions are assumed continuous outside of S). The direct mathematical resolution of variational problem (4.100) in Q, with given data discontinuous at S is perfectly acceptable since none of our mathematical constraints oppose this discontinuity in x, over S. We may then verify that the solution u of the global problem (4.100) in Q, satisfies over S, and a.e. t E [0, T], the continuity condition of the projection of the neutron current j defined by:

j = D grad u across S, onto the normal v. In effect consider, for simplicity(83), the equation:

(4.101) ~~ - div(D gradu) = 0, in Q x ]0, T[

with the Dirichlet conditions ulaax]o. T] = 0 and the initial condition:

(4.102) u(O) = Uo E L2(Q) ,

and we assume that the function D(x, t) is subject to a discontinuity at the boundary S

(80) Formally, since Ix €I V with Ix denoting the characteristic function of X. (81) See Chap. lA, §5. (82) With the discontinuities bounded so that the given functions remain in L ""(0 ~ tf x [0, T]). (83) Instead of equation (4.91). But the proof remains valid for equation (4.91).

546

(4.103) . { Dl (x, t) , I.e.: D(x, t) = )

D 2 (x, t ,


X E Q 1 (84) ,

X E Q2'

In variational form, the problem may be written:

(4.104) u { ( ~Ut ' v) + (D grad u, grad v) = 0, a.e. t E [0, T] ,

u(o) = Uo for all v E V = H6(Q).

If, in (4.104), we take v E '@(Q), then we obtain (4.101), in the sense of distribution in Q and more precisely (a.e. t E [0, T]):

(4.105) t~~VdX - t div(Dgradu).vdx = 0.

Note that (due to Green's formula), we have:

- r div(Dgradu).vdx = - ~ r div(Digradu).vdx Ju · J~

(4.106) -f D 1 V • (grad u) v dS + r D 1 grad u grad v dx s Ju.

+ f D2 v.(gradu)vdS + r D2 gradugradvdx, r J~

(4.107) 1-t div(D gradu)vdx = t D gradu grad vdx

+ Is (D2 v.gradu - D1v.gradu)vdS.

By comparing (4.105) (taking (4.107) into account) and (4.101), we must have:

(4.108) Is (D2 v.gradu - Dl v.gradu)vdS = 0, "Iv E V, a.e. t E [0, T] ,

i.e. the transmission conditions:

(4.109) D1v.gradul u• =D2 v.gradul u2 over S,tE[O,T].

The preceding application of Green's formula to obtain (4.106), and therefore

(4.109), if formal since it is not evident that the trace of Di ~~ on either side of S

exists (notably if it is not evident that a.e. t E [0, T], i = 1,2, div Digradu(t) E L2(Q;) which should imply Diou/ov(t) E H-l/2(oQ;)); we shall note that a.e. t E [0, T], u(t) ¢ H2(Q): otherwise (4.109) may not be satisfied over S. For these problems, called transmission problems, in the stationary case see

(84) With, for example D, E 'C°([O, TJ, 'C°(!2,)), i = 1,2,0, bounded.


Chap. II, §8.2. Prop. 4 and Chap. VII, §2.4. To make (4.109) precise here, we may -~ ~ .

proceed as in §4.2, by setting ~ = (~o, 0 = (u, - D grad u), u the solutIOn of _ _ def _

(4.104). Then ~ E H(div, QT) and ~i = ~IQi E H(div, QiT)' QiT = Qi X ]0, T[,

i = 1,2; therefore (see Chap. IXA) [ and [i have normal traces Yn[ and Yn[i in H- 1/2 ( OQT) and H- 1/2 ( OQiT ). The application of Green's formula (permissible in QT and QiT for [ and [i) as in §4.2, leads (with (4.104)) to:

L <Yn[i, Yol/li)i - <yi: Yol/I) = 0, \fl/l E W(O, T; Hb(Q), L2(Q)) , i

with I/I i = I/II Q., <')i (resp. <,») denoting the duality H- 1/2(oQiT)' Hl/2(oQiT ), ~ . .

i = 1,2 (resp. H- 1/2(oQT), H 1/2(oQT)' and Yol/I' = I/I'IOQ'T' Yol/I = I/IIOQT' From which again (with v normal to S), for all h = y~TI/I, (ST = S x ]0, T[) the trace over ST of 1/1 E HMQT)' therefore h E Hb~2(ST):

1 2 / ou ou) <v(~ - ~ ), h) = \ D2 ov - Dl ov' h = 0

where <. ) denotes the duality (H b~2(ST ))', H b~2(ST)' this gives the sense of (4.109). This method allows us to treat the case where S depends on time. 0

Review. We have treated in this example of neutron diffusion (Sect. 7 of §4), a problem much more complex than those which we may tackle with the methods of the preceding chapters. Amongst the new possibilities brought about in this Chap. XVIII for this application, note: - the treatment of operators A(t) depending on time through the given data D, 1:t

and i and which are:

{dependent on a new parameter, the energy E of the neutron (85),

integro-differential, the integration being over the parameter E ;

- the treatment of given data D, 1:t , i discontinuous with respect to x, - the use of given initial data Uo depending on, apart from x, the parameter E. The methods used here have allowed us to construct the solutions u through approximations, these constructive procedures are susceptible to being used for numerical calculation of the solutions. The constraints imposed on the data indicated by (4.93) do not pose difficulties in applications. They are for the purpose of assuring that the bilinear form { d (t.; u, v) - K(t;u,v)} is V-coercive with respect to H (equation (4.99)), and continuous

over V x V (see (4.95) and (4.97)). These two conditions have been used for all the solutions of §3 and therefore of §4. We have seen that u(t) depends continuously on Uo and fin a suitable topology. We may ask ourselves if u( t) is also continuous with respect to the other given data of

(85) See Chap. lA, §5.


problem (4.100), the given functions D, Lt , Ls and VX(E)Lf occurring in (4.91). This will be deduced in Sect. 8.

8. A Stability Result

The present section establishes a result of continuity with respect to the coefficients occurring in an equation of first order in t. We are given V c H as in §4.4, but complex (86); we assume a family ofsesquilinear forms a6 (t; u, v) over V x V depending on the parameter e E IR (to fix ideas) with

(4.110) {for all e E IR, u, V E V

t -+ a6(t; U, v) is measurable over ]0, T[ ,

and

(4.111) la 6(t; u, v)1 ~ M6.11 u 11.11 v II M6 constant> 0 independent of t .

Further, we assume that there exists A E IR such that

(4.112) { Rea6(t;u,U) + Alul 2 ~ 1XlluIl 2 ,

1X constant > 0 independent of t and of e, "VU E V .

Besides, let a(t; u, v) satisfy, as in §4.4, the conditions analogous to (4.110), (4.111), (4.112) above. We assume

(4.113) { la 6(t;u,V) - a(t;u,v)1 ~ <p(e)lIull.llvll, "VU,VE V,

where <P( e) -+ 0 as e -+ + 00 .

Then let u6(resp. u) be the solution, for givenfin L2(0, T; V') and Uo in H, of the problem

(4.114) Ii) :t (u 6(.), v) + a6(.; u6(.), v) = (f(.), v)

in the sense of ~'(]O, T[)

ii) u6(0) = Uo ,

(resp. of the problem)

Ii) :t (u(.),v) + a(.;u(.),v) = (f(.),v)

(4.114)' in the sense of ~'(]O, T[)

ii) u(O) = Uo .

Then we have

(86) This is not necessary, but is assumed here for generality.

for all v E V

for all v E V


Theorem 7. As 8 -> + 00

{ i) u6 -> u in '6'0([0, T]; H) , (4.115)

ii) u6 -> u in U(O, T; V) .

Proof Denote by w6 = u 6 - u. From (4.114), (4.114)" we deduce that w6 satisfies:

(4.116) Ii) :t(W6,V) + a6(.;w6,v) = a(.;u,v) - a6(.;u,v)

for all v E V in the sense of £0'(0, T) ,

ii) w 6(0) = ° . From (4.116), we deduce the 'energy' equality:

(4.117) 11 i' :2IW6(tW + Jo a6(a; w6(a),w 6(a))da

= I [a(a; u(a), w6(a)) - a6(a; u(a), w6(a))] da .

By using (4.112)-(4.113) and assuming A = 0, we deduce:

(4.118) -lw6(tW + rx Il w6(a)11 2 da ~ <1>(8) Ilu(a)11 1 i' i' 2 0 0

549

we bound the right hand side of (4.118) by using the Cauchy-Schwarz inequality; denoting by I the left hand side of (4.118), we obtain

(4.118)' I ~ <1>(8) (I Ilu(a)11 2 da yl2 (I Il w6(a)11 2 da yI2

rx . 2 and since a.b ~ :2 a2 + ~ b2 ,

Thus, we obtain:

1 rx i' 2 i' (4.119) :2IW6(tW + :2Jo Il w6(a)11 2 da ~ ~<1>2(8)'Jo Ilu(a)11 2 da = C.2(8).

From (4.119), we deduce:

{

i) lim suplw 6(t)1 = ° 6 - 00 'E(O. T)

ii) lim iT Ilw6(t)11 2dt = 0. 6- 00 Jo

(4.120)

In the case where A "# ° in (4.112), we shall obtain the result with the help of Gronwall's lemma (see Lemma 1 of §5).


From which we have Theorem 7. o Mathematical Example 6. Let Q be an open set of IR". QT = Q x JO, T[; we are given

(4.121)

and

(4.123)

{ arjEL:(QT) for ~.~: 1.2 •...• n. OEIR. aij E L (Q T ) for l,j - 1.2, ... ,n .

IX > 0, independent of x. t (and 0)

a.e. (x. t) E Q T

supess sup larj(x. t) - aij(x, t)1 ~ <1>(0) where <1>(0) -+ 0 as 0 -+ + 00; (x.t)e!h i. j= 1 •...• n

then if we set

(4.124) 1 8 f i 8 au oj) a (t; u. v) = L.. aij(x. t).;--;-dx

i.j= 1 U uXj uX i

n i au oj) a(t;u,v) =.~ aij(x.t)-;--;-dx. u,vEH1(Q).

',j= 1 U uXj uXi

the conditions (4.110) to (4.113) are satisfied for u. v E V. a closed subspace of H1(Q) with H~(Q) c: V c: H1(Q).

Remark 10. This type of result gives the dependence of the solution u as a function of the coefficients of the operator in L'XJ(QT ) or in ~O(QT)' 0

Remark 11. Suppose that we know that for all t E [0. TJ. u(t) E W1.P(Q) (Sobolev space constructed over LP(Q)) with p > 2, then Theorem 7 remains true if (4.123) is replaced by

(4.125) {supeSSSUPllarj( .• t) - aij( .• t)IILq(U) ~ <1>(0) -+ 0 as 0 -+ 00

te[O. Tji.j

with! = ! - !. q 2 p

In effect as a8 (t; u, v) and a(t; u, v) are given by (4.124) we shall have(87):

I la8(a; u(a). w8(a)) - a(a; u(a), w8(a))1 da

(87) By using the same notation as in the proof of Theorem 7.


B . h H"ld' I' . h 1 1 1 y usmg teo er mequa Ity WIt - + - + - = q p 2

i I(afj - aij)l~u II~w8IdX ~ Ilafj - aijIIU(Q)·II~u «(1)11 ·IIO",w8 «(1)11 Q uXj uX i UXi LP(Q) uXi U(Q)

II aU II IIow8 II ~ €f>(O)' -«(1) . -«(1) ; oXj LP(Q) oX i U(Q) the analogous inequality to (4.118)' is then:

afT ~ C.€f>2(O) +"2 0 Ilw 8 11 2 d(1

( where Illu«(1)lllp = ~ II ~u «(1)11 ) J uXj LP(Q)

form which we again have the result of the theorem. o Remark 12. In the framework of Remark 11, we see that we may approximate a problem with aij E L 00 (Q T) by a family of problems depending on the parameter 0 with coefficients afj for example of class C(j 00 in x in Q, the convergence holding in the sense of LP(Q). 0

Remark 13. We take the notation of Mathematical example 6. We are given: a~(t; u, v)(resp. ao(t; u, v)) satisfying (4.124) with the hypotheses (4.121) to (4.124), and let:

a8(t; u, v) = a~(t; u, v) + L b~ (x, t)uvdx

and a(t;u,v) = ao(t;u,v) + L bo(x,t)uvdx

with, moreover {b~ and bo E LOO(QT)}' and:

supesslb~(x,t) - bo(x,t)1 ~ 4>(0) where 4>(0) --+ 0 as 0 --+ + 00.

Then of course, the result of the stability theorem is applicable. I n particular, this is applicable to the problem of the application example of neutron physics (equation (4.91)). We deduce the continuity of the solution of (4.91) with respect to the given data D, L 1> L s and v XL f . 0

Remark 14. We shall verify without difficulty some analogous results for evolution equations of second order in t (hyperbolic, well-posed in the sense of Petrowski) and for transport equations (88).

But we must insist on the fact that in these results, T is finite.

(88) An example of the transport equation is given in Chap. lA, §5.1, (5.1).


Otherwise, we shall be cautious with phenomena of the following type: consider:

~~ - Au + a.u + ° in 0 x ]0, T[, 0 bounded

au av (x, t) = ° over L = ao x ] 0, T[ ,

u(x,O) = 1

where the solution is simply u«(x, t) = e-«t. We have u« -+ Uo = 1 in L2(0 x ]0, TD ifcx -+ 0; but ifcx > 0, u« is in U(O x ]0, 00 D whereas Uo = 1 is not in L2(0 x ]0, 00 D. 0

Remark 15. The study of composite media (or of porous media, or of media with chaotic structure) leads to problems of stability with respect to the coefficients which are much more delicate; we must consult Bensoussan, Lions and Papanicolaou [1]. 0

§5. Evolution Problems of Second Order in t

1. General Formulation of Problem (PI)

1.1. Given Spatial Data

We are given two complex, separable Hilbert spaces V and H in the setting of §3, Remark 9; II again denotes the norm in H, II II, II II. the norms in V and V' respectively. The notation is the same as that used in §3 of this Chap. XVIII. We shall denote by U(X), 1 ::;; p::;; + 00 the space LP(O, T; X) when there is no fear of confusion (with X = V, H or V').

1.2. The Families of Operators A(t), B(t), C(t)

1.2.1. The family a(t; u, v), t E (0, TI. We are given a family ofsesquilinearforms {a(t; u, V)}tE[O, T] continuous over V x V with:

(5.1) a(t; u, v) = ao(t; u, v) + at(t; u, v)

(ao is the 'principal part' of a, at the 'rest'). And we assume

i) t -+ ao( t; u, v) is once continuously differentiable in [0, T] ,

ii) ao(t; u, v) = ao(t; v, u) for all u, v E V

(5.2) (i.e. ao(t; u, v) is Hermitian) ,

iii) there exists A. and a. E IR (independent of t) such that

ao(t; u, u) ~ IXilul1 2 - A.luI 2, IX > O,for all u E V, t E [0, T] .


If we denote :t ao(t; u, v) = a~(t; u, v), u, V E V, from (5.2)i), it follows that

(5.3) la~(t;u,v)l::;; collulI.llvll foralltE[O,T],u,vEV (89).

Further we assume that:

{i) t -+ a1(t; U, v) is continuous in [0, T], for ail u, v E V

(5.4) ii) la1(t;u,v)l::;; c1IlulI.lvlforailtE[0,T],u,vE V (89).

Thus we define: A(t) = Ao(t) + Al(t) E !e(V, V') for each t E [0, T] with

{ i) Ao(t) E !e( V, V') for each t E [0, T] (5.5)

ii) Al (t) E !e( V, H) for each t E [0, T] .

We denote by

(5.6) { D(Ao(t)) = {u;u E V,v -+ ao(t;u,v) continuous over V for the topology of H} .

553

1.2.2. The family b(t; u, v), t E (0, TI. We are given a second family of sesquilinear forms continuous over V x V, let b(t; u, v) be:

(5.7) b(t;u,v) = bo(t;u,v) + b1(t;u,v)

where

(5.8) {i) bo(t; u, v) = bo(t; v, u), tE [0, T], u, v E V

ii) bo(t;u,u) ~ Pollull 2 , Po> 0forailtE[O,T],uE V.

Further

{i) t -+ b(t; u, v) is continuous in [0, T]

(5.9) ii) Ib1(t;u,v)l::;; c2I1ulI.lvlforailtE[0,T],u,vE V.

We denote by B(t) = Bo(t) + Bl(t) E !e( V, V'), the operator defined by b, in a manner analogous to A(t) defined by a. We have

(5.10) {i) Bo(t) E !e( V, V')

ii) B1(t)E!e(V,H),

with analogous remarks to those made for A (t) ...

1.2.3. The family c( t; u, v), t E (0, TI. Let {C( t) },elO, TJ be a family of operators of !e(H) with

(5.11) { C( t) is Hermitian and there exists a constant y > ° such that

(C(t)u, u) ~ ylul 2 for ail u E H, t E [0, T]

(89) Denoting by co. C 1 some constants> O.


and

{the function t ~ (C( t )u, v) is once continuously differentiable in

(5.12) [0, TJ for all u, v E H

(i.e.; t ~ (C(t)u, v) E ~1([0, TJ)for all u, v E H).

From (5.12), note that there exists a constant C3 > 0 with

(5.13) I(C'(t)u, v)1 ~ c3 lul.lvl for all u, v E H and for all t E [0, TJ .

Note also that from (5.11), C(t) is invertible and that C(t)-l has the same properties as C( t ):

i) t ~ (C(t)-lU, v) E ~1([0, T]) for all u, v E H

and there exists a constant C3 > 0 with

(5.14) ii) I (:t C(t)-lu, v) I ~ C3 1ullvl for all u, v E H

and for all t E [0, T] .

We have, in effect, from the identity of the resolvent:

(5.15) C(t + h)-l - C(t)-l = C(t + h)-l [C(t) - C(t + h)]C(t)-l .

We deduce(90)

(5.16)

From the scalar interpretation of this formula, that is to say

(5.16)a) (:t C-1(t)u, v) = - (C'(t)C-1(t)u, C-1(t)v),

We deduce (5.14), taking into account (5.11) to (5.13). In everything which follows, we shall set:

(5.17) c(t; u, v) = (C(t)u, v), t E [0, T], u, v E H .

(For some examples of operators A, B, C, see §6 later) (9 1) •

(90) In what follows we use (for typographical reasons) the notation C - I (t) to denote C(t) - I •

(91) The hypotheses made here are too restrictive for certain applications. For example they are not suitable for the problem

a2 u - - Llu = f in Q x ]0, T[ at2

a2 u au - + - = 9 on r x ]0, T[ , at2 av

au (au ) u(x, 0) and - (x, 0) given - (x, 0) must be defined on r . at at

But it is not difficult to adapt the methods which follow to problems of the type above. We can find a general presentation in J. L. Lions [1], Chap. 8, p. 159.


1.3. The Exact Problem

We are given

(5.18)

(5.19)

and we pose:

Problem (PI)' Find u satisfying

(5.20)

(5.21)

(5.22)

Remark 1

U E L2(V), u' = ~~ E L 2(V), :t [C(.)u'] E L 2(V')

f :tC(,;U'(')'V) + b(.;u'(.),v) + a(.;u(.),v) = (f(.),v)

tfor all u E V in the sense of.@'(]O, T[),

{i) u(O) = UO

ii) u'(O) = u 1 .

i) First of all introduce the space:

(5.23) We ( V) = { v; V E L2( V), :t [C(.)v] E L2( V')} .

We easily verify (exercise) that, equipped with the norm:

(5.24) v -+ Ilvllwc = ["vI12(Yl + \\:t C(.)V[2(Y'lJ/2 ,

555

We( V) is a Hilbert space which is none other than the space W( V) used for first order problem if C(t) = I. Then let

(5.25)

We ( V) is a Hilbert space for the norm

(5.26) u -+ Ilullw, = (lIuII12(vl + lIu'lI~yI2 and (5.20) is equivalent to u E We = We< V). ii) It is immediate that

(5.27)

so that condition (5.22)i) has a sense. We shall show that we have

(5.28)

so that condition (5.22)ii) also has a meaning.


iii) Verification of(5.28). 1°. Note firstly that if C(t) operates from V to V (therefore by transposition from V' to V'), C(t) and C- 1 (t) being defined on V with

(5.29)

then

(5.30)

{the functions t -> (C( t) u, v), t -> (C - 1 (t) u, v) E <6' 1 ([0, T])

for all u E V', V E V

{the space WA V) coincides algebraically and topologically with

the space W( V) ,

from which we have (5.28) in this case. In effect, on the one hand the algebraic inclusion W( V) c We ( V) is immediate. On the other hand, we shall show:

(5.31)

d set for v E WAV), dt [C(t)v(t)] = wet).

We have

C(t)v(t) = Vo + I w(O") do- with Vo E V' .

We deduce that

(5.32) vet) = C- 1 (t)VO + C- 1(t) I w(O") dO" ,

from which it follows that

(V'(t),g)=(:t[C- 1 (t)VO],g) + (:t [C- 1(t)] t wdO", g) + (C- 1(t)w(t),g)

- (C(t)C- 1 (t)VO' C- 1(t)g)

- ( C(t)C- 1(t) I wdO", C- 1(t)g) + (w(t), C- 1(t)g)

= (z(t), C- 1 (t)g) where z E L2( V'), for all 9 E V.

If, in this last equality, we replace 9 E V by gEL 2( V), we deduce:

II(V'(t),g(t))dtl ~ Coligllu(v), Co constant forallgEL 2 (V),

which shows that

(5.33)

from which we have (5.31).

§5. Evolution Problems of Second Order in t 557

It now remains to verify the equivalence of the norms. It is immediate that Ilvllwc:::; a11lvll w (a constant). Note that

Ilv'lli2(v'l = IIC- 1(,),C(.)v'lIi2(v'l :::; Y111C(.)v'lli2(v'l

:::; Y11l dd [C(.)v] - C(.)VII Z

t L2(V'l

:::;y{lI:tC(·)V[(V'l + IIV lli2(Vl]

from which we have II v II w :::; Yzil v II WC'

Thus (5.30) is proved and (5.28) holds in this case from § 1, Theorem 1. 2°. To show (5.28) when C(t) is only assumed to satisfy (5.11)-(5.12), we firstly establish:

(5.34) the space ~([o, T]; V) is dense in Wc( V) = Wc(O, T; V) .

For this, we start by extending C(t) to the whole of IR by setting, for example:

{ C(O)IH if t < °

C(t) = C(t) if t E [0, T]

C( T)IH if t > T.

(5.35)

Then by using the same method as for the proof of Lemma 1 of § 1, we are reduced to establishing that ~(IR; V) is dense in Wc( - 00, + 00; V) (obvious notation), which does not pose any difficulty and is left to the reader. Finally, we define the same extension as for W( V) (see Lemma 2 of § 1), again let P: Wc(O, T; V) --+ Wc( - 00, + 00; V) with Pu = u over [0, T]. The proof of (5.28) then proceeds as in Theorem 1 of § 1, the fundamental inequality (replacing (1.25)) being the following: For all cp E ~(IR; V)

- = z (5.36) sup (C( t )cp( t), cp( t)) :::; C. II cp II W,(IR, V) . tE IR

To establish this, we start from

(C( t )cp(t), cp( t)) = f 00 ddO' (C( 0' )cp( 0'), cp( 0')) dO'

= 2 Re ft (~[C(O')CP(O')], CP(O'))dO' - 00 dO'

- Re foo (C'(O')cp(O'), cp(O')) dO' ,

which, by using the Cauchy-Schwarz inequality, gives

(C(t)cp(t), cp(t)) :::; c[foo IIddO' C(O')Cp(O') [(V'l + foo IICP(0')lli2( Vl dO'],



We conclude (as in Theorem 1, §1) by noting that (C(t)u, u) ~ ylul 2 for all u E H.

o We shall now prove

Theorem 1. Under the hypotheses (5.1) to (5.12), and (5.18), (5.19), there exists a unique solution to problem (Pd(92).

2. Uniqueness in Problem (Pl)(92)

2.1. Integration by Parts Formula

Note again that from (5.34), we deduce

(5.37) the space ~([O, T]; V) is dense in Wc ( V) .

We then have

Proposition 1. Let C(t) be given and satisfy (5.11)-(5.12) and u E Wc( V) then

2Re f: (:t [C(t)U'(t)],U'(t))dt = c(T;u'(T),u'(T))

- c(O; u'(O), u'(O)) + Re f: (C'(t)u'(t), u'(t))dt .

(5.38)

Proof We note that if u E ~([O, T]; V) we have

2 Re( :t [C(t)u'(t)], U'(t)) = :t (C(t)u'(t), u'(t)) + Re(C'(t)u'(t), u'(t)) ,

from which we have (5.38) as u E ~([O, T]; V), the result follows by using (5.37).

o In the same way, we obtain

Proposition 2. Let ao(t; u, v) satisfy (5.3) u, v E Wc( V); then

i) r {ao(t;u(t),v'(t)) + ao(t;u'(t),v(t))}dt

= ao(T; u(t), v(T)) - ao(O; u(O), v(O)) - r a~(t; u(t), v(t))dt ;

(5.39)

ii) if u = v,

2 Re r ao(t; u(t), u'(t))dt = ao(T; u(T), u(T))

- ao(O; u(O), u(O)) - f~ a~ (t; u(t), u(t))dt .

(5.39)'

(92) See (5.20), (521) and (5.22).


We again have

Proposition 3. Let u E We (V), then:

(S.40) (:t[C(')U'(')]'v) = :tC(.;U'(')'V) in !,)'(]O,TD

for all v E V.

The proof of Proposition 3 is made in an analogous fashion to that of formula (1.S0).

Remark 2. We may interpret (S.21) as follows:

(S.21)' :t [C(.)u'(.)] + B(.)u'(.) + A(.)u(.) =f(·),

this equality holding in the sense of L2( V'). This follows from the fact that the set of functions of !')(]O, TD ® V (that is to say the set of finite linear combinations of functions of the form cp ® v) is dense in !,)(]O, T[; V) (which is dense in L 2( V)) and from (S.21), (S.40). We may give another vectorial interpretation, by replacing (S.21)' by a system of first order equations, of the following type: set

We have:

u(.) = {u(.), C(.)u'(.)}

g(.) = {O,f(.)} .

(S.21)' immediately gives:

(S.21)" d_ (0 _C- 1(.»)_ _ dt u(.) + A(.) B(.)C- 1(.) u(.) = g(.)

in the sense of Wc(V) xL 2( V'). It remains to show that we are under the conditions of application of §3 (conditions (3.3) and (3.2S) or (3.S)) to justify the term 'parabolic' already used in Remark 1 (proof left to the reader) (Ref. Lions-Magenes [1]). 0

2.2. The Gronwall Lemma

The following lemma will be very useful to us.

Lemma 1 (Gronwall). Let: cp be a function E L 00(0, T), cp( t) ~ 0, a.e. t E [0, T], J1. be a function ELl (0, T), J1.(t) ~ 0, a.e. t E [0, T]. We assume

(S.41) cp(t) ~ f~ J1.(cr)cp(cr)dcr + C, a.e. t E [0, T], C = constant.

Then

(S.42) cp(t) ~ C exp (f~ J1.(cr)dcr) a.e. t E [0, T] .


Proof We note that <p E L 00 (0, T), f..I. ELI (0, T) implies that <p f..I. ELI (0, T), def it

and consequently F(t) = Jo <pf..l.drr + C is absolutely continuous therefore

F'(t) = <p(t)f..I.(t) a.e. in [0, T]. From (5.41) we deduce

F'(t) F(t) ~ f..I.(t) a.e. in [0, T] ,

from which

Log (F~t») ~ I f..I.(rr)drr,

with the same constant C as before, and consequently:

F(t) ~ C. exp I f..I.(rr)drr;

(5.42) then follows from (5.41).

Remark 3. If C = 0, ¢(t) = 0, a.e. in ]0, T[.

2.3. Proof of Uniqueness

o

o

Let u be a solution of problem (PI) corresponding to Uo = UI = ° and f = 0. Since the problem is linear, it is sufficient to show that u = 0. If t is fixed in [0, T], it is possible to apply integration by parts formulae (5.38) and (5.39) to the restriction of u to ]0, t[. Taking into account the continuity properties of u and u', we obtain from (5.21)' (or (5.21) with (5.40»:

c(t; u'(t), u'(t» + ao(t; u(t), u(t» + 2 I bo(rr; u'(rr), u'(rr»drr

(5.43) = - 2 Re I al(rr; u(rr), u'(rr»drr + I a~(rr; u(rr), u(rr»drr

- Re I (C'(rr)u'(rr), u'(rr»drr - 2 Re I bl(rr; u'(rr), u'(rr»drr;

M denotes various constants in the following, we have by using (5.4), (5.5), (5.13) and the inequality 2ab < + a2 + b2 :

(5.44) 1- 2Re I al(rr;u,u')drr + I a~(rr;u,u)drr - Re 1 (C'(rr)u', u')drrl ~ M 1 <1>(rr)drr,

where

(5.45) <1>(rr) = lu'(rrW + Ilu(rr)11 2 ,


and by using (5.9)ii):

I I 2 Re b1 (0'; u'( 0'), u'( 0')) dO' I ~ C2 I II u'( 0') III u'( 0')1 dO'

~ Po I Ilu'(u)1I 2 du + M f~IU'(uWdU thus

(5.46) Ithe right hand side of (5.43)1 ~ 2M f~ <P(u)du + Po I Ilu'(u)11 2 du

on the other hand by using (5.2)iii), (5.8)ii), (5.11), we obtain

(5.47)

left hand side of (5.43) ~ ylu'(tW + IX II u(t)112 + 2Po I Ilu'(u)11 2 du - A.lu(tW ;

from (5.45), (5.46), (5.47), we deduce

(5.48) inf(y, IX), <P(t) + Po f~ lIu'(u)1I 2du ~ Alu(tW + 2M I <P(u)du.

Now, from u(t) = f~ u'(u)du, which is valid in H, we deduce

(5.49) lu(tW ~ t I lu'(uWdu ~ T f~ <P(u)du,

which (by substituting in the right hand side of (5.48)) gives

(5.50) <P(t) ~ Ml I <P(u)du.

The Gronwall lemma (Remark 3) implies

(5.51) <P( t) = 0, t E [0, T] ,

from which we have the uniqueness.

3. Existence of a Solution of Problem (PI)

We use the Faedo-Galerkin method.

3.1. Stage no. 1. Approximate Problem (Plm )

Let {Vm}meN* be a family of finite dimensional vectorial subsapces satisfying (2.1). As in §3, Sect. 3, there exists two sequences

(5.52) { u~ E Vm, u~ -+ U O in V

u~ E Vm , u~ -+ u1 in H.


We note

(5.53) dm = dim Vm, {Wjm }, j = 1, ... ,dm, a basis of Vm .

The approximate problem is then:

Problem (P1m). Find

(5.54)

satisfying

(5.55)

dm

um(t) = L gjm(t) Wjm j= 1

{

:t c(t; u~, Wjm ) + b(t; u~, Wjm ) + a(t; Um' W jm ) = (f(t), Wjm )

1 :::; j :::; dm

um(O) = u~, u~(O) = u~ .

The system (5.55) is a 2nd order linear system, which may, as we know, be reduced to a 1st order system and admit a unique solution. We therefore have

Lemma 2. There exists a unique solution to problem (P 1m ) satisfying

3.2. Stage no. 2. A Priori Estimates

We start by multiplying both sides of equation (5.55) by gjm(t) and we sum in j from 1 to dm : we obtain:

(5.57) (:t [C(t)u~(t)], U~(t)) + b(t; u~(t), u~(t)) + a(t; um(t), u~(t)) = (f(t), u~(t))

from which by integrating from 0 to t both sides of (5.57) after having taken the real parts (multiplied by 2):

(5.58)

c(t; u~(t), u~(t)) + ao(t; um(t), um(t)) + 2 I bo(a; u~(a), u~(a))da

= c(O; u~, u~) + ao(O; u~, u~) - 2Re f~ al(a; um(a), u~(a))da

+ f~ a~(a; um(a), um(a))da - Re I (C'(a)u~(a), u~(a))da -2Re f~ bl(a;u~(a),u~(a))da + 2Re I (f(a),u~(a))da;


M denoting various constants, we note that:

(S.S9) lao(O; u~, u~)1 :( M II u~ 112 :( Mil U O 112

I I2 Re(f(a), u~(a))dal :( IT If(aWda + I lu~(aWda, and by preceding as in the proof of uniqueness (inequalities (S.43) to (S.48)), by noting that:

(S.60) lum(t)1 2 :( 2[lu~12 + t I lu~(aWda] :( M[lluOl1 2 + I lu~(aWda],

we arrive at

(S.61) inf(y,1X)cI>m(t) + Po I II u~(a) 112 da :( quo, u1,f) + M f~ cl>m(a)da ,

with

(S.62)

From (S.61), we firstly deduce that

(S.63) cl>m(t) :( C1(UO, u1,f) + Ml I cl>m(a)da,

which, by using the Gronwall lemma, shows existence and uniqueness of a constant K (independent of m) such that

(S.64) {

SUP II um(t) II :( K IE [0. TJ

sup lu~(t)l:( K , IE [0. TJ

then

(S.6S)

We have therefore obtained

Lemma 3. The solution Um of problem (P 1m ) lies in a bounded set of LOO(O, T; V) (and therefore of L2(0, T; V)); u~ lies in a bounded set of LOO(O, T;H) and of L2(0, T; V).


3.3. Stage no. 3. Passage to the Limit

From Lemma 3 and the hypotheses of continuity made on the forms a(t; u, v), b(t; u, v), we deduce that

(5.66) { A(. )um lies in a bounded set of L OO( V) and of L2( V) ,

B(. )u;" lies in a bounded set of L 2( V') .

From which, from the properties of weak compactness of the unit balls in the spaces which occur, we deduce

Lemma 4. We may extract from the sequence {Um} mE N*' a subsequence {ull } possessing the following properties:

i) ull --+ u weakly in U( V),

ii) u~ --+ X = u' weakly in L2(V) and weakly * in LOO(H),

iii) A(.)ull --+ A(.)u weakly in L2(V'),

iv) B(. )u~ --+ B(. )u' weakly in L 2( V'),

v) C(.)u~ --+ C(.)u' weakly in U(H).

Consider then qJ E 0J(]0, T[) and v E V. From (2.l)ii), there exists {Vm} mE N*' Vm E Vm for all m such that Vm --+ v strongly in V. We introduce

(5.67)

we particularly have

(5.68) {i) t/Jm --+ t/J strongly in L 2( V)

ii) t/J;,. --+ t/J' strongly in L2( V) .

This set, from (5.55) we deduce:

(5.69)

f: (C(t)u~(t), t/J~(t))dt + f: [b(t; u~(t), t/J1l(t))

+ a(t; ull(t), t/J1l(t))] dt = f: (f(t), t/J1l(t))dt ,

and from Lemma 3, (taking account of (5.68)) we deduce, by letting J1 --+ 00.

(5.70)

which is (5.21).

-f: (C(t)u'(t), t/J'(t))dt + f: [b(t; u'(t), t/J(t))

+ a(t; u(t), t/J(t))] dt = f: (f(t), t/J(t))dt


3.4. Stage no. 4: u is a Solution of (PI)

To show that u is a solution of (P 1)' it remains to show that u E Wc ( V) and that equations (5.22) hold.

Verify that u E Wc ( V). We already know that u E U( V), u' E L 2( V). From (5.70), the mapping

t/J -+ (:t [C(.)u'], t/J ) = - tT (C(t)u'(t), t/J'(t))dt

= f: (f(t) - A(t)u(t) - B(t)u'(t), t/J(t))dt

is continuous over !:0(]0, T[; V) equipped with the topology of L 2( V) therefore, (by density) over L2( V).

Thus :t [C(. )u'] E L 2( V'), which is what must be satisfied.

Verification of(5.22)ii). Let <p E ~l([O, T]) be zero in a neighbourhood of T, with <p(0) # O. Then we set

(5.71)

where Vm E V is such that Vm -+ v strongly in V. From (5.55), we deduce

- f: c(t; u~(t), t/J~(t))dt + f: b(t; u~(t), t/J/l(t))dt

+ f: a(t; u/l(t), t/J/l(t))dt = c(O; u;, t/J/l(0)) + f:(f(t), t/J/l(t))dt ,

(5.72)

and by passage to the limit (J.I. -+ 00), we obtain:

(5.73)

- f: c(t; u'(t), t/J'(t))dt + f: b(t; u'(t), t/J(t))dt

+ f: a(t; u(t), t/J(t))dt = c(O; ul, t/J(O)) + f: (f(t), t/J(t))dt .

Moreover, from (5.21)' we deduce:

(5.74)

- f: c(t; u'(t), t/J'(t))dt + f: b(t; u'(t), t/J(t))dt

+ f: a(t; u(t), t/J(t))dt = c(O; u'(O), t/J(O)) + f: (f(t), t/J(t))dt .


From the comparison of (5.73) and (5.74), we deduce

(5.75) (C(O)u'(O), v) = (C(0)u 1, v) for all v E V

from which C(O)u'(O) = C(O)U 1, and C(O) being invertible in H, u'(O) = u 1. 0

Verification of(5.22)i). We may note for example that:

(5.76) f: (u~(t), t/I,.{t))dt = - (u~, t/I,JO)) - f: (u,,(t), t/I~(t))dt , from which by passage to the limit, (which is permissible):

(5.77) f: (u'(t), t/I(t))dt = - (UO, v)qJ(O) - f: (u(t), t/I'(t))dt ;

moreover,

(5.78) f: (u'(t), t/I(t))dt = - (u(O), v)qJ(O) - f: (u(t), t/I'(t))dt ,

from which, by comparison of (5.77) with (5.78)

(5.79) (UO, v) = (u(O), v) for all v E V,

which implies u(O) = uO, from which we have the result. Thus

Lemma 5. The function u is the solution of problem (PI)'

3.5. Stage no. 5. Strong Convergence

Thanks to the uniqueness of the solution, the entire sequence {um } satisfies Lemma 4 and not only a subsequence. It is then possible to show the following strong convergence results:

(5.80) {i) um(t) ~ u(t) strongly in V, t E [0, T]

ii) u;"(t) ~ u(t) strongly in H, t E [0, T]

iii) u;" ~ u' strongly in L 2( V) .

For the verification of (5.80), we refer to Remark 5 later.

4. Continuity with Respect to the Data

4.1. Energy Equality

o

By using Remark 2, take the scalar product (in the duality [L 2( V'), L 2( V)]) of (5.21)' with u' E L 2( V).


Taking account of formulae (5.38)-(5.39)', we obtain the energy equality:

(5.81)

567

X(T) = X(O) + f: a~(O"; u(O"), u(O"))dO" - 2 Re f: al(O"; u(O"), u'(O")) dO"

- 2 Re f: bl(O"; u'(O"), u'(O")) dO" - Re f: (C'(O")u'(O"), u'(O")) dO"

+ 2 Re f: (f(0"), u'(O"))dO"

where we set for t E [0, T].

(5.82) { X(t) ~ c(t; u'(t), u'(t)) + ao(t; u(t), u(t))

+ 2 I bo(O"; U'(O"), u'(O")) dO" (93).

Of course, (5.81) is still true with Treplaced by t. Taking account of the continuity of u and u' we therefore have, for all t E [0, T],

(5.83)

X(t) = X(O) + I a~(O"; U(O"), u(O")) dO"

- 2 Re I adO"; u( 0"), u'( 0")) dO" - 2 Re I bl (0"; u'( 0"), u'( 0")) dO"

- Re I (C'(O")U'(O"), u'(O"))dO" + 2 Re I (f(0"), u'(O")) dO" .

4.2. Continuity Result

Theorem 2. Let u be the solution of problem (Pd corresponding to given data u 0, u 1, f, and u * be the solution of the problem with data u *0 , u * I ,f *. Then there exists a constant M (T) > 0 such that

(5.84)

{

i) II u-u* II LOO(V) ~ M(T)[II uo-u*o II + lul-u u I + II f-f* II L2(H)]

ii) II u' -u*'11 LOO(H) ~ M( T)[ II Uo -u*o II + luI -u*11 + II f-f* II L'(H)]

iii) II u'-u*'11 L2(V) ~ M(T)[ II uo-u*o II + lul-u*11 + II f-f* II L'(H)] .

(93) We shall again use (see (5.85) and (5.88» the notation X(t, u(t» for X(t) to indicate the dependence on the function u.


Proof. Set W(t) = u(t) - u*(t), g(t) = f(t) - f* (t); then we have in L 2( V'):

:t[C(')W'(,)] + B(.)W'(.) + A(.)W(.) = g(.),

from whiCh we deduce, thanks to (5.83) by setting

def [' X(t, W(t)) = c(t, W', W') + ao(t, W, W) + 2 Jo bo(u; W'(u), W'(u)du,

that:

X(t, W(t)) = X(O, W(O)) + I a~(u; W, W)du

(5.85) - 2 Re f~ a1(u; W, W')der - 2 Re f~ bl(u; W', W')du

- Re f~ (C'(u) W', W')du + 2 Re I (g(er), W'(u))der .

By continuing as for a priori estimates, we obtain:

(5.86)

cP(t, W(t)) + Po f~ II W'(u)11 2du ~ C[cP(O, W(O))

+ f: Ig(erWder] + C l f~cP(er; W(u))du

with

cP(t; W(t)) = I W'(tW + II W(t) II 2, C, C 1 constant.

By using the Gronwall lemma, we obtain

cP(t, W(t)) ~ C[cP(O, W(O)) + f:lg(er) 12 der] expCl T

~ M l (llW(O)1I 2 + IW'(OW + IIgllL2(H»)

from which we have the theorem.

Remark 4. The hypotheses made on f are not the most general possible. Theorem 1 remains true if we choose

(5.87)

or even

(5.87)'

In effect, it is sufficient to note that

I f>Re(f(er),u;,,(u))dul ~ 2 I If(u)1 I u;"(u) I du ~ f~lf(er)l[l + lu;"(u)12]der

~ Df(er)l[l + cPm(u)]der (with (5.62)),

o


in the case of (5.87), and (5.63) becomes:

(5.63)' m(t) ",-; C1(UO,U1,f) + I(M1 + If(u)l)m(u)du;

Gronwall's lemma is applicable with JL( u) = M 1 + If (u) I, JL E U . In the case of (5.87)' it is sufficient to note that

12 I Re(f(u), u:"(u))du 1 ",-; C f~ II feu) II; du + ~o I II u:"(u) 112 du

and (5.63) is still true. In both cases, we have a theorem analogous to Theorem 2.

Remark 5. Verification of strong convergence results (5.80) We introduce Wm = u - Um' From (5.58) and (5.83) we deduce by addition:

(5.88)

569

o

X(t, Wm) = X(O, Wm(O)) + I [a~(u; Wm, Wm) - 2 Rea1(u; Wm, W:..)] du

- 2 Re I [bdu; w:.., W:..) + ~ (C'(u) w:.., W:..)] du

+ 2 Re f~ (f(u), W:")du + 2Ym(t)

where Ym(t) = Re c(O; u1, u~) + Re ao(O; uO, u~) + Re I a~(u; u, um)du

- 2 Re r a1(u; u, u:")du - 2 Re I [b1(u; u', u:..)

+ ~(C'(U)U',U:")]du + 2Re I (f,u:")du

- [Re c(t; u'(t), u:"(t)) + Reao(t; u(t), um(t))] .

We note that thanks to the weak convergence of Lemma 4 and from (5.83)

(5.89) Vt E [0, T], I Ym(t)1 -+ 0 when m -+ + 00 .

By setting

we note (as for a priori estimates) that

m(t) + I II W:"(u)1I 2 du ",-; Co [,,!(O) + t m(u)du + Zm(t)]

(5.90) where

Zm(t) = I Ym(t)1 + 21 f~ (f(u), W:"(U))dUI, Co constant.


Since for all t E [0, T], lim Zm(t) = 0 and ~m(t) is bounded independently of m

and t, we obtain, by setting

I/!(t) = lim ~m(t), t E [0, T], I/! E L1 ,

and by noting that

lim I ~m(Ci)du ~ I lim ~m(Ci)dCi = I I/!(Ci)dCi ,

the inequality

(5.91) I/!(t) ~ Co I I/!(Ci)dCi, t E [0, T]

which implies I/!(t) = 0, a.e. t E [0, T]. Besides ~m(t) ~ 0, t E [0, T] implies lim ~m(t) ~ O. Therefore, it follows from all this that

(5.92) lim ~m(t) = 0, t E [0, T].

From which we have (5.80)i)-ii); (5.80)iii) then follows from (5.90)-(5.92), from

Lebesgue's theorem, and from lim ~m(O) = lim Zm( T) = o. 0

5. Formulation of Problem (P2)

5.1. Hypotheses and Problem (P2)

The spatial data are those of Sect. 1.1 of §5. The operators A(t), C(t) are given as in Sect. 1.2. of §5. The operator B(t) = 9i(t) is only assumed to satisfy

(5.93) ii) t --+ fJ(t; u, v) = (9i(t)u, v) is continuous in [0, T] {

i) BI(t) E !l'(H, H)for every t E [0, T]

for all u, v E H; u, V E H ;

then, there exists a constant B > 0 (B independent of t) such that 'tit E [0, T]

(5.94) IfJ(t; u, v)1 ~ Blullvl for all u, v E H .

The problem considered is:

Problem (P2). Find u satisfying

(5.95) u E <c°([O, T]; V), u' E <c°([O, T]; H)

{ ddt c(.; u'( . ), v) + fJ(. ; u'( . ), v) + a(. ; u( . ), v) = (f(.), v)

(5.96)

for all v E V in the sense of f0'(]0, T[),


(S.97) {i) u(O) = Uo

ii) u'(O) = ul ,

UO given in V, ul in H, and f satisfying for example (S.19).

571

Remark 6. Problem (PI) is parabolic in nature; problem (P2) will be said to be hyperbolic but this classification will be inappropriate in certain cases, particularly Example 4, §6 and the examples of §6.S.6. D

5.2. Uniqueness in Problem (P2). The hypotheses made on the data are actually the following:

(S.98)

(S.99)

(S.100)

i) A(t) = Ao(t) + Adt)

with

ii) {Ao(t) = A6(t) coercive after addition of AI AI(t) E 2(V, H)

iii) Ao of class ~I from [0, T] into 2(V; V')

Al E LOO(O, T; 2 (V, H».

!!4 E 2(H; H) and !!4 is of class ~o in t

{C(t) = C*(t), C(t) ~ yJ, y > 0

C is once continuously differentiable in 2(H) .

Problem (P2 ) is relative to the equation

(S.101) Pu = f where

(S.102)

Remark 7. The formal adjoint of P is then

1 P* - d Cd d * *

- dt dt - dt!!4 + Ao + A I

d d d [(d!!4*)] = dt C dt + (- 91*) dt + Ao + At - ~ ,

(S.103)

and we see that P * has the same properties as P if

(S.l04)

(S.10S)

AI(t) E 2(V, H) n 2(H, V'), Al being L oo in t

!!4 is ~ I in t in 2 (H, H)

(i.e. t ~ (J(t, u, v) is ~I([O, T]) for all u, v E H). D

We shall now give a uniqueness theorem with supplementary hypotheses (S.104)-(S.1OS), hypotheses which are not necessary to establish the existence of a solution of problem (P2 ).


Theorem 3. Under the supplementary hypotheses (5.104)-(5.105), the solution of problem (P2 )(94) is unique.

We shall give two proofs of this uniqueness theorem.

Proof 1. We note that every solution of (P2) satisfies (5.101). However here u' E U(H) and not L2(V) and consequently the method followed in Sect. 5.2 to show the uniqueness is no longer useful. In this case we use the following procedure (due to Ladyzhenskaya [2]). For this, note that if cp E W(V, H) = {cp: cp E L2(V), cp' E L2(H)} satisfies cp(T) = 0, we have (starting from (5.101) or from (5.96)):

(5.106) f: {a(t; u, cp) + 13 (t; u', cp) - c(t; u', cp')} dt

= f: (f, cp) dt + c(O; ul, cp(O)) ;

(5.106) is established by using the density of 9t)([0, T]; V) in W(V, H). Assume therefore that u satisfies (5.106) with Uo = u 1 = f = O. We shall choose cp in the following way. Let s E ]0, T[; we set:

(5.107) cp(t) = { - r u(O") dO"

o if t ~ s,

and cp E W(V, H) with cp(T) = O. From (5.106), we deduce:

if t ~ s

r {a(t; cp' (t), cp(t)) + f3(t; u' (t), cp(t)) - (C(t)u'(t), u(t)} dt = 0 .

Taking the real part (multiplied by 2) of this equation, we obtain r {:t ao(t; cp, cp) - a~(t; cp, cp) + 2Rea1(t; cp', cp)

from which

+ 2 Re f3(t; u', cp) - ~ c(t; u, u) + (C'(t)u, U)} dt = 0, dt

ao(O; cp(O), cp(O)) + c(s; u(s), u(s)) = r {2 Re a1 (t; cp', cp) - a~(t; cp, cp)

+ 2 Ref3(t; u', cp) + (C'(t)u, u)} dt.

By using the supplementary hypothesis (5.104) Al(t) E LOO(!£>(H, V')), we have

la1(t;cp',cp)1 = la1(t;u,cp)1 ~ C·lul·llcpll,

(94) See (5.95), (5.96) and (5.97).


and, by using the supplementary hypothesis (5.105):

t P(t; u'(t), <p(t)) dt = - t [P(t; u(t), u(t)) - P'(t; u(t), <p(t))] dt

(where P'(t; u(t), <p(t)) = (9i'(t)u(t), <p(t)), so that, denoting by Mo, M o, M, M 1

'suitable' positive constants, we have

12 Re t P(t; u'(t), <p(t)) dtl ~ Mo [t lu(tW dt + f~ lu(t)111 <p(t) II dtJ

and then

ao(O; <p(0), <p(0)) + c(s; u(s), u(s))

~ Mo[ t [lu(tW + lu(t)111 <p(t) II + II <p(t) 11 2]dtJ.

From which by the coercivities:

(5.108) II <p(0) 112 + I u(sW ~ M f~ (II <p(t) 112 + lu(tW) dt .

Now, if we introduce

der f' V(t) = 0 u(O") dO" ,

we have c/J (t) = v(t) - v(s) for t ~ s, therefore

II V(S) 112 + lu(sW ~ 2M[ t II v(t) 112 dt + ~ t lu(tW dtJ + 2Msil V(S) 11 2 ,

and

(1 - 2Ms) II V(S) 112 + lu(sW ~ Ml [t II V(t) 112 dt + DU(tW dtJ;

therefore, for s ~ So, where 1 - 2Mso > 0, we have

(5.109) II v(s) 112 + lu(sW ~ M2 t [II V(t) 112 + lu(tW] dt, M2 = constant.

The Gronwall lemma implies u = 0 in the interval [0, so]. We use the same reasoning again starting from So, from which, step by step, u = 0 in [0, T]. 0

Proof 2. Assume the existence of a solution of problem (P2 ) (which we shall show later). Then there also exists a solution for the adjoint problem (Pi) [which consists of solving

(5.110) { P*u = f u(T) and u'(T) given] ,

since thanks to (5.104)-(5.105) the coefficients of P* have the same properties as the


coefficients of P from Remark 7 (invariance of hypotheses by passage to the adjoint). Therefore u, with

(5.111) {pu = 0

u(O) = u'(O) = 0,

must satisfy

(5.112) f: (Pu, cI» dt = 0 for all cI> E L2(V) .

Take gEL 2 (H) arbitrary and define cI> by

(5.113) {p*cp = g

cp(T) = cp'(T) = O.

Since :t (C( . )u') E L 2(V,), we can integrate by parts in (5.112), and we arrive at

f~ (u, g) dt = 0 for all g E L2(H) ,

from which u = O.

5.3. Existence Theorem for a Solution of Problem (P2)

Theorem 4. We make the hypotheses:

{10 of Sect. 1 which concern V and H ,

(5.114) 2° (5.98) to (5.100) for A, fJI, C ,

3° UO E V, u l E H, fE L2(H) ,

o

Then, there exists a solution of problem (P 2) (in particular satisfying (5.95».

Remark 8. The problem of uniqueness under only hypotheses (5.114) is open. 0

Remark 9. As for problem (PI)' we can use the Galerkin method; Um then denotes the Galerkin approximation of order m, we are led to the following a priori estimates:

(5.115) { ~ a bounded set of LaO (V), u;.. E a bounded set of LaO (H) (95)

dt (C(. )u;") E a bounded set of L2(V') ,

from which we deduce the existence of a solution u in the Hilbert space

(5.116) ~ = {u; u E L2(V), U' E L 2(H), :t [C(. )u'] E L 2(V')} ,

(9') This is due to the fact that the operator fJI of problem (P2 ) is different from the operator B of problem PI'


- . which is 'larger' than the space J¥c( V) of problem (PI)' But U E J¥c does not imply • U E 'C°([O, T]; V) and u' E 'C°([O, T]; H), the functions of the space J¥c are only scalar continuous (u E 'C~([O, T]; V) and u' E 'C~([O, T]; H)). The difficulty remains therefore of obtaining continuous solutions in V (and in H for u'). For this, we shall use an approximation of problem (P2 ) by a problem (P2., (e > 0) of the same kind as problem (Pd, whose solutions u. are in w,,(V), then pass to the limit as e --+ O. The method which we shall use, which consists of approximating a 'hyperbolic' problem (see Remark 6) by a 'parabolic' problem (P2.) (analogous to (Pd, is called parabolic regularisation (ref. Lions-Magenes [1], vol. 1). 0

We therefore introduce

5.3.1. Problem (Pl.)' We set

(5.117) {g,J.(t) = e[Ao(t) + AJ] + g,J(t), e > 0, given, P.(t; u, v) = e[ao(t; u, v) + A(U, v)] + P(t; u, v) for all u, v E V,

and we consider

Problem (Pl.). Find u., satisfying

(5.118)

(5.119)

(5.120)

U. E w,,(V)

{ :t c(.; u~( . ), v) + P.(· ; u~( . ), v) + a(. ; u.( . ), v) = (f(.), v)

for all v E V in the sense of

{i) u.(O) = UO

ii) u~(O) = u l .

Thanks to the introduction of the term e(Ao(t) + Al)U~ - which is an 'artificial viscosity' term - problem (P2.) is a particular form of problem (Pd, therefore

(5.121) Ve > 0, there exists u. which is the unique solution of problem (P2.) •

Remark 10. The interesting part of the method is that the term 'viscous' implies that u~ is more regular. 0

5.3.2. Proof of the Existence Theorem. First Part. By introducing the function

<p.(t) = I u~(tW + II u.(t) 112 ,

we deduce from the equality of energy associated with problem (P2.) the inequality

{ <p.(t) + e (t II u~(o') 112 dO' ~ C(uo, ul , f) + M I (t <p.(O') dO'

(5.122) Jo . Jo where C(uo, ul , n, M 1 are constants independent of e .

576

From which:

(5.123)


{

i) UE remains in a bounded set of L 00 (V)

ii) u~ remains in a bounded set of L oo(H)

iii) Je u~ remains in a bounded set of L 2 (V)

We can therefore extract a subset from uE-again called UE for simplicity-such that

(5.124) {i) UE --+ U weakly * in L oo(V)

ii) u~ --+ U' weakly * in L oo(H) .

Since Jeu~ remains in a bounded set of L2(V), then Je[A o(' )u~ + A.U~] remains in a bounded set of L 2 (V') and

(5.125) lim e[Ao(. )u~ + A.U~] = ° weakly in L 2(V') .

Therefore from (5.93)-(5.94), we deduce that ,

d -d (C(.)u~)=J- A(.)uE - ~(.)u~ - e(Ao(·) + U)u~

t .

--+ J - A(.)u - ~(. )u' weakly in U(V') .

Moreover, since C( . )u~ --+ C(. )u' weakly * in L oo(H), we have

(5.126)

Further

(5.127)

d dt [C(. )u'] = J - A(.)u - ~u' .

{ UE(O) --+ u(O) weakly in H for example

u~(O) --+ u'(O) weakly in H ,

from which we deduce (5.81). Thus the existence theorem is proved subject to establishing the continuity of u and u' in V and H respectively.

5.3.3. Proof of the Existence Theorem. Second Part

i) From the preceding section, it follows that:

(5.128) {u E CCO([O, T]; H) (\ Loo(O, T; V)

U' E CCO([O, T]; V') (\ LOO(O, T; H),

since u E L2(V), U' E L2(H), ~ C(. )u' E L 2(V'). dt

Introduce the following space: let X be a reflexive Banach space, we denote by

(5.129) {

CC.([O, T]; X» the space of functions t --+ J(t) Jrom [0, T] --+ X

which are scalar continuous in X

(i.e. t --+ <J(t), x') is continuous Jor all x' E X) .


Then from (5.128) it follows in particular that

(5.130) {u E ~s([o, T]; H) (l L'x,(O, T; V)

u' E ~s([O, T]; V') (l C'(O, T; H) .

We see in part iii) the importance ofthe following lemma which we shall now prove:

Lemma 6. Let X and Y be two Banach spaces, X ~ Y with continuous injection, X being reflexive. We have:

(5.131) ~s([O, T]; y) (l L 00(0, T; X) c ~s([O, T]; X) .

Proof of Lemma. Let f E ~s([O, T]; y) (l L OO(X). a) We show that

(5.132) '<It E [0, T], f(t) E X, sup Ilf(t) Ilx ~ constant. tEIO. TJ

For this, we can assume that f is defined over IR with properties analogous to those of f over [0, T].

Let Pn be a regularising sequence of functions of £&(IR), even, with t Pn(t) dt = 1.

Since f E L 00 (IR; X), f * Pn satisfies

(5.133) II f * Pn(t) II x ~ M = constant for all t E IR .

It follows that for all arbitrary fixed t we can find a sequence v -+ 00 such that (X' being reflexive)

(5.134) { f,! Pv(t) -+ j(t) weakly in X

Ilf(t) Ilx ~ M .

But for arbitrary y' in Y', t -+ <f(t), y') is continuous, therefore

(5.135) <f * Pn(t) - f(t), y') = (Pn * <f, y') )(t) - <f, y' )(t)

tends to zero as n -+ 00.

Therefore in particular

f * Pv(t) -+ f(t) weakly in Y ,

and from the uniqueness of the limit

j(t) = f(t) from which we have (5.132) .

b) We show that f E ~s([O, T]; X). If tn -+ t, as Ilf(tn) Ilx ~ M, we can extract tv such that f(t v) -+ ~ weakly in X. Since f(tv) -+ f(t) weakly in Y it follows that f(t) = ~ from which we have the lemma. D

The conclusion of i) is then the following: from (5.130) and Lemma 6, it follows that

(5.136) U E ~s([O, T]; V), u' E ~s([O, T]; H).


ii) We shall now show that the function

(5.137) t -+ «1>;.(t) = c(t; u'(t), u'(t» + ao(t; u(t), u(t» + Alu(tW

is continuous over [0, T]. This follows from

Lemma 7 (Energy equality). The function u found in Sect. 5.3.2 satisfies

1 «1>0(t) + 2 Re r P(u; u', u') du + 2 Re r a l (u; u, u') du

(5.138)

= «1>0(0) + r [ -c'(u; u', u') + a~(u; u, u)] du + 2 Re f~ (f, u') du .

We assume Lemma 7 whose proof is very technical and fairly long. The principal difficulty is the following. Starting from the equation

d dt [C(t)u'(t)J + .sf(t)u'(t) + A(t)u(t) = f(t) ,

we must multiply by u' which, via integration by parts, formally gives (5.138). This is formal because u' E L2(0, T; H) (for example) which is not the dual space of L2(0, T; V')! To justify (5.138), we must approximate u' by functions with values in V which are obtained by double regularisation and uses technical lemmas. For this type of approximation and the proof of Lemma 7, it suffices to adapt the proof given by Lions-Magenes [IJ (Vol. I, pages 275ff, to which we refer the reader) to our situation. From Lemma 7 it follows that the function t -+ «1>0(t) is continuous over [0, TJ, therefore so is t -+ «1>;.(t), since we already know that t -+ u(t) is strongly continuous in H. iii) The fact that u (resp. u') is scalar continuous in V (resp. H) with the continuity of t -+ «1>;.(t) allows us to finish. In effect, let tn -+ t and set

(5.139)

we have

Cn = c(tn; u'(tn) - u'(t), U'(tn) - U'(t»

+ aO(tn; U(tn) - U(t), U(tn) - U(t» + Alu(tn) - U(tW ;

Cn = «1>;.(t") + «1>;.(t) + [C(tn; U'(t), U'(t» - C(t; U'(t), U'(t))]

+ aO(tn; U(t), U(t» - aO(t; U(t), U(t» - 2 Re c(t; u'(tn), u'(t»

- 2 Re [ao(t; u(tn), u(t» + A(U(tn), u(t))]

- 2 Re [c(tn; u'(tn), u'(t» - c(t; u'(tn), u'(t))]

- 2 Re [ao(tn; u(tn), u(t» - ao(t; u(tn), u(t))] .

§5. Evolution Problems of Second Order in f 579

Now t -> ao(t; u, v) (resp. e(t; u, v» having, from the hypotheses, a derivative in the sense of bounded measurable distributions, satisfies a Lipschitz condition of order 1. More precisely, C denoting various constants:

(5.140)

{ i) le(t;u,v) - e(t';u, v)1 ~ C.lt - (1Iul.lvl, \;ft, (E [0, T], \;fu, v E H,

ii) lao(t; u, v) - ao(t';u, v)1 ~ Cit - t'lll u 11.11 vii, \;ft, t' E [0, T] , \;fu, v E V.

Thus there exists a constant C I > ° such that we have

(5.141)

and

(5.141)'

{ Ie (t n; U' (t), u' (t» - e (t; u' (t), u' (t» I ~ C I I t n - t I le(tn; u'(tn), u'(t» - e(t; u'(t.), u'(t»1 ~ Cllt. - tl

{ ao(t.; u(t), u(t» - ao(t; u(t), u(t»1 ~ Cllt. - tl ao(tn; u(t.), u(t» - ao(t; u(t.), u(t»1 ~ C lit - t.1 .

From the expression for ¢. and from (5.141 )-(5.141)' and from continuity results i)-ii), it follows that

'n -> 24>,,(t) - 2[e(t; u'(t), u(t» + ao(t; u(t), u(t» + A(U(t), u(t»] = ° . Now, from the coercivity hypotheses

,. ;::: al(lu'(t.) - u'(t)I Z + Ilu(t.) - u(t) liZ) ;::: 0, (a l > 0)

from which we have the continuity of t -> u(t) strongly in V and of t -> u'(t) strongly in H, which proves Theorem 4. 0

5.3.4. Strong Convergence Results. We shall now prove

Theorem 5. Let u. be the solution of problem (Pz.), u that of problem (Pz). Thenfor all t E [0, T],

(5.142)

Proof

{i) u.(t) -> u(t) strongly in V.

ii) u~(t) -> u'(t) strongly in H .

i) Thanks to the uniqueness of the solution u of problem (Pz), it is the whole sequence {u.} which satisfies (5.124) and not only a subsequence (see § 1, Sect. 4.4). ii) Moreover from a priori inequalities (5.123) it follows that

(5.143) {

for all fixed t E [0, T]

u.(t) -> ¢? weakly in V

u~(t) -> ¢tl weakly in H .

By reasoning analogous to that done to show that u(o) = uo, u'(O) = ul (see (5.75) and (5.79», we verify that

(5.144) ¢? = u(t) , ¢tl = u'(t) .


iii) Additionally from the two energy equalities corresponding respectively to (P2.) and to (P2)' we deduce by setting

J¥. = u - u.

the equality for all t E [0, T]

(5.145)

c(t; W~(t), W~(t)) -. ao(t; J¥.(t), J¥.(t)) + 2e I (ao(u; u~, u~) + A!u~12) du

+ 2 Re I {3(u; w~, W~) du + 2 Re I a1 (u; J¥., W~) du

= I [ -c'(u; w~, W~) + ao(cr; W., J¥.)] du + Z.(t) ,

where

(5.146)

Z.(t) = 2[c(0; u1, u1 ) + ao(O; uo, Uo)]

- 2 Re I [{3(cr; u~, u') + {3(u; u', u~)] du

- 2 Re I [a 1 (u; u., u') + a 1 (u; u, u~) du

+ Re I [- c'(u; u~, u') + c'(u; u', u~) + ao(u; u., u)

+ ao(u; u, u.)] du + 2 Re I (f(cr), u' + u~) du

- 2 Re(c(t; u~(t), u'(t)) + ao(t; u.(t), u(t)) .

We note that, thanks to (5.124), (5.143), (5.144)

(5.147) { for all t E [0, T]

IZ.(t)1 -. 0 when e -. O.

Now by the same technique as for a priori estimates, we obtain from (5.145), by setting t/I.(t) = ! W~(tW + II J¥.(t) 11 2,

{ t/I.(t) ~ c f~ t/I.(cr) dcr + IZ.(t)! ,

C constant > 0 independent of t and e .

(5.148)

Since {t/I.(t)} is bounded independently of e and t, we set

- def - -t/I(t) = lim t/I.(t), t/I EL l (0, T) (96) ,

(96) Recall lim = lim sup.

§6. Problems of Second Order in t. Examples

therefore since

lim I 1/1.(0") dO" ~ I lim 1/1.(0") dO" = I ~(O") dO" ,

from (5.147) and from (5.148) we deduce

(5.149) ~(t) ~ c I ;j;(0") dO" ,

581

which implies I//(t) = ° almost everywhere in t. Moreover, since I/I.(t) ~ 0, lim 1/1 .(t) ~ 0(97) and therefore

(5.150) for all t E [0, T] lim I/I.(t) = ° , .~ 0

from which we have Theorem 5. o


We shall now give several examples of problems which are second order in t belonging to the framework studied in §5. First of all in Example 1, we look again at the study of the model wave problem in an open set Q, which has already been considered in the preceding chapters: 1) to recall (and make precise) the spaces in which the solutions are found, and in what sense the different equations are satisfied (evolution equations and initial conditions); 2) to insist on the fact that for an effective numerical solution of such a problem in an arbitrary open set Q (nonetheless sufficiently regular), it is this variational method which will be used. But obviously the method developed in §5 allows us to treat more difficult problems, as we shall see in the examples which follow.

1. Mathematical Example 1

Let Q be an open bounded set of [Rn with boundary F; we take

(6.1)

(6.2)

(6.3)

v = HMQ) , H = L2(Q) , V' = H-l(Q) (98),

n I au aiJ a(t; u, v) = L ~ . ~ dx ,

i=l a uXi UXi

B(t) = 0, C(t) = I (identity) ,

(97) Recall lim = lim inf. (98) These spaces will be taken a priori complex.


(6.4) Uo given in H6(Q) , { fgiven in L 2(0, T; L2(Q)) = L 2(QT)' QT = Qx]O, T[,

u1 given in L 2(Q) ;

then Theorems 3 and 4 of §5 give the existence and uniqueness of u satisfying:

i) u E ~O([O, T]; H6(Q)) , u' E ~O([O, T]; L2(Q)) ,

(6.5)

ii) ~:~ - Au = fin the sense of E'}'(QT ) ,

iii) u(x,O) = UO(x) almost everywhere in Q ,

iv) ~~ (x, 0) = u1 (x) almost everywhere in Q ,

v) ulr = 0 Dirichlet condition contained in the fact that

u(t) belongs to H 6(Q) .


We again consider the situation (6.1)-(6.2)-(6.3) and we take:

(6.6) P,(t; u, v) = ea(t; u, v)

then Theorem 1, §5 gives the existence and uniqueness of u,(t) satisfying(99):

(6.7)

i) u, E ~O([O, T]; H6(Q)) , u~ E ~O([O, T]; L2(Q))

ii) ~2t~' - eAu~ - Au, = f in the sense of E'}' (] 0, T [ x Q)

iii) u.(x, 0) = Uo(x) almost everywhere

iv) aa~' (x, 0) = u1(x) almost everywhere

v) u.lr = 0;

the term - eAu~ represents (in a mechanical interpretation of the problem) a viscous term. From Theorem 5, §5, problem (6.4) is the limit of the 'viscous' problems (6.7) as e ..... 0 in the following sense:

(6.8) { u,(t) ..... u(t) strongly in V

for all t E [0, T] u~(t) ..... u'(t) strongly in H .

(99) We set AU, == u~. at

§6. Problems of Second Order in t. Examples 583


Let Q be an open bounded set of IR" with boundary r sufficiently regular (100).

Let V be a closed subspace of HI (Q) with

(6.9) Hb(Q) eVe Hl(Q) (inclusion in the large sense) .

We again take H = L2(Q) and we note that V' (antidual of V) is not a space of distributions, except if V = H 6 (Q). The family a(t; u, v) is assumed given over V by

"f au au (6.10) a(t; u, v) = i, ~ 1 Q aij(x, t) oXi a~ dx, u, V E HI (Q)

where the functions aij are in L 00 (QT)' QT = Q x ]0, T[. The functions under consideration here have complex values. We assume that

a(t; u, v) = a(t; v, u) for all u, v E HI (Q), the function t --+ a(t; u, v) being once continuously differentiable in [0, T] with

{a(t;u,u) + A.luI2~exlluIl2, ex> 0,

(6.11) for suitable A. ,

where lul 2 = L lu(xW dx, II u 112 = Igrad ul 2 + lul 2.

We then consider B(t) given by

(6.12) B(t)u(x, t) = b(x, t)u(x, t)

U E V

where b is given in Loo(QT) and assumed such that t --+ b(., t) is once continuously differentiable with values in Loo(Q), the dual of Ll(Q) equipped with the weak * topology. Then for u, v E L2(Q)

Ii) t --+ f3(t; U, v) = L b (x, t)u(x) v(x) dx

(6.13) is once continuously differentiable;

ii) 1f3(t; u, v)1 ~ f3lul.lvl, u, v E U(Q) .

Finally we define the operator C(t) by

(6.14) C(t)u(x, t) = c(x, t)u(x, t)

where the function c is given in L 00 (Q T ) and assumed such that t --+ c(. , t) is once continuously differentiable in L 00 (Q) equipped with the weak * dual of L 1 (Q) .

Further, we assume that c(x, t) E IR+ = ]0, + 00 [ with

(6.15) c(x, t) ~ y > 0 .

(100) Lipschitz is sufficient.


Thus, by setting

c(t; u, v) = In c(x, t)u(x)v(x) dx

we have

(6.16) ii) c(t; u, u) ~ ylul 2 for all u E L2(0) {

i) Ic(t; u, v)1 ~ c.lul.lvl for all u, v E L2(0)

iii) t ~ c(t; u, v) is once continuously differentiable.

Therefore if we assume

(6.17) {

i) fgiven in L 2(OT) = L2(0 X ]0, T[),

ii) UO given in V

iii) u 1 given in L 2(0) ,

we can apply Theorems 3 and 4 of §5. We deduce: there exists a (unique) u with

(6.18)

. i) u E ~O([O, T]; V), U' E ~O([O, T]; L2(0)) ,

ii) :t c(. ; u'(.), v) + P(.; u'( . ), v) + a(.; u'(. ), v) = (f(.), v)

for all v E V, in the sense of ~'(]O, T[) ,

iii) u(O) = uO, u'(O) = u1 .

We shall interpret the problem which we have resolved. Set

(6.19) ( 0) deC 0 ou A x, t':l u = - L:;- aij(x, t):;-.

uX i, j uXj uXi

Then, u satisfies in the sense of distributions in ~'(OT):

. 0 (ou) ou ( 0 ) 1) ot c(x, t) ot + b(x, t) ot + A x, t, ox u = f,

(6.20)

ii) u(x, 0) = UO(x) , ~~ (x, 0) = u1(x) almost everywhere in 0,

and the boundary conditions which correspond on the one hand to

(6.21) U E ~O([O, T]; V) ,

and on the other hand to the (formal) equation for all v E V.

(6.22) (A (x, t, :x) u, v) + (b(X, t) ~~, v) + (:t C ~~, v) = < f, v) .


Then by using (formally) Green's formula:

with

( A (x, t, : ) u, v) = / - ~u , v) + a(t; u, v) uX \ uVA H-1i2(r),H1/2(r)

au n a ~ = L aij(t, x) cos (v, Xi) ~ U~ ~j=l U~

585

(where v is the outward normal to r, cos (V, x;) the ith direction cosine of this normal) gives (formally)

(6.23) \ :~ , v) = 0 for all v E V .

The rigorous interpretation of these boundary conditions can be made in a fashion analogous to the case of evolution problems of first order in t (see §4.4): we set

def, au.-~o=C(t)u(t), ~i=-Laij~' l=lton; ~=(~O'~l'···'~n)·

j uXj

From (6.18) and the hypotheses made, we have [E L2(QT)n+l, with

- a (a) divx,t~ = at (C(t)u'(t)) - A x, t, ax u = - b(x, t)u' + f E L2(QT) .

Therefore [E H (div, QT) and [has a normal trace on the boundary aQT: Yn[ E H -1/2(aQ T) (see Chap. IXA). Green's formula (which is now allowable) with [ E H(div, QT) and", E Hl(QT):

([, gradx,t"') + (div [, "') = <Yn[, Yo"'>

(with the bracket <, > denoting the duality H l /2(aQT)' H- 1/2(aQT)) gives:

( C(t) u' (t), ~~) - ~ ( taij ::j' ;~) + (- b(x, t)u' + f, "') = <Yn[, Yo"'> .

By comparison with (5.106) (or (6.18)) we therefore obtain the condition, with (J E L 2(aQT) defined by (Jlax{o) = - C(O)uI, (Jlax{T) = C(T)u(T), (Jlax)o, T[ = 0,

<Yn[ - (J, Yo"'> = 0, V", E W(V, H) (101) .

As for the equations of first order in t (see §4.4), the Neumann boundary conditions can be expressed by:

\ a~:t)' w) = 0

for all W E HAil (r T)' the restriction to r T = r x ] 0, T [ of an element

v E W(V, H) with vlax{o} = Vlax{T} = O.


For other details on the sense of such boundary conditions of (weak) solutions of the Cauchy problem (6.18) (with nonhomogeneous boundary conditions), see Lions-Magenes [1]. Case a). We take V = H 6(Q). Then, (6.23) is always satisfied and the boundary conditions are all contained in (6.21); these are Dirichlet boundary conditions. Case b). We take V = HI (Q). Then, (6.23) gives formally

(6.24) au

aVA = 0 over T .

These boundary conditions are Neumann conditions. Case c). We assume that Vis the space of v E HI (Q) such that vlro = 0 where To is part of T (closed, with capacity> 0); we denote by Tl the complement of To in T (T = To u Td; then the boundary conditions (6.22)-(6.23) give together with being an element of V;

(6.25)

These are mixed type conditions. We meet these examples in particular in wave problems. We have, particularly in electromagnetism (102) and acoustics(103), the equation

1 a2 au ~( ) -a 2 u + bo(x, t) -a - Llu = 0 v x, t t t

with V2(X, t) a function with positive values representing the square of the velocity of wave propagation which is therefore a continuous function of x and of t; bo(x, t) is interpreted in the physical model as a dissipation term. We can put this equation in the form of (6.20):

a( 1 au) au _ (104) at v2 (x, t) at + b(x, t) at - Llu - 0

with

b(x, t) = bo(x, t) - aa (~») . t v (x, t

The conditions which the given data v(x, t) and bo(x, t) must satisfy in order to fulfill the conditions of this Example 3 are deduced from those given for equation (6.20). The cases where v(x, t) and bo(x, t) are constant have been treated before (see Chap. XVII B).

(102) See Chap. lA, §4. (103) See Chap. lA, §1. (104) By assuming V2(X, t) regular.



Let Q be an open set of IRn with boundary r (sufficiently regular). We denote by H(L1; Q) the maximum domain of the Laplacian (i.e.):

(6.26)

equipped with the norm 1111, defined by:

(6.27) II u 112 = lul2 + I L1uI2, U E H(L1; Q), lui = (L lu(xW dx y/2 ,

H(L1; Q) is a Hilbert space. We then take H and V as follows:

(6.28) {i) H = L 2(Q) ,

ii) H5(Q) eVe H(L1; Q) ,

V is a closed subspace of H(L1; Q) equipped with the norm II II. We now consider a(t; u, v) given by

(6.29) a(t; u, v) = L a(x, t)L1u. L1v dx ,

where the function (x, t) ~ a(x, t) has the following properties:

(6.30)

i) a(x, t) E IR for all t and almost everywhere in x ;

ii) for all t E [0, T], a(. , t) is in L oo(Q) ;

iii) for every function <p ELI (Q), the function

t ~ L a(x, t) cp (x) dx

is once continuously differentiable in [0, T]

(i.e.: t ~ a(t, .) is once continuously differentiable from

[0, T] ~ L oo(Q) equipped with the topology

of the weak dual of L 1 (Q» ;

iv) a(x, t) ~ If. > ° . Then, we have a(t; u, v) = a(t; v, u) and we satisfy the conditions of application of Theorems 3 and 4 of §5, with B = 0, C = I, I = identity. Therefore, for f given in L 2(0, T; H), UO given in V, u1 in H, there exists a unique function u with

588

(6.31)


i) u E ~([o, T]; V), u' E ~O([o, T]; H) ,

ii) d22 (u( . ), v) + a(.; u( . ), v) = (f(.), v)

dt

(in the sense of g&'(]0, T[)) for all v E V,

iii) u(O) = uo, u'(O) = u1 .

We now interpret the problem which we have solved. 1 ° The function u(x, t) is the solution in an open set QT = Q x ]0, T [ of

(6.32) iP ot 2 u(x, t) + J(a(x, t)Ju(x, t)) = f(x, t)

in the sense of distributions in QT'

2° The initial conditions are:

(6.33) ou

u(x,O) = UO(x) , ot (x, 0) = u1(x)

almost everywhere in Q.

3° The boundary conditions correspond on the one hand to:

(6.34) u(t) E V, Vt E [0, T] ,

and on the other hand to the formal equation.

(6.35) (J (a( . , t)Ju( . , t)), v) = a(t; u(t), v)

for all v E V and all t E [0, T]. Note that thanks to Green's formula and if the surface integrals have a sense, (6.35) is equivalent to the formal relation

(6.36) { t (:n (a(x, t)Ju))v dr = t a(x, t)Ju :n v dr

for all v E V.

In fact (see Lions-Magenes [1]), if r is regular enough (a sufficiently differentiable manifold of dimens\on n - 1) and Q is an open bounded set, for u E H(J; Q), we can define You E H- 1/2 (r)[you = ulr , trace of order zero] and Y1U E H- 3 / 2 (r)

[Yl u = :~ 1/ trace of order 1 J It is then possible to use the theory of distributions to give a sense to (6.36), by taking v regular enough and by replacing the integrals by suitable dualities. We then have the following examples (which are formal but which may be justified particularly by operating in a manner analogous to Mathematical example 3):

4.1. V = H~(n)

All the boundary conditions are in (6.34).


These are Dirichlet conditions.

4.2. V = H(A; U)

All the boundary conditions correspond to (6.36), these are (formally):

(6.37) a

Au = 0 and an Au = 0 over r .

4.3. V = {u E H(A; U); 1'0U = O}

The boundary conditions correspond i) on the one hand to (6.34), ii) on the other hand to (6.36). We obtain

(6.38) u = 0 and Au = 0 over r .

4.4. V = {u E H(A; U), I'IU = O}

The boundary conditions correspond to part of (6.34) and part of (6.36). We obtain:

(6.39) au = 0 an '

a an (a.1u) = 0 over r .

5. Application Examples

589

5.1. Hyperbolic Problems with Coefficients Which are Irregular in t. Generalities

We have seen in §5, that the mixed problem

(6.40)

with

(6.41)

(6.42)

a2u a ( au) at2 - aXi aij(x, t) aXj

= f(x, t) ,

u(X, 0) = UO(x) ,

au at (x, 0) = u1 (x)

XED, t E (0, T)

and u being subjected to suitable boundary conditions over r x ]0, T [, r = aD, has a unique solution under ellipticity and symmetry hypotheses on aij(x, t) and with the regularity hypothesis

(6.43) a;;j(x, t) E LOO(DT) , DT = D x ]0, T[, Vi, j.

That a hypothesis of type (6.43) is necessary to solve the problem in standard Sobolev spaces has been recognised, in particular, by Hurd and Sattinger [1].


It has been discovered by Col om bini, De Giorgi and Spagnolo [1], that, at least for the case where the aij(x, t) do not depend on x, hypothesis (6.43) is useless, provided that we consider a space offunctions which are very regular in x (Gevrey functions or real analytic functions). Later we shall give one of the most simple results obtained, see the ref. Colombini, De Giorgi and Spagnolo [ll

An example in one space dimension. We consider the equation

(6.44) 02U 02U ot2 - a(t) ox2 = 0 in Q x ]0, T[, Q = ]0, 2n[,

with

(6.45) ° au 1 u(x, 0) = u (x), at (x, 0) = u (x) ,

(6.46) au au

u(O, t) = u(2n, t), ox (0, t) = ox (2n, t)

(solutions periodic in x). We assume that

(6.47) a(t) ~ ao > 0, a.e. in ]0, T[ ,

and

(6.48) a E Ll(O, T)

(without supplementary regularity hypotheses). We shall assume that the given Uo and u1 are 'very regular' in the following sense. We define:

(6.49) a = the space of 2n periodic functions, which are real analytic.

A function r[J of a can be expanded in a Fourier series

+00

(6.50) r[J = L r[Jk eikx k= - 00

and

(6.51 ) ~ r[J E a ~ <=> {there exists M and (j > 0 such that I r[Jk I :::;; Me- 6lkl Vk.

[For the remainder of this no. 5.1, the reader can also, if they wish, take the characterisation by Fourier series as the definition of a.] It is easy to equip a with the topology of a locally convex vector space; we shall not do this here as it requires some technical details. We shall prove the following result, due to Colombini, De Giorgi and Spagnolo:

Theorem 1. We assume that (6.47), (6.48) hold and that

(6.52) uo, u1 E a (defined in (6.49) or in (6.51» .


Then problem (6.44), (6.45) and (6.46) has a unique solution u such that

(6.53) U E ~1([0, T]; a) .

Remark 1. It is obviously ambiguous to introduce ~ 1 functions in t with values in a without making explicit the topology of a; but this ambiguity disappears in the representation (6.56) of the solution u. D

Proof The principle is standard, but the a priori estimates are not. 1) We expand UO and u1 in a Fourier series:

(6.54)

and, from (6.51), there exists M and () > 0 such that

(6.55) ! ! + !P ! ::::::: Me- 6lkl "'k. ak k "::: v

We look for u(x, t) in the form

(6.56)

and, subject to convergence, the uk(t) are given by

(6.57) u~ + Pa(t)uk = 0 ,

( where ~~ = cP) with

(6.58)

2) It remains - this is essential- to obtain a priori estimates. Generally, this results in the following problem: we consider over [0, T], the differential equation

(6.59) V" (t) + a(t)v(t) = 0 ,

and we must bound !v(t)! + !v'(t)! with the help of!v(O)!, !v'(O)! and a(t)(we assume v to have complex values). . A standard estimate is given by the Gronwall lemma:

(6.60) !v(t)! ~ (!v(O)! + !v'(O)l) exp f~ (1 + !a(s)l) ds.

Applied to (6.57) it gives for !uk(t)! a factor exp(ck2) which is not compensated by (6.55); (6.60) is therefore without interest here. The other standard estimate is the energy estimate: it involves a'(t); therefore we cannot use it here. We make the hypothesis [we shall see in point 3) of the proof how to use this in the case (6.57), (6.58)]:

(6.61) {a(t) = P(t) + y(t) , P, P' E Ll(O, T), P(t) > 0, Y E Ll(O, T).

We introduce the 'partial energy'

(6.62) E(t) = P(t)!v(tW + !V'(t)!2 .


We then have

(6.63) 1 It (WI IYI) jE(t) ~ JE(O) eXP 2: 0 T + J7i ds.

The proof is easy; we deduce from (6.62):

E'(t) = [P'lvl 2 + 2pRe(vv') + 2 Re(v'v")](t) .

Using (6.59) we deduce

E'(t) = [P'lvl 2 - 2 Re(yv'v)](t)

~ [I~I (PlvI2) + ~ (Plvl 2 + IV'12)}t) ,

from which, in particular

(6.64) E'(t) ~ ! (IP'(t)1 + Jl'.!QL) E(t) , 2 P(t) J7i(i)

from which we have (6.63). 3) We now apply (6.63) to the situation (6.57) with

lX(t) = k2a(t)

and we have a decomposition of type (6.61) dependent on k. We introduce (see (6.47))

(6.65) a(t) = a(t) in ]0, T[, ao somewhere else in IR .

We have

(6.66)

and we introduce

(6.67)

a(t) ~ ao a.e. over IR .

Pk E ~oo over IR, with support in [ - I~I' 0] ' Pk ~ 0, Pk(t) ~ Colkl ,

f Pk(t) dt = 1, f'P~(t)' dt ~ C1lkl

(such functions exist). We then define:

(6.68) lbk = a * Pk' (t)

Ck = a - bk , Pk = k2bk , Yk = k2Ck ,

therefore IX = Pk + Yk' We verify easily, for large enough k,

(6.69)


We have b~ = ii * P~ and since

(6.70) f+OO f+oo

_ 00 ii(s)p~(a) da = ii(s) _ 00 p~(a) da = 0 ,

we can write

f+OO

b~(s) = _ 00 [ii(s - a) - ii(s)] p~(a) da ,

therefore

(6.71) Itlb~(S)ldS~ C3 1kl sup f+oo'ii(S - a) - ii(s)lds. o lal"l/lkl -00

We can write

Ck(s) = ii(s) - (ii * Pk)(S) = f:: (ii(s) - ii(s - a)) Pk(a) da ,

from which

(6.72) It I Ck(s) I ds ~ C4 sup f+oo lii(s - a) - ii(s) I ds . o lal" 1/1kl - 00

We now apply (6.63) with:

(6.73) Ek(t) = Pbk(t)luk(tW + luatW .

The exponential in (6.63) becomes

exp!It(lb~1 + IkIM)dS; 2 0 bk A

Since bk ~ ao and from (6.71), (6.72) it is bounded by

therefore

(6.74)

exp [Cs1k l sup f+oo lii(s - a) - ii(s)1 dSJ, lal"l/lkl -00

{ jEk(t) ~ jEk(O) exp(lkl "'(Ikl) where "'Ok!) --+ 0 if Ikl --+ 00 .

Using (6.69) we have:

and therefore, from (6.55):

(6.75) jEk(O) ~ C7e-361k1/4 (for example) .

We then define from (6.74), (6.75) that (for example)

(6.76)

593


which shows the convergence of the series (6.56) in the space a, and that of the derivative of the series in t. 0

5.2. Equipartition of the Energy for Wave Type Equations

We consider the Hilbert spaces V, H, V' in the usual setting (V <:+ H == H' <:+ V') (real or complex) and we use the usual notation. We consider A, an isomorphism from V -+ V' given by a positive, symmetric (resp. hermitian) bilinear (resp. sesquilinear) form a(u, v) such that (A + 1) is coercive over V. Then let u be the solution of

(6.77) { ~2~ + Au = 0

U;O) = UO E V,

From equation (5.138), the 'energy'(105)

u'(O) = u1 E H .

(6.78) 1

E(t) = 2 [lu'(tW + a(u(t), u(t))]

is constant:

(6.79)

We introduce the two terms (average energy)

(6.80) 1 rt

K(t) = 2t Jo I U'(U)2 I du, t > 0,

(6.81) 1 rt

L(t) = 2t Jo a(u(u), u(u)) du, t > O.

From (6.79), we have:

(6.82) K(t) + L(t) = Eo .

We shall prove:

Theorem 2. (Equipartition of energy.)

lim K(t) = lim L(t) = -21 Eo . t-oo t-oo

(6.83)

Proof. As we know, from the general theory, for all T > 0, the solution u of (6.77) satisfies u E ~O([O, T], V); u' E ~O(O, T; H), we can, in (6.77), take the scalar

(lOS) See Chap. lA, §1, §2, §4, Chap. IB, §2.


product with u (~~~ and u are in duality} by integrating from 0 to t, we obtain

(6.84) (u'(t), u(t)) - (u\ UO) - I lu'(oWdO" + I a (U(O"), u(O")) dO" = O.

From which

(6.85) K(t) _ L(t) = (u'(t), u(t)) - (u 1, UO) . 2t

From (6.82) and from (6.85), we deduce

1 1 K(t) = -2Eo + -[(u'(t), u(t)) - (u 1, UO)] .

4t (6.86)

But, from (6.79)

(6.87) u E L 00([0, + 00 [; V), U' E L 00([0, + 00 [; H)

from which it follows

(6.88) I(u'(t), u(t)) - (u 1, uO)1 :s; constant

thus

(6.89) K(t) = ~Eo + oG), from which we have the result.

5.3. Simple Remarks on the Dissipativity

Consider an evolution equation

(6.90) du dt + Au = 0, u(O) = UO ,

where u may be a vector.

o

A function E(t) = c1>(u(t)) where c1> is a (positive) functional over the space of solutions (or the state space) is called invariant if

(6.91) dE(t) = 0 V ° dt ,u ,

and the problem is called (in a slightly ambiguous manner) dissipative if we cannot find a positive invariant function, but if we can find a function c1> such that

E(t) = c1>(u(t)) satisfies

(6.92)


Example. Consider the problem: find u such that

(6.93) a2u 0 au l' at2 - Au = 0, u(O) = u , at (0) = u In o c IRn

with, for example, u = 0 over ao (which we can write as a system). Then

(6.94) E(t) = ~ In [ut(x, tf + I rxul2] dx (106)

is invariant, for arbitrary t. If now we consider the problem: find u such that

(6.95) a2u au 0 7fi2 - Au + a(x) at = 0, u(O) = u , ut(O) = u1 ,

with a(x) ~ 0

then:

(6.96) d 1 2 - E(t) = - aUt dx ~ 0 . dt 0

We say that the problem is dissipative. The type of consequence which we generally deduce from the dissipativity is the exponential decrease in t of the solution as t -+ 00, if there is 'strict' dissipativity (in the example above a # 0, a ~ 0).

Remark 2 1) A problem may be dissipative and reversible in the following sense. We have a group {G(t)} which has exponential decrease as t -+ + 00 and exponential increase as t -+ - 00. In the case (6.95), there is reversibility. 2) There is no connection, in general, between the property of dissipativity and of regularisation. In effect, take for example (6.95) with a = constant and set

a

(6.97) U = e2t w.

Then, w is the solution of

(6.98) a2

Wit - Aw - -w = 0 4

and equation where there is no regularisation (and the transformation (6.97) changes the conditions at infinity in t, but does not change the regularity properties). Therefore dissipativity does not imply regularisation in general. But we can,

OU (106) We denote here by Ut the derivative-.

ot


however, have dissipativity and regularisation simultaneously. If we consider in QclR

(6.99) au - - Au = 0 for (x, t) E (Q x IR +) , at '

u(O) = Uo for x E Q and t = 0, and u = 0 over aQ , then

(6.100) :t In [Vu[ldx ~ 0,

u decreases exponentially in t as t -> 00 and there is regularisation. o Remark 3. A study of the decrease of u(x, t) as t -> 00, u the solution of (6.93), depending on the properties of a, is made in ref. Rauch [1]. 0

5.4. Hyperbolic Equation with Point Effect

We now give a simple example of a situation which occurs in the theory of optimal control, in particular in stabilisation. In the open set Q = ] Xo, Xl [ c IR, we consider the equation

(6.101) azu a1u ai2 - axl + u(x, t)J(x - b) = f(x, t)

where J(x - b) is the Dirac mass at the point b, b E Q with the usual initial conditions

(6.102) au

u(x, 0) = Uo(x) , -(x, 0) = u1(x) , X E Q , at

and equally some standard boundary conditions,for example

(6.1 03) u(xo,t)=u(x1,t)=0, t>O.

Note that (6.101) can also be written:

(6.101)' azu azu ai2 - axl + u(b, t)J(x - b) = f(x, t) .

Variational formulation. We take

(6.104) {i) V = HJ(Q) , H = LZ(Q)

ii) a(u, v) = In:~ :~ dx + u(b)v(b) .

Problem (6.101), (6,102), (6.103) is written

(6.105) (un, v) + a(u, v) = (f, v), '<Iv E V;

(6.106) u(O) = uo, u'(O) = u1 .


As in 1 dimension: HMQ) c eeO(.Q), (6.104) defines a continuous bilinear form over V and the general theory of § 5 applies. We deduce the existence and uniqueness of the solution of problem (6.101), (6.102), (6.103), (6.104), reformulated in (6.105), (6.106), for given data uO, ul,f satisfying (5.18), (5.19).

Remark 4. We can more generally consider the case of a Dirac mass at b(t) = a given function of t, b(t) E Q, and consider the equation:

(6.107) i}2u i}2u i}t2 - ox2 + u(x, t)!5(x - b(t)) = f(x, t) ,

or, which comes to the same thing

(6.107') 02U 02U i}t2 - i}x2 + u(b(t), t)!5(x - b(t)) = f(x, t) .

If b(t) is piecewise eel and satisfies Ib'(t)1 :::; 1, except at a finite number of points, we can also resolve the corresponding problem, with conditions (6.102) and (6.103) (see Bamberger, Jaffre and Yvon [1].) 0

5.5. Boundary Conditions Corresponding to Derivatives with Respect to Time. Application to a Vibrating String Coupled to a Harmonic Oscillator. Application to a Problem of Propagation of Seismic Waves

General problem. Let Q be an unbounded, open set of [Rn which is the complement of the bounded set @; let r be the boundary, which we assume regular, and let Ii be the normal oriented exceptionally towards the interior of Q.

r

Fig.!

-+ n

We look for a solution u in Q, for t > 0, of the wave equation:

02U (6.108) - - Au = 0

ot2

with the initial conditions:

(6.109) u(x,O) = UO(x) , ~~ (x, 0) = ul(x) , X E Q ,

and the boundary conditions

(6.110) aU 02U an =a8t2" + fJu -f, (X,t)E(r x [R+),


where IX and p are constants> 0 and where f is a function given on r, for t > 0, and with some hypotheses made precise below.

Example 1. Vibrating string and harmonic oscillator (Ref. Nussenzveig [1].) We return to the study of the vibrating string whose physical framework has been presented in Chap. lA, §2 and which has been treated in Chap. XV, §4. However, we modify the boundary condition as x = o. We assume that the extremity at x = 0 of the vibrating string is 'coupled' with a harmonic oscillator. In this example:

a = ]0, + 00 [, r = {a} ,

and (6.110) is taken here in the form:

A =~ (107)

ox2

(6.111) ou (0 2 ) 2y ox = ot2 + w2 u(O, t) where y and ware known constants

( 1 w2 ) therefore IX = 2y' P = Ii.! = 0 . o

Remark 5. The solution of Maxwell's electromagnetic field equations (see Chap. lA, §4) in the complement (assumed occupied by a vacuum) of a perfectly conducting ball leads to a problem analogous (but this time in a c 1R3 ), to that treated in this Example 1. The resolution of the Schrodinger equation (see Chap. lA, §6) in a c 1R3, where a is the exterior of a ball B whose boundary oa is assumed impenetrable to the wave function (which is expressed mathematically by u = 0 over oa, u being the wave function), the ball B being taken with centre 0 and radius 1, leads also to use of methods analogous to those of this example 1 and of the remainder of this Sect. 5.5. These two examples are treated in Nussenzveig [1]. 0

Example 2. Seismic waves (Ref. Ewing-Jardetzky [1], Achenbach [1], Rodean [1]) Physical introduction. Energy is released rapidly in a homogeneous ball B with centre 0 and radius Ro containing a perfectly elastic isotropic material. This ball is placed in a perfectly elastic, homogeneous isotropic material which occupies the domain a = 1R 3 \B. We shall denote by (n;), i = 1, 2, 3 the outward normal components of a (that is to say the 'inward' to B). We assume that the release of energy in B has the sole effect of creating at surface of separation between B and a, denoted by oa, a radial tension with spherical symmetry ero(t) depending only on time and assumed continuous in the present problem. The tension ero(t) acting at the boundary of the elastic body filling a produces an unknown displacement field (assumed radial), which we write

(6.112) - _ x· V (x, t) = v(r)e, that is V;(x, t) = v(r)~, r = Ixl .

r

(107) Note here that the complement of Q in IR is] - 'X, 0] which is not bounded. But r is bounded as in the general problem. Otherwise the hypothesis 'Q bounded' is not essential in the general case.


It is easy to verify that such a field in a homogeneous, isotropic perfectly elastic body is irrotational; this implies in particular on the one hand the existence of a

--> potential q> (starting from the existence of V)(108) such that

--> ~ (6.113) V = grad q> from which

and on the other hand the simplification of the dynamical equation seen In

Chap. lA, §2(109)

(6.114)

where a is a real, known constant, representing the speed of propagation of the dilation, which is expressed as a function of the elasticity coefficients by

(6.115) 2 A + 211 a = ;

Po

we shall denote here by b the speed of propagation of the rotation (II 0) of the medium which is expressed by:

(6.116)

b is assumed 'known'. Equation (6.114) may be written

thus

(6.117)

11

Po

o

0,

since q> need only be defined as a function of t. If we introduce

(6.118) u(r, t) = rq>(r, t) ,

we see that the equation (6.117) is written

(6.119) o.

(108) The open set Q is here simply connected (see Chap. lXA). (109) See equation (2.54) of Chap. lA, §2, in which we neglect the right hand side. that is to say the density of the material, in front of the intensity of the seismic force. (110) Or of distortion. See Chap. lA. §2.5.2.

§6. Problems of Second Order in I. Examples 601

The boundary condition on the sphere oQ of radius R o is (see Chap. lA, §2)

(6.120) o"r(Ro, t) = (Tij(x, t)ninjlo.Q = (To(t) (111)

by denoting (TiiX, t) the stress in the elastic body occupying Q and (Tr(R O , t) the value of the radial stress in the elastic body at Ro. Now the classical constitutive law(11Z) of elasticity for the body occupying Q implies that on oQ

that is to say

(6.121)

1 + 2 (113) (Tr = (Tijnin j = ItV/,/ /lvi.jnin j

_ 1(OV 2v ) 2 OV - It - + - + Ji-or R o or

Taking account of relation (6.119), the boundary condition (6.120) finally becomes:

(6,122)

(where Ro, b, a, Po are given constants and (To a given function of t), which is a boundary condition of type (6.110). By assuming a = 1 (which is equivalent for example to dilating the space), the problem posed becomes finally: find u(r, t) defined for r E ]Ro, + CXJ [ = Q,

t E [R+; such that

OZu oZu i) Jt2 - orz = 0 on Q x [R + ,

.. OU II) u(r, 0) = 0; at (r, 0) = 0, r E Q ,

This problem is of the general type described above by (6.108), (6,109) and (6.110). o

Study of general problem (6.108), (6,109) and (6.110), Variational formulation

(111) With the summation convention over repeated indices and (n), j = I to 3 denoting the components of the (outward) normal n of the sphere aQ. (112) See Chap. lA, §2. (113) With the notation of Chap. lA, Appendix "Mechanics".


Let v E H1 (Q); multiply (6.108) by v; applying Green's formula, it becomes (paying attention to the orientation of the normal n)

G:~ , v) + f :> dr + L Vu V v dx = ° , by using (6.110), it becomes:

(6.123) (jj2u) (a 2u) at2 ' v + a atZ' v r + a(u, v) = (f, v)r ,

where we have set

(6.124)

and

(6.125) a(u, v) = L VuVvdx + f3 L uvdr .

We choose the spaces:

The precise result is then: If we assume that the function I and the given initial conditions are such that:

(6.126) { IE U(O, T; L2(T)) , Uo E V, u1 E V (115) ,

there exists a unique function u such that

(6.127)

(6.128)

(6.129)

(6.130)

(6.131)

U E U)(O, T; V) ,

au at E LOO(O, T; H),

a ~You E LOO(O, T; L2(T)) , Yo = tracev -+ vir, at

equation (6.123) (116) holds "Iv E V,

u(O) = Uo , au(O) = u1. at

(114) We assume the spaces are taken real. (115) This condition can be improved, but 'u l E H' will be insufficient, unless we introduce weak solutions, which is possible but will not be detailed here. (116) With (6.127), (6.128), (6.129), it is convenient to write (6.123) in the form:

~(au,v) + 1X~(~You,v) + a(u, v) = (f,v)r. dt at dt at r

& a Note that a priori Yo- does not have a sense for (6.128), but -You is perfectly defined.

& &


As usual, we deduce from (6.123) that:

a2u 7Ji2 = Au E LOO(O, T;H-l(Q))

so that (6.131) has a meaning.

603

Abridged proof The principle is again the same, by the energy method. We approximate (6.123) by a system of finite-dimensional differential equations, then we establish a priori estimates; replacing v by u' in (6.123) (we write instead of a/at), it becomes

(6.132)

We deduce:

(6.133)

1 d 2 dt [lu'1 2 + alu'li- + a(u)] + (f, u')r (117) •

lu'(tW + alu'(t)li- + a(u(t)) = lu l l2 + alyu11i- + a(uO)

+ 2 f~ U«(J), u'«(J))rd(J .

It follows that, in particular:

lu'(t)li- ~ C1 + ~ i' If«(J)lrlu'«(J)lrd(J , a a Jo

from which we deduce that

(6.134) c I' lu'(t)li- ~ -..3. If«(J)Ii-d(J . a °

Using (6.134) in (6.133), we have

(6.135)

The result then follows.

Remark 6. The solution can also be obtained by use of the Laplace transform in t. o

Remark 7. The estimates (6.134) and (6.135) can be thought of as singular asymptotic phenomena as a -+ 0. This is not the case except for small difficulties arising from the regularity off Let us make this a little more precise. Assume we have the additional regularity

(6.136)

(117) We denote a(u, u) by a(u).


Then, we can write (6.133)

IXlu'(t)li- + lu'(tW + a(u(t)) = CI + 2(f(t), u(t))r - 2(f(0), uO)r - 2 t (f'u)rdo-

~ C2 + 2If(t)lr lu(t)lr + 2 t If'(o-)Irlu(o-)Irdo- .

But Ivli- ~ a(v) + klvl2

(we also have Ivli- ~ a(v)/fJ, but this introduce fJ artificially as a singular parameter) and consequently we deduce some uniform estimates in IX as IX -+ O. If we denote by u the solution of (6.126) to (6.131) and if (6.136) is satisfied, then, as IX -+ 0

{

Ua -+ U weakly star in L 00 (0, T; V) ,

oUa OU . at -+ ot weakly star III L 00 (0, T; H) ,

where u is the solution of

(6.l37) ! ~:~ -Llu = 0, u(x,O) = UO(x), ~~ (x, 0) = ul(x),

OU on = fJu - f on r, t > 0 .

If f does not satisfy (6.136), the preceding result is again true but in a weaker topology, the solution u of (6.l37) being taken in a weak sense. 0

5.6. Application: Vibration of a Thin Plate

We return to the problem stated in Chap. lA, §2, but instead oflooking for a local formulation of type IA (2.87), we shall use a variational formulation which belongs to the framework of application of Theorems 3 and 4 of §5.

5.6.1. Variational Formulation of the General Problem. We have seen in Chap. lA, §2 that, with the Love-Kirchoff theory: the equations of motion are reduced to

(6.l38) Q (118)

- and the constitutive law

(6.l39)

(!l8) Recall that, conventionally the Greek indices describe the set P,2}; g denotes gravitational acceleration. The notational conventions are those set out in Chap. lA, Appendix "Mechanics".


(where MaP denotes the stress tensor and w the deflection, which is the principal unknown). The boundary conditions are made precise in the different physical situations envisaged, but we shall now establish that they follow naturally from the variational form obtained from relations (6.138) and (6.139). We are led to introduce a set ifIIad , of 'admissible deflections', (which we shall make a posteriori explicit in each concrete situation) for which we first impose the necessary regularity for the following integration by parts, then all or part of the boundary conditions which must be satisfied by the solution sought. Therefore let

(6.140) V E ifIIad ;

Relation (6.138) implies

(6.141) In MaP,apvdx = In Ph(g + ~:~)VdX' from which, by integrating the left hand side twice by parts:

(6.141)' 1 r Mapaa2aV dx- r Map~nadr+ r Map,avnpdr J.a Xa Xp Jr axp Jr

= In Ph(g + ~:~)VdX, '<Iv E ifIIad ;

therefore, taking account of the identity:

(6.142) av av av _ - --a = ~a np + ~rp where r = k 3 1\ n , Xp n ar

we finally obtain:

1 r ph aa2~ vdx = r MaP a a2av - r pghvdx J.a t J.a Xa Xp J.a

(6.143)

+ t Fvdr + t vlt :~ dr, '<Iv E ifIIad ,

after having set

(6.144) { F ~_ ~,.,n, + ;, [M.,n.',l.

vIt - Mapnanp .

Remark 8. By comparing (6.144) with relations (3.54) to (3.59) of Appendix "Mechanics" Chap. lA, we see that F and vIt may be interpreted respectively as forces and couples and therefore equation (6.143) is nothing more than another expression of the principle of virtual work for which the set ifII ad of admissible deflections defines the set of 'virtual movements'. 0

We have therefore obtained a variational formulation (formal at present) of problem (6.138), (6.139), (6.140) and the initial conditions:


find the deflection w satisfying:

w E i1Itad , and for all v E i1Itad :

(6.145) i ph ~2: v dx - i M ~p :l 02~ dx n ut n uX~uxp

= In - pghv dx + L F. v dr + L.H· ~~ dr ,

to which it is convenient to add the initial conditions:

(6.146) o ow 1 w(x, 0) = w (x); Tt(x,O) = w (x) .

It remains to make precise the set i1Itad as a function of different possibilities of the boundary conditions.

5.6.2. Definition of the Set d/I ad of Admissible Deflections. The space i1It ad will be chosen equal to a closed vectorial subspace of H2(Q) satisfying

(6.147) H6(Q) c i1Itad = V c H2(Q)

and V will depend on the concrete boundary conditions imposed (see Chap. lA, Appendix "Mechanics", §3.6) in a sense which will be made precise in §6.5.6.3.

Example 3. Clamped plate The boundary conditions are

(6.148) wi r = 0 and ~: I r = 0 ;

we shall take

(6.149) i1Itad = V = H5(Q) ,

so that the boundary conditions will be contained in the fact that w belongs to The variational formulation (6.145) reduces to finding w satisfying

{ w(t) E HMQ) = V, and Vv E H~(Q) = V

(6.150)

i ph 0:l2: vdx - i M~p:l 02~ dx = - ghi pvdx. n ut n uX~uxp n

Example 4. Simply supported plate. The boundary conditions are expressed by

(6.151) wi r = 0 and .H = 0

(deflection and stress couple is zero along n. The set i1Itad is then a little less restrictive and becomes

(6.152)

The fact that w belongs to i1Itad , only implies the condition wlr = 0 here. The variational formulation is the same as in (6.150) except that the test functions and therefore the solution w(t) need only be in H2(Q) n HJ(Q) instead of H~(Q).

§6. Problems of Second Order in l. Examples 607

Example 5. Free plate. The boundary conditions are written

(6.153) F = 0 and uN = 0 on r

(force and stress couple zero along r). The variational form again remains the same, but this time:

(6.154)

does not imply any boundary conditions on v and therefore w(t).

Example 6. A restoring force proportional to w exerted on r. This might correspond to an elastic support and is produced by:

(6.155) F = - kw and .A = 0 along r with k > 0; again we have

(6.156)

but the variational formulation becomes: find w such that: w(t) E HZ(Q) and '<Iv E HZ(Q):

(6.157) ! f azw f azv ph-a Z vdx - M~fJ~a a dx

Q t Q x~ xfJ

+ k t wvdr = - gh L pvdx

Example 7.

(6.158)

A restoring couple exerted on the boundary r. We then have

wlr = 0 and uN = - k ~:, [k > 0] over r ;

we must take here

(6.159)

and the variational formulation returns to looking for w such that:

! w(t) E HZ(Q) (\ H6(Q) and '<Iv E HZ(Q) (\ H6(Q)

(6.160) f azw f azv f aw f ph-a Z vdx MafJ~a a + k -a vdr = - gh n pvdx.

Q t Q Xa xfJ r n "

Example 8. Half-clamped plate. We assume

(6.161 ) r = r 1 u r z, meas r 1 > 0 (119); r 1 (\ r Z 0,

the plate being clamped at r 1 and free at r z.

(119) meas for measure.


The boundary conditions are then

(6.162) {i) wiT, = 0

ii) F = 0 and .Jt = 0

owl and -;-un T,

=0

over r2 ,

and o/iad must be chosen in the following manner

(6.163) o/iad = V = {v; V E H2(Q), vir 1 = 0 , ;~IT' = 0 }

so that V is a closed vectorial subspace of H2(Q) with:

(6.164)

The variational formulation is as in (6.150), only the definition of V has changed.

Example 9. Plate partially clamped in a vibrating body. This is the problem which has been described in Chap. lA, §2.5.7; it follows from the preceding case by replacing (6.l62)i) by

(6.165) owl wiT, = cp and -;- = '" un T,

(cp and", given over rd. This example can be treated by several methods; in particular(120) by using a lifting of the nonhomogeneous boundary conditions (6.165), we return to the case of homogeneous boundary conditions (6.162) of Example 8.

5.6.3. Application of Theorems 3 and 4 of §5. In all of the examples given above, the variational formulations proposed are again formal since the functional framework has not been made completely precise; we shall do this in this section in order to be in the framework of application of Theorems 3 and 4 of §5. In the concrete example considered

{ M~fJ is given by (6.139) (121)

(6.166) p = constant ;

with the notation of §5, we are led to take:

(6.167) { H = L2(Q), V satisfying (6.147) ,

C = ph. I (l identity of H) .

('20) We can also use the method of transposition (see Lions [I] or Lions-Magenes [1]). (121) We can treat the case of anisotropic material in the same way (see Chap. lA, §2) with a law of the type


In the case of Examples 3, 4, 5, 8, we define a(t; u, v) = a(u, v) by

{ u, V E H2(Q) ,

(6.168) i a2 v a(u, v) = - Map:;--;- dx .

n uXauXp

In the case of Example 6, we define a(u, v) by

{u, v E H2(Q),

(6.169) a(u,v) = - r MaP,;, a2~ dx + k r uvdF, In uXauXp Jr

and in the case of Example 7, by;

(6.170) { u, V E H2(Q) ,

i a2v i au a(u, v) = - Map:;--;-dx + k -;-. vdF . n uXauxp run

609

By using, on the one hand, the trace theorems in Sobolev spaces(122), and on the other hand the properties of the coefficients of elasticity (123), we verify that a(u, v) satisfies the hypotheses of Theorems 3 and 4 of §5. Therefore for

(6.171) { WO given in V, w1 given in H ,

(here the termf = - pgh is assumed constant (124» ,

we obtain the result: there exists a unique u satisfying

i) WE rcO([O, T]; V), w' E rcO([O, T]; H),

ii)

(6.172)

:t(CW'(.), v) + a(w(.), v) = (f(.), v) for all v E V in the sense of ~'(]O, T[),

iii) w(O) = wO, w'(O) = wi,

where

(6.173) (f, v) = In fv dx .

This now makes precise the sense to give to (6.150), (6.157), (6.160).

(122) See Chap. IV, §4 and Lions-Magenes [I], Vol. I. (123) See Chap. lA, §2. (124) We can, without difficulty, consider a right hand side more general than f E L2(0, T; H).


In each case, the problem to be resolved is then: find w satisfying:

Ii) w E <C°([O, T], V); ~~ E <C°([O, T], H) (6.174)

ii) ~:~ + (X2,12w = - gin Q x ]0, T[

with

(6.175) 2 (A + 2p.)h2

(X =~-,

iii) the boundary conditions of Examples 3 to 8, iv) the initial conditions

(6.176) aw

w(O) = WO E V and &(0) = Wi E H (125) ,

which contains, in particular, problem (2.87) of Chap. lA, §2.

5.7. Application to Elasticity with Time Dependent Coefficients

Example 10. We return to the model presented in Chap. lA, §2, equation (2.57).

(6.177)

a2U i a . 3 . 3 p(x, t)-a 2 - -a (Jij = J;(x, t) In Q c IR , I = 1,2, ,

t Xj

(Jij = aijkh(x, t)Ekh(U) (126\ aQ = r 0 uri' ron r I = 0 ,

u = ° on r o ,

(Jijnj = Fi(x, t) on r l , i = 1,2,3,

u(x,O) = UO(x) ,

au at" (x, 0) = ul(x).

a2u For the quasi-static case, we make --ail = ° in the preceding equations and

suppress the initial conditions; Q being an open set of 1R 3 , we take:

(6.178)

(125) These are therefore in the framework of Mathematical Example 4 (Sect. 4). We can also treat the case where the coefficients A and jl, the characteristics of the material, depend on x and t.

aUk (126) Recall that: aijkhekh(U) = a ijkh - due to the symmetry in k, h of a ijkh .

oXh


equipped with the scalar product

(6.179) (U, v) = (U i , vJU(Q) .

Now for U = {u l , U2, u3}, Ui E ~'(Q), i = 1,2,3, we recall that

(6.180) £hk(U) = ~[DhUk + Dkuh] (127), Dk = a:k ' k = 1,2,3 .

We introduce the following space [£(Q)]3:

(6.181) [£(Q)]3 = {u E H; £kh(U) E U(Q), h, k = 1, 2, 3} .

From Korn's inequality(128), if Q is bounded and regular (which we assume here), then

For u, v E [£(Q)r we set

(6.182) ((U, v)) = (u, v) + (£kh(U), £kh(V)) ;

[£(Q)]3, equipped with the scalar product (6.182), is then a Hilbert space and the closure of [~(Q)]3 in [£(Q)]3 coincides with [H6(Q)]3. Then let V be a closed vectorial subspace of [£(Q)]3 with

(6.183) [H6(Q)r eVe [£(Q)]3 .

We shall introduce the bilinear form a(t; u, v) over V. Recall first of all that the coefficients of elasticity aijkh which have been defined (Chap. lA, §2) have the following symmetry and positivity properties

(6.184)

(6.185)

If

(6.186)

then we set

{ .~) aijkh = ajikh = aijhk 11) aijkh = akhij

{ aijkhXijXkh ~ XijXij

Xij E IR, Xij = X ji , i, j = 1, 2, 3 .

(6.187) a(t; u, v) = (uij(u), £i/V)) = L aijkh£hk(u)£ij(v)dx ,

the bilinear form a(t; u, v) is symmetric (thanks to (6.184)), continuous over V x V

(127) With the usual notation for the mechanics of continuous media this is written:

(128) See Chap. VII, §2.

612

and satisfies:

(6.188)


{there exists A ~ ° and IX > ° such that

a(t:u, u) + Alul 2 ~ IXllul1 2 for all u E V

for almost all t E [0, T] ,

which follows from the study made in Chap. VII, §2. Recall again that the operator associated with a(t; u, v) here will be

(6.189) {Au = ((Au)I' (Auh, (Auh) ,

(Au)i = - a~. (aijkh(t, .)ekh(U)) . J

We shall make the following supplementary hypotheses:

(6.190) { for all u, v E V ,

t --+ a(t; u, v) is once continuously differentiable over [0, T] .

Moreover, assume that the function p has the following properties:

{ i) p(x, t) ~ Po > ° , (6.191) ii) t --+ p(., t) is of class rei over [0, T] with values in L 00(0) .

Then, we can write

(6.192)

there exists, from the existence and uniqueness theorem for problem (P2) (§5), a unique solution u E L2(0, T; V) to

d 1) dt c(.; u(.), v) + p(.; u' (.), v) + a(.; u(.), v) = (g(.), v)

(6.194) in the sense of ~' (] 0, T [) for all v E V,

2) u(O) = UO (given in V), ~~ (0) = u1 given in H ,

with

(6.195) { g = go + gl E L 2 (0; T; V') , go E L2(0, T; H), gl E L2(0, T; V'), g; E L2(0, T; V') .


Then the elasticity system (6.177) reduces to (6.194) by taking

(6.196)

(6.197)

v = {u; U E [E(Q)] 3, uil ro = 0, i = 1,2, 3} ,

f = (fl,J2,J3) E L2(0, T; H) ,

613

(6.198) F = (F 1 ,F2,F3)EL2(0, T;U(rd3) , a; EL2(0,T;L2(rd 3)

and by noting that g defined by

(6.199) { <g(t), v) = LJ;(t)VidX + L Fi(t)vidr 1 ,

for all v E V,

satisfies (6.195). The verification of the fact that (6.194) leads to (6.177) is analogous to that made in Chap. VII, §2.

5.S. Application to Linear Viscoelasticity. Material with Short Memory

Example 11. We return to the situation of Chap. lA, §3. Let Q c 1R3 , with boundary r = r 0 u r 1, roll r 1 = 0, we look for U = (u1 , U 2 , u3 ) satisfying

(6.200)

in QT = Q x ]0, T[, i = 1,2,3,

ii) {u = 0 on r 0

(Jijuj(x, t) = Fi(x, t) on r 1

... 0 OUi 1 2 3 1Il) ui(x,O) = Ui (x), at (x, 0) = Ui (x), i = 1, , ,X E Q .

We restrict ourselves here, as in Chap. IA §3, with the case of coefficients independent of t, but there is no extra difficulty, as in the elasticity case, in assuming dependence on t (see remark later). Assume that

(6.201)

(6.202)

(6.203)

(6.204)

Po > 0, Po E VX'(Q) ,

aULh = aU~k = a}lLh = a12ij' I = 0, 1, i, j, k, h = 1, 2, 3 ,

aULh E L OO(Q) , I = 0, 1, i, j, k, h = 1, 2, 3 ,

{ alJ~hXijXkh ~ IX(I) XijXij , Xij E IR, i, j = 1,2,3 ,

IX(I) > 0 (l = 1, 2) constants independent of x .


Then for u, v E (H I (Q))3, we define:

(6.205)

Therefore for V, a closed vectorial subspace of (L 2 (Q))3 with

(6.206)

we are under the conditions of application of the existence and uniqueness theorem of problem (PI)' Therefore for g given with

(6.207) g E U(O, T; V') ,

there exists a unique u E W satisfying

(6.208)

d i) dt c(u'(.), v) + b(u'(.), v) + a(u(.), v) = <g, v) ,

in the sense of ~'(]O, T[), for all v E V,

ii) u(O) = U O given in V, au (0) = u1 given in H . at

Problem (6.200) corresponds to the choice of

(6.209)

and to g defined by

f <g(t), v) = (f(t), v) + f Fi(x, t)vi(x) dT

(6.210) 1 wherefis given in L 2(0,Tf; H), FE L2(0, T; L2(T 1)3).

Remark 9. The case of coefficients depending on t. It is sufficient to take:

(6.211) Po(x, t) ~ Po > 0, Po E rgl([O, T]; L "'(Q)) ,

(6.212) alJ~h E L')(Qr) satisfying (6.202) and (6.204) ,

with moreover

[aIJ~h(x,t) + :iaINh(x,t)]XijXkh ~ rxXijXij,XijEIR

(6.213) {a l7Jh E rgl([O, T]; U"(Q)) ,

al]Jh E rg2([0, T]; L:XC (Q)) .


Then we satisfy the conditions of the existence and uniqueness theorem for problem (P 1) with

. -i [ 0 a 1 ] aUk aVi a(t,u,v) - aijkh(x,t) + ;;-aijkh(x,t) :;,:;,dx, n ut uXh uXj

(6.214) i 1 aUk aVi i a )) d b(t; u, v) = aijkh(x, t):;, :;,dx - ;;-(Po(X, t UiVi X, n UXh uXj nut

c(t; u, v) = In Po(X, t)uividx .

5.9. Very Weak Solutions of Hyperbolic Evolution Equations

Consider

(6.215)

with

(6.216)

(6.217)

U" + Au = I,

u(O) = UO ,

u'(O) = u1 ,

o

where A is an isomorphism from V --+ V' given by a symmetric bilinear form a(u, v) which is V-coercive with V ~ H ~ V', (H is a "pivot" space). We have seen that for

(6.218)

then

(6.219)

{IE L2(0, T;H),

UO E V,

u1 E H ,

{ U E L 00 (0, T; V) ,

u' E L 00(0, T; H) ,

so that the "energy" E defined by

(6.220) 1

E(t) = 2" [lu'(t)1 2 + a(u(t), u(t))]

is bounded over [0, T]. We shall show that we can obtain some "very weak" estimates of the energy, that is to say with u, u' given in "very large" spaces, by taking the scalar product of (6.215) with A -ou' (0 < () < 1). This no longer is a question of the "energy" in the physical sense of the term.

Remark 10. This also reduces to changing the pivot space, that is to say replacing H by a "larger" space. 0


Therefore take (formally for the moment) the scalar product of (6.215) with A -6 U',

with () E IR; we obtain

1 d 6 1-6 6 6 2 dt[IA -"2u'1 2 + IA-2-uI 2] = (f, A -6U') = (A -"2f, A -"2u') ,

from which we set 6 1-6

(6.221) E6(t) = IA -"2 u'(tW + IA-2-U(tW ,

(6.222)

or also 1+6 1-6 1+6 1-6

(6.223) E6(t) = E6(0) + 2(A -~/(t), A-2-U(t)) - 2(A --2-/(0), A-2-UO)

it 1+6 1-6 - 2 Jo (A --2-1', A-2-u)ds;

we then deduce the following results corresponding respectively to (6.221) and (6.222)

1-6 6 if UO E D(A-2-) , u 1 E D(A -"2) ,

6

(6.224) 1 E L 2(0, T; D(A -"2)) ,

1 1-6

h UE L 00(0, T; D(A-2-)) , t en 6

U' E L 00(0, T; D(A -"2)) ,

and

(6.225) 1 1-6 6

if UO E D(A-2-) , u 1 E D(A -"2) ,

1+6 f,f' E L2(0, T;D(A-2-) (129) ,

then we have the same conclusions as in (6.224) . o Remark 11 i) In (6.225) 1 is more regular in t and less regular in the space variables than in (6.224). ii) We have made a "formal" demonstration. It only remains to justify that we can take the scalar product of (6.215) with A -6U'.

Now this is possible with the Galerkin approximation of the problem (here the Fourier method) from which we have the result. 0

(129) In the framework of parabolic equations, see Chap. XV, §3, Rem. 7, for the corresponding results with the spaces D(AS); the difficulty in using this method lies in the characterisation of the spaces D(AS) with the help of the Sobolev spaces. For the more general case of nonhomogeneous boundary conditions, we refer to Lions-Magenes [1], Vol. 2.


Example 12. We return here to an example considered in Chap. XV. Let Q = ]0,1[, f = 0, UO = 0, u1 = b(x - b), bE Q. Then u1 EH-«(Q) if

IX > 1/2. Take A = - ::2 with some boundary conditions (Dirichlet, Neumann,

etc.). We have, for 0 ~ 0,

D(A O) c H20(Q) ;

we can then apply (6.224) if we choose 0 with 0 > 1/2. Let 6 > 0 be arbitrary and choose 0 = (1/2) + 6, then

(6.226) !u E LOO(O, T;Ht-'(Q)) u' E LOO(O, T; H-t-'(Q)),

with 6 arbitrarily small.

5.10. An Example of an Evolution Problem Which is Second Order in t on the Boundary (See Chap. lA, § 1: Motion of Water in a Canal Under the Hypothesis of Small Perturbations.)

Setting of problem. We look for u(x, t) = u a solution of

(6.227) Au = 0 in Q x ]0, T[ = Q T

n

r :---.-___ --

Fig. 2

D

where Q c 1R3, unbounded, with boundary r 1 U r2 as in Fig. 2, and with the conditions

(6.228) iJ2u + au = 0 ]0 [ '" ot2 ov over r 1 x ,T = '" 1 ,

(6.229) AU = 0 over r 2 x ] 0, T [ = r 2 , ov and the initial conditions:

(6.230)

au ! u(x,O) = UO(x) , at (x, 0) = u1 (x) ,

UO and u1 given on r 1 (in suitable function spaces indicated below) .


For the solution of this problem, we must introduce a number of Junction spaces; firstly:

(6.231)

where in this section (this is a little different from the usual notation) Hloc(Q) denotes the space of ep such that ep E HI ((!)), I:/(!), an arbitrary open bounded set as in Fig. 3. We define

(6.232)

We therefore obtain a norm and V is complete for this norm (this is a Hilbert space).

Fig. 3

Remark 12. Here r l and r 2 are two varieties of dimension 2, which are regular to

infinity, for example with equations X3 = gj(x I , x 2), i = 1, 2, gj, ~gj E L 00 (1R2), uXj

then V c L6(Q).

If we introduce W = {ep; ep E L6(Q), rep E (L2(Q))3}, then eplr! is not necessarily in U(F d, so that V c W strictly. 0

Remark 13. Locally eplr! is in Hl.!c2(Fd.

We also need the following space:

(6.233) E = {ep I ep E HI!2((!)), I:/(!) as above Llep = 0 in Q,

2 oep } eplr!EL (Fd,Jv = OonF2 ,

which is a complete (therefore Hilbert) space for the norm

(6.234) IlepilE = (t! ep2 dFI y!2

Besides let v E V with

(6.235) Llv = 0 in Q.

o

We can then define :vl which is in Hl~/!2(FI) locally on Fl' We denote by F the v,r!


space of traces ~v I ,equipped with the norm uV r,

(6.236) avl IlgilF = Ilvllv, -;- = g. uV r,

We shall show

Theorem 3. We assume that

(6.237)

(6.238)

Uo is the trace on FI ofuo E V, Lluo = 0;

u l E L2(FI) .

Then there exists a unique u such that

(6.239) {

U E U)(O, T; V) ,

U:,E LOCl(O~ T;E), u'lr, E LOCl(O, T;L2(Fd) ,

u Ir, E L (0, T;F) .

and u satisfies (6.227), (6.228), (6.229).

Remark 14. The variational formulation is

(6.240) (u", v)r, + a(u, v) = 0, Vv E V,

where

619

o

o

Existence proof Everything rests on an a priori estimate (we then apply Galerkin for example). Take v = u' in (6.240), it becomes

~[lu'lf, + a(u)] = 0, dt

(lu'lf, = (u', u')r" a(u) = a(u, u)), therefore

(6.241) lu'(t)lf, + a(u(t)) = lUI If, + a(uO) .

[Note that we are given u(O) on Fl> or in Q, it doesn't matter which. We can therefore consider u(O) = UO in Q]. We deduce from (6.241) that lu'(t)lr, ~ c, therefore

(6.242) lu(t)lr, ~ C = C T if t ~ T,

and since lu(t)1 ~ C, we therefore have Ilu(t)llv ~ c. We therefore have an a priori estimate for u in L 00(0, T; V) and for u'lr, in LOCl(O, T; L 2(F I )). But we therefore have:

(6.243) 1 Llu' = 0 in

cu' -~ = 0 on ill'


and therefore - by definition of E -

(6.244) U' E Loo(O, T; E).

Besides we have, by definition of F:

aul ;,- E L 00(0, T; F) uV r,

and therefore unlr, E Loo(O, T; F).

Proof of uniqueness. Let u be a solution of the problem with UO = 0, u1 = O. Let s be such that 0 < s < T and let

(6.245) wet) = f u(o) da, t ~ s; 0 for t ~ s .

We therefore have

(6.246) w' = - u if t < s, 0 if t > s ,

and, by taking v = win (6.240), it becomes

J: (un, w)r, dt - J: a(w', w)dt = 0 ;

therefore since u'(O)lr, = 0 and w(s) = 0:

is 1 is d - (u', w')r, dt - - -d a(w(t»dt = 0,

o 2 0 t

therefore

1 [ IS d IS d ] "2 Jo dt lu(t)Ii-, dt - Jo dta(w(t»dt = 0 ,

therefore lu(s)Ii-, + a(w(O» = 0 ,

therefore w(O) = 0 i.e. J: u(a)da = 0, for arbitrary s, therefore u = O. 0

§7. Other Types of Equation

1. Schrodinger Type Equations

1.1. Formulation of Problem

We again consider two complex Hilbert spaces V and H such that V is included and dense in H. As the spaces are assumed complex, we identify H with its antidual H' such that V <+ H <+ V' where V' is the antidual of V.

§7. Other Types of Equation 621

We are given the operators A(t), t E [0, T], by the family of continuous sesquilinear forms, a(t; u, v), t E [0, T], over V, satisfying:

(7.1)

(7.2)

{ for all u, v E V, the function t ~ a(t; u, v) is once continuously

differentiable in [0, T], T < + CIJ .

{i) a(t; u, v) = a(t; v, u) for all u, v E V and all t E [0, T]

ii) there exists A E IR and a > ° independent of t E [0, T] such that

a(t; v, v) + Alvl2 ~ allvl1 2, for all v E V (130).

We then consider:

Problem (71). Find u satisfying

(7.3) d dt (u(.), v) + ia(.; u(.), v) = (f(.), v) in the sense of

(7.4) u(o) = Uo ,

Uo and f being "suitable" data (see later). We remark that in quantum physics, we have f = ° (except for the study of elementary solutions, in which case f is a Dirac distribution). Also, our interest in the casef =f. ° studied here resides in the comparison of mathematical properties of the Schrodinger equation with the properties of the diffusion and wave equations. We remember that in Chap. XVIIB, §4 other problems than the Schrodinger problem, are also relevant to Stone's theorem, and therefore to an equation of type (7.3). We draw attention to the fact that, in (7.3), the sesquilinear form a(., u, v) satisfies the coercivity inequality (7.2)ii), therefore the operator A(t) is bounded below(131). We have seen in Chap. XVIIB (and A), §4, that this is the case for the Schrodinger equation (with the potentials v considered) and for the real transformed wave equation (4.10)(132). The study below, which is made by using only the coercive form a(t; u, v) therefore does not distinguish between the two cases. It is only in §7.1.4.2. that we are restricted to the case A(t) = - Ll + v. We shall prove

Theorem 1. We assume that the hypotheses (7.1)-(7.2) hold. Let Uo and f be given with

(7.5) Uo E V,

(130) We note that (7.2)ii) implies that the quadratic form art; v, v), and therefore the operator A(t) are bounded below by a constant Ao independent of t; that is to say that

(131) See Chap. IXB.

art; v, v) ;;, Ao(v, v) for all v E V and all t E [0, T] ,

(A(t)v, v) ;;, Ao(V, v), Vv E D(A(t)), Vt E [0, T] .

(132) Note that in the case of the Schrodinger equation, A(t) = A = ( - .1 + v), and that in the case of the "transformed wave equation" A(t) = A = ( - .1)1/2 (also bounded below).


(7.6) f L 2(0 T H) df = f' E L2(0, T,· V') . E ,; , dt

Then there exists a unique solution to problem (n)(133), satisfying

(7.7) U E W(V) = {v; V E L 2 (V), v' E L 2 (V')} .

Remark 1. Since u satisfies: u E W(V), then u E ~O([O, T]; H), therefore (7.4) has a sense. Note however that Uo is given in V. 0

Remark 2. If fE L 2(H),f' E L 2(V')(134), thenfE ~([O, T]; V') (in fact, see (1.61) and Lions-Magenes [1], we have a stronger result) such that

(7.8) f(0) E V' . o

1.2. Proof of Uniqueness

Let u be a solution of problem (n) withf = Uo = O. Then from (7.3) and from the integration by parts formula of Sect. 1 of §3, we deduce

(' 1 i Jo a(<J; u(<J), U(<J» dO" + 2Iu(t)12 = 0, t E [0, T] .

From which by taking the imaginary part:

(7.9) U(t) = 0 for all t E [0, T] .

The uniqueness is therefore proved.

1.3. Proof of Existence

We shall use the Galerkin method with variable bases as in the preceding sections, but we can also use the method of parabolic regularisation. Therefore let {V m}m El'J* be a family of finite dimensional vectorial subspaces (dim Vm = dm) forming a Galerkin approximation of V (see Sect. 1 of §2). Then there exists

(7.10)

and {Wjm } j = 1, ... , dm denotes a basis of Vm. The approximate problem is then:

drn

Problem (Pm)' Find um(t) = L gjm(t) Wim' satisfying j= 1

(7.11) { :, (U.:, Wi') + ia(l; ".(1), Wi.) ~ U(I), Wi,) ,

um(O) - UOm .

We shall establish two types of a priori approximation.

(133) The problem given by (7.3) and (7.4). of

(134) We again set - == 1'. Ot


1.3.1. First Type of a priori Approximation. Multiply (7.11) by gjm(t) and sum from 1 to dm , it becomes

from which by taking the real part of both sides (multiplied by 2):

d dt lum(tW = 2 Re(f(t), um(t)) ;

thus

lum(tW = IUom l2 + 2 Re t (f(u), um(u))du ,

from which we deduce

lum(tW ~ IUoml2 + f: If(uW du + f~ lum(uW du ,

and Gronwall's lemma implies:

(7.12) lum(tW ~ C[luol2 + f: If(uWdu J, t E [0, T].

We have therefore established

Lemma 1. The sequence {um} remains in a bounded set of L'x'(H).

In fact, the preceding estimate is insufficient for passing to the limit in the problem (1tm ) as m ~ 00.

We must therefore have estimates of Um in V!

1.3.2. Second Type of a priori Estimate. We now multiply (7.11) by glm(t), and sum from 1 to dm , then we take the imaginary part of the result (multiplied by 2); it becomes:

a(t; um(t), u~(t)) + a(t; u~(t), um(t)) = 2 Im(f(t), u~(t)) ,

from which by using (7.1),

(7.13) :t a(t; um(t), um(t)) - a'(t; um(t), um(t)) = 2 1m (f(t), u~)).

Integrating (7.13) we obtain

(7.14) { a(t; um(t), um(t)) = a(O; UOm' UOm )

+ t a'(u; um(u), um(u))du + 21m t (f(u), u~(u))du. But

f~ (f(u), u~(u))du = (f(t), um(t)) - (f(0), UOm ) - t (f'(u), um(u))du .

624

Therefore


It (f(a),u~(a))dal ~ IIf(t)II*·lIum(t)ll + Ilf(O)II*lluoll

+ t 1If'(a)II*lIum(a)llda (135)

and (7.14) gives (taking account of (7.2), and denoting various constants by C)

{ ocllum(t)112 ~ C(lluol1 2 + t lIum(a)11 2da + t 1If'(a)ll;da)

(7.15) + I)" II Um (t)12 + 211 f(t) II * lIum(t)ll

but from (7.12)

and 0( 2

2I1f(t)II*lIum(t)ll ~ 21Ium(t)112 + ; Ilf(t)II; .

Since f E W(H, V') implies f E CC([O, T]; V') and (as for W( V, V')) the mapping f --+ f(t) is continuous from W(H, V') into V', we have

IIf(t)ll* ~ C[f:1f(tWdt + f: 11f'(t)lI; dtl thus

21If(t)II*"um(t)11 ~ ~IIUm(t)ll2 + 2~[LT [If(tW + 1If'(t)II;]dt],

and from (7.15) we deduce

(7.16)

Ilum(t)11 2 ~ C{lluoll 2 + LT(lf(aW + 11f'(a)IIHda} + C tIIUm(a)1I2da;

from which by applying Gronwall's lemma

(7.17) Ilum(t)11 2 ~ C[lluoll 2 + LT (If(aW + 11f'(a)II;)daJ, t E [0, T].

We have therefore proved.

Lemma 2. The sequence {um } then remains in a bounded set of L <Xl (V).

(135) Here we assume implicitly (this gives no problems), that Iluom II ~ Iluo II.


We then extract u" -+ u weakly * in L oo( V) and we verify as in Sects. 3 and 5 of §6, that u(o) = Uo and

{ - f: (u(t), v)cp'(t)dt + if: a(t; u(t), v)cp(t)dt = f: (f(t), v)cp(t)dt

(7.18) for all v E V and cp E 92l(]0, T[) .

Thus

(7.19) du

iA(.)u + dt = f, u(O) = Uo ,

du and we deduce dt E L 2(V').

Theorem 1 is therefore proved. o Remark 3. We have proved more than Theorem 1: that there exists a solution satisfying:

(7.20) { U E Loo(O, T; V)

U ' E L 00(0, T; V') .

In fact, we can show that

(7.21) { U E CCO([O, T]; V)

U ' E CCO([O, T]; V') . o

Remark 4. The hypotheses made on A (.) and f are not the most general which allow us to solve a problem of type (n) «7.3) and (7.4)). We can take:

(7.22)

with

(7.23)

(7.24)

! i) Ao(.) E Ll(O, T; 2'(V, V'», :t Ao(.) E Ll(O, T; 2'(V, V')),

ii) A6(t) = Ao(t), t E [0, T] ,

iii) ao(t; u, u) + Alul 2 ~ IXllul1 2 , IX > ° for all u E V,

1 d 1. » A 1 (.) E L (0, T; 2'(H, H)), dt Ad.) E L (0, T, 2'(H, H .

and f satisfying

(7.25)

Then for Uo E V, there exists a unique solution to the corresponding problem (n).

o We compare the Schrodinger equation under the conditions of Theorem 1, with the diffusion equation under the conditions of problem (P) seen in §3.1.4., (see (3.21),


(3.22) and (3.23» from the point of view of the regularity of the given data and of the solution. In the two cases, we assume that the sesquilinear form a, which is continuous over V x V, satisfies the conditions: i) condition of uniform boundedness: there exists a constant M = M(T) > 0 such that:

la(t; u, v)1 ~ M Ilullllvll for all u, v E V;

ii) condition of uniform coercivity over V with respect to H: there exist A., a constants > 0 such that

a(t, u, u) + A.lul 2 ~ allul1 2 , "It E [0, T], Vu, v E V.

We summarise the differences in a table

Equation t --+ a(t; u, v) Initial Right Solution condition hand side u

Uo I

Diffusion Measurable H IE L 2(V') W(V)

Schrodinger ~1 V {IE L2(H) f' E U(V')

W(V)

(Continuously and also Rem. 3 differentiable) UE~O([O, T]; V)

u' E ~O([O, T]; V')

We are led to making more restrictive hypotheses than in the case oithe Schrodinger equation, on a(t; u, v), Uo and I to obtain the solution u in W( V). In many ofthe mathematical aspects studied here, the Schrodinger equation is very close to equations of second order in t (the wave equation). We have studied some aspects of the wave equation in Chap. XVIIB, §4.

1.4. Application Examples

1.4.1. Generalities. As stated at the beginning of this §7.l, the physical examples treated here will always have I = O. Therefore recall problem (n) with 1= O. The vectorial form of (7.3) is then:

(7.3)v d d/u(.» + iA(.)u(.) = 0 in the sense of L 2 (V') (136) •

(136) (7.3). denotes the vectorial form of (7.3).


A priori estimates a) By multiplying (7.3)v by u(.), then taking the real part, we obtain the conservation of the norm:

(7.26)

b) We "formally" multiply (7.3)v by u' and take the imaginary part of the result. We obtain, after integration:

(7.27) a(t; u(t), u(t)) = a(O; un' un) + L a'(a; u(a), u(a))da .

This equation is called the "energy equation". This procedure is formal, since u' E L 2 (0, T, V'), and this space is not here dual to L 2(0, T, V'). To justify the energy equation, we must proceed for example as has been stated in §5.5.3, that is to say by double regularisation (there are other regularisation procedures.) From the conservation of the norm, we deduce the existence of a family of unitary operators in H:

(U(t, s))o,",s.tG such that u(t) = U(t, s)u(s) .

We generally call this the Green's operator of problem (7.3)v (137), or the propagator, or resolvent. It satisfies the following properties:

(7.28)

and

i) U(s, s) = I the identity mapping in H ,

ii) U(t,s)U(s,r) = U(t,r)forO ~ r,s and t ~ T.

... {(t, s) -+ U(t, s)x 111) for all x E H, the mapping [] [ ]

D T = 0, T x 0, T -+ H

is continuous ,

iv) for all (s, t) EDT' U (t, s) V = V, and the mapping

{ (t, s) -+ U (t, s)x is continuous for all x E V

DT -+ V

U(t, s) E Y(V, V)

(this follows easily from the energy equation and Gronwall's lemma); v) further U (t, s)D(A(s)) c D(A (t)) (138)

Finally for all s E [0, T] and all x E D(A(s)), the mapping

{ t -+ U (t, s)x is continuously differentiable.

[0, T] -+ H .

These properties are generally obtained by methods of perturbing the (semi-) group(139). We see that we easily recover these properties here. Further, we know in

(137) See the preceding chapters on evolution problems, and particularly Chap. XV, §I, Remark 8. (138) See §3.1.2 - the definition of D(A(t)).

(139) See particularly Chap. XVIIB, §I, Remark 3 and Yosida [I], Kisynski [I], Simon [I].


what sense u(t) = U(t,O)uo satisfies the Schrodinger equation if Uo E V, Uo ¢ D(A(O)) (equation (7.3U On the other hand, to extend the family U(t, s) to IR + x IR +, we must consider the spaces W(O, T; V, V') for all T > 0. The conservation of the norm implies that we cannot have u in L2(0, + 00; H) (we therefore have to work with local spaces to use the L 2 norm).

1.4.2. The Schrodinger Equation for a Particle Subject to a Potential Which is Variable in t. We have to solve the Schrodinger equation (in \R3)

(7.29) du i- = A(t)u

dt

that is to say equation (7.3)v (equivalent to (7.3)) in the case where

A(t) = - L1 + v(t)

that is to say when the potential v depends on time(140). We meet this situation in quantum physics when we consider a single particle as part of a system, and we make the approximation of replacing the action of the system on this particle by a time dependent potential. We often consider the following problem: 1) the potential v(t) is constant for t < 0, and only really depends on time for t > ° (we then say that the "potential perturbator" acts from t = 0); 2) the state u(t) is stationary for t < 0, that is to say of the form

;,;(t) = e~iEot<po, t < ° with Eo and qJo respectively the eigenvalue and eigenfunction of the (self-adjoint)

def operator Ao = A(O) = - L1 + vo.

We then have to solve problem (7.29) for t > ° with potential v(t) depending on time, and with initial condition:

(7.29)' u(O) = Uo = qJo .

In order to apply Theorem to problem (7.29) with (7.29)', we have to verify hypotheses (7.1) and (7.2) for the sesquilinear form: a(t; u, v) made precise below, with a(t; u, v) = (A(t)u, v).

1.4.2.1. Verification of hypotheses (7.1) and (7.2). We assume that

1 v(t) = V<p)(t) + VOO(t)

with v(p) and v(OO) rea~ and moreover, for all t E \R ,

V<P)(t) E U(\R3), P = 2 + ex, ex > 0, VOO (t) E L (1) (1R3)

(7.30) (141)

(140) We use the 'natural' system of units here (see Chap. lA, §6). (141) We note that for PI < P2' we have (with the obvious notation) LP' + L 00 c U' + L 00, these being dual spaces of U' n L I and LP'· n L I (p; and P; conjugates of PI and P2), and for all q, U n L I = {uLm, 1 ~ m ~ q}, therefore LP\ n LI c U' n LI,


We further assume that

{the mappings t ---+ V<P)(t) and t ---+ v(co)(t) are continuously

(7.31) differentiable from [R into LP([Rn) and L CO ([Rn) respectively (142) .

629

For each t E [R, we therefore have an operator A(t), already encountered in Chap. XVIIA, §4 (but we made the supplementary hypothesis p ~ 2). The operator A(t), with domain D(A(t)) is self-adjoint for all t. We show that conditions (7.1) and (7.2) for the application of Theorem 1 are valid by taking H = L 2([R3), V = HI ([R3): We have for all u and

i 3 au au i (7.32) a(t; u, v) = I ;;-- ;;-- dx + (V<p) + V<CO))uu dx [KlJ ;= 1 uX; uX; [KlJ

Condition (7.1) is immediately verified for the term in v(co). For that in VIP) it is proved by using the Sobolev embedding theorem (see Chap. IV, §4): HI ([R3) is contained in U([R3) for 2 ~ q ~ 6, with continuous injection. Consequently u.U E U/2([R3), therefore uu E U'([R3)(143), from which we have (7.1),

We now prove condition (7.2)ii). The only term which presents any difficulty is the

term r v(p) lul 2 dx. Holder's inequality gives: J[KlJ

(7.33)

but from the preceding embedding theorem (since 2 ~ 2p' < 6), we have:

(7.34) 2 2 [" 1 au 12 2 ] IIull L 2P ~ CllullHI = C L.,.;;-- + lulu . , uX j L2

As in Chap. XVIIA, §4, by changing x to px, therefore u to up with up(x) = u(px),

we can take the coefficient of 1~12 as small as we wish(144). ax;

Consequently, for all I: > 0, there exists a constant C(I:) such that:

(7.35) Ilull~2P < { ~ I:;; 12 ] + C(I:)lu1I, .

From which we have condition (7.2)ii). o (We emphasise that for p ~ 2 we have the particularly simple property that D(A(t)) = H 2 ([R3) is in fact independent of t).

(142) We assume here for more generality that the potential v(t) effectively also depends on time for t < O.

1 I (143) With - + - = 1.

p' P

(144) We have Ilupll = p-n/21Iullu, tdx upl\u = pll(::)l, = pl-TII::IL here with n = 3.


We have therefore proved that under hypotheses (7.30), (7.31), problem (7.29), (7.29)' has a unique solution from Theorem 1. From the preceding section, we have also proved that there exists a Green's operator (or propagator)for this problem(145). Consequently, iffor t = 0, the initial condition Uo is in D(Ao), the solution of (7.29), (7.29)' is given by u(t) = U(t, O)uo, and the mapping t E IR --. u(t) E H is differentiable: we have a strong solution of the Schrodinger equation (7.29). If Uo ¢ D(Ao), Uo E V, the expression for u(t) above is a solution of the Schrodinger equation (7.29) in the variational sense.

Remark 5. We remark that we can also consider potentials v(t) which are more general than the potential [V<P)(t) + v<oo)(t)] which occur in (7.29). We point out in particular, the "Rollnik" potential v(x, t), that is the potentials which are such that:

(7.36) r Iv(x, t)llv(y, t)1 dxdy < + 00, Vt E IR . J [J;l3 X [J;l3 Ix - yl

This type of potential is used in particle physics and nuclear physics. A particular example of this is the Yukawa potential (ref. Roman [1]) which is

11 00 , v(r) = - e-'· dJ.l(a'), a > 0

r •

where J.l is a suitable positive measure with support [a, 00].

We refer for such a case to Simon [1]. D

Example of application of problem (7l') given by (7.3), (7.4), with A (t) of type (7.30). Physical presentation At time t < 0 an electron in the midst of an atom is subjected to a "Kato" potential U(x) independent of time, which we assume for simplicity is equal to(146):

-Zq U(x) = --, t < 0, X E 1R3 ,

4nlxl

where Z is a positive integer, Zq represents the "effective" charge of the atomic nucleus. We assume, as at the beginning of this Sect. 1.4.2, that the electron is in a stationary state(147), and is represented by the wave function

u(x, t) = e-iEo'cpo(x) for t < 0

where Eo and CPo are respectively an eigenvalue and an eigenfunction corresponding to the operator

(145) With the hypotheses (7.30), (7.31), we can replace the time interval ]0, T[ of the preceding section by [J;l.

(146) In the 'natural' system of units (see Chap. lA, §6) with a mass m = 1/2. (147) See Chap. lA, §6.


such that u satisfies the Schrodinger equation (7.29)

au i at = ( - L1 + uo)u for t < 0

- Zq/4n with vo(x) = U (x) = ~---=--

Ixi

631

At time t = 0, a particle IY., with charge Q (with Q = 2q), leaves the nucleus with a speed Vo which we assume constant, Vo E 1R3. The potential to which the electron is subject then becomes, at time t > 0:

(Zq - Q) Q v(x, t) = - - , t > 0 .

4nlxI 4nlx - votl

We look for the wave function of the electron, denoted u(x, t) E L 2(1Rn, for t > 0 which is the solution of:

(7.37) { i~~ = [ - L1 - Z!n~IQ - 4nlx ~ votl}'

u(x,O) = CPo(x) , CPo E H2(1R3) .

t > 0

This problem is a particular case of the following general problem: find the wave function u(x, t) of an electron subject at time t > 0 to the potential:

(7.38) v(x, t) = U(x) - Qo , with Qo = 4Q , Ix - XII n

U being a "Kato" potential, and t E IR + --+ Xl E 1R3 a given continuous function describing the movement of the particle (assumes known) of charge Q which "perturbs" the electron for t > 0, and the initial conditions u(x, 0) = uo(x) being given. We therefore propose to solve

Problem. Find the function u(x, t) which is the solution of:

(7.39) { i~~ = A(t)u

u(x,O) = uo(x)

t E IR+ ,

Uo E V given,

with A(t) an operator given by

(7.40) A(t) = - L1 + U _ Qo Ix - xII'

that is to say a problem of type (7.29). We shall make precise the domain of this operator for all t E IR + .

We remark that - Qo is a Kato potential for all given t, that is to say that it Ix - XII


decomposes for all t into:

(7.41 )

with

(7.42)

def v(x, t) =

Ix - Xtl

Consequently, for all t, the domain of A(t) is

(7.43)

Conditions (7.30) are therefore satisfied for all t E IR +.

Lastly, we prove the following lemma:

Lemma 3. It is possible to choose the decomposition (7.41) in such a way that the mappings:

(7.44) {t E IR+ ~ f~2) E L2(1R3)

t E IR+ ~ f~OC!) E LOC!(1R 3)

are continuous.

This is sufficient to prove that the inequality (7.2)ii) is satisfied for T = + 00 (the proof of (7.2)ii) follows from formulae (7.33), (7.34), (7.35) using only the continuity of (7.44)). Further, we deduce from Lemma 3 the continuity for all u and v E Hl (1R3) of the mapping: t ~ a(t, u, v), where

(7.45) a(t; u, v) = r [~aau aav + V (x)u(x)v(x) - I ~o I u(x) V(X)] dx . J ~3 , Xi Xi X X t

However, it is not obvious that this mapping is once continuously differentiable in [0, T] (T > 0), (nor that the stronger hypothesis (7.31) over IR+ be satisfied). Nevertheless we can conclude with the help of Remark 4 (Sect. 1.3.2 of this § 7). Formally differentiating (7.45) with respect to t: we obtain:

(7.46) '( ) Q i (),,(Xi-Xit)X;t-()d at; u, v = - 0 u x L., 3 V X X ~3 i Ix - Xtl

by assuming that the given mapping t f---+ X t is once continuously differentiable from IR + into 1R3. The fact that (7.46) has a sense for all u and v E H 1 (1R 3 ) is due to the following Hardy inequality:

(7.47) i lu(x)12 i la 12 --2-dx ~ 4 -u(X) dx ~ Cllull~1 (C constant). ~3 Ixi ~3 ar


Proof of (7.47). It is sufficient to prove the inequality for u E ~ (1R3) (we then obtain the general case by extending through continuity)(148). For u E ~(1R3), we have:

f oo d foo ( au ) lu(xW = - 1 dA. lu (..1.xWdA. = - 2Re 1 u(A.x) Xi aXi

(..1.x)dA.,

therefore

r ,u(x112 dx = _ 2 Ref 00 dA. r U(A.x) ~(~)(A.X)dX JIR3 x 1 JIR3 Ixl Ixl aXi

foo dA. r u(x) a - 2 Re 1 A.2 JIR3 Vi ar u(x)dx

r u(x) a - 2 Re JIR3 Vi ar u(x)dx ,

from which we obtain the result via the Cauchy-Schwarz inequality. 0

We then have, by using (7.46) and (7.47):

(7.48) la'(t;u,v)l:::; Qolx;1 r 1~(X)IV(~~ldX:::; Qolx;IClluIIHlllvIIHl. JIR3 x - X,

We deduce immediately from (7.48) and from Lemma 3 that the family of operators (A (t))t E IR+ E 2'(V, V') defined by:

(7.49) (A (t)u, v) = a(t; u, v), Vu, V E V,

is such that:

(7.50) dA(t) 00 I

A(t) E rcO(IR+, 2'(V, V')), ----ctt E L (0, T, 2'(V, V)), VT> o.

Consequently, conditions (7.23) and Remark 4 are true (with Ao(t) = A(t)), which allows us to solve problem (7.39). We therefore only have to prove Lemma 3.

Proof of Lemma 3. The "natural" choice

f~2)(x) = I 1 I Ys" f,oc'(x) = I 1 1(1 - Ys.) x - x, x - x,

(148) In fact, we have a little more, that is

I ~I ,;;; CIIull~1 Ixl L'

where ~1 denotes the (Beppo-Levi over R3 ) space which is the completion of ~(R3) for the gradient norm


with Ys, the characteristic function of the ball St with centre X t radius R (R constant > 0), or even

1 f (2) _ 1 t (x) - Ixllx - xtl and f;"(x) = I I x - X t

are not convenient to realise these conditions. On the other hand, the choice which is associated with the decomposition of the

f · 1 1 (b . der b unctIOn x --+ ~ = ~ y settmg r = Ixl) y:

(7.51 ) f(2)(X) = {1/r r::::; R pooJ(x) = {O r ::::; R R/r2 r? R , (r - R )/r2 r > R ,

(we obtain the functions f~2)(X) and f~oo)(x) occurring in (7.51) by replacing r by I x - X t I), gives the desired continuity. This is what we shall now show. 1) Proof that the mapping t E [R + --+ f~oo) E L 00 ([R3) is continuous: Take, for t = to, xto = 0. Set rt = Ix - xtl. Let a be a positive constant less than R/4 such that Ixtl < a (it is sufficient to choose t close enough to to for this inequality to hold). To calculate II f~:) - f~oo) II, we decompose the space [R3 into three parts:

a) S = {x,r < R - a}; we then have:f:oo)(x) - f::)(x) = 0, XES;

b) SO = {x, R - a ::::; r ::::; R + a}

sup If::)(x) -f:OO)(x)l::::; sup {! - ~,! -~ - (~- ~)}. XE sO X E so r r r r rt rt

But in So, we easily show the inequalities(149)

1 R aiR 2a - - - ~ - - - ~ -:----:---;;-r r2 '" (R - af' rt rf '" (R - 2a)2

therefore

(7.52) sup If::)(x) - f:oo)(x) II ::::; C RA2 (C constant) ;

xeS O

c) S 1 = {X, R + a ::::; r}

(7.53)

lsup If~:) - fl oo ) (x) I ::::; sup {! - ~ - R(4 - 4)} XES' XES' r rt r rt

a Irf - r21 C' ::::; + R ::::; a- (C' constant) . r(r - a) r2(r - a)2 R2

We obtain by definition

(7.54) Ilfl:) - f: OO) IlL::::; a ~~ (Co constant) ,

(149) Rough estimates are sufficient.


from which we deduce the stated continuity. 2) Proof that the mapping t E ~ + -+ Il2) E L 2 (~3) is continuous. The expression Il/l;) - Il2) IIIz is the sum of three terms: a) The first term II corresponds to integration in S

(7.55) II = IsI~ -~12 dx = Is ~~ + Is ~; - 2 Is ~: ' from which

(7.56) II ~ 4n(R - a) + 4nR - 2f dx = 8nR - 4na - 21. s rrt

635

Set I xtl = at> (at ~ a); take X t as the axis Oz, and let (8, q» be a system of spherical coordinates of the point M (with cartesian coordinates x).

z

We have the relation:

(7.57)

from which

I = f ~ r2 dr dw = fR -a f f r dr sin 8 d8 dq> Srrt C 8E(O.1t) ((JE(o.21t)jr2 + a; - 2ratcos8'

we immediately obtain (by setting f.1. = cos 8)

I = 2n fR -a f + 1 -----;~=r=d~r=df.1.=== o - 1 P + a; - 2rat f.1.

2 fR-a = ...!!.. [ - I r - at I + I r + at I] dr

at 0

2n fa, = - [ - (at - r) + (at + r)] dr

at 0


from which:

(7.58) I = + 2nat - 4n(a + at) + 4nR = 4nR - 4na - 2nat ;

(7.56) gives finally.

(7.59) 11 ~ 4n(at + a) ~ 8na .

b) The second term 12 corresponds to an integration in So, whose measure tends to o with a, of a bounded function in SO (independent of a). c) The third term 13 is given by:

13 = I" (~ - ~)2dX = f. R2(rt ;/)2dX, JSI r rt r>R+a r rt

from which

13 = f. R 2 (at - ~r~t cos ())2 dx r> R + a r r t

f. f+1 (a - 2J-L)2 ~ 2nat R2 2( _ )4 drdJ-L

r>R+a -1 r r at

again from which

(7.60)

we have therefore proved the continuity of the mapping

tEIR+ --+ftEU(1R3). o Finally, we have proved that the hypotheses (7.1) and (7.2) are satisfied for all t E IR + and therefore from Theorem 1, that problem (7.39) with A(t) given by (7.40) (with the hypotheses following (7.38)) has a unique solution u(t).

1.4.2.2. Potential perturbation periodic in time. The case of a potential perturbation whose time dependence is periodic is frequently met in practice - we cite two examples: a) Example of application to the Schrodinger equation of a charged particle subject to the action of an electromagnetic field: Physical introduction: electron subject to an imposed electromagnetic field which is sinusoidal in t(t > 0). We consider an electron with charge - q, mass m, placed in a given (real) potential U(x) and subject to the action of a given electromagnetic field. We assume that the electrom~netic field is described by a potential vector(150) of the form (sinwt) A(x) (with A (x) E 1R3 and WEIR given). Then the Hamiltonian operator, called here A (t), relative to this particle is given from formula (6.38) of

(150) See Chap. lA, §4.


Chap. IA (in the S.I. system of units) formally by:

(7.61) A(t) = 2~( - ihV - ~sinwt 1y + U,

where i = Fl, h is Planck's constant, e the speed of light, and V the gradient in 1R3.

This formula is again written formally, by neglecting(151) the term in ~, and by e

assuming that the potential vector satisfies the condition (called the Coulomb -+

gauge(152»): div A = 0,

(7.61), h2 iqh -+

A(t) = - -2 A + -sinwt( A . V) + U , m me

-+ 3 a -+ where A· V = L Ar ;--, with A j , j = 1,2,3 denoting the components of A.

j = 1 UX j

The evolution equation of the wave function u of the electron will be:

(7.62) au

ih at = A(t)u ,

therefore of type (7.29), which is, again by passing to the "natural" system of units, -+ -+

with h = e = 1,2m = 1 and by replacing qA by A :

au -+ (7.63) at + i( - A + U - isinwt A . V)u = O.

We shall solve this equation with the initial condition:

(7.64) u(X, 0) = uo(x) . D

We can put equation (7.63) in the "weak" form (7.3). We assume that the given potential U which occurs in (7.63) is a Kato potential (153), that is to say satisfies (7.30) (here with p = 2). However, we give a slight variation here as far as the choice of spaces is concerned, which leads to analogous results. We set:

(7.65) a(t; u, v) = ao(u, v) + a 1 (t; u, v)

(151) Note that there will be no particular mathematical difficulties in keeping this term, see the hypotheses made later on A: we can simply replace the potential U in (7.61) (independent of time) by U

2

+ -Q-A 2 sin2 wt which depends continuously on time. But formula (7.61) will be more easily 2mc2

amenable to a direct variational statement of the evolution problem which we shall study. (152) See Chap. lA, §4. (153) See Chap. XVIIA, §4.


with

(7.66) ao(u, v) = [ Vu Vvdx + [ V(x)uvdx, J~3 J~3

(7.67) 3 1 au a l (t; u, v) = - i L sinwt Aj(x)-a· vdx . j=l ~3 Xj

--+ We make the following hypotheses about the given functions V(x) and A (x) = [A I (x), A 2 (x), A3(X)].

(7.68) {V(X) = Vdx) + V 2 (x) , with VI E L2([R3) , V 2 E LOO([R3) ,

(7.69)

and in the applications considered, A j is real. The functional spaces considered here are:

i) H = L 2([R3, C) n LtVll ([R3, C) ,

ii) where LtVll([R\C) = {u,L3IVllluI2dX < + CIJ}. (7.70)

We set

iii) lulH = (lu ll2 + L31VlllUl2dX y12;

H is a Hilbert space for this norm;

(7.71)

We note that if u E H 1 ([R3), we also have(154) u E U([R3) with:

therefore,

luIL4(~3) ~ c .11 U IIHl(~3) ,

from which it follows that if u E HI ([R 3) then u E H. In effect it is sufficient to prove that u E Ltvl l([R3). Now

L3IVll.IUI2dX ~ (L3IVlI2dX)l/2luli.4(~3) ~ c.llull~l(~3)' from which we have the result. Consequently, the norm:

(7.72)

is equivalent to the usual Sobolev norm of HI ([R3).

(154) We again use here the Sobolev embedding theorem (see Chap. IV) already used in the preceding Sect. 1.4.2.1.


Here we consider V equipped with this norm. The functional framework H and V having been made precise, we verify that:

(7.73) la(t;u,v)1 ~ Milullllvll,

and that

(7.74) Rea(t;u,u) + Alul 2 > IXilul1 2 (IX = t,A i= 0).

We can therefore apply Theorem t to problem (7.63), (7.64) in its variational form and we obtain, for every time interval ]0, T[: for Uo E HI (IR 3), then there exists a unique u such that

u E L2(0, T;H I (1R 3 )) and u E <c°([0, T]; H),

is the solution of our problem (7.63), (7.64) in the form (7.3), (7.4) (problem (n)) with f= O. b) Harmonic oscillator subject to a periodic electric field. Physical introduction. A particle of mass m, electric charge Q, is displaced along the x-axis, x E IR, and subject to a force - mW6 x for all t, and, starting from t = 0, to an electric field Esinwt directed along the x-axis. We assume that at time t < 0, the wave function u(x, t) representing the state of the particle is of the form:

iEot

I/Io(x,t) = e--h-<po(x), t <. 0,

where Eo and <Po (x) are respectively an eigenvalue and eigenfunction corresponding to the operator Ao:

L1 t Ao = _112_ + -mw2 x 2

2m 2 0

At time t > 0, the potential to which the particle is subjected becomes:

t . 2mw6x2 + QExsmwt.

We look for u(x, t) at time t > 0, the solution of:

X E IR ,

u becomes, at time t = 0, equal to <Po. By setting E = 1,2m = t,11 = 1, Q = 1 (155), we must solve the problem:

(7.75) {~~ = ( - L1 + w6x42 + xSinwt)u, t ~ 0,

u(O) = Uo given,

(155) See Chap. lA, §6.


def where uo(x) = CPo(x), which is again a problem of the form (7.29), (7.29)', with

x2 v(x, t) = w~4 + xsinwt . o

We verify that conditions (7.1) and (7.2) are realised in an adequate functional framework. Firstly, H is the space of (complex valued) square integrable functions L 2 (IR), V is defined by:

(7.76) V={u,uEH,ao(U,u) ~ t~:~:dX + 81tlxl21ul2dX

+ 82 tlUl2dX < + 00 }

{ 2 du 2 2} = U, U E L 'dx E L , XU E L ,

with 81, 82 strictly positive constants. The space V is equipped with the structure of a Hilbert space with the norm:

(7.77)

We now define the family of sesquilinear forms

(7.78) a(t; u, v) = f~~: ~: dx + f~ (~W~X2 + xsinwt )UVdX .

By using the inequality:

(7.79) IXX2 1

x ~ 2 + 21X' IX an arbitrary positive constant,

we verify that the family of sesquilinear forms a(t; u, v), t E IR +, is continuous over V x V, "It E IR +.

Further by remarking that:

(7.80) a(t; u, v) = f~ (~: ~: + w~X: uv )dX + sinwt f~ xuvdx ,

we see that condition (7.1) is verified for all t > O. It then remains to verify condition (7.2). Set

(7.81) adu, v) = t (:: :: + w~:\v )dX .

We have the inequality

(7.82) a(t;u,v) ~ adu,u) -It x,u,2dxl,


and by using (7.79), we obtain:

(7.83) a(; u, v) ;:: a1 (u, u) - f~ ~()(X2 + ~ }u 12 dx .

1 IY. From which we obtain, by choosing IY. sufficiently small so that 4w6 -"2 > 0, the

"coercivity" inequality:

(7.84) a(t;u,u) + ~luI1;:: f~[I:~12 + (~W6 - ~)x2IuI2JdX. Therefore Theorem 1 can again be applied; we therefore obtain the existence of a unique solution u(x, t) (with (7.7)) of problem (7.75) for all t > 0, since Uo = <Po E V.

1.4.2.3. Perturbation v(t) of limited duration T. In certain physical situations, we use a perturbation v(t) of limited duration (sudden impulse given to the nucleus of an atom; atom subjected to an electromagnetic field of brief duration, etc .... ). We are then in the situation of (7.29) with

lOA (t) = - A + vo , t ~ ° where Vo is a potential which does not depend on time t.

2° A(t) = - A + Vo + v(t) , t > ° , where D(t) satisfies: D(t) = 0, '<It > T, T being a given positive constant. Set v(t) = Vo + D(t). Then, if conditions (7.30), (7.31) are verified by v(t) for all t ;:: 0, problem (7.29) with the given initial condition

u(o) = Uo

(u o being for example an eigenfunction of the operator ( - A + vo) and therefore corresponding to the value at t = ° of a stationary solution of(7.29) for t ~ 0), can be treated with the methods of Sect. 1.4.2.1.

Review of Section 7.1. We have shown existence and uniqueness of the solution of (Cauchy) evolution problems related to the Schrodinger equation for time dependent potentials in some general cases considered in quantum physics. We insist on the fact that for the variational method used here, we can obtain a (quantitative) numerical solution of these problems for all time intervals. On the other hand, the qualitative study, particularly for the behaviour as t -> 00, of the solution of the considered equation, does not follow from the method used here. However, we can make the following remarks, in the most simple case where the perturbation v(t) has a limited duration T (Sect. 1.4.2.3). 1) The Green's operator U(t, s) is such that for t > T,

U(t, T) = e-iAo(t-T) with Ao = - A + vo ,


therefore, by (7.28)ii),

U(t, 0) = U(t,T)U(T,O) = e-iAO(t-T)U(T,O) , "It> T.

Therefore the effect of the perturbation is expressed by the operator U (T, 0), called the transition operator. We note that, generally, this operator U( T, 0) does not commute with the operator Ao: this means that starting from a stationary state:

u(t) = e-iEotcpo, t < 0,

with Eo and CPo respectively an eigenvalue and eigenvector of Ao, the state u(t) = U(t, O)CPo, t > T, is not, in general, a stationary state. 2) We can also ask if, starting from a stationary state as before for t < 0, the state u(t) will be stationary for all t > 0, that is to say if we can have

A(t)u(t) = Etu(t) , "It > ° (156)

with Et an eigenvalue of the operator A(t). We therefore would have:

au i at = A (t) u(t) = Et u(t)

therefore

u(t) = exp ( - i I Esds ) CPo = A(t)cpo

with A(t) = exp ( - i I Esds } which implies:

A(t)cpo = Etcpo, "It > 0.

This is only possible if the family of self-adjoint operators A (t), t > 0, have CPo as a common eigenvector. 3) With the help of the energy equality (sec (7.27», we can also obtain some qualitative information about the behaviour for t > T of the solution of the problem under consideration (for example in 1.4.2.3.) From (7.27), the (average) jump in energy due to the perturbation at time t(t > 0) is given (for Uo such that (uo, un) = 1, with Uo E D(Ao) by:

def ft LiEt = (u(t), A(t)u{t» - (uo, Aouo) = ° (u(a), v'(a)u(a»da (157)

from which we have the jump in energy due to the perturbation for all t > T:

LiEt = LiET = f: (u(a), v'(a)u(a»da .

In numerous practical cases, the operator Ao = - Li + Vo has its spectrum a(Ao) comprised on the one hand of negative eigenvalues with finite multiplicity, corresponding to 'bound states' uo, and on the other hand of a continuous positive

(156) Such an evolution can be called adiabatic. (157) Corresponding to states called scattering states.


spectrum (157). Then starting from a stationary (bound) state CPo with Ao CPo = Eo CPo, Eo < 0, we finally obtain, after perturbation, a bound state u(t)(t > T),

that is to say such that

(u(t), Aou(t)) < ° if Eo + LlET = Eo + f: (u(u), v'(u)u(u))du < ° . In the opposite case, we say that we have a "scattering" state, (or also that the particle is not bound, as a consequence of the energy introduced by the perturbation). 0

2. Evolution Equations with Delay

2.1. Setting of Problem

2.1.1. Operational Cauchy Problem with Point or Dense Data. To fix ideas let X be a real or complex Hilbert space, and F(X) a space offunctions with values in X with

(7.85) F(X) C+ <C°([O, T];X) , T > 0.

We consider

i) t -+ w(t) a measurable function in ]0, T[ ,

(7.86) .. ) {der } 11 Eo = t; I/I(t) = t - w(t) > ° a.e. ,

iii) E 1 the complement of Eo in ]0, T[ ,

and we set (for example):

Problem (R). Find u E F(X) satisfying (in a sense to be made precise):

1') au I at + A(t)u(t) + B(t)u(I/I(t)) = f(t) a.e. t E [0, T]

ii) u(O) = Uo given in X ,

iii) u(t) = u(t) a.e. t E E1 = I/I(E1) ,

where A(t), B(t) are unbounded operators.! and u given functions. Condition (R)ii) is point Cauchy data. Condition (R)iii) is dense Cauchy data.

2.1.2. General Data. We again consider V, H, V' in the setting of Sect. I of §5, a family of operators A(t), t E [0, T] defined by a family of sesquilinear forms


a(t; u, v), t E [0, T], continuous over V x V and satisfying:

{i) la(t;u,v)1 ~ Mllull.llvll,

(7.87) M a constant independent of t E [0, T], u, V, E V ,

ii) t -+ a(t; u, v) is measurable in ] 0, T[ for all u, v E V .

We again assume

(7.88) { there exists A E IRl and rx > ° (A, rx constants) such that

Rea(t;u,v)+),luI2~rxlluIl2 forall UEV.

2.1.3. Case of Constant Delay. We introduce a family of operators B(t) E 2'(V, V'), t E [0, T] with

(7.89) ! t -+ II B(t) 112'(v, V') is bounded and measurable in ]0, T[

sup II B(t) 112'(v, V') ~ f3 , IE[O, TJ

and we assume

(7.90) w(t) = W o = constant> ° . The corresponding problem (Ro ) is the following.

Problem (Ro ). Let

(7.91) {11o E L2( - wo,O; V)

fEU (0, T; V' )(resp. L 1 (0, T; H))

be two given functions and

(7.92) Uo E H be given.

We look for u satisfying

1 i) U E L2( - W o, T; V) (l 'C°([O, T]; H) ,

ii) u'(t) + A(t)u(t) + B(t)u(t - wo) = f(t), T> t > 0,

iii) u(O) = Uo

iv) u = 11o, t E ] - W o, O[ .

Problem (Ro) is a parabolic problem with constant delay W o' The solution of problem (Ro) is immediate under the hypotheses (7.87) to (7.92) and there is existence and uniqueness of the solution u. In effect, we may note that (Ro) reduces over ]0, W o [ to the point Cauchy problem: find UO E L 2 (0, wo; V) (l 'Co ([0, wo] ; H) satisfying

{ UOO/(·) + A(.)uO(.) = go(.) ,

(7.93) u (0) = Uo ,

where


Then, from Theorems 1 and 2 of §3, there exists a unique Uo satisfying (7.93). We then define u1 such that:

(7.95)

Ii) u1 E L2(WO' 2wo; V) n '6'°([wo, 2wo]; H),

ii) d;t1 + A(.)u 1 = gl, gl(t) = f(t) - B(t)u°(t - wo),

iii) u1 (wo) = UO(wo) (given Cauchy point data at wo) ,

by a new application of Theorems 1 and 2 of §3. Step by step we therefore construct u, the unique solution of (Ro) whose restriction to [nwo, (n + l)wo], (n EN) is equal to un the solution of a Cauchy problem analogous to (7.95) where g is replaced by

gn, gn(t) = f(t) - B(t)un~ 1 (t - WO) ,

the given Cauchy data at t = nwo being un (nwo) = un ~ 1 (nWO)'

2.1.4. The Case of Variable Delay. We are given

{ t -> w(t)_a measurable. funct_ion in ]0, T[ such that the set (7.96)

£1 - {t E ]0, TL tjJ(t) - t - w(t) ~ 0 a.e.} ,

has non zero measure strictly less than T, and we set

(7.97) - '6'0 = inf tjJ(t) , - 00 ~ - '6'0 < o. tEEl

If we now consider problem (R) with for example suitable given ii, it is now necessary to have stronger hypotheses on the operator B(t) to obtain a unique solution. The study of the case w(t) = Wo however suggests transforming the problem with 'thick data' (R) into a point Cauchy problem. Formally, for the moment, we introduce: - the operator M ° defined (for measurable u with values in V)

(7.98) { u(tjJ(t)) if tjJ(t) > 0 a.e.,

Mou(t) = . o If tjJ(t) ~ 0 a.e.;

- the function 1 defined by

(7.99)

Then by setting:

(7.1 00)

- {f(t) if tjJ(t) > 0 a.e., f(t) = f(t) - B(t)ii(tjJ(t)) if tjJ(t) ~ 0

M(t) = B(t) 0 M o ,

problem (Ro) becomes

a.e.


Problem (ii). Find u E L 2 ( - ceo, T; V), whose restriction to ]0, T[ (again denoted u) satisfies:

! i) u E L2(0, T; V) 11 ceO([O, T]; H),

ii) u(O) = uo(uo given in H) ,

iii) ~~(.) + Au(.)u(.) + (M(.)u)(.) =1(.) (158)

(the given data B(t), Ii, f being such that IE L 2 (0, T; V') (resp. L 1 (0, T; H) + L2(0, T; V')).

We shall show more precisely


. i) the injection of V into H is compact,

ii) a(t; u, v) satisfies (7.87), (7.88) .

iii) B(t) E ft' (H, H), II B(t) II Y(H. H) ~ {J

({J a constant independent of t) ,

iv) f E L 2 (0, T; V')(resp. L 1 (0, T; H)) ,

v) Uo E U( - ceo, O;H), Uo E H .

Then problem (R) has a unique solution.

2.2. Proof of Theorem 2

2.2.1. Uniqueness. If U1 and U2 are two solutions of problem (R), then w = U1

- U 2 satisfies:

1 d 2" dt Iw(t)12 + Rea(t; w(t), w(t)) + Re((M(t)w)(t), w(t)) = 0 .

From which we deduce:

1 fl 2" lw(t)12 + IX ° IIw(O") 112 dO"

(7.101) ~ f~I(MW(o")'W(O"))ldO"+A.LIW(O"WdO" (159)

~ sup IMw(O")I. fl Iw(O")1 dO" + A. fl Iw(O"W dO"; o.::s:;O'~t Jo Jo

(158) Consequently, we shall also use the notation M(. )u(.) or simply Mu to denote the function (M (. )u)(.): t ..... (M(t)u)(t).

(159) With the notation Mw(u) ~ (M(u)w)(u).


but, from (7.98)-(7.100)

(7.102) sup IMw(a)1 ~ p sup Iw(a)1

from which it follows that

(7.103) r 1 r Jo I (Mw)(a) I Iw(a)1 da ~ 4suplw(a)12 + Ct Jo Iw(a)1 2da.

Therefore for 0 ~ t ~ tl ~ T, we deduce from (7.101) to (7.103):

1 1 i'l -41(w(td21 ~ -4 sup Iw(aW ~ C.t 1 Iw(a)12da 0'::::; a ~ tl 0

(7.104)

and Gronwall's lemma implies that w(t) = O.

2.2.2. Existence

A) Approximate problem We consider the Galerkin approximation Vm of V and the approximate problem (with the notation of §3):

dm

Problem (Rm). Find Um' um(t) = L /ljm(t) Wjm with j= 1

i) (:tUm(t), UJm) + a(t;u",(t), Wjm ) + (M(t)um(t), UJ",)

= (j(t), UJm), 1 ~ j ~ dm ,

ii) um(O) = UOm ' uo", -+ Uo strongly in H .

Problem (Rm) is here a finite dimensional operational problem in contrast to those considered previously because of the term (M(t)um(t), UJm). If we denote ~( «I) (I») ITbd h . d . h /lmtt} = /llm"'" /ldm,m E 11\\ m t e vector assocIate WIt Um' fio = «UOm , W1m ), (UOm , Wm), ... ,(UOm ' Wdm,m)) associated with UOm ' then problem (Rm) is written in the form:

-+ Problem (Rm)

(7.105)

where

(7.106)

{ d -----. -----. -+ ~ dt/l",(t) + dm(t)/lm(t) + .Hm(/lm)(t) =f",(t) ,

-+ -+ /lm(O) = /lo,

{

d m(~ = (a(t; wim , UJ",L, i,j = 1, ... ,d", ,

.H m(/lm)(t) = (M(t)um{t), UJ",), j = 1, ... , dm ,

r (t) = (fm(t), UJm), j = 1, ... , dm .


Since the operator /Z. --. .A m(/Z.) is linear and continuous from L 00 (0, T, IRdm) into itself with:

--;;;t _

we show (see Artola [1]), that (R".), and therefore (Rm), has a unique solution which satisfies Um E ~o ([0, T]; Vm), u~ ELI (0, T; V m).

B. a priori inequality B.!. First type of a priori estimate ~ We start from equation i): multiply by J.ljm(t) and sum from 1 to dm - by taking the real part of the result - we obtain

1 d -2dt lum(tW + Rea(t;um(t), um(t)) + Re«Mum)(t), um(t)) = Re(f(t), um(t))

from which we deduce that if 1 E L 2 (0, T; V') (which holds for f E L 2 (0, T; V' )), we have:

1 r 1 r 2 lum (tW + a Jo II um(a) 112 da ~ 21uom l2 + Jo 111(a)II * Ilum(a) II da

+ f3 sup lum(a)l. rtlum(a)lda + A rtlum(aWda; O~O"~t Jo Jo

we then continue as for uniqueness and, c denoting a constant, we obtain:

from which

(7.107) {I aft ft -4 sup lum(a)1 2 + -2 Ilum(a)11 2 da ~ C(T;uo,f) + c lum(aWda O~O'~t 0 0

C( T; uo,f) = cons tan t depending on T, uo,f .

It follows from Gronwall's lemma and from (7.107) that we have:

(7.108) { Um E a bounded set of L 00(0, T;H) ,

Um E a bounded set of L 2 (0, T; V) .

B.2. Second type of a priori inequality The a priori inequalities (7.108) are insufficient to pass in general to the limit in the term «M(t)um)(t), Ujm) since, except in particular cases, U --. M(.). U is not conti-


nuous from L 2(0, T; H} into itself. From (7.108), we deduce:

(7.109) { i} A (.) Um remains in a bounded set of L 2 (0, T; V') (160)

ii} M (. }um remains in a bounded set of L <Xl (0, T; H) c L 2 (0, T; V')

and therefore from the approximate equation it follows that

dUm 2 (7. 110} dt remains in a bounded set of L (0, T; V') .

In fact (Artola [I]) we can show that the "derivative of order}" }' E ]0, 1/4[ of um"

remains in a bounded set of L 2 (0, T; H).

Passage to the limit. To pass to the limit, we shall use a compactness result(161) (see Lions-Magenes [1], Lions [I]), and use hypothesis i} of the theorem which has not yet been used here. This result is the following:

(7.111) {if the injection from V -+ H is compact, then the injectionfrom W(V, V'} -+ L2(0, T;H} is compact.

Therefore, as Um remains in a bounded set of W( V, V'}, we can extract a subsequence from Um (again called um ) which converges to U in W(V, V'} weakly and therefore in L 2 (0, T; H) strongly. Then (free to make a new extraction of a subsequence), {um(t)} converges almost everywhere in H strongly. Now, if we note that

{if Vm -+ v almost everywhere in H strongly, then

(7.112) MVm -+ Mv almost everywhere in H strongly,

we shall have, from the Lebesgue theorem:

{ Vcp E ~(]O, T[), Vm E Vm -+ V E l' (1' dense in V)

(7.113) (T (t Jo ((Mum)(t), vm}cp(t}dt -+ Jo (Mu(t), v}cp(t}dt,

so that (the other terms behave well), U satisfies

(7.114)

-f: (u(t), cp'(t}v}dt + f: a(t; u(t}, cp(t}v}dt

+ f: (Mu(t), qJ(t}v}dt = f: (J(t), qJ(t)v}dt

for all v E 1', qJ E ~(]O, T[) .

It is then easy to show (see §5) that u is the solution of (R).

(160) Here we use, as in the remark following Lemma 4 of §3, a special basis, the vector description of Rm in L 2 (V') being:

d _

-um(.) + P~' A(.)um(.) + P~' M(.)u .. (.) = P~'f(.). dt

(161) Here we shall meet some "nonlinear" reasoning.


Remark 6 1. HfE Ll(O, T;H), thenJE L2(0, T; V) + Ll(O, T;H). The a priori estimates follow from

(7.115) If:(f(O"),U(O"))dO"I ~ bo~~~,lu(0")12 + C(b)(f:lf(O")ldO"Y,

where b is chosen small enough so that b + ~ < ~.

2. H t -+ I/!(t) is of class <'6'1, and suitable (for example ° < I/!(t) for all

t E JO, T[, I~~ I > Wo > 0), then

(7.116) M(.) E Y(L2(0, T;H), L2(0, T; H))

and the hypothesis of compactness of the injection of V into H is superfluous to pass to the limit. 3. Problem (R) occurs as a perturbation of problem (P) considered in §3, by an operator M with the following properties: X being a Banach space, - the operator M(.) is continuous from L'" (X) into L'" (X). - it is "local" in the following sense. First of all, if t is fixed in JO, T[, it is possible to identify L'" (0, t;X) with the subspace of L'" (0, T; X) defined by

{u; U E L"'(O, T;X), u(s) = ° a.e., s > t} .

In these conditions, if U E L'" (0, T; X) we can define its restriction to J 0, T[ by

(7.117) rtu(s) = {U(S) a.e., s E JO, t[ ° elsewhere ,

therefore

(7.118) { rtU E L"'(O, t;X)

II rtu IIL",(o, T;X) ~ II U II LOO(O, T;X) ;

we then set

Definition 1. Let X be a Banach space. The bounded operator

ME Y(L"'(O, T;X), L"'(O, T;X))

will be called type "L" (local) if M satisfies:

(7.119) { there exists a constant Jl = Jl( T) > ° such that for all to E J 0, T[

II rtoMu IIL",(o, T;X) ~ JlII rtou II LOO(O.T; X) •

- Therefore the operator M is of type "L" in L'" (0, T; H).


- Further, the operator M is regular for convergence almost everywhere in ]0, T[:

(7.120) {if Un --+ u almost everywhere in X strongly,

MUn --+ Mu almost everywhere in X strongly.

4. Theorem 2 extends to the case of operators M with properties (7.119), (7.120). With supplementary regularity hypotheses on the sesquilinear form a(t; u, v) (for example a(t; u, v) Hermitian, with t --+ a(t; u, v) having a derivative in the sense of distributions, measurable and bounded for all u, v E V) and on the given data

(uo E VandfE L2(0, T;H) (resp. fE L2(0, T; V'), f' E L2(0, T; V')),

Theorem 2 extends to the case M E 2 (L 00 (0, T; V), L 00 (0, T; H)) of type" L" with Uo given in L2( - ~o, 0; V). The solution u of the corresponding problem (R) is then in W( V, H). 5. Numerous variations are possible. We refer for these to Artola [1]. 6. Some integral operators belong to the framework of the operators M; for example, for (s, t) E ]0, T[ x ]0, T[, {K(t, s)} with K(t, s) E 2(H, H). We define M for u E LOO(H)

(7.121) Mu(t) = L K(s, t)u(s)ds .

If K(. ,.) E LOO(]O, T[ x ]0, T[; 2(H, H)) then

(7.122) {

ME 2(L 00 (H), L 00 (H))

and

M is of local type .

Besides if {un} is bounded in L 00 (H) and Un --+ u almost everywhere in H then MUn --+ Mu almost everywhere in H. We shall therefore have a variant of Theorem 2 for operators M which do not correspond to retardation.

3. Some Integro-Differential Equations

3.1. Mathematical Example of lst Order

3.1.1. Setting of Problem. We consider V, H, V' as in the preceding paragraphs and we are given a family of sesquilinear forms a(t; u, v) which are continuous over V x V and satisfy:

{ t --+ a(t; u, v) is measurable in ]0, T[ for all u, v E V

(7.123) I a(t; u, v) I ~ c. II u II . II v II (C = C( T) = constant) for all u, v E V;

(7.124) {there exist constants r:x. = r:x.(T) > 0,..1. = ..1.(T) E IR such that

Rea(t;u, v) + ..1.lul 2 ;::: r:x.llul1 2 for all u, v E V.

Besides, we are given a family of sesquilinear forms k(s, t; u, v) defined for (s, t) E ] 0, T[ x ]0, T[ with

652

(7.125)


i) for all (s, t) E [0, T] x [0, T], k(s, t;u, v) is continuous over

V x H, (s, t) -+ k(s, t; u, v) being measurable in

]0, T[ x ]0, T[ for all (u, v) E V x H ;

ii) there exists ko = ko (T) > ° such that

Ik(s, t; u, v)1 :::;; ko lIull.lvl for all (u, v) E V x H.

This family of sesquilinear forms defines a family of operators K(s, t) continuous from V to H, by

k(s, t; u, v) = (K(s, t)u, v), Vu E V, V E H;s and t E [0, T] .

In these conditions, for u E L 2 (0, T; V), we can define a function M u by

(7.126) Mu(t) = I K(t, u)u(u)du for almost all t E ]0, T[ ,

the integral being taken in H. Note now that the operator M: u -+ Mu is such that:

(7.127)

In effect, for all qJ E L2(0, T;H) from (7. 125)ii),

II: (MU(t),qJ(t»dtl :::;; I: I~ ko(T)llu(u)lldulqJ(t)ldt

( rT )1/2 ( rT )1/2 :::;; Tko(T) Jo Ilu(u)11 2 du Jo IqJ(tWdt ,

from which

(7.128)

Since the mapping u -+ Mu is linear, the result (7.127) follows.

3.1.2. Statement and Proof of Result. We have the following theorem which is a variant of Theorem 2.

Theorem 3.

(7.129)

With the hypotheses of Sect. 3.1.1, let

{i) f given in L2(0, T; V') ,

ii) Uo given in H .

Then, there exists a unique u satisfying

(7.130)

i) UE W(V,H),

ii) :t(U(.), v) + a(. ;u(.), v) + (Mu(.), v) = (f(.), v)

in the sense of .@'(JO, T[) for all u E V,

iii) u(O) = Uo .


Proof

A. uniqueness. From (7.130) we deduce (equality of energy) for W = U 1 - U2

where U; (i = 1,2) is the solution of (7.130):

(7.131) ~I W(tW + r a(cr; W(cr), W(cr))dcr + f~ (M(W(cr», W(cr))dcr = 0;

now r (MW(cr), W(cr»dcr = J: f: k(cr;s; W(cr), W(cr»dsdcr,

and we deduce from (7.125)

Ir (MW(cr), W(cr))dcrl ~ ko f: (f: II W(s) II ds }W(cr)ldcr

and

(7.132) I r (M W(cr), W(cr» dcr I ~ eko T r II W(cr) 112 dcr + C(e) r I W(crW dcr ;

choosing e such that eko T = rx./2, we deduce from (7.131) and from (7.124)

(7.133) ~IW(tW + ~f~ II W(cr)1I 2 dcr ~ C 1 f~ IW(crWdcr (C 1 = constant)

from which we have uniqueness by Gronwall's lemma.

B. Existence. We use a Galerkin method and it is sufficient to establish the a priori inequality for the approximation of order m.

(7.134) Um remains in a bounded set of L 00 (0, T; H) and of L 2 (0, T; V) ,

and the passage to the limit can be carried out thanks to (7.128). Now (7.134) follows from the analogous inequality to (7.132) on um • 0

3.1.3. An Example ofthis Situation. We take (with 0 an open set of IR"):

{V = HA(O), H = £1(0),

(7.135) f a(t;u, v) = a(u, v) = n gradugradv dx ,

then

(7.136) " f au k(s, t;u, v) = L k;(s, t;x)-;-vdx ;=1 nUX;

where

k;(s,t;x)eLOO(]O,T[x]O,T[xO), i = 1,2, ... ,n;


we therefore obtain

(7.137)

satisfying

(7.138) { ~~ - Au + I K(t, a)u(a)da = f

U(X,O) = Uo(X), Uo given in L 2 (Q), in Q, t E ] 0, T[ ,

where

(7.139) " a K(t, a) = Jl kiaXi .

Such operators K (t, s) occur in mechanics, particularly with

(7.140) ki(t,s;X) = ki(t - s;x), i = 1,2, ... ,n

and M is therefore a convolution.

3.1.4. Remarks i) Under hypothesis (7.140), it is sufficient to consider p -> K(p), p E ]0, T[ with

(7.141) K E U(O, T; ~(V, H)).

Equation (7.138) is then suitable to (non-constructive) solution by Laplace transformation(162) since it can be written, by suitable prolongments ii andJof u and/, in the form

(7.142) dii -dt - A ii + K * ii = f + Uo <8l f> .

ii) In reality, for example in diffusion problems in material with long memory in linear viscoelasticity (ref. Duvaut-Lions [1]) we need a theorem analogous to Theorem 3 with M E ~(L2(0, T; V), L2(0, T; V')). This case, which needs some supplementary regularity hypotheses on A and K, is considered in the examples below (Sects. 3.2 and 3.3).

3.2. Mathematical Example 2: lst Order Problem (continued)

3.2.1. The Problem. Note first of all that if we assume we have a(t; u, v) as in Sect. 3.1 and k(s, t; u, v) continuous over V x V with properties analogous to (7.125), we shall have (by obvious modification of (7.128)):

(7.143)

and the theorem will again be true provided that we have

(7.144) ko small compared with ~ .

(162) See Chap. XVI.


We shall obtain a result without restnctlOn (7.144), but with supplementary regularity hypotheses on a(t;u, v) and k(s, t;u, v). We assume, with the spatial data of Sect. 3.1, that we are given a sesquilinear form a(t;u, v) which is continuous over V x V with

(7.145)

(7.146)

(7.147)

a(t; u, v) = a(t; v, u)

{ t -> a(t; u, v) admits (for all u, v E V) a derivative in the sense of

bounded measurable distributions,

c = c(T) denotes various constants> ° : la(t; u,v)1 ~ cilullllvil

Ia'(t; u, v)1 ~ cIIullllvll

for all t E ]0, T[ and for all u, v E V,

finally, (for simplicity)

{ a(t; u,u) > cxllul1 2 (cx = cx(T) > 0) (7.148)

for all t E ] 0, T[ and all u E V .

Besides, we consider k(s, t; u, v), a family of sesquilinear forms which are continuous over V x V, and defined over [0, T] x [0, T] with

(7.149) Ik(s, t;u, v)1 ~ ko Ilull.llvil , {

there exists ko = k(T) constant> ° such that

(7.150)

for all (s, t) E [0, T] x [0, T] and for all u, v E V.

(s, t) -> k(s, t; u, v) is differentiable for all u, v E V

and there exists kl = kdT) > ° with

1~~(S,t;u'V)1 ~ k11Iull.llvll,

1~~(S,t;u'V)1 ~ kJilullllvll,

for all (s, t) E [0, T] x [0, T] and for all u, v E V.

We again define M by (7.126)(163) and we note that

{ME .!l'(L2 (0, T; V), U(O, T; V'))

(7.151) II M II~W(v). L2(V')) ~ ko T .

We shall establish

(l63) Note that in this Sect. 3.2, we define by k(s, t; u, v) = < K (s, t) u, v>, with u and v E V «, > denoting the duality V', V), a family of operators K(s, t) which are continuous from V to V'.

656

Theorem 4.

(7.152)


We are given, with hypotheses (7.145) to (7.151):

{i) fE L2(0, T;H),

ii) Uo E V.

Then there exists a (unique) u which satisfies:

(7.153)

(7.154)

U E JY(V, H) ,

{:t(U(.)'V) + a(.;u(.),v) + (Mu(.),v) = (f,v),

for every v E V in the sense of ~' (]O, T[) .

3.2.2. Proof of Theorem 4

A. Uniqueness. We shall use the procedure already employed for second order hyperbolic equations (see §6). We set, for fixed s in ]0, T[

(7.155) { I/I(t) = - f u(o")da

I/I(t) = ° if s ? t

if s :::; t ,

so that 1/1 E L2(0, T; V), 1/1 ' E L2(0, T; V), I/I(s) = 0. Therefore from (7.154), we deduce (for u the solution of (7.154) with Uo = f = 0):

(7.156) f: (u'(t), I/I(t))dt + f: a(t;u(t), I/I(t))dt + f: (Mu(t), I/I(t))dt = 0,

or even

(7.156)'

2Re{f: (u'(t), I/I(t))dt + t a(t;I/I'(t), I/I(t))dt + t ((Mu)(t), I/I(t))dt} = 0,

Note that

(7.157) 2Re t (u'(t), I/I(t))dt = - 2 t lu(tWdt

and that

(7.158) 2 Re t a(t; 1/1 ', 1/1) dt = - a(O; 1/1(0), 1/1(0)) - t a'(t; 1/1 (t), I/I(t)) dt ;

from (7.156)" (7.157) and (7.158) we deduce:

(7.159) a(O;I/I(O),I/I(O)) + 2 f~ lu(t)12dt = - f~ a'(t; 1/1, I/I)dt - 2Re f~ (Mu, I/I)dt .


Now set

(7.160) w(t) = f~ u(a) da ;

We have t/J(t) = w(t) - w(s) so that from (7.148) and (7.159) we deduce

(7.161)

IX II w(s) 112 + 2 f~ lu(tW dt

~ C I IIw(t) - w(s)1I 2 dt + 21I Re(Mu,t/J)dtl

~ C1 I II w(t) 112 dt + 2cs II w(O) 112 + 21 I Re(Mu, t/J)dt .

Besides

2Re I (Mu, t/J)dt = 2Re I [I k(t, a;t/J'(a), t/J(t»da Jdt

and

a ak aa [k(t, a; t/J(a), t/J(t»] = aa (t, a; t/J(a), t/J(t» + k(t, a; t/J'(a), t/J(t» ,

so that

I k(t, a; t/J' (a), t/J(t» da = k(t, t; t/J (t), t/J(t»

(I ak - k(t, 0; t/J (0), t/J(t» - Jo aa (t, a; t/J(a), t/J(t» dO" .

Therefore

1 Re f~ (Mu, t/J)dtl ~ ko f~ II t/J(t)11 2 dt + ko II w(s) II I II t/J(t) II dt

+ k1 (IIIt/J(t)lldty ~ (~ko + k1 s)IIIt/J(t) 112 dt + k~SIIW(S)112.

The Ci being diverse constants, we deduce

(7.162)

and taking account of (7.161), we obtain

(7.163)

657


From Gronwall's lemma, we obtain uniqueness over the interval [ 0, :4]. then we

start again until we reach T .

B. Existence. We use the Galerkin method with fixed basis for simplicity. The approximate system is:

(7.164) { (u~(t), Wj) + a(t; um(t), Wj) + (Mum' Wj) = (f, Wj)

m n

Um = L gim(t) Wi' UOm = L lXim Wi --+ Uo in Vas m --+ 00 . i= 1 i= 1

To establish the a priori estimates, we take the scalar product with u~ in (7.164). We obtain (by taking the real parts and integrating from 0 to t);

(7.165)

f I l(u~(uW du + ~a(t; um(t), um(t)) = ~a(o; UOm ' UOm )

l + I a'(u;um, um)du + I (Mum, u~)du + I (f, u~)du ; besides,

f~ (Mum' u~)du = f~ [J: k(u, r;um(r), u~(u))dr JdU = I (cPm(u), u~(u))du,

(by setting cPm(u) ~ J: K(u, r)um(r)dr = Mum(u)); an integration by parts gives:

f~ (cPm, u~)du = (cPm(t), um(t)) - I (cP~(u), um(u))du ,

(7.166) with cPm(O) = 0 ,

ra ok (cP~(u), um(u)) = k(u, u;um(u), um(u)) + Jo ou (u, r;um(r), um(u))dr ;

from which as

l(cPm(t),um(t))l:::;;; koIIUm(t)lIf~IIUm(U)lldU:::;;; ellum(t)1I 2 + c(e)IIIUmIl 2 dU,

we deduce from (7.166):

(7.167)

f I f: (<1> •• o~) du I " '110. ('>II' + (c(,) + k.) J: 110.(0) II' do

l + kl(I Ilum(u)lIdu y :::;;; ellum(t)1I 2 + cde) I lIum(u)1I 2 du.


Finally, from (7.165), we obtain:

~IIU~(ITWdlT + (~- c}IUm (t)11 2 ~ c(uo,f) + c'IIUm 11 2 dlT

(7.168)

Therefore

(7.169)

<I. C chosen < "2'

{ um remains in a bounded set of L OCJ (0, T; V)

u~ remains in a bounded set of L OCJ (0, T; H) .

The passage to the limit does not present any particular difficulties. For the regularity U E reO ([0, T]; V), we can proceed as follows: Firstly if fEU (0, T; H) has a derivative!, E L 2 (0, T; H), then by differentiation of (7.164) and taking account of (7.169), we easily obtain that u~ E a bounded set of L2(V), therefore forfE W(H,H), u' E L2(V). Therefore we have the analogue of (7.165) with Um replaced by u. Therefore let {In} be a sequence of functions In E W(H, H) converging to f in L 2 (0, T; H) and Un the solution of the problem associated with In satisfying Un (0) = Uo' Then for wn, p = Un + p - Un' we have (7.165) with Urn replaced by wn, p from which we have the analogue of (7.168):

(7.168)' 11 II , 2 (<I.) 2 "2 ° IWn,pl dO' + "2 - 2 II wn,p(t) II

~ C f: Ifn+ p - 1n1 2 dIT + c'I Ilwn,pl12dlT.

This shows that {un} is a Cauchy sequence in reo ([0, T]; V) from which we have the result. 0

3.2.3. A Particular Case concerning Linear Viscoelasticity. We consider a material obeying the viscoelastic law of behaviour with long memory (see Chap. lA, §3).

(7.170) lTiit) = aijkhCkh(U(t» + I bijkh(t - s)ckh(u(s»ds

where U is a displacement field and where the coefficients aijkh are the coefficients of instantaneous elasticity; they depend on x and t. The coefficients bijkh( x, t) take account of the effects of the memory of the material. We assume

(7.172) aijkh E L OO(Q r ) Vi,j, k, h, Q r = Q x ]0, TL Q c [R3 ,

660

(7.173)

(7.174)


1 t -+ a;jkh(·, t) has a weak derivative in L 00 (Q) equipped with the

aa··kh weak dual topology of L1(Q) with -t- E Loo(QT)

! bijkh E L oo(Q}y ,

t -+ b;jkh(., t) is differentiable with values in L 00 (Q)

ab··kh the weak dual of L 1 (Q) with _'1_ E L 00 (QT ) . at We then find ourselves in the situation of application of Theorem 4, by taking for example:

(7.175)

Then for

(7.176)

i) V = [Hb(Q)]3, H = [LZ(Q)]3 ,

ii) a(u, v) = In a;jkhckh(u)c;iv)dx ,

iii) k(t, s; u, v) = b(t - s; u, v) = In b;jkh(X, t - s)ckh(u)cij(v)dx .

{ i) Uo = (U01 , UOZ , U03) given in [H6(Q)P ,

ii) f = U;, fz, fJ) given in [LZ(Q)P

there exists a unique U = (u 1 , Uz, u3 ) satisfying:

(7.177)

i) U E ~O([O, T]; V), U' E LZ(H) ,

.. au; a ft a b ( ))] d _ I" (164) n) -a - -a a;jkhCkh(U) - -a [ijkh(X, t - S)Ckh(U S S - Ji t Xj 0 Xj

i = 1, 2, 3

iii) uil r = 0 (r = aQ) i = 1,2,3,

iv) ui(O) = UOi i = 1, 2, 3 .

In the case of an isotropic material (see Chap. lA, §3), the preceding model reduces to:

(7.178) ! ~~ -rdu + Iy(t - O")L1u(O")dO" =finQ

ulr = 0

u(O) = Uo given in H 6(Q) ,

Q being a regular open set of W(r = aQ),j E L 2(QT) given and y a real function of

([64) If u is a velocity field, relation (5.177)ii) can be interpreted as a Navier-Stokes like viscous diffusion equation (see Chap. lA, § 1).


class <6'1 over [0, T] to fix ideas. Then:

(7.179) there exists a (unique) uL 00 (V), U' E L 2(QT ) satisfying (7.178) .

Note that (7.178) is susceptible to treatment by using the Laplace transformation (see Chap. XVI). In the absence of instantaneous elasticity, we have:

~~ - f>(t - cr)L1u(cr)dcr = f·

In the case where y is such that Re f > ° (y the Fourier transform of y), we can, by Laplace transformation, reduce the problem to an elliptic problem. By the variational method, we can resolve such an equation, for example if ( - 1)ky(k) ~ 0, k = 0, 1,2 by using a special function like a Levin function (see Artola [1], Raynal [1]).

3.3. Mathematical Example 3. 2nd Order Problem

Statement of results. We take the hypotheses of Sect. 3.2. We obtain the following theorem with applications to dynamic problems without friction for viscoelastic materials with long memory (see Chap. lA, §3).

Theorem 5. We assume that the hypotheses (7.145) to (7.150) hold. We are given

(7.180) UO E V, u1 E H, fE L2(0, T; H).

Then there exists a (unique) u satisfying:

(7.181)

(7.182)

(7.183)

U E <6'°([0, T]; V), U' E <6'°([0, T]; H) ,

{:t22 (u(.),v) + a(.;u(.),v + «Mu)(.),v) = (f(.),v) ,

in .@'(]O, TD for all v E V,

u(O) = uO, u'(O) = u1 •

3.3.2. (Quick) Proof of Theorem 5

A) Uniqueness We work as for Theorem 4 (uniqueness) with the following variation:

instead of "IX II w(s) 112 + 2 f~ lu(tW dt" in the left hand side of (7.161) we shall

have IX II w(s) 112 + lu(sW; the rest of the proof then leads to an estimate of the type:

(7.184) (1 - cs)llw(s)11 2 + lu(sW:::;; C[f:IIW(S)112dS + f:IU(sWdS]

(with C and c constant), from which we have uniqueness.


B) Existence. As in Theorem 4, we take the scalar product with u~ and (7.168) is replaced by

(7.185)

1 ~ lu~(t)12 + (~ - e) II Urn(t) 112 ~ C(Uo, u1,f) + C' I [II Urn 112 + lu~12] dO"

C(UO,u1,f) = c[lluOl1 2 + lu l l2 + f:lfI2dO"].

From which we have the a priori estimates (see Gronwall's lemma) which allows us to pass to the limit. For the regularity, we proceed as in §5 by using the double regularisation procedure (see Lemma 7).

3.3.3. Example of this Situation. Dynamic Problem for Material with Long Memory. We use the notation of the example of Sect. 3.2.3.; we obtain, for example with V = (H6(Q))3 and H = (L2(Q))3: for un, u1,f satisfying (7.180), there exists a unique u = (u 1 , U2' u3) satisfying:

(7.186) i = 1,2,3

iii) udr = 0 (r = aQ) i = 1,2,3 ,

Equation (7.186)ii) is the dynamic equation for a viscoelastic material with long memory.

4. Optimal Control and Problems where the Unknowns are Operators

4.1. Introduction

The theory of "uncoupling optimal systems" in optimal control leads to partial differential equations where the unknowns are operators. We meet some similar situations in physics. The Liouville-von Neumann equation is an example. It is then natural to ask, in each specific example, if there is an equivalence. This is the case for Liouville-von Neumann equations(165), as we shall now show.

(165) See Chap. lA, §6 and XVIIB, §5.


4.2. Liouville-von Neumann Problem

4.2.1. The Liouville-von Neumann Equation and Pair of Schrodinger Equations. In the complex Hilbert space H, we are given a family of unbounded operators A (t) with:

(7.187) { D(A(t)) is dense in H, A(t) is closed,

A(t)* = A(t) (166) .

We can make various hypotheses on the structure of A(t); we shall assume here (which is sufficient for the applications) that A(t) is defined by the usual triplet:

(7.188) {

V C H C V',

(A(t)u, v) = a(t; u, v), a(t; u, v) continuous over V x V,

a(t;v,v) + ..1,lvl2 ~ allvl12.

We shall also assume that:

(7.189) t --+ a ( t; u, v) is ~ I, V U, V E V .

In this situation we know how to solve the Schrodinger equation:

(7.190) i ~~ + A(t)u = 0, with: u(O) = UO ,

the solution being taken in a more or less "strong" sense as UO is more or less "regular". If, for example, UO E V, then the solution u is in L 00(0, T; V), is scalarly continuous with values in V, and du/dt E L 00(0, T; V'). If UO E H, we shall have a "weak" solution.

Remark 7. The operator u --+ i du/dt + Au is formally self-adjoint. We therefore can reverse the sense of time in (7.190): the problem:

(7.191) dv

i dt + A(t)v = 0, v(s) = u l , t < s,

has exactly the same properties as (7.190).

Remark 8. If we have:

(7.192) A(t) = A independent of t ,

then

(7.193)

{G(t) = eitA } is a group ofisometries in H (Stone's theorem).

Pair of Schrodinger equations. Let h be given in H.

(166) We can generalise this a little by assuming that only the principal part of A(t) is self-adjoint.

o

o


We define y(t), t :( s, by:

(7.194) { dy

i dt + A(t)y = 0,

y(s) = h.

Let QO be given with:

(7.195) QO E 5l'(H; H) .

Formally, y(O) is defined in H by (7.194), therefore QOy(O) has a sense and we consider(167):

(7.196) { i ~P + A(t)p = 0,

P(;) = QOy(O).

Then p(s) is defined uniquely in H and the mapping:

(7.197) h --> p(s)

is linear and continuous from H --> H. Therefore:

(7.198) p(s) = Q(s)h .

Consequently, if y is the solution of:

(7.199) dy

i dt + A(t)y = 0 on [0, + 00 [ ,

and if p( t) is defined by (7.196), then there exists a family of operators Q(t) E 5l'(H; H) such that we have the identity:

(7.200) p(t) = Q(t)y(t).

The value Q(s)h is given by the solution of (7.194), (7.196) and by (7.198).

Properties of Q( t). We shall verify,formally, that Q is the solution of:

(7.201) .dQ / Cit + [A(t), Q] = 0,

(7.202) Q(t) = QO •

In effect, if we admit (this is the formal aspect which must be justified) that we can differentiate (7.200), it becomes:

(7.203) {.dP .(dQ) (.dY) /-=/ - y+Q /-

dt dt dt

= i( ~;)y + Q( - A(t)y).

(167) The notation {y, p} is that of optimal control theory, as we indicate briefly below.


Therefore:

(7.204) dp ( dQ ) i dr + A (t) p = i dt + [A (t), Q] y = 0 ,

and as y is arbitrary (we can start from yes) = h arbitrary), (7.201) follows. Since p(O) = Q(O)y(O) = QO yeO), we deduce (7.202), noting that yeO) goes over the whole space H (reversibility). We shall see in Sect. 4.2.3 how we can easily study Q( t) from formulae (7.194), (7.196), (7.198). The existence of a solution for the Schrodinger equations (7.194), (7.196) established in §7.1 (under suitable hypotheses) implies the existence of a solution to the Liouville-von Neumann equation in the case of a time-dependent Hamiltonian.

4.2.2. Solution Properties. Taking the system of Sect. 4.2.1. We therefore have:

dy i dr + A(t)y = 0,

(7.205) dp i dt + A (t) p = 0 ,

yes) = h, p(O) = QO yeO) ,

and Q(s) is defined by:

(7.206) pes) = Q(s)h .

Let h be another element of H and let {y, p} be the solution of the corresponding system (7.205). We have:

(7.207) 0= J: (i~~ + A(t)P,Y)dS = i(p(s),.Y(s» - i(p(O),'y(O»

+ J: ( p, i ~~ + A (t).Y ) dt ,

therefore (p(s), y(s» = (p(O), .Y(O», i.e.

(7.208) (Q(s)h, h) = (QOy(O), y(O».

Therefore:

(7.209)

(7.210)

if(QO)* = QO then Q(s)* = Q(s) '<Is;

ifQo ~ 0 then Q(s) ~ 0 '<Is.

We have in addition:

(7.211)

In effect:

(7.212) .(dY ) I dr' y + aCt; yet), yet»~ = 0 ,


therefore by taking the imaginary part:

(7.213) d dt ly(tW = 0

therefore

(7.214) ly(t)1 = Ihl .

Then (7.208) gives:

(7.215) I(Q(s)h, h)1 :::; IQol:£(H)ly(O)lly(O)1 = IQol:£(H)lhllhl


Reversibility of the Liouville-von Neumann equation. We can "reverse the solution" by inverting the sense of time. We consider (J < s over IR and the pair of equations:

(7.216) { i ~~ + Ay = 0, t > (J

y((J)=h,

(7.217) { i ~p + Ap = 0

p(:) = Rly(S).

Then p((J) is defined in a unique manner and we have:

(7.218)

and

(7.219)

p((J) = R((J)y((J)

{ i ~~ + [A (t), R] = 0, t < s ,

R(s)=RI.

This is therefore the retrograde Liouville-von Neumann equation - and it has the same properties as the equation in the sense of t increasing:

(7.220) the Liouville-von Neumann equation is reversible. o

Equation for the inverse Q(s) -I of Q(s). We consider the system analogous to (7.205) but with "exchange" of boundary conditions:

(7.221) dp _ i dt + A(t)p = 0,

p(s) = k, ji(0) = Mp(O) .


If we assume that Q ° is invertible and that we take:

(7.222)

then:

(7.223) y=y, p=p.

On the other hand, we can consider, starting from (7.221), the mapping:

(7.224) k ~ y(s)

from H ~ H, therefore

(7.225) .Y(s) = T(s)k

and we verify that T( s) also satisfies:

(7.226) dT

i dt + [A (t), T] = 0 ,

with:

(7.227) T(O) = M.

On the other hand, from (7.225), (7.223), (7.222), we have:

(7.228) y(s) = T(s)k = y(s) = h = T(s)Q(s)h

and consequently:

1 if QO is invertible, the operator Q(s) is invertible for all sand

Q(S)-I satisfies the same Liouville-von Neumann equation: (7.229)

d i dt (Q - I) + [A (t), Q - I] = 0 .

667

We can verify this last point directly: we multiply (7.201) to the left and right by Q( t) -I , it becomes:

(7.230) iQ(t)-1 d~~t) Q(t)-I + Q(t)-IA(t) - A(t)Q-I(t) = 0

i.e., since dQ(t)-1

dt

the last equation in (7.229). o Representation of the solution when QO is of range 1. Assume that QO is given in H by:

(7.231) QOk=(k,cp)t/I, cp,t/lEH.

Let cP( t) - resp. '1'( t) - be the solution of:

(7.232) dcP

id( + A(t)cP = 0, cP(O) = cp,


resp.

(7.233) d'l'

i(ft + A(t)'I' = 0, '1'(0) = t/J.

We then have the property:

(7.234) Q(s)k = (k, lP(s))'I'(s) , Vs > O.

Proof From the construction (7.205), (7.206), we have:

Q(s)h = p(s),

and p must be the solution of:

dp i dt + A(t)p = 0, p(O) = QOy(O) = (y(O), cp)t/J ;

therefore, comparing with (7.233):

p(t) = (y(O), cp)'I'(t)

so that (7.234) is equivalent to:

(7.235) (h, lP(s)) = (y(O), cp) .

To verify (7.235), multiply the first equation in (7.205) by lP(t), and integrate by parts; it becomes:

i(y(s),IP(s)) - i(y(O),IP(O)) + t(Y'i~~ + AlP )dt = 0

therefore (y(s), lP(s)) = (y(O), IP(O)) i.e. (7.235).

We shall deduce from (7.234):

(7.236) {if QO is a trace operator, then Q(s) is also a

trace operator V s > 0 and

TrQ(s) = TrQo .

Proof If QO is a trace operator, it has a representation of the form:

(7.237)

where:

(7.238)

QOh = L Aj(h, CPj)t/Jj, j

{ L'\Aj\ = TrQo < 00

tPj' 'I'j are orthonormal sequences.

o

We assume firstly that the sum if finite (we then pass to the limit in the formulae obtained). From (7.234), and with the obvious notation:

(7.239) Q(s)h = L Aih, IPj (s))'I'j(s) . j


It is sufficient to verify that, for all s > 0, the sequences <Pj(s) and <Pj(s) are orthonormal - and it is sufficient to verify this for <Pj(s). From (7.214), we have:

(7.240)

If Y is the solution of the Schrodinger equation with y(O) = h, we have:

d A

dt(Y'Y) = 0

so that (y(s), Y(s)) = (h, h) and therefore:

(7.240)'

We shall now show:

( {if QO is Hilbert-Schmidt, then Q(s) is also Hilbert-Schmidt and (168) :

7.241) IIIQ(s)111 = IllQolI1 .

Let {hj} be an orthonormal basis of H. We have to calculate IIQ(s)hjI2 (if this j

series converges). Denote by {Yj, Pj} the solution of (7.205) for h = hj' We have:

LlQ(s)hjI2 = Llpj(s)1 2 •

d But dt Ip(tW = 0 therefore IPj(sW = Ipj(OW = IQoYj(0)1 2 and therefore

L IQ(s) hjI2 = L I QOYj(OW

(this is correct for all finite sums and we then pass to the limit). We note that from (7.240), (7.240)', an 'orthonormal basis moves along an orthonormal basis': if the {hj} form an orthonormal basis of H, then the {Yj(s)} form an orthonormal basis 'is, and therefore:

from which we have (7.241). o 4.2.3. The Time Dependent Case. If A (t) = A, we can express everything with the help of semigroups. We have

(7.242)

This allows us to deduce regularity properties in t after having defined the infinitesimal generator of the group t --> Q(t) (which is a group in !e 1 (H), in !e 2(H), in !e(H) and in fact in !eP(H), 'ip). The domain of the infinitesimal generator can be defined as in Chap. XVIIB, §5.

( )1/2

(168) If QO is Hilbert-Schmidt, its Hilbert-Schmidt norm is given by IIIQoll1 = IIQ°ipj I2 where J

ipj is an orthonormal basis.


4.2.4. Nonlinear Case. We consider a priori the system:

dy i dt + Ay + Mp = 0 where

(7.243) .dp I dt + Ap = 0,

y(s) = h, p(O) = QOy(O) ,

and assume that the problem is well posed. Then:

(7.244) p(s) = Q(s)y(s) .

We differentiate (7.244), which gives

but

from which:

(7.245)

i dp = i dQ y + Q (i dY ) dt dt dt

Q( i~~) = - Ay - Mp = - Ay - MQy,

{ i ~; + [A(t), Q] = Q(t)MQ(t)

Q(O) = QO

We therefore obtain a Liouville-von Neumann equation with quadratic nonlinearity.

5. The Problem of Coupled Parabolic-Hyperbolic Transmission

5.1. Statement of the Problem

This occurs in the study of linearised, incompressible, non stationary flow in a domain Q 1 bounded by a perfectly elastic thick wall in which we assume small displacements.

Fig. I


(We can imagine, for example, in a very schematic manner, the flow of blood along the veins(169)). We denote by (/> the velocity vector of the flow in Q l' by n the pressure and by w the displacement vector in Q2'

We then have in Q1 (with fJ. constant):

(7.246)

(7.247)

and in Q2

(7.248)

a(/> - - fJ.A(/> = fl - Vn, fl given in Q1 x ]0, T[ at

div (/> = °

a2 w at2 - A w = g, g given in Q2 x ]0, T[ .

In (7.246) and (7.248), fl and g are vectors fl = UL ... ,f~ }, g = {gl' ... ,gn} (170), with n = 2 or n = 3 in applications. The boundary condi-tions are

(7.249) over r2 , for example w = ° (the solution which follows is independent of the boundary conditions imposed on r 2) and on r 1 the following transmission conditions:

aw (/> =-at (7.250)

(since (/> is a speed and w a displacement), and

iJ(/> aw fJ.- - nv =-

av iJv (7.251 )

where v = normal to r l' unitary, directed outwards from Q 1

(to fix ideas) and where a(/>/av = {iJ(/>l/aV, ... ,iJ(/>n/iJv}, and likewise for aw/iJv. We add the initial conditions.

(7.252)

(7.253)

(/>(x,O) = (/>O(x) , X E Q1 ,

iJw w(x,O)=WO(x), Tt(x,0)=w 1(x), xEQ2'

5.2. New Formulation of Problem

We introduce the following new unknowns:

(7.254)

(7.255)

(169) See Fung [1] for a bibliography on this subject. (170) We also have Q 1 and Q 2 c [Rn, and <P and w with values in


Then

(7.256) w(x, t) = I u2(x, a)da + WO(x),

and the system (7.246), (7.247), (7.248) becomes:

div u l = 0 10

(7.257)

where

(7.258)

The boundary conditions are:

(7.259) over r I

(7.260) u2 = 0 over r 2 ,

with the initial conditions

(7.261)

If we solve the system (7.257), (7.259), (7.260), (7.261), then the solution of the problem is given by (/) = u l and w is given by (7.256).

5.3. Variational Formulation

Notation. We set:

(7.262)

(7.263)

Q = Q I n Q2

V = {v; V E (H6(Q»n, divv = 0 in QI} ;

V is a closed subspace of (H 6(Q»n; this is therefore a Hilbert space for the norm induced by that of (H6(Q»n. For v E V, we set:

v = {Vi, v2 }, Vi = the restriction of v to Qi ,

We also set

(7.264) {VI = {Vi; Vi E (HI(Q»n, divv l = 0 in Qd V 1 = {v 2 ; v1 E (HI(Q»n, v1 = 0 on r1 } .

Then


For u, v E V, we set

(7.265)

which definesfE L2(0, T; V') if WO E V2 and if

{Ii, g} E L2(0, T;(U(Qd)" X (L2(Q2))") .

We then consider the following variational problem: find u satisfying

(7.266)

(7.267)

(7.268)

( au) 1 1 ((' 2 2 ) at'V +a1(u,v)+a 2 JoudO",v = (f,v),

{ u ~ L"'(O, T;(L2(Q))") , u1 E L2(0, T; Vd,

I u2 (0")dO" E LOO(O, T; V2 ), u1 = u2 on r 1 .

'<Iv E v,

Note that the condition u1 = u2 over r l' must in fact be expressed by:

L u1 dO" = L u2 dO", for almost all t .

673

We shall show later - by a simple variant of general methods - that this problem has a unique solution. Before that, we verify that if u = {u 1 , u2 } is a regular solution of (7.266), (7.268), then we have (7.257), (7.259), (7.260), (7.261). Naturally (7.260), (7.261) follows from the definitions; it only remains therefore to verify (7.257), (7.259). Taking v = {v 1 , O}, where v1 E (.~(Qd)" with div v1 = 0, then, as for the linear Navier-Stokes equations, we have the first equations (7.257), (7.259). Multiplying scalarly the first (resp. second) of equations (7.257) by v1 (resp. v2 )

where v = {v 1 , v2 } E V (therefore v1 = v2 over r 1) and using Green's formula we have:

(7.269)


(7.270)

Adding (7.269) and (7.270) and taking account of (7.266), (7.265), we obtain - since v I = v2 over r I -

(7.271 )

where

(7.272) X = jJ. - - nv - - u2 dO" - ~ ou l 0 if owo

ov ov ° ov Relation (7.271) is satisfied for every function VI which is the trace over r I of an element of V. Since div VI = 0 in QI' we have

(7.273)

We know (see Chap. IV, §4) that if h is given in (H 1/2(rl ))" with

(7.274) r hv drl = 0 Jrl then there exists v t in (H I (Q d)" with v t = hover r l ; then we construct v2 E (H I(Q2))" with v2 = VI over r l and V 2 = 0 on r2 ; we can therefore take VI = h in (7.271), which therefore holds for all VI = h satisfying (7.274); thus there exists a constant .Ie such that

X = .lev

by changing n to n + .Ie (which is permissible), we obtain the second condition (7.259). D

ex k tlx

- !

Fig.!


5.4. Solution of the Variational Problem

We take v = u in (7.266). It becomes

(7.275) 1d 2 1 2 11 2 2dtlu(t)1 + a1 (u ) + a2(w,u ) = (f ,u )1 + (g,u h

where Ivl2 = Iv1 1i + IV21~ = (V 1,V1 )1 + (v2,v2)2,a 1 (cI>,cI» = a1 (cI».

675

Using (7.255) and the fact that the form a2 (u2, v2) is symmetric, we can write (7.275) in the form

(7.276)

from which

(7.277)

(7.278) lun belongs to a bounded set of Loo(O, T;(L2(Q))n) ,

Wn belongs to a bounded set of L 00(0, T; V2 ) ,

u ~ belongs to a bounded set of L 2 (0, T; Vd ,

from which we deduce the existence of a solution of problem (7.266), (7.267), (7.268).

5.5. Solution of the Variational Problem: Uniqueness

Let u and u be two solutions - if we set

u=u-u, we have:

(au) 1 1 (it 2 2) _ (7.279) at'V + a1(u ,v) + a2 Jo u du,V - ° VUE V.

If we formally set v = u in (7.279), then

(7.280) ~:tIU(t)12 + a1 (u 1(t)) + ~:ta2(f~ u2dU) = ° from which we immediately deduce that u = 0; but this assumes that u2 (u) E V2 -

and we only know that I u2(u)du E v2·

We must then replace v by a regularised approximation of u; for the technical details, we refer to 1. L. Lions [2], pp. 126--128. We then deduce the uniqueness.


6. The Method of "Extension with Respect to a Parameter"

We return to the 1st order evolution equation:

du (7.281) dt + A(t)u = f,

with

(7.282)

where

(7.283)

u(O) = 0,

this problem has a unique solution(l7l) with

(7.284) U E L2(0, T; V) ,

(7.285) du 2(0' ') dt E L ,T, V .

Remark 9. (7.285) follows from (7.284) and (7.281), (7.283). We can therefore content ourselves by saying that there exists a unique solution with (7.284). D

We shall give a new proof of this result, by an important method, linked to (i) the theory of perturbations, (ii) some topological methods. This method is moreover very useful in non linear problems. We start from the operator Aofor which problem (7.281), (7.282), (7.283) is assumed solved. Therefore:

(7.286)

the equation

(7.287)

with

(7.288)

has a unique solution in L 2 (0, T; V).

Example 1. Suppose that

(7.289)

A(t) = - La~i ( aij(x, t) a~j) + ao(x, t)

aij E LOO(QT) ' QT = Q x ]0, T[, ao E LOO(QT) ,

~>ij(X, tKej ~ IX L eJ, IX > 0 a.e., Vej E IR ,

ao(x, t) ~ IX a.e .

(171) We always assume that (in the case of "reals" to fix ideas)

(A(t)v,v);;:' (XllvI1 2 , (X> 0,

(if(A(t)v, v) ;;:. (X II vl12 - ).lvI 2, we are reduced to (*) by changing u to eA'u), and that hypothesis (3.3) is verified.


We can take

(7.290) Ao = - ,1 + I (or - ,1);

the operator

(7.291) o

Po = ot + Ao

is therefore the heat operator and for the resolution of(7.287) we dispose of "all" the methods of spectral decomposition in space, Laplace transformation in t, semigroups, potential and probability theory and energy methods. 0

We now introduce the family of operators:

(7.292) o

Pa = ot + (1 - 8)Ao + 8A(t) , ° ~ 8 ~ 1 .

For 8 = 0, we return to (7.291) and

(7.293) o

PI = ot + A(t)

which is the operator to be inverted to solve (7.281) and (7.282). The method of extension with respect to a parameter 8 consists of showing that starting from () = 0, for which the problem is solved by hypothesis, we can solve step by step until () = 1. 0

We set:

(7.294) { W = {v; V E L2(0, T; V), v' = ~~ E U(O, T; V'), v(O) = ° } L2( V) = U(O, T; V) .

We consider Pa as an operator from W -+ U( V'):

(7.295)

The working hypothesis is

(7.296)

We shall show:

Lemma 4. If Pao is invertible, then Pa is too for 8 E [()o - y, ()o + y] n [0, 1], where y is > ° independent of ()o'

Starting from POl, we shall therefore reach P;; I step by step until 8 = 1, from which we have the desired result.

Proof 1) The principle is obvious: we note that

Pa = Pao + (() - ()o)(A(t) - Ao),


so the equation

is equivalent to

(7.297) u + (8 - ( 0 )P;'/(A(t) - Ao)u = P;'/ f,

the equation to be solved in W, or, from Remark 9, in L2(0, T; V) = L2(V). 2) We have the following a priori estimate: if gEL 2( V'), and if w is the solution of

(7.298)

then

(7.299)

This is immediate: we deduce from (7.298) that

(7.300) ( ~;, w ) + (1 - 80HAo w, w) + 80 (A(t)w, w) = (g(t), w) ;

but

(1 - 80HAow,w) + 80 (A(t)w,w) ~ [0(0(1 - ( 0) + 80ocJllwl12 ~ oc 1 11w1l 2 ,

oc 1 > 0 independent of 80 E [0, 1 J; therefore (7.300) gives

(7.301) 1 I' rt

2lw(tW + OC 1 Jo IIw(u)11 2du:::; Jo Ilg(u)II*llw(u)lldu

(where II II * = norm in V'), from which we have (7.299). 3) But

II A(t) - Ao 1I.~(L2(V);L2(V')) :::; C2

(the C i are constants independent of 8), so that, if we set

(7.302) K6 = P;l(A(t) - Ao), o 0

we have:

(7.303)

Therefore

(7.304) {I + (8 - ( 0 )K6o is invertible in L2( V) for

18 - 80 lC3 < 1 .

Remark 10. We can represent the solution U6 of

(7.305)

by

(7.306) 00

U6 = L (- W(8 - ( 0 )nK'8;;lPf6o' n;O

D

D

Review of Chapter XVIII 679

Remark 11. As indicated at the beginning of this section, the method which we have indicated is classical. It was introduced by J. Leray and Schauder to solve nonlinear elliptic problems; the stationary analogue of the above method is obvious: to solve

(7.307) Au =/, we consider:

(7.308) (1 - O)Aou + OAu =/ and we assume that the problem is solved for 0 = 0; we can use this idea when A is nonlinear, starting from Ao linear. In the present (simple) context, the method has been used by Ladyzhenskaya [2], Ladyzhenskaya and Visik [1], J. L. Lions [1], p. 203. 0

Remark 12. Curiously enough, it is not obvious how to adapt this method to all of the second order problems in this chapter. 0

Review of Chapter XVIII

We have seen in this chapter that variational methods are often among the most natural to treat evolution problems in physics and mechanics. These variational methods enable us to treat problems in a much larger range of situations (time dependent coefficients for example) than the preceding methods (diagonalisation (172), Laplace, semigroup). Further, they have numerous generalisations in different domains, such as variational inequalities and nonlinear problems. Finally, these are constructive solution methods: they are the most adaptable to numerical calculation. However, we obtain the solution as the limit of a sequence, and in general we do not have a simple explicit formulation of this solution(I73). On the other hand the method of semigroups supplies some supplementary regularity properties in t. Finally, certain problems (particularly first order hyperbolic problems such as transport or electromagnetic problems) cannot be a priori treated by variational methods.

(172) We have seen, moreover, that the method of diagonalisation can be considered, in certain cases, as a particular variational method. (173) Note, on the other hand, that the explicit forms of Chaps. XV, XVI, XVII may be very useful, but are not generally constructive.

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