Mathematical Analysis and Numerical Methods for Science and Technology || Evolution Problems: Cauchy Problems in ℝn

Chapter XIV. Evolution Problems: Cauchy Problems in \R II

Introduction

In numerous evolution problems in physics and mechanics the spatial variable runs over the whole space [R", the most usual cases being for n ~ 3. The best way of having an understanding of the set of properties of this system of equations is to study the resolution of the following problem: we look for the solution u of the equation (resp. equations) governing the problem at time t, when we know u at the initial time, as well as certain of its derivatives with respect to time. Such a problem is called a Cauchy problem.

1. In § 1, we shall study this Cauchy problem, for linear ordinary differential equations (that is to say where the only derivatives of the unknown functions are taken with respect to time). This problem is very simple; however the solution of general systems of partial differential equations will often be carried out by reducing them to this simple case; finally, some of the results found will be generalised to particular systems of partial differential equations.

2. In §2 we shall study the Cauchy problem for the heat equation which belongs in the framework of diffusion equations, whose model is:

(1) Lu = [:t + A (x, :x ' t) ] u = f, X E [R" ,

where A is a linear differential operator (of 2nd order) operating over the spatial variables x, and which is 'elliptic' (see the introduction of §2). When the operator L is invariant(1) under translation in x, we look for the Fourier transform of equation (1). This transforms Au into a multiple A. u. The equation obtained by transforming (1):

(2) L.u = 1, is an ordinary differential equation with a parameter, and we may treat this with the methods of § 1. In the case where:

(3) f = b(x)b(t), b = the Dirac function,

every solution E(x, t) of (1) is called an elementary solution: their study allows us to

(1) See Chap. VIII, §2.

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

2 Chapter XIV. Cauchy Problems in IW

fully understand the properties of equation (1): when there exists an elementary solution, we can write a solution of (1) (by assuming zero initial value) in the form

(4) u = E(x, t) * f(x, t) . (x, t)

Writing distributions like functions, formula (4) becomes(Z)

(4a) u(x, t) = r dy rt E(x - y, t - s)f(y, s)ds . JIlO Jo

If the integration in the time variable does not cause any difficulty in this formula (naturally this must be justified), it is not necessarily the same for the integration in y. In the hyperbolic cases, this is a different matter, as we can find an elementary solution E(x, t) whose support in x is a compact set for each positive fixed t. But this is not the case in diffusion problems. Consequently, the problems to be studied are the following: 1) the properties of E(x, t) concerned with the decay as Ixl-4 00; 2) the hypotheses to be made on f (belonging to a suitable function space) so that there exists a unique solution; 3) the study of the dependence of the solution (when we have found a class where it exists and is unique) as a function of the given data; 4) the study of the regularity in x and t of u(x, t); 5) with E(x, t) there is associated a semigroup {G(t)} given by:

(5) G(t)uo(x) = to E(x - y, t)uo(Y) dy ;

the study of the properties of this semigroup; 6) the irreversibility of the problem, the smoothing action of the evolution.

3. The subject of §3 is the wave equations in [Rn. The model problem is the search for the solutions of:

(6) L z u = [:tZZ + A (x, :x ' t) ] u = f

with suitable initial conditions, A being a 2nd order elliptic(3) operator. In the case where x E [R, the classical methods allow us to study several properties of these problems, thanks to the existence of characteristic curves x(t), along which equation (6) reduces to an ordinary differential equation. Likewise, in diffusion problems, the Fourier transform of (6) with respect to the variable x(x E [Rn), when the operator is invariant under translation in x, will allow us to reduce the study of these equations to that of ordinary differential equations; the study of elementary solutions (corresponding again to f = <5(x)<5(t))

(2) For a given fIx, t) and an elementary solution E(x, t), zero for t < O. (3) In the same sense as for equation (1), that is following the terminology of Chap. V, §2.1.3, O-strongly elliptic.

§l. The Ordinary Cauchy Problems in Finite Dimensional Spaces 3

provides the principal properties of these problems. We shall study, as in §2, the properties mentioned above, but the results will be very different. In particular, the supports of the elementary solutions relative to the variable x (for fixed t) do not cover the whole of [R0, as they do in the case of diffusion equations. We shall deduce from this the support of the solutions u, and therefore the propagation of initial conditions, in particular that of singularities (since the evolution of the solution with t, which is called here the propagation, will not have a smoothing action on the solutions).

4. In §4, we shall introduce the first ideas concerning the Schrodinger equation in [R0. Two simple model problems will be presented: the case of a free particle and the case of a harmonic oscillator.

5. In §5 we shall treat Cauchy problems in the whole of [R0 for which the evolution equation is of the form:

where (d * u) is a convolution. (x)

They will be treated, taking account of the convolution (when the coefficients of ;; are independent of x), by a Fourier transform in the spatial variable x. Hyperbolic systems give rise to propagations with finite velocity, as in the case of waves in §3. Finally in §6, we shall give Ovsyannikov's theorem which allows us to solve nonlinear evolution problems. The reader may also refer to Chap. V (linear differential operators). The Cauchy problem is studied there in §3, in the case of a linear differential operator P with constant coefficients. This study enables us to separate two particularly important classes of differential operators: parabolic operators for which the Cauchy problem is well-posed in [I", for example, and hyperbolic operators for which it is well-posed in !?,&'. We gave some examples: the heat flow operator, the wave operator, the Schrodinger operator. In this present chapter, we make a direct study of each of these examples, explaining the results, and extracting the concept of 'semi-group' which will be taken up in greater detail in Chap. XVII. We shall refer constantly to results on the Fourier transformation in Y', which is treated in the Appendix "Distributions".

§ 1. The Ordinary Cauchy Problems in Finite Dimensional Spaces

In this Sect. 1, we recall the elementary results on first order linear differential systems of finite dimension (that is, the spaces of values of functions of t which are fini te-dimensional).

4 Chapter XIV. Cauchy Problems in [hl"

1. Linear Systems with Constant Coefficients

Let

{

E en (resp. ~n)

Uo given in E

f: t ..... f(t), t > 0 be a suitable function with values given in E .

Let A be an endomorphism of E whose matrix, in the canonical basis of E, is also denoted by A. The usual Cauchy problem is:

(1.1)

find a function u: t ..... u(t) with values in E satisfying

du dt(t) + Au(t) = f(t) , t > 0

u(O) = O.

If we denote by (ai) i = 1, ... , n, j = 1, ... , n, the coefficients of the matrix A relative to the canonical basis and set:

u(t) = {ul(t), . .. , un(t)} , Uo = {UOI ' .•• ,UOn } (4)

f(t) = {/l (t), ... ,f,,(t)} ,

(1.1) becomes:

(1.2) { ddUi (t) + t aijuj(t) = J;(t) , i = 1, ... , n

t j= I

Ui(O) = UOi , i = 1, ... , n ,

which is a differential system.

Suppose firstly f = 0; then (1.1) becomes

{ ~~ (t) + Au(t) = 0, t > 0 ,

u(O) = Uo . (1.1 )'

It is well known (see for example Arnold [1]) that the system (1.1)' has a unique

(4) We write here (and more generally in the following chapters) the 'vectors' u(t) and f(t) in the form of 'row vectors' for typographical convenience. The custom of right multiplying an n x n matrix by a column vector would lead to writing u(t) and f(t) in the form of column vectors (which is in fact the usual custom, particularly in numerical approximations).

§ 1. The Ordinary Cauchy Problems in Finite Dimensional Spaces 5

solution given explicitly by

(1.3)

where e - AI is the following matrix:

(1.4) 00 (_ tAlk (5)

e- AI = L ---k=O k!

Remark 1. As the matrix A has constant coefficients, the function t -+ u(t) given by (1.3) is of the class C(J 00 with values in E. D

Remark 2. If we set

(1.5) G(t) = e- AI ,

the mapping t -+ G(t) satisfies, for all t, t' E \R

(1.6) G(t + t') = G(t). G(t') , G(O) = I (I the identity in E) (6) ,

which one expresses by saying that the family {G(t)} I E IR forms a group of operators. It follows from this that the solution u of (1.1)' given by (1.3) is also defined for t < o. We say that we are able to 'reverse the direction of time'(?) (if t denotes time) in (1.1)'. Generally this does not take place, as we shall see later, in infinite dimensions (see for example §2 relating to diffusion equations).

Remark 3. The mapping t -+ G(t) also satisfies the (matrix) differential equation

(1.6)' {

dG dt (t) + A. G(t) = 0

with:

G(O) = I. D

Let! be a continuous function in t with values in E (we denote by C(JO(\R, E) the space of functions continuous with respect to t with values in E and by C(Jk(\R, E) the space of k-times differentiable functions with continuous derivatives only up to order k having values in E.) Then (1.1) has a unique solution of class C(J 1, that is to say u E C(J 1 (\R, E), given by:

(1.7) u(t) = e-Aluo + I e-A(t-U)!(a)da ,

(5) e- A1 is defined here as the limit in the sense of the norm in 2'(E) of the series having general term (-tA)k

(which is convergent as A is bounded: we are here in a finite dimensional space). We then have k!

00 tk II A Ilk Ile-A111!t'(E)';;; L --,- = exp(tIIAII)·

k~ 1 k.

(6) The point occurring in (1.6) denotes matrix multiplication. (7) That is to say solve problem (1.1)' for t < O.

6 Chapter XIV. Cauchy Problems in IRn

which, taking into account (1.5), may be written

(1.8) u(t) = G(t)uo + I G(t - a-)f(a)da .

Remark 4. If f is not a continuous function, for example iff E Lloc{lR; E)(8), then (1.8) defines a continuous function over IR such that u(O) = uo, having a derivative du dt E Lloc(lR; E) equal almost everywhere to - Au(t) + f(t). 0

Remark 5. Iff E ~OO(IR; E) (that is to say that f is in the class ~oo having values in E), then u defined by (1.8), is a solution in the class ~oo of (1.1). We have seen above that G(t)uo is always in the class ~ cr). Expression (1.8) shows us that it is the regularity of the second term of the second member which therefore defines the regularity of u. More precisely it is the regularity of f, since G is in the class ~oo (for finite-dimensional E). 0

2. Linear Systems with Non Constant Coefficients

We shall now consider the following problem: find u(t) satisfying the following conditions:

(1.9) { ~~ (t) + A(t). u(t) = f(t)

u(O) = Uo;

Uo and f being given. The function t --+ A(t) has values in .5£ (E) (.5£(E) is here the space of linear mappings from E to E: see Chap. VI) or matrix valued under our convention (which identifies the endomorphism A(t) with its matrix in the canonical basis). To fix ideas, we shall suppose that f and A are continuous functions of t defined over R Generalising the solution of(1.1) by (1.7), (1.8) (with (1.5), (1.6), (1.6)'), we show that the system (1.9) has an n x n resolvant matrix R(t, r) which possesses the following properties:

1) it satisfies the differential equation

(1.10) { .) a . I at R(t, r) + A(t). R(t, r) = 0, WIth:

ii) R(r, r) = I (I identity) ;

2) it satisfies the functional relation

(1.11)

(8) LJ'oc(lR; E) is the space of (classes of) measurable functions for the Lebesgue measure over the line,

such that f If(t)1 dt exists for ali compact [a, b] c IR.

§ 1. The Ordinary Cauchy Problems in Finite Dimensional Spaces 7

In particular R(t, r) is invertible and

(1.12) R- 1(t,r) = R(r,t);

3) the Cauchy problem (1.9) is solved by the formula:

(1.13) u(t) = R(t,O).uo + L R(t,r)·f(r)dr;

4) R (t, r) is differentiable in t, and therefore from (1.12) in r, hence it is continuously differentiable in t and r. Now if A(t) E 2'(£) has an inverse B(t), then B(t). A(t)x = x for all x E £. The relation

implies

from which

(1.14)

with

(1.15)

B(t). A(t) = I

B'(t).A(t) + B(t).A'(t) = 0

B'(t) = - B(t).A'(t).B(t)

B(t) = A(t)-1 .

Therefore by applying (1.14) (replacing t --+ A(t) by r --+ R(t, r)), we find that

(1.16) a Or R(t, r) - R(t, r). A(r) = O.

5) If A is independent of t, then setting

(1.17)

we have:

( 1.18)

G(t) = R(t,O) ,

{R(t,r) = G(t).G(r)-1

G(s + t) = G(s). G(t)

and (1.13) reduces to (1.8).

G(t - r)

Remark 6. Differential systems of order m > 1, for example of the type(9)

dm m-l dk

-d m u + L Ak(t) -d k u + Ao(t)u = f in] - T, T[ (10) t k = 1 t

( 1.19)

with: u(O) = uo,o,u'(O) = UO. 1, ... ,U(m-l)(0) = UO,m-l,

may be reduced to first order systems. Suppose for example that Ak(t) and f are continuous functions of t in ] - T, + T[.

(9) The Adt) are operators (Ak(I)E 2'(£)). (10) We may take here T = + 'L.


We set:

(1.20)

Equation (1.19) may then be written

(1.21) u~ + Am- t (t)um + Am- z(t)um- t + ... + Ao(t)u t = f with

(1.21)' U~=Uk+t' k= 1, ... ,m-1.

Let A(t) be the m x m matrix

(1.22) A(t) ~ (

and 11 and J the vectors

° °

I

° ° I

(1.23) ii = (ut,uz, ... ,um), J= (0,0, ... ,f).

Then (1.19) becomes

(1.24) dii -- = A(t).ii + f. dt

We are therefore brought back to the study of problem (1.9). o Remark 7. When n = 1, equation (1.9) reduces to a first order linear differential equation

(1.25) { ~~ + a(t)u(t) = f(t) in] - T, + T[

u(O) = Uo ,

whose solution is given by (1.13) with

(1.26) R(t, r) = exp {f -a(S)dS}, Itl, Irl < T.

§2. Diffusion Equations

o

We have seen in Chap. I that numerous applications in physics and mechanics lead to diffusion equations. In the majority of cases the problem is to deal with a space variable x which belongs to a domain Q of [Rn. Some examples have been

§2. Diffusion Equations 9

collected in Chap. IB, for diffusion problems. But the study of these examples with Q = IRn is often useful in order to understand fully certain essential properties of these problems, properties which are sometimes more simple in this case. All of these examples in IRn lead to a Cauchy problem of the following type (restricted in this presentation to second order linear differential operators): Let A be a differential operator which is second order in the spatial variables, which is elliptic(ll) and whose coefficients may depend on x E IRn and on the time t:

( a) n a ( a) n a A = A x, -;-, t = - . ~ -a ajj(x, t) -a +.L bj(x, t) -a + c(x, t)I . UX 1,)=1 Xj Xj 1=1 Xj

We then use the following notation: if v is a function (or a distribution) from IRn+ 1 -+ C of two variables (x, t), x E IRn, t E IR, v(t) (or sometimes v(., t)) denotes the function (or by abuse of notation the distribution) x -+ v(x, t). We look for a distribution or a function u satisfying:

{au A f, f . f' d"b' at + u =, a gIVen unctIon or Istn utlOn

u(.,O) = uo, Uo being a given distribution over ~'(IR')

or a given function of x .

The method of solution used here allows us to solve this problem when the coefficients aij' bj, c are independent of x. We shall study a model problem: the heat equation in IRn.

1. Setting of Problem

Let Q = IRn and IR+ = ]0, + oo[ = {t > O}. We set Q+ = IRn x IR+. We look for a function u: x, t -+ u(x, t) (or a distribution u E ~ '( Q +)) satisfying

(2.1) {(i) ~~ - Llu = 0, in Q X 1R+(12)(13)

(ii) u(x,O) = uo(x) , Uo given;

(:t - LI ) is called the heat operator. (See Chap. V).

Naturally it is necessary to specify the sense which we give to the derivatives in (2.1) and to the Cauchy initial condition uo.

(II) More precisely, in the sense of Chap. Y, §2, Definition 2 (see also §3.6 of this Chap. Y), A is a O-strongly elliptic operator:

(12) See Chap. lA, § 1 for the physical origin of this problem. • iJ2

(13) The symbol LI denotes, as usual (see Chap. II), the Laplacian: LI = L -2' X = (XI' ... , Xn). i= i OX;

10 Chapter XIV. Cauchy Problems in [R"

Generally speaking, equation (2.1) is meant in the sense of distributions in Q+ (i.e. in the sense of ~'(Q+)); Uo is a given distribution. This will allow us to treat the largest possible category of given data. However, if this problem has a solution, there is no uniqueness unless we make supplementary hypotheses about the behaviour, as x goes to infinity, of u and of the initial condition Uo. As the operaror -.d commutes with transiations(14), the action of the operator -.d on a function (or distribution) may be represented by a convolution operator (see Appendix "Distributions", §2.4); for example if

d2 .d (j"*' = dx2 = n = 1,

where * denotes the convolution. Thus, it is advisable to transform problem (2.1) by making a partial Fourier transform with respect to the spatial variable x = (x l' x 2 , ••• , x") E IR" (see Appendix "Distributions"). In order to do this, we are therefore led to restrict the class in which we look for u, and where we are given uo, and to use the space 9"(IR") of tempered distributions (instead of ~'(IR")), which amounts to making a growth assumption, to infinity in x, on Uo and u.

2. The Method of the Fourier Transform

In the following, we shall denote by

(2.2) j = .?FAn (15)

the Fourier transform of a distribution f E 9"(IR") with respect to the space variable x E IRn. We shall also denote sometimes by lx, instead of J, a distribution of the variable x when this is useful to our understanding. Let then

(2.3)

By the Fourier transform in x, (2.1) is equivalent to

(2.4) {i) ~~ (t) + lyI2 U(t) = 0

ii) u(O) = Uo

(14) Recall that (- .1) also commutes with rotations of [R" (and also with all the operations on the time variable). See Chaps. II and V. (15) With, recall (see Appendix "Distributions") j defined by the duality .'I"([R"), .'I'([R"), the definition of the Fourier transform in .'I'([R") throughout this chapter being given by

j(y) = r e-'Y.X!(x)dx. JR'


where y = (Yl,"" Yn) is the dual variable of x = (Xl"'" Xn) and where ly(tW = IYll 2 + ... + IYnI 2 ; u(t) denotes, for fixed t, the Fourier transform of u(., t) = u(t). To solve (2.4), we associate with (2.4)i), the ordinary differential equation depending on the parameter y, Y E ~n,

(2.5)

whose general solution is v(y, t) = Ce-IYI2t, where C is constant 'with respect' to t and therefore, in fact, a function of y. Set

(2.6) R(y, t) = e-IYI2t, Y E ~n, t E ~

and note that R is a function of the class CI} 00 in (y, t) in ~~ x ~t and that

(2.7) for t ~ 0, R(t):y --+ R(y, t) is in (!)M (16) .

Then for all t, let wy(t) be a distribution of .@' (~n).

In (2.4) we can make the change of unknown from ~::!1'(~n) into itself (and not necessarily from 9"(~n) into itself) u(t) --+ w(t) defined by

(2.8) {U(t) = R(y, t)w(t) w(t) = R -1 (y, t)u(t) = elyI2tu(t) .

In effect, R(y, t) "# ° for all (y, t) E ~~ X ~t and the multiplicative products in (2.8) have a sense, R(t) and R -1 (t) being CI} 00 in y. As each solution of (2.4) must necessarily be a continuous function of t with values in .@'(~n), if u(t) E ~'(~n) is a solution and satisfies (2.8), then necessarily w(o) = Uo and w satisfies:

(2.9) {i) e-IY12t ~ w(t) = °

dt

ii) w(o) = Uo .

d Now (2.9)i) is equivalent to - w = 0, which implies that w is a distribution

dt independent of t and (2.4) therefore has a solution in ~'(~n):

(2.10)

This solution is the only possible one as the preceding method shows that if Uo = 0, then w(o) = 0, and (2.9) gives w = 0. The above reasoning demonstrates the following general proposition:

(16) See Appendix "Distributions", §3, for the definition of the space C'M'


Proposition 1. For Uo given in 0J'([Rn), the Cauchy problem has a unique solution in 0J'([Rn) given by (2.10) and defined for all t E IR.

Remark 1. The hypothesis Uo E Y"([Rn) does not appear in Proposition 1. Recall however that the condition Uo E Y"([Rn), that is to say Uo E Y"([Rn), is imposed on us by the use of the Fourier transform and that we are interested in problem (2.1), which is equivalent to (2.4) in Y"([Rn), but not in 0J'([Rn). Thus Proposition 1 implies uniqueness of the solution of problem (2.1) in Y"([R"), but we are not guaranteed that there are not other solutions of (2.1) in 0J' ([Rn). 0

Return therefore to problem (2.4). Since from (2.7), R(t) E (!)M for t ~ 0 only, u(t) given by (2.10) is in Y"([Rn) only for t ~ O. Therefore we have

Proposition 2. For UOy given in Y"([Rn), the Cauchy problem (2.4) has a unique solution,for t ~ 0, in Y" ([Rn) given by (2.1 0).

We may return to problem (2.1) via an inverse Fourier transform, which has unique solution in Y" ([Rn) given by

(2.11) u(t) = .? y- 1 (R(t)uo) = .? y- 1 (R(t)) * Uo = R(t) * Uo x x

where * denotes the convolution with respect to x, where.? y- 1 (R(t)) E (!)~ (l 7) and x

where R(t) is the inverse Fourier transform of R(t). From the calculus made in the Appendix "Distributions" - formula (3.17) - , we have:

(2.12) _ _ _ (n)nI2 lxi'

.?y(R(t)) = .?(R(t)) = t e-"4t if t > 0

and

- 1 - - 1_~ R(t) = .? -1 (R(t)) = --.? (R(t)) = e 41 if t > 0 .

y (2n)" y (2Ft)" (2.13)

Note the following result:

Lemma 1. We have the property

(2.14) R(t) --> 1 in Y"([Rn) as t --> 0+ .

Proof For all cp E Y'([Rn)

<R(t),cp> = 1 e-IYI'tcp(y)dy J R"

I<R(t) - 1, cp>1 ~ 1 (1 - e-IY1't)lcp(y)ldy + 21 Icp(y)ldy. J Iyl ~ A J Iyl > A

(17) See Appendix "Distributions", §3 for the definition of the space 0~ ..


For t; > 0 given, we may choose A large enough so that

2 r Icp(y)ldy < ~ JIYI > A 2

and having made this choice, we choose t close enough to zero such that

r (1 - e-IYI2t)lcp(y)ldy ~ ~; from which we have the result (2.14). 0 JIYI '" A

Since fi' is an isomorphism from 9"(lRn) to 9"(lRn), we have

(2.15)

and:

(2.16)

Therefore u(t), defined by (2.11) satisfies

(2.17) u(O) = Uo

and is the solution of (2.1). To summarise, we have obtained

Theorem 1. For Uo E 9" (lJ;n, problem (2.1) has a unique solution u(. , t) E 9" (IRn) given by

(2.18) 1 I y 12

U(.,t) = r= e-4t *uo ' (2'1/ ntt (x)

Remarks 2

1) Suppose that Uo E 9'(lRn). Then Uo E 9'(lRn) and also u(t) E 9.'(IR") (u(t) is then

the function: y -+ e-IYI2tuo(y)). We deduce that u(t) E 9'(IR") is the function:

1 1 Ix-~12 (2.19) x -+ u(x,t) = r::: e--4t- u (~)d~ (18).

(2'1/ 7T.t)" w 0

2) Suppose Uo ELl (IR"). Then Uo E Bo(lRn) (see Appendix "Distributions", §3.1); sincey -+ e-IYl2tisin e(lRn) for t > 0,u(t)EL1(1R")(t > 0); from which itfollows

1 Ix 12 that u(., t)EBo(IR") for t > 0. But for t > 0, x -+ r::: e-4t E Ll(lRn), so

(2'1/ 7T.tt that (2.18) implies that u(., t) E Ll(IR") with u(x, t) given by (2.19). We can again note that we have:

(2.20) 1 r Ixl 2

(2fot JlRn e-4t dx = 1 for all t > 0

(and by passing to the limit as t -+ 0, for all t ~ 0). Likewise, we may show directly that u(., t) -+ Uo in L 1 (IRn) as t -+ 0.

(18) Recall that (2.18) may be written in the form (2.1lJ) under the hypothesis that UOx is an integrable function, which is the case here (see generalisation in 6) of this remark).

14 Chapter XIV. Cauchy Problems in [R;"

Therefore we have:

(2.21 ) {i) u(., t) E Ll([R1n) for all t ~ 0, u(., 0) = Uo

ii) u(., t) E Bo([R1n) for all t > 0 .

Further (a property of the convolution product in L 1 ([R1n), see for example Brezis [1], p. 66), taking into account (2.20), we have

(2.22) Ilu(.,t)IILl(~") = r lu(x,t)ldx ~ Iluolle(~"), t~O. Jw which implies that for all t ~ 0, the mapping

Uo -+ u(., t) is continuous from L 1 ([R1") into itself (19) •

3) Suppose that Uo ELI ([R1n). Then Uo ELI ([R1n) as does u(t). Moreover yju(t), lyI 2u(t), YiYjU(t) are in L2([R1") for all t > O. So that u(t), which is again given explicitly by (2.19), satisfies:

au a2u u(t), ax(t), L1u(t), ax.ax. (t) E U([R1"), Vt > O.

I I J

(2.23)

Thus u(t) E H 2([R1"), Vt > 0 and we also have (by (2.10) or by Young's inequality)

(2.24) II u(., t) II L2(W) ~ II Uo II L'(W)' Vt > 0 .

In fact, for t > 0 and for all s E [R1, (1 + IYI2)'/2.u(t) is in L2([R1n) and therefore u(t) E HS([R1n) for all s (20), Vt > O.

Therefore u(., t) is, for t > 0, of class Cf} 00 in [R1"; it is similar to the 'preceding ~ases. 4) Suppose Uo = <5 E 9"([R1n). We then have Uo = 1 and uy(t) = R(t) where R(t) is the mapping:

Y -+ R(y, t) = e - tlyl'

for t ~ O. We therefore have:

u(., t) =:F -1 (R(t)) and therefore u(., t) is the mapping:

1 1 x 12 X -+ u(x, t) = e -4<, t > 0 ,

(2Ft)n (2.25)

u(.,O) = <5 .

The solution obtained in this case plays an important role which we shall examine in Sect. 3 of this §2(21).

5) More generally, every distribution Uo E 9" may be written (this is the 'structure theorem' of distributions of 9"; see Schwartz [2])

Uo = L DP((1 + x 2)qfpq) p,q

where the sum is finite and where fpq E L2([R1") for example.

(19) It is even a contraction (see Chap. VI, § 1.6.2 and XVII). (20) For the Sobolev space H '(IR"), see Chap. IV. (21) See also Chap. V.


Then (2.18) gives

u(., t) = fo L DP(e-I~~2 * (1 + X 2 )qfpq) (2 nt)n p,q (x)

and we can easily state that - due to the extremely rapid decay in x of e-lxI2/4t,

V't>O and of each of its derivatives - the function x, t-+e-lxI2/4t * (1 + x 2 )qfpq(x) is (x)

rc 'fJ in x, t for t > 0 therefore so is u( x, t). 6) The decay as Ixl-+ OCJ of e-lxI2/4t is, to some extent, 'unnecessarily' rapid to assure that we may calculate the convolution product in x of Uo with e-lxI2/4t. In fact we may define the convolution product (2.18) with some Uo which are not in!f' and the formula (2.18) will once more give one solution of the problem. Here is an elementary example. Suppose that

uo(x) = ealxl, IX > 0 ;

such a function is not in !f', but the function

u(x, t) = e --4-t -eal~1 d~ 1 i Ix-~12

(2fo)" ~.

is well defined and gives a solution of the corresponding Cauchy problem. The study of the 'largest' class, such that (2.18) has a meaning, has led to the work of Tacklind [1] and of Widder [1], [2]. 0

3. The Elementary Solution of the Heat Equation (see Chap. V, §2)

By introducing the Heaviside function

(2.26) Y(t) = {I if t > 0 o ift < 0

we may write R(t) in the form

(2.27) R(t): y -+ R(y, t) = Y(t)e - tlyl'

so that R(t) is defined for all t E IR, and with support in {t E IR; t ~ O}. Adopting this definition of R(t) from now on, we have (in the sense of g&' (IR~ x IRt ) (22))

(2.28) d ~ ~ dt R(t) + lyl2 R(t) = b(t) = 1, ® b(t) .

(22) There is no consensus regarding notation, the second term of (2.28) consists of the distribution .5(t) = the Dirac mass at the origin, for all y; it is therefore a distribution of y and t defined by

Oy ® .5(t), q» = r q>(y, Oldy, 'v'q> E .@'(~~ X ~,) .

JR"

16 Chapter XIV. Cauchy Problems in 1R1"

Then by inverse Fourier transform with respect to y, E(x, t) given by

- Y(t)_~ E(., t) = :Fy - 1 (R(t)): x --+ e 4,

(2Ft)" (2.29)

satisfies, in the sense of £C'(IR~ x IR,):

(2.30) a at E(t) - LlE(t) = b(x) ® b(t) = b(x, t) (23) •

The function t --+ E(t) is continuous from IR --+ Y"(IR~) and zero for t < 0; it is the only elementary solution having this property.

Solution of Cauchy problem (2.1) knowing E. The Cauchy problem (2.1) may be written (by extending u(x, t) by zero for t < 0) in the sense of £C'(IR~ x IR,):

au (2.31) at - Llu = Uo ® b(t) .

Then, from the results of Chap. V, a solution of (2.31) is given by:

(2.32) u = E * (uo ® b(t) = E(t) * Uo (x. t) x

provided that the convolution in (2.32) has a meaning. Now the support in IR~ x IR, of the elementary solution E is the half"plane Q+ = Q+ = IR~ x IR,+ (where IR,+ = ~,+ = [0, + UJ [). Because of this, for (2.32) to have a meaning, it is necessary to make a hypothesis on the growth of Uo towards infinity. Since for t > 0, E(t) E (i)~, the natural hypothesis is Uo E Y"(IR").

4. Mathematical Properties of the Elementary Solution and the Semi group Associated with the Heat Operator

We shall summarise the properties of the distribution E(x, t) which is defined by (2.29); 1) Positivity of E. Note firstly that this distribution is positive:

(2.33) E(x, t) ~ ° for all x E IR", t"# ° . This has the following consequence: If Uo is given in Y" with Uo ~ 0, then u(t) (the solution of problem (2.1), given by (2.32)), is also a positive distribution (u(t) ~ 0) (24)

2) Semigroup. If we define the operator G(t) in Y" (IR") by

(2.34) G(t) = E(t) *, t ~ ° , x

then the solution of problem (2.1) may be written in Y"(IR"):

(2.35) u(t) = G(t)uo .

(23) £(t) is the notation introduced in § 1.1. (24) It follows that if Uo is a (positive) tempered measure, then u(t) is also a (positive) tempered measure.


This formula (2.35) should be compared with the formula (1.3) of § 1 which gave a solution to the finite dimensional Cauchy problem. Indeed we verify with the help of the Fourier transform (formula (2.10)) that we have:

(2.36) {G(t + s) = G(t) 0 G(s) , s;?; 0, t;?; 0 ,

G(O) = I .

From Remarks 2, for all t;?; 0, G(t) operates continuously from Y'(~n) --+ Y'(~n); Ll(~n) --+ Ll(~n), L2(IW) --+ L2(~n) and more generally from Y"(~n) --+ Y"(~n). We say that the family of operators {G(t)}t~O forms a semigroup in Y'(~n), Ll(~n), L2(~n), Y"(~n). Semigroups are defined and studied, in Banach spaces, (the case of L 1 (~n) or L2(~n)) in Chap. XVII. 3) 'Irreversibility' of problem (2.1). It is important to note that G(t) does not operate continuously on these spaces for t < O. Therefore {G(t)} does not constitute a group. The fact that {G(t) L ~ 0 constitutes a semigroup and not a group as in finite dimensions is linked to the 'irreversibility' of diffusion problems. Mathematically, this signifies that we cannot 'reverse the sense of time' in problem (2.1) (resp. (2.4)). More precisely: The operator R(t) of multiplication by e- c1yl ' is, we have said, invertible in f0'(~n), but R-1(t) which is defined by multiplication by eclyl2 does not operate in Y"(~n) since y --+ eclYl2 is of class C(} co for fixed t, but ¢ (DM for t > O. The result of this is that the Cauchy problem for (2.4) with initial condition at t = to, to > 0 has, for t > to a unique solution:

(2.37)

We say, like Hadamard, that the Cauchy problem for (2.4) (and by inverse Fourier transform for (2.1)) is well-posed in the sense of t ;?; to ;?; 0 (the sense of t increasing) in the space Y"(~n). Conversely, the 'backward' problem which would consist of finding v(t) for t ~ to knowing v(to), v(t) satisfying (2.4) is ill-posed in Y"(~n) since it does not, in general, have a solution(25). In other words, for problem (2.1) (resp. (2.4)), the present allows us to predict the future, but not the past! 4) Vaguely, but intuitively enough, the character of 'irreversibility' is linked to the smoothing property of the operator G(t): starting at Uo, a distribution in Y" for t = 0, we obtain for arbitrarily small t > 0, a solution u(., t) which is C(} co in x; 'in the inverse direction', by inverting the direction of time, if this would be possible, we see that starting at to > 0 from a C(} 00 function we 'arrive' at t = 0 at a distribution; such a 'loss of regularity' (infinite!) over an arbitrarily small interval corresponds to the ill-posed nature of the backward problem(25).

(25) For the ill-posed character of the backward problem, see Chap. V.

18 Chapter XIV. Cauchy Problems in IR"

5) We may proceed further. Starting from Uo E g", u(., t) is for fixed t > 0, analytic in x (and also analytic of a particular type(26»). 6) The properties of irreversibility, of being a semigroup but not a group, of 'regularisation' by passage from t = 0 to t > 0, are properties linked to the

parabolic nature of the operator [:t - A]; the terminology comes from the fact

that, by Laplace transformation(27) gives p - ~2 and that p - ~2 = ° is the equation of a parabola in the (~, p) space. 7) We must note one difficulty linked to the physical modelling by equation (2.1). If we start from Uo = b (which corresponds to a 'very large' heat source concentrated at the origin for a very brief instant), we obtain a temperature > 0 at some points arbitrarily distant for some arbitrarily small time - which does not conform to the physical reality - in fact: (i) we obtain, certainly, a temperature> 0, but very small due to the term

e -lxI 2 /4c. , (ii) the propagation at finite velocities of phenomena having nonetheless a diffusive character may only be modelled by nonlinear partial differential equations (these are the models introduced by Kolmogorov et al. [1], see also Lions [2] and Raviart [1]). 8) The elementary solution E(x, t) is invariant under all rotations of the space (it only depends on the Euclidean norm Ixl in /Rn). This is linked to the invariance of the operator A under rotation(28).

Remark 3. The nonhomogeneous Cauchy problem. consists of finding u(., t) satisfying:

(2.38) { at _ Au = f at u(o) = Uo

This is the problem which

where Uo is given over /Rn and f is given for t ~ 0, zero for t < O. Suppose that, for all t ~ 0, f(., t) E g" (/Rn) and that the mapping t -> f(t) is continuous from t ~ 0 into g"(/Rn). Then if Uo is given in g"(/Rn), we have

Theorem 2. The nonhomogeneous Cauchy problem (2.38) is well-posed in g' '(/Rn)for t ~ O.

The solution of (2.38) is then given by

(2.39) u(t) = E(t) * Uo + E(t) * f(t) . x (x, t)

Having made the preceding hypotheses on f, we introduce the operator G(t)

(26) See Widder [ll (27) See the parabolic operators in Chap. V, §2 and 3. (28) See Chaps. V, VIII and IX.


defined by (2.34) and the corresponding solution of (2.38) is given by:

(2.40) u(t) = G(t)uo + I G(t - a)f(o) da ,

a formula which is similar to formulae (1.7) and (1.8) of § 1.1, in the case of finite dimensions. We are, by analogy with the finite dimensional case, tempted to set:

G(t) = e- At with A = - L1 .

We shall see (in Chap. XVII) what meaning it is suitable to give to such a representation. To establish (2.39), we may use the general properties of the elementary solution (see Chap. V) or the method used for the homogeneous equation, a variant of the method of variation of constants. Indeed, (2.38) is equivalent in Y"(lRn ) to

(2.41 ) {au -~t (t) + AIYI2U(t) = f(t)

u(O) = Uo .

Then, by making the change of unknown defined in g&'(lRn ) by:

(2.42) u(t) = e - Iyllt v(t) where v(t) E g&' (IRn) ,

we see that the distribution v(t) must satisfy

(2.43) e- Iyl t - v(t) = f(t)

dt {

1 d -

v(O) = Uo .

The unique solution of (2.43) is

(2.44) v(t) = Uo + I elyllr](r)dr ,

from which

(2.45)

By introducing R(t) = Y(t)e -Iyl\ t E IR, (2.45) may be rewritten as:

u(t) = e -Iyllt Uo + R(t) * ](t) t

from which, by the inverse Fourier transform with respect to y, we have formula ~m 0

Remark 4. Self-similar solutions It is interesting to recover the elementary solution by the use of solutions called self-similar, that is to say, by invariance of the heat flow equation under a group of transformations.


For simplicity we restrict ourselves to the space dimension n = 1. If E is a solution tempered in x of

(2.46)

then,

(2.47) for all k E IR, Wk(X, t) = IkIE(kx, k2 t)

is again a solution of (2.46). But (2.46) has, as we have seen, only one tempered solution, therefore by uniqueness:

(2.48) E(x, t) = IkJE(kx, k2 t) for arbitrary x E IR, t > 0, and k =F O.

1 If therefore, we take k = Jt

(2.49)

and by the change of variable ~ = .it (2.50)

1 E(x, t) = JtfW where f(~) = E(~, 1) .

Note that f(~) must be an even function of ~ in virtue of (2.48) (with k = - 1). Thus

aE a2E , at - ax2 = 0 ¢> 21' + ~f' + f = 0 .

We have therefore to find even solutions of

21" + U' + f = 0 .

The operator (2 dd~22 + ~ dd~ + 1) is invariant by changing ~ ~ - ~, so that every

solution may be written

1 1 f(~) = 2: [f(~) + f( - ~)] + 2: [f(~) - f( - ~)]

i.e. the sum of an even solution and an odd solution. Therefore the dimension of the space of even solutions is 1 and since e-~2/4 is an

even solution, we see that

§3. Wave Equations 21

But we must have:

f~oo E(x, t)dx = 1; therefore f~oo f(~)d~ = 1 ;

from which C = 1 r:: and 2..,; 7r

(2.51) 1 1 _~

E(x, t) = r:: r. e 4/. 2..,; 7r ..,; t

which agrees well with (2.29) (for t > 0) when n = 1. o

§3. Wave Equations

Certain problems cited in Chap. I may lead to the study of Cauchy problems in the spatial domain Q = IRn of the following type: A denotes a differential operator, second order in the spatial variables, which is elliptic (29), and we look for a distribution or a function u satisfying

a2 u at2 + Au = f, X E IRn, t > 0

(3.1)

au 1 at (.,0) = u ,

where: fis a given function or distribution, UO and u 1 are distributions over!!)' (IRn) or given functions of x.

1. Model Problem: The Wave Equation in 1R"

1.1. Setting of Problem

We look for u satisfying

a2 u i) 8t2 - Au = 0, X E IRn, t > 0

(3.2) ii) u(x,O) = UO(x) ,

iii) ~~ (x, 0) = u1(x);

UO and u 1 are given functions or distributions.

(29) In the sense indicated in §2.

22 Chapter XIV. Cauchy Problems in [J;l"

Problem (3.2) is the homogeneous Cauchy problem for the wave equation. We shall see that this problem is (contrary to the Cauchy problem for the heat flow equation) always well-posed without its resolution needing a hypothesis of growth to infinity on its solutions. This is due to the hyperbolic nature of the wave

. h· f h 02 11 d h equatIOn, t at IS to say 0 t e operator On = ot2 - A once more ca e t e

d'Alembertian (for hyperbolic operators see Chap. V, §2).

1.2. The Wave Equation for Dimension n = 1

1.2.1. Classical Solutions. We shall start by obtaining several properties of the usual classical solutions(30) in dimension n = 1. The problem is to find u(x, t) of class I(j 2 satisfying

(3.3)

02U a2u i) fu2 (x, t) - ax 2 (x, t) = 0, X E IR, t > 0

ii) u(x,O) = UO(x) ,

iii) ~~ (x, 0) = u1(x) ,

where U O and u 1 are two given functions respectively of class I(j 2 and I(j 1.

We may associate with the partial differential equation (3.3)i), the polynomial P(X, T) = T2 - X 2 (31); in the (X, T) plane the curves ofthe equation T 2 - X 2 are hyperbolae which are transformed by rotation of the axes by n/4 (T1 = (X - T)/2, X 1 = (X + T)/2) into hyperbolae(32) with equation T1 . X 1 = constant. This suggests, therefore, taking new variables (~, ,,) defined by

(3.4) { < ~ ~(X + 'i

" = 2 (x - t) .

We shall set u(x, t) = v(~, ,,). Then equation (3.3)i) becomes:

a2 v a~o" = 0

and all the solutions are given in the classical sense(33), by

(3.5) v(~,,,) = cp(~) + 1/1(,,),

where ¢ and 1/1 are arbitrary functions of class I(j 2.

(30) By a classical solution, we mean a solution which is sufficiently regular. This notion will be made precise later.

(31) This polynomial P(X, T) and the differential polynomial p[~, ~J have been studied in Chap. V. ax at

[ 02 02 ] (32) From which comes the term hyperbolic for the operator 2 - -2 ' see Chap. V. at ax

(33) And besides all the solutions which are distributions, tjJ and", then being distributions over IR.


We deduce that

(3.6) (X + t) (X - t) u(x,t) = cp -2- + !/I -2- .

Condition (3.3)ii) implies

(3.7) u(X, 0) = cp(~) + !/I(~) = UO(X) ,

and condition (3.3)iii) implies

(3.8) au 1 (X) 1 (X) 1 - (X 0) = - cp' - - -!/I' - = u (X) . at' 2 2 2 2

From (3.8) we deduce, (a being an arbitrary constant in IR):

(3.8)'

so that

(3.9)

Thus the (unique) solution is given by:

(3.10) 1 [ ] 1 fX+I u(x, t) = 2 UO(x + t) + UO(x - t) + 2 x-I u1(u)du .

Formula (3.10) is sometimes called 'D'Alembert's formula'(34).

Remark 1. From formula (3.10) and that which precedes it, we obtain: 1) Problem (3.3) has a unique solution of class CC 2 which moreover has the property of 'continuous dependence' on the data, that is to say that if uO, u1

converge uniformly towards zero over all compact sets of IRx' then u -+ 0 uniformly over all compact sets of IRx x iR ,+.

2) The value of u(x, t) at the point (xo, to) (for to > 0) only depends on the values of UO(x) and of u1 (x) in the set L (xo, to), the intersection of t = 0 with the 'cone' CC(xo, to) with the equation

(3.11)

(34) Denoting by U 1 a primitive of ut, (3.18) may be written

This formula can be interpreted as the propagation of two 'waves' of speed 1, in the positive x direction and the negative x direction.

24

l:(xo. to)

Fig.!

Chapter XIV. Cauchy Problems in ~.

x

We summarise property 1 by saying, like J. Hadamard, that the Cauchy problem is well-posed. Property 2 has a less general character; it deals with the hyperbolicity of the wave operator. 3) The lines x + t = constant, x - t = constant are called the 'characteristic lines' of the partial differential equation (3.3)i). The cone ~(xo, to) is called the characteristic cone with apex (xo, to). In the present case, the 'surface' r(xo, to) of the characteristic cone is formed by the two lines with equations

x + t = Xo + to and x - t = Xo - to .

The coordinates (e, '1) defined by (3.4) are, up to a coefficient of 1/2, the characteristic coordinates; (3.5) shows that u(x, t) is the sum of two terms, each of which is constant along one of the two families of characteristics. Further, it is clear that if, for example, the given initial data Uo or one of its derivatives has a singularity at a point, the corresponding solution u(x, t) or its derivatives will have singularities which propagate along the characteristic lines passing through the singularity of uo. 0

The importance of the characteristic cone is again emphasised by

Remark 2. The nonhomogeneous Cauchy problem. Consider the problem of finding u satisfying

(3.12)

This is the nonhomogeneous Cauchy problem with Cauchy conditions equal to 0 (the general nonhomogeneous Cauchy problem (i.e. with uo, u1 =I 0) then leads to the superposition of (3.3) and (3.12)). Suppose, to fix ideas that f is zero for t < 0 and of class ~ 1 in (x, t) for t ~ 0, with compact support in x. Then by taking the coordinates (e, '1) defined by (3.4), and setting f(x, t) = g(e, '1); u(x, t) = v(e, '1), we have:

iJ2v oe 0'1 = g(e, '1) ,


from which

v(e, 17) = - roo roo g(e', 17 1 )deld17 ' •

Returning to the coordinates (x, t), we obtain (since f is zero for t < 0)

u(x, t) = ~ If f(x ', t')dx' dt' , J(x, I)

(3.13)

where A (x, t) represents the hatched region of Fig. 2, that is to say the set:

A(x, t) = {(x', t') E ~(x, t), t ~ t' ~ O} . o

(x. t)

x - t x + t x

Fig. 2

Remark 3 1) Ifuo, u 1 are only continuous, formula (3.10) has a meaning. We say that it then defines a generalised solution of (3.3), that is to say a solution in the sense of distributions in IRx x IRt+ = Q +. Indeed, if qJ E ~ (Q +), with Q + = IRx x ] 0, + 00 [

= IRx x IRt+, u given by (3.10) satisfies

(3.14) I r (02qJ 02qJ) JQ+ U ot2 - ox2 dxdt = O.

We note that all solutions of (3.12), zero for t < 0, satisfy in the sense of distributions in ~1(lRx X IRt) = ~I(Q), Q = IRx x IR"

(3.15)

2) Likewise, if f is only of class ~o, formula (3.13) defines a solution of (3.12) in the sense of distributions, in ~I(Q+) or in ~I(Q). Indeed we have, for example, for all qJE~(Q+)

o

1.2.2. Utilisation of the Fourier Method. Recall problem (3.3) and suppose for the moment that UO E 9" (IR), u 1 E 9" (IR) and look for u(t) E 9' I (IR) satisfying (3.3).

26 Chapter XIV. Cauchy Problems in [f;l"

By using the Fourier transform in x, we are led to

(3.16) Ii) ~:~ + y2 U = 0

ii) u(O) = ao , ~~ (0) = a 1 (35)

As in §2, we associate with equation (3.16)i) the ordinary differential equation depending on the parameter y E IR.

(3.17) d 2v 2 dt 2 + y v = 0,

whose general solution is given by:

(3.18) v(y, t) = Cocos(IYlt) + C1 sin(lylt),

where Co and C 1 will be functions (or distributions) of the variable y. By substitution in (3.16), we obtain (according to the method called 'variation of constants') the unique solution in .@'(lRy ), given by

(3.19) u(t) = cos(lylt)aO + sin(IYlt) a1 (36)

Iyl

Note that the functions cos (yt) and sin (yt) are in (!)M(IR) and that the products y

occurring in (3.19) are well-defined for any t positive or negative. The system (3.16) is therefore reversible here, and the Cauchy problem is well-posed in both the forward sense (t increasing) and the backward sense (t decreasing). Returning to the variable x, there therefore exists a unique solution to problem (3.3) in Y)/(IR), where U O and u 1 are given in Y"(IR), this solution is given by

(3.20) ( sinyt) u(t) = ffy- 1(cosyt)*u o + ffy-1 -y- *u 1 .

Note that the functions y --> cos yt, Y --> sin yt are, for fixed t, entire holomorphic y

exponential functions. Therefore, from the Paley-Wiener theorem(37), their Fourier transform (direct or inverse) has compact support. We have moreover calculated

(35) For typographical convenience, we shall use the notation UO and ul from now on, to denote the Fourier transforms of UO and u I.

sin /I (36) In consideration of the evenness of the functions cos /I and -f}-' we may suppress the absolute

values here. (37) See Appendix "Distributions", §3.

§3. Wave Equations

these Fourier transforms in the Appendix 'Distributions' (§3) and obtained.

(3.21)

1 .

{!F ; 1 (cos yt) = "2 [b(t) + b( - t)]

I7C -1 (sinyt) _ 1 :f/' y -y- -"2 XI-t, +t]

where XI-t, +t] the characteristic function of the compact interval [ - t, + t];

so that (3.20) may be written:

10 1 1 (3.22) u(x, t) = "2 (b(t) + b( - t)) * u +"2 XI-t, +t] * U •

27

When UO and u 1 are continuous functions, we note that (3.22) agrees with (3.10).

Remark 4. The tempered elementary solution for the wave equation for n = 1 We must solve in .@'(lRxx IRt )

(3.23) 02E1 02E1 ----at2 - ox2 = !5(x, t) = !5(x) ® !5(t) ,

or, which is the same thing(38), the system (3.3) with

UO = 0, u1 = !5(x) .

Via the Fourier transform in x, (3.23) is equivalent in Y"(IR) to

(3.24) 02 £1 2 A ----at2 + y E 1 = 1 (y) ® !5(t) ,

which gives:

(3.25) A sinyt

E1 : (y, t) -+ Y(t) --; Y

so that

{

I 1 -

E1 : (x, t) -+ "2 Y(t)XI-t, +t](x) = ~ if t ~ Ixl

elsewhere

is the tempered elementary solution of the wave equation(39). Note that the support of E 1, denoted supp E 1, is the cone given by equation

(3.26) t ~ lxi, x E IR ,

called the forward light cone.

(38) For E1 with support in IRx x IR,+; see, for this, Chap. V, 33, Prop 2. (39) We thus recover the result given in Chap. V, §3.


We denote by the singular support of a function u(x, t) the smallest closed set outside of which u is of class ff,} 00. Here the singular support of El (x, t) is the set of the two 'rays':

{X + t = 0, 0 t;;:;, . x-t=O,

Note that this coincides with the boundary of the support of E l' Generally, the . '. a2u 2 a2 u _

wave equatIOn IS. atz - c ax 2 - 0

where c is a (constant) speed. The 'surface' of the characteristic cone with apex 0 is then defined by:

The forward light cone is then ct > 1 x I. Note that the forward light cone is therefore the part of the characteristic cone in which t > O. Finally, we remark that El is the unique solution of (3.23) whose support is given by (3.26) (forward light cone). It is this which is used for the resolution of the Cauchy problem. There are however other elementary solutions with supports in the three other quadrants of the (x, t) plane (see Chap. V). D

1.2.3. The Cauchy Problem with Data Given in ,@'(lRx). We now suppose

(3.27)

We look for u E !;0'(Q)(Q = IRx x IR,) with support contained in the half-plane {(x, t); X E IR, t ;;:;, O} = IRx x IR,+ = Q + and satisfying, in the sense of distributions in ,@'(Q) the partial differential equation

(3.28) a2 u a2 u

DIu = at 2 - ax 2 = u 1 ®J(t) + uO®J'(t).

Then

(3.29) u = El * (u 1 ® J(t) + UO ® J'(t)) (x, ,)

is a solution of (3.28). We have shown in Chap. V, §3.2, that the solution thus found is the only one possible. Further, if UO and u 1 tend to zero in ,@'(lRx), then so must u(t) for all t E IR. The Cauchy problem in ,@'(IRJ is therefore well-posed for the wave equation for n = 1. We shall now generalise this to the case of arbitrary n.

1.3. The Wave Equation in IR~

We now recall problem (3.2).


1.3.1. Utilization of the Fourier Method. By taking the Fourier transform in x, we obtain the system equivalent to (3.2) in 9" (IRn)

(3.30) {i) ~:~ (t) + lyI2a(t) = 0, lyl2

ii) a(O) = ao , ~~ (0) = a 1 .

Just as in the case of dimension n = 1, (3.30) has a unique solution in 2&'(IR;) given by

(3.31 ) A A D sin I y I tAl u(t) = cos(lylt)u + -- u .

Iyl

sin IYlt . The functions y --> cos I y It, y --> --- WhICh are the solutions of (3.30),

Iyl corresponding respectively to the initial conditions ao = 1, a 1 = 0, uD = 0, a 1 = 1 are called the fundamental solutions of equation (3.30)i). They are in (i)M(lRn) and further, they are entire, holomorphic, exponential type functions, therefore their direct or inverse Fourier transform has, for fixed t, compact support due to the Paley-Wiener theorem(40). By inverse Fourier transformation we deduce from (3.31) that

(3.32) u(t) = u(.,t) = ff;l(coslylt)*u D + ff;l(~sinIYlt)*ul. (x) Iyl (x)

We then deduce the following consequences. 1) The Cauchy problem is well-posed in the space 9"(lRn):

if UD, u 1 E 9"(lRn), then u(t) E 9"(lRn)

and the mapping t f---+ u(t) is continuous from t ~ 0 --> 9" (IRn) (and even infinitely differentiable from t ~ 0 --> 9"(lRn)). 2) We may change t to - t; this leaves the first term of (3.32) invariant and, in the second term, changes u 1 to - u 1. Consequently the Cauchy problem is well-posed in 9" (IRn) in both the forward and backward senses. 3) Since, for each fixed t, the distributions ff; 1 (cos I y I t) and ff y- 1 «sin I yl t)/Iy I) have compact support (we give below their explicit representations), formula (3.32) has a meaning - and gives a solution to the Cauchy problem - for uo, u1 in 2&' (IRn), which we shall see is unique. The Cauchy problem, in the forward and backward sense, is well-posed in the space 9"(lRn) and equally, as we shall see later, in 2&' (IRn), 6"(lRn) and 2&(lRn). Conversely, it is ill-posed in L 2(lRn). Indeed if Uo and U 1 E L 2(lRn) then u(t) E L 2(lRn)

'it > 0, but we do not necessarily have au (t) E L2(lRn) for t =I O. We may remedy at

this difficulty, and this is fundamental for applications corresponding to a natural mathematical framework for modelling systems with finite energy, by taking

(40) See Appendix "Distributions", §3.S.


Uo E Hl(lRn) and u1 E L2(lRn): we verify easily with the help of (3.31) that:

u(t) E Hl(lRn) Vt, i.e. r (1 + y2)lu(y, tWdy < + 00 and aau E L2(lRn) Vt. J~" t

We shall return in Chap. XVIIB to problem (3.2) taken in the framework H 1 (IRn) x L2(lRn) (the method used occurs also in Chap. XV; the use of the Fourier transformation 'diagonalising' the operator - LI). 4) We can always write (3.2)i) as a 1st order system in t:

{ au _ v = 0 at ' av - - Llu = o· at '

if we introduce

w = {u,v} , A = (~LI ~I), Wo = {UO,u 1} ,

then problem (3.2) is equivalent to

{ ~; + Aw = 0,

w(O) = Wo ,

and formula (3.32) becomes

(3.32)' w(t) = E(t) * Wo = G(t)wo (x)

where

If we set:

E = (fZ'(lRn)f, or (9(lRn))2, or (fZ(lRn))2, or (9"(lRn))2 ,

we see that {G(t)} is a group in E. 5) All this shows the essential differences between the parabolic case (§2) and the hyperbolic case (§3). Some of these are summarised in the table below:

Parabolic case

Irreversibility (semigroup) Smoothing action in x as t increases Infinite speed of propagation Elementary solution whose support is the whole of IR~ for fixed t > O. The elementary solution is regular.

Hyperbolic case

Reversibility (group) No smoothing action Finite speed of propagation (waves) Elementary solution with compact support in x for fixed t > O. The elementary solution is a distribution Pf (see later).


Remark 5. No difference appears in the expression (3.32) of the solution due to the parity of n. In return some differences will appear in the explicit calculation of the inverse Fourier transforms. D

1.3.2. Elementary Solution of the Wave Equation(41). We look for an elementary solution (if it exists), E = E. (we use the index n to denote the space dimension) of the problem:

{

DE = <5(x) ® <5(t)

(3.33) E = 0 for t < 0 and the intersection of the support of E

with the hyperplane t = to is compact for all to ~ 0 .

If such an elementary solution exists, it is unique. Indeed, let F be another elementary solution with properties analogous to (3.33) (the supports do not coincide a priori). Then

D (E * F) = (D E) * F = F = E * (D F) = E . (x, I) (x, I) (x, I)

It remains to calculate E (we shall see that it exists) and for this one way is to use the Fourier transformation in x. We find for E.(t):

(3.34) 02 - -ot 2 E.(t) + lyl2 E.(t) = 1 (y) ® <5(t) ,

whose solution is given by

(3.35)

and, consequently;

(3.36) E.(x, t) = Y(t)~ -1 (Si~;flt).

1.3.3. Calculation of ~y- 1 (sin ~It») = R(t). As we have announced in

Remark 5, the calculation depends on the parity ofn. We obtain (see L. Schwartz [1] and [2]): The case n = 1

(3.37) { ! if t 2 ~ x2

R(t) = 2

o otherwise

(this is the result established above in Remark 4). The support of E.(x, t) is then

CR(x, t) = {(x, t), t ~ Ixl} .

(41) See Chap. V.


At time t = lxi, the value of R(t) at a given point x, passes suddenly from 0 to 1/2, then stays afterwards, at this value for t > Ix!. The case n even, let n = 2p, p ~ 1

1 (42)(43)

R(t) = 3 Pf------2nn;1 r(2 -~) [(t 2 IXI 2)+J_n;_1

(3.38)

where r is the gamma function, a +, = sup (a, 0) and where Pf denotes the

distribution defined by [2 1 12J + that is to say that / Pf( 2 1 2)+' cjJ) is the t -Ix \ t - Ixl

finite part of the divergent integral:

i cp(x, t) n ----n--=-l dxdt, cp E ~(lRx x IRt ) ,

t 2 ;'lxl 2 (t 2 _ IxI2)-2-

see Appendix "Distributions" and Schwartz [2l The support CR(x, t) of En(x, t) is then

CR(x, t) = {(x, t); t ~ Ixl} .

An observer placed at x therefore sees at time t, such that t = lxi, the value of R(t) increasing suddenly at t. Afterwards when t > lxi, the value of R(t) in x decreases, and tends towards zero without ever becoming zero. We may note that if we set n = 1 in formula (3.38), we recover formula (3.37). The case n odd, let n = 2p + 1 ~ 3. Let p.(t) be the distribution in IR~ formed by the mass + 1 distributed homogeneously on the surface of the sphere I x I = t. Then

1 [1 d](P-l) (3.39) R(t) = 1.3.5 ... (2p _ 1) t dt (t 2P - 1 p.(t)) , n = 2p + 1 .

It is natural to denote by ()r-t the measure carried by the sphere Ixl = t with surface density + 1 such that

(3.40)

where On = 2nn/2 / r(n/2) is the area of the unit sphere of IRn(r is the Eulerian function). We also have therefore

(3.41) { 1 [1 d ](P-l)(()r_t) R(t) = nP2P+1 t dt -t-

n = 2p + 1, p ~ 1

(42) See the definition of Pf (finite part) in Appendix "Distributions". (43) Other expressions for R(t) exist in the literature (see Schwartz [1]. Lax-Phillips [1]. Treves [I]).


[1 d ](P-1l 1 d

where - -d denotes the operation - - iterated (p - 1) times. t t t dt

The support of En(x, t) in IR~ is then the sphere r = Ixl = t. This case is therefore very different from the case n· = 1 or n = even. Here, an 'observer' placed at x, sees arriving at t = lxi, a wave whose value is different from o (the larger lxi, the smaller R(t)). Then, as soon as t > lxi, the value of R(t) drops back to 0 in x. A particularly important case is that where n = 3. In this case, (3.41) gives

(3.42) 1 R(t) = tf.1(t) = _1 Jr - t , n 3 4nt

r = Ixl = (xi + x~ + xn1/2 .

We shall verify formula (3.42) by calculating the Fourier transform of Jr - t • Since Jr - t is a measure with compact support in x for given t, the Fourier transform in x of Jr - t is given by

(3.43)

Jr - t being invariant under rotation around the origin, we make the calculation in spherical coordinates p, e, ¢ by choosing the vector y in such a way that it is the polar axis; then

x.y = r.pcose with r = (xi + x~ + xn1/2, p = (yi + y~ + ynl/2.

Thus, (3.43) may be written:

(3.44)

which confirms formula (3.42).

Remark 6. We have performed the calculation of the elementary solution for n = 1 and n = 3. Formulae (3.38) and (3.39) may be evaluated directly. As the calculations are fairly complicated, we may use a method called 'the principle of descent' due to Hadamard. This is the object of the following Sect. 1.3.4. It allows us, knowing the elementary solution in IR~+ 1, to obtain the elementary solution in IR~. Before verifying the principle of descent, we make more explicit the formulae giving the elementary solutions in the cases n = 1, n = 2, n = 3. We have in all these cases:

(3.45) En(x, t) = Y(t)R(t)


with (for all <p E fi)(~~ X ~I)):

i) n = 1, (E1' <p) = ~ r <p(x, t)dxdt, JI> Ixl

(3.46) .. ) 2 1 I <p(x, t) d d 11 n = ,(E2 , <p) = -2 x t,

n t> Ixl Jt 2 - Ixl2

... ) - 3 (E ) - ~ i <p(x, Ix!) d III n - , 3, <p - 4 I I x .

n [R' x o

Remark 7 1) In all these cases, the elementary solution E studied above has its support contained in rt' n + 1 = {( x, t) E Q, t ~ I x I} which is the 'forward light cone' of ~n + 1.

However, in the case where n is odd and different from 1, this support is exactly equal to t = I x I which is none other than the singular support of the elementary solution (i.e. the smallest closed set, outside of which it is of class rt'OO), that is to say the boundary of the forward light cone rt'.+l. For all the other values of n, this support is the entire forward light one rt'.+ 1. 2) Further, outside of the boundary of the forward light cone, the elementary solution En(x, t) is an analytic function of (x, t) (44). 3) For n = 1, Ed x, t) is a function (constant over its support rt' 2). - For n = 2, E2 (x, t) is a Radon measure, which is absolutely continuous with

respect to the Lebesgue measure. - For n = 3, E3 (x, t) is a measure distributed over the boundary of the forward

light cone rt'4. - For n = 4, E4 (x, t) is no longer a measure. It is the derivative in the sense of

distributions of a measure distributed over the boundary of the forward light cone rt' 5.

This irregularity of En(x, t) increases with n, and beyond n = 4. 0

Remark 8. The elementary solution E1 and self-similar solutions(45) Setting E1 = E we look for E(x, t) the solution of

a2 E a2 E -2 - -2 = 0 for all t > 0 , at ax

aE E(x,O) = 0, at (x, 0) = <5(x) ,

E(x, t) with compact support in x for 'fixed' t > 0 .

Since E(kx, kt) has the same properties for k > 0, we see that

E(kx, kt) = E(x, t) ,

(44) Therefore the singular support of the elementary solution is for all n, the boundary of the forward light cone. (45) Compare with Remark 4 of §2.

§3. Wave Equations

and therefore, if ~ = x/t, E(~, 1) = f(~), we have:

E(x, t) = f(O ,

so that finally, f must be the even solution of

(~2 _ 1)1" + 2~f' = 0

35

where we look for distributional solutions. There is again 'uniqueness' of even solutions and we find

f( ~) = c. X, X = the characteristic function of] - 1, 1 [ .

We choose C in such a manner that iJE/iJt(x, t) ~ l5(x) as t ~ 0; if qJ E ~(IR), we have:

f + ooiJE f+oo~ --;-(x, t)qJ(x)dx = - -f'(~)qJ(t~)td~. -00 ut -00 t

Therefore, we must have: -f:: ~f'(~)d~ = 1 .

Now f'(~) = C[I5(~ + 1) - 15(~ - 1)], from which C = 1/2. Therefore

E(x, t) = ~ if Ixl < t, 0 otherwise. o

1.3.4. Hadamard's Principle of Descent. We shall firstly explain the (very simple) idea. Writing all x E IRn in the form x = (x', xn), x' E IRn-l, Xn E IR and noting that En is the solution (with suitable support) of

lfwe integrate in Xn (which, naturally for distributions, needs to be made precise -and is the object of the proposition below), and if we set

.!l'(x', t) = f En(x', xn, t)dxn

(the integral must have a meaning since we are integrating over a finite interval (46»), then

as

(46) Iff E G'(IR), then f f(x)dx = <f, I).

36 Chapter XIV. Cauchy Problems in ~.

As !l' has compact support for fixed t > 0 and !l' = 0 for t < 0, we have: !l' = En-I. Putting this in the form of distributions:

Proposition 1. Let En be the elementary solution of the operator On = :t22 - LI

with (supp En) contained in the forward light cone of ~n + 1.

For x E ~n, we set x = (x', xn ), x' = (Xl' ... , Xn- l ) E ~n-l. Let !l' be the linear form defined by

(3.47) (!l',<p) = (En,<p®lxn)' <pE~(~~;-l X ~,).

Then:

Proof 1st stage:

In effect, firstly [supp En n supp( <P ® 1)] is a compact set, therefore (!l', <p) given by (3.47) has a meaning. On the other hand, if <P --+ 0 in ~(~n), then (En' <P ® lxn) --+ 0 in C, therefore!l' is a distribution. 2nd stage:

supp!l' c {(x', t), Ix'i :::; t} = C(jn.

In effect, if <P E ~(~n) is such that supp <p n C(jn = 0, then supp(<p ® 1) n C(jn+ 1

= 0 also and consequently

(3.48) (!l', <p) = 0 for all <p of this type, from which we obtain the result .

3rd stage:

!l' = En-I.

For all <p E ~(~~;-l X ~,)' we have

now

therefore

(3.50) (On-l!l',<p) = (En,On(<p®lxJ) = (On En,<p®lxn)

= (<p ® lxnHO) = <p(0) . o

We now apply the principle of descent to calculate the elementary solution E2 and recover formula (3.46)ii) with the help of (3.46)iii). For all <p E ~(~~ X ~,)' we have:

(3.51 ) 1+ 00 dt i (E3 , <p) = -4 <p(x, t)dcr(x) , o nt Ixl;'

where dcr is the surface measure of the sphere with equation Ixl = t.


In effect:

1 i <p(x,lxl) 1+ 00-<E3'<P)=~ dx= t<p(t,t)dt,

4n 1R3 I x I 0

where

fp(t,1) = 4 1 2 r <p(x, t)dO'(x)

n1 J Ix 1= <

= average value of <p over the surface of the sphere with centre 0, radius 1;

from which we obtain (3.51). If we replace <p by 1/1 (8) 1, < E 3, 1/1 (8) 1) still has a meaning. Now

r _ <p(x, t)dO'(x) = ~xl =, I/I(x l , x 2 , t)dO'(x) + ~xl =, I/I(x l , X2, t)dO'(x) J I x I - , J ~3 ;. ° J ~3 .;; ° = 2 i tl/l(x l , x 2 , t) dx dx

2 2 2 I 2, IX'12.;;,2Jt -X I -X2

and therefore (by setting x' = (XI' X2))

(3.52) <E3,<p) = <E3,I/I(8)I)

= ~ dt I/I(x', t) = <E2 ,I/I) 1 1+ 00 i dx' 2n 0 Ix'12 .;;,2 Jt 2 - Ix'I 2

from Proposition 1. o 1.3.5. Regularity and Integral Expression of the Solution of the Cauchy Problem. It follows first of all from the uniqueness of the elementary solution that we have

Proposition 2. For all distributions f E f!fi' (IR~ X IR,) with

suppf c Q+, Q+ = {(x, t); x E IRn, t ~ O} ,

there exists a unique solution u E f!fi'(IR~ x IR,) with supp u c Q+ satisfying

(3.53) On u = f ; u is given by

(3.54) u = En *f. Proof In effect, if W an arbitrary solution of (3.53) with supp W c Q +,

En * On W has a meaning and is equal to: En * f Therefore

En * f = On(En * W) = {} * W = W

and from (3.54), W = u. 0

The Cauchy problem (called 'homogeneous') in f!fi'(IR~ x IR,) is then to find u E f!fi'(IR~ x IR,), suppu c Q+ satisfying

(3.55) { On u = u l (8) (}(t) + Uo (8) (}'(t)

uo, UI given in f!fi'(IR~) .

38 Chapter XIV. Cauchy Problems in iR"

Note that the distribution

f = u1 <8l <5 + Uo <8l <5' has for support ~n X {O} ,

therefore supp f c Q + .

It then follows from Proposition 2 that there exists a unique u satisfying (3.55). We now examine the regularity of the solution of the 'non-homogeneous' Cauchy problem in ~~ x ~t+:

az at 2 - Llu = f

u(x, 0) = UO(x)

au 1 at (x, 0) = u (x), X E W .

Let m be an integer ~ 2 and v an integer ~ (n - 3)/2. We suppose that

(3.56) {

UO is of class ~m+v+2 over ~n u 1 is of class ~m+v+ 1 over ~n

f is of class ~m + v over ~~ x ~t+ .

Then, we have (see H6rmander [1]):

(3.57) u is of class ~m over ~~ x ~t+ = Q + .

In the case of the homogeneous Cauchy problem, u is given by:

(3.58) u(x,t) = (Mu1)(x,t) + (:tMUO)(X,t),

o

where M is an averaging operator which we make explicit in the following manner for n = 1, 2, 3. lfn = I

til f+t (3.59) (Mv)(x, t) = 2 v(x + tOd~ = 2 v(x - ~)d~ , I~I < 1 -t

and (3.58) gives again the result which we have obtained when n = 1. lfn = 2

(3.60) (MV)(X,t)=!.-fi V(x+tOd~=~fi v(x-~) d~. 2n JI~I < 1 J1 - 1~12 2n JI~I <t Jt Z _ 1~12

lfn = 3

(3.61) (Mv)(x,t) = !.-fi v(x + t~)d(J(~) = -I-fi v(x - Od(J(O 4n J I ~ I ~ 1 4n t J I ~ I ~ t

(where d(J is the surface measure of the sphere). To verify formula (3.58), it is sufficient to make explicit the convolution u = En * f wheref = u1 <8l <5(t) + UO <8l <5'(t) over qJ E ~(~~ X ~t) of the form qJ = qJl ® qJz,


(i.e. <p(x, t) = <Pl(X)<P2(t)) by applying the definition of the convolution in the sense of distributions, see Appendix "Distributions". These formulae show notably that the results (3.56) may be improved in the particular cases n = 1, 2, 3; particularly:

for n = 1 Uo E rc m+1(1R), u1 E rcm(lR) ~ u E rc m+1(1R x IRr)

for n = 2 and 3 Uo E rc m+1(lRn), u 1 E rcm(lRn) ~ u E rc m(lRn+1 x IRr).

(v E rcm(lRn) implies Mv E rcm(lR" x IRr) for n = 2 and 3). This is directly due to Remark 7.

Remark 9. We establish that in space dimension n = 1, formula (3.58) which reduces to

if u 1 = 0, does not perform any regularisation in (x, t). In dimensions greater than 1, neither is there any regularisation in (x, t) which is not evident from formula (3.58). But we may note by Fourier transformation that

A( ) AO sintlyl Al u y, t = (cos Itly)u (y) + -IY-I- u (y).

There is therefore no regularisation. For example, if UO and u1 are in L2(1R) and therefore so are 14° and 141, as the functions cos(tlyl) and sin(tlyl)/lYI are multipliers of L 2, u(t) remains in L2. Conversely, Iyl' cos t Iyl and Iyl·-l sin tlyl (s > 1) are not

multipliers of L 2, and therefore the derivatives of u (and in particular ~~ (t)) are

not necessarily in L2(1R). 0

Remark 10. I nvariants of the wave equation. Consider the wave equation

iJ2u 02U 2 - 0 ot2 - C ox2 - , (E) in IRx x IRr .

The reader can verify that the following quantities are invariant in t(47)

12 = r ou ou dx . J~ ot ox

On the other hand if we set v = ~~, we have ~:~ - c2 ;:~ = 0 (differentiating (E)

with respect to t).

(47) Assuming that these quantities have a meaning.


We see therefore that

13 = r (~lovI2 + e2lovl2)dX = r (~I02UI2 + e21~12)dX Jill 2 at 2 ax Jill 2 ot2 2 oxot

and

are also invariant. More generally, by differentiating equation (E) n times with respect to t, we thus obtain a sequence of invariants of the wave equation. We recall some interesting physical interpretations of these invariants constructed from the wave equation. 1 n the case of the vibration of a string, or of a rod, ... , the energy of the vibrating system is (proportional to) the quantity 11; thus a state of the system with finite

energy corresponds to initial conditions UO and u1 such that ~:o E L2(1R) and

u 1 E L2(1R). Take the case of aeousties(48). The equations are (with e and Po constant)

1 el2 ~~ + Po div v = 0

ov Po at + gradp = 0 ;

u is the displacement, v is the velocity, p the (variation of) pressure, from which we deduce

Jp = o.

The speed v satisfies the equation

1 02V 2"-;-2 - grad(divv) = o. e ut

Since the displacement u satisfies

we therefore have

The quantity

u(t) = I v(r)dr ,

1 02U 2"-;-2 - grad(divu) = o. e ut

(48) See Chap. lA, § 1 and also Sect. 1.4.1.


is therefore invariant (amongst others): it is interpreted as an energy (kinetic + internal). 0

Remark 11. Naturally the study of §3.1.3 may be generalised to the problem

1 iPu -- - - Llu = 0 X E 1R 3 , t ~ 0 , c2 at 2 '

(3.62) u(x,O) = UO(x) , X E 1R3 ,

au 1 3 at (x, 0) = u (x), X E IR ,

with coli (this equation is equivalent to (3.2) by changing the scale c --. ct). It is useful to write the equivalent of formulae (3.58) and (3.61): the solution u(x, t) of problem (3.62) is given by

(3.63) 1 (1 a ) u(x,t) = ~(MUl)(X,t) + ~atMuO (x,t)

with

(3.64) (Mv)(x, t) = -41 f r v(x - ~)da(~) . net JI~I = ct

o

Remark 12. The preceding formulae allow us to verify that if the initial conditions (UO, u 1 ) have compact support, then the solution at time t also has compact support, which is a fundamental property of hyperbolic operators (see Chap. V, §3). More precisely, in 3 dimensions, ifuo and u1 have their support included in a ball of radius ro (which we suppose centred at the origin), then the solution u(t) at the moment t, has its support included in the shell ct - ro ~ Ixl ~ ct + roo At the limit (r ° --. 0) if the initial condition (UO, u 1) is a distribution with point support (Dirac mass), the solution u(t) at the moment t is a distribution with support over thespherelxl = ct(note that ifuo = Oandu1(x) = <>(x),u(t) = R(t) where R(t) is defined by (3.42) for c = 1); c appears therefore as the speed of propagation in equation (3.62). 0

1.4. Applications

1.4.1. Applications to Acoustics. Consider the following acoustic problem in 1R3 (the equation of small perturbations for a perfect compressible gas, see Chap. lA, § 1):

(3.65)

.) 1 ap d' 0 I 2" -;- + Po IV V =

C ut

.. ) av d 0 11 Po at + gra p =

iii) v(x,O) = VO(x)

iv) p(x,O) = pO(x)

42 Chapter XIV. Cauchy Problems in JR"

where p = p(x, t) is the (variation ofthe) pressure of the gas, v = v(x, t) E 1R3 is the velocity of the gas, e is the speed of sound, and Po the average density of the gas. By combining equations (3.65)i) and (3.65)ii), we obtain

1 a2 p (3.66) 2~ - Ap = O.

e ut

Moreover equations (3.65)i) and (3.65)iii) give the initial condition

(3 67) ap 0) 2 d' ° . at (. , = - poe IV v .

The 'pressure' p is therefore the solution of the wave problem (3.2) with initial conditions (3.65)iv) and (3.67). If at moment t = 0, the flow of the gas is (irrotational) such that curl Vo = 0, Vo = grad cpo, then at each moment t we have: curl v(t) = 0, v(t) = grad cp(t) with cp the velocity potential, the solution of the wave equation (3.66), with cp(O) = cpo

acp and Tt(O) = cpt = _po/po.

The speed of sound e appears as the speed of propagation of the wave equation satisfied by the 'pressure' and the velocity potential (49).

1.4.2. Application to Electromagnetism. Maxwell's equations in a vacuum may be written (50)

a2 A i) at 2 - A A = j , X E 1R 3 , t ~ 0

ii) A(x, 0) = A O(x) ,

. a2 cp I) at 2 - A cp = p ,

ii) cp(x,O) = cpO(x) ,

iii) ~~ (x,O) = cpl(X) , X E 1R3 ;

where A and cp are respectively the vector potential and the scalar potential, p is the charge density, and j the current density.

(49) Note that the hypothesis of small perturbations leads to (3.65) requiring a stricter mathematical framework than the spaces!JP' or [/": we shall look a priori for solutions with finite (acoustic) energy. See Remark 10.

(SO) See Chap. lA, §4, equations (4.67); f{J and A are chosen such that af{J + div A = 0 which at

necessitates f{J 1 + div A 0 = O.


In the case where P = j = 0 (absence of charge and of current) the problem in <p reduces to the wave problem (3.2), and the problem in A consists of a system of equations which uncouple for each component into the same wave equation. Each component of the potential vector is therefore the solution of the wave problem (3.2) with a velocity of propagation which is the speed of light in the vacuum, taken equal to 1 (see Chap. lA, §4) in the natural system of units chosen here.

1.4.3. Application to Elasticity. The problem of the motion of a continuous elastic medium, isotropic and homogeneous(51) occupying all of 1R 3, is given by

(3.68)

i) ~:~ - (oCL1u + pgraddivu) = 0

ii) u(x,O) = Uo(x) , x E 1R3

iii) ~~ (x, 0) = u1(x), X E 1R3

where u = u(x, t) E 1R3 represents the displacement field in the elastic medium, and the parameters rx and p are given by rx = /l/Po, P = (A. + Ill/Po where Po is the density of the medium and A. and /l are the Lame coefficients of elasticity(52). The existence and uniqueness of the solution of problem (3.68) in one of the spaces ~3, ff3, ff 13, ~'3 results directly from the study of the Cauchy problem for the wave equation in these spaces: by application of the divergence operator to (3.68)i), and by setting () = div u, we obtain () as the solution of the problem

02 () • o(} . ot2 - (IX + P)LJ() = 0, (}(O) = (}o = dlVUo, ot (0) = (}l = dlVU 1 ,

which allows us to obtain u as the solution of the nonhomogeneous problem:

02U ° OU 1 ot2 - rxLJu = pgrad(), u(O) = u , ot (0) = u .

We thus see two different propagation speeds appearing, one C1 = (rx + P)1/2 = ((A. + 2/l)/PO)1/2 for (), the other C2 = rx 1/2 = (/l/PO)1/2; it is possible to make

this precise by decomposing the vector field u into

u = v + w with div v = 0, w = grad <p •

Such a decomposition does not necessarily exist for all u E ~3, ff3, ffl\ ~13, and is not unique in ff ' 3 and ~'3. However we may show the existence and uniqueness of this decomposition in (L2 (1R 3 ))3 (see Chap. IX and Ladyzhenskaya [1]) with <p E W1(1R 3) (the Beppo-Levi space over 1R 3). Without limiting ourselves to this framework, we may see that if UO and u 1 admit the decomposition:

UO = VO + gradqJ0, u1 = v1 + gradqJl

with divvo = 0, divv 1 = 0,

(51) See Chap. lA, §2, equations (2.48). (52) See Chap. lA, §2, p. 37.


then, if we denote by v and qJ the solutions of the Cauchy problems:

82 v - - aAv = 0 8t 2

(3.69) v(O) = Vo with div Vo = 0

~~ (0) = Vi with div Vi = 0

(we shall verify easily that we have divv(t) = 0, 'It) and:

82 qJ 8t 2 - (a + P)AqJ = 0

(3.69)' qJ(O) = qJ0

8qJ (0) = 1 8t qJ

we obtain the solution of problem (3.68) with the decomposition:

u(t) = v(t) + grad qJ(t) with div v(t) = 0, 'It E ~ .

Thus the initial decomposition of u(O) and 8u (0) implies the decomposition of u(t) 8t

for all tE~. The problem in fjJ, (3.69)', leads to the wave equation (3.62) with a propagation speed Ci == (a + P)1/2 = ((A. + 2p,)/PO)1/2. The problem in v leads for each component in v to the wave equation (3.62) with a propagation speed C2 == ai /2

= (p,/PO)1/2. There is therefore superposition of two propagations: the incompressible part of the solution (v) propagates at the speed C2 , the other part propagates at the speed c i (53).

2. The Euler-Poisson--Darboux Equation

Introduction. Consider the wave problem in ~3 in the classical framework

82 v i) 8t 2 - Av = 0

(3.70) ii) v(x,O) = 0

iii) ~~ (x, 0) = g(x) ,

where 9 is given in a space of functions or of distributions E over ~3 (54).

(53) See Chap. lA, §2.5. (54) For example E = ,€2(1R3) so that (3.70) is satisfied in the classical sense.


If SZ the unit sphere of 1R3 (SZ = {x E 1R3; JI xf = I}). then, from (3.58), (3.61),

the solution of (3.70)i) is given by:

u(x, t) = tu(x, t) with

u(x, t) = -41 f g(x + ~t)dO"(O (55)

11. S2

where dO" is the element of area over SZ. We see that u is the solution of the singular Cauchy problem:

. oZu 2 au 1) ot Z + tat - Ju = 0

(3.71) ii) u(x, 0) = g(x)

... ) au ( 111 at x, 0) = 0 .

Equation (3.71)i) is a particular case of the Euler-Poisson-Darboux equation. The problem is singular since the Cauchy data is given at time t = 0, and the coefficient

of ~~ in (3.71)i) then becomes infinite.

In general, k being a given real parameter, the singular Cauchy problem in IRn for the Euler-Poisson-Darboux equation is:

. oZu k au 1) ot Z + tat - Ju = 0, X E IR n ,

(3.72) ii) u(x,O) = uo(x) (uo given in a space E over IRn),

... au 111) at (x, 0) = 0 .

For k = 0, the corresponding problem (3.72) is a (regular) Cauchy problem for the wave equation. Under conditions general enough on Uo (for example E = LZ(lRn) or Y"(W); see particularly Carroll [1] and Ames [1]) we may demonstrate the uniqueness of the solution of problem (3.72) in C6'Z(IR, E) for k ?: O. For k < 0, there is not, generally, uniqueness of the solution of (3.72): if uk(x, t) is a solution of (3.72h, Uk(X, t) + t1-kUZ_k(X, t) is also a solution of (3.72h where U2 - k(X, t) denotes a solution (for arbitrary uo) of (3.72h-k' (We note that (3.72) is invariant by changing t to - t, and therefore that problem (3.72) is well-posed in the backward sense (and therefore in IR) if it is well-posed in the forward sense (IRt)).

(55) This formula being generalised by convolution for g E !lii'([J;l3).

46 Chapter XIV. Cauchy Problems in Ihln

A particularly important and interesting problem is the one corresponding to the value of the parameter:

(3.73) k = n - 1 for x E W

(which is the situation of problem (3.71) above, k = 2 if n = 3!). In this case, the corresponding problem (3.72) has in fact for a solution (denoted un - 1 to recall (3.73)):

un-1(x, t) = .,({(uo; x, t) where .,({(uo; x, t) = ~ f uo(x + et)du(e)(56) is Qn sn-l

the average of Uo taken over the sphere sn - 1 = {x E IRn; I x I = 1} with Qn nn/2

= 2 r(n/2) = the area of the sphere sn-I in W.

Generally, we may, by using the method of 'generalised descent' analogous to that used for the wave equation, show that for k = n, n + 1, ... ; n E ~ the solution Uk

of the corresponding problem (3.72) is written

uk(x,t) = k+l-n uo(x + et)[1 - leI 2]-2-de Q i k-n+1

Qk+l I~I<I 2 ).12

h - n h . d - d dJ; ) (57) were Q). - rp,/2) (we ave set. e - del e2'" '>n ,

a formula which may be extended to all real k > n - 1, then for k < n - 1 byanalytic continuation) for k distinct from odd negative integers - 1, - 3, ... , these values of k play an exceptional role(58). The exceptional character of k = - 1 is

verified, in an elementary manner, by the following remark: if ~~ (x, 0) = 0 and if u

is the 'regular' solution of (3.72)i), then ~ ~u (x, t) -+ 02~ (x, 0) as t -+ 0 such that t vt ot

[~:~ - ~ ~~ ] -+ 0 if t -+ 0 and then (3.72)i) gives in the limit Llu(x, 0) = O. There

can be a regular solution only if Uo is harmonic. For these results, we refer to Weinstein [1], [2], [3] and to Diaz-Weinberger [1].

Remark 13. Recall the case of n = 3 and of problems (3.70) and (3.71), and consider the linear mapping u -+ v, let X" defined by:

{ u(x, t) = ! v(x, t)

V E ~2(1R, ~), zero for t = 0,

(56) This formula is extended by the convolution of Uo with the measure jl, = da( e) concentrated over Qn tn - I

the sphere tSn- 1 for Uo E !Zi'(lhln); as in the case of the wave equation (for n = 3), we have here Uo E ~m(lhln) => un-I E ~m(lhln X Ihl,). (57) For other expressions using the Fourier transform, and also for the elementary solution of (3.72) (with Uo = .5), see for example Carroll [I]. (58) Recall that for k < 0, there is no uniqueness of the solution of (3.72).


with inverse X ,- 1 defined by

v(x, t) = tu(x, t)

X, transforms the singular problem (3.71) into the regular problem (3.70). For the systematic study of the existence of operators - (analogous to X,) called transmutation operators - which transform a singular problem which is second order in t into a regular problem which is second order in t, we refer to J.-L. Lions [1] (Chap. XII and the bibliography of this work). For a study of Euler-Poisson-Darboux variational problems in spaces which are weighted to cancel the exceptional values of k, we refer to Langlais [1]. D

Remark 14. Some considerations analogous to those preceding may be developed by replacing the Laplacian by the elasticity operator (see Chap. lA, §2 anC: in this Chap. XIV, (3.68))

Ei(u) = eulu i + P ~ [ ± oUj ] OXi j= 1 OX j

i = 1, 2, ... , n with

U = (u 1 , u2 , ... , Un) a mapping from IRn -+ IRn.

In a precise manner, we are interested in the following singular Cauchy problem: we look for u = (u 1 , •.. , un) satisfying:

i = 1,2, ... , n

(3.74)

i = 1,2, ... , n .

As in the case of the Euler-Poisson-Darboux equation, we may show, particularly for k = n - 1, the uniqueness of the solution of problem (3.74) in rc 2 (1R, En) for some general spaces E, (E = L 2(lRn), Y'(lRn) for example-see for example Carroll [1] and Ames [1]; note again the invariance of problem (3.74) by changing t to - t; the solution of (3.74) is given for k = n - 1 by the expression:

where a = ~, b = .;;+Ii and where

For more details, we refer to Diaz [1]. D


3. An Application of §2 and 3: Viscoelasticity

Introduction. We wish to study the longitudinal displacements of small amplitude U(x, r)(59) of a 'long, stiff' metallic wire under a viscoelastic constitutive law (see Chap. lA, §3 and Levine [1]). We take first of all in the case of ~~ the viscoelastic constitutive law of Kelvin-Voigt for a viscoelastic material with short memory, given in Chap. lA, §3.4.1; it expresses the spherical and deviatoric tensors us, u D as functions oftensors corresponding to displacements by formulae (3.16) of Chap. lA, that is

{US = 3K(t: s + ()ee S )

(3.75) u8 = 2IL(t:8 + ()ge8)

which leads to the constitutive law(60):

1 Uij = At:Il Jij + 2ILt:ij + K ()eellJ ij + 2IL()g (eij - e~, J ij )

(3.76)

= [ At:1l + ( K()e - 2;()g )euJJij + 2IL(t:i + ()A) ;

if we set 2IL()

A' = K ()e - T' IL' = IL()g ;

we see that (3.76) becomes:

(3.77)

Return now to the law of motion:

(3.78)

and replace uij by expression (3.77), and t:ij by

1 t:ij = :2 (u i• j + uj,J .

We therefore obtain the equivalent of the Navier equations of elasticity(61)

(3 79) iJ2 U ( , d d' I a . p or2 - IL AU - 1\ + IL)gra IVU - IL or AU

- (A' + IL') :r graddiv U = f, where

(3.80) p ~2r~ - (A + 2IL)graddiv U + IL curl curl U - (A' + 21L') :r graddiv U

a + IL' or curl( curl U) = f·

(59) Here we denote time by T; we shall change the scale of time in (3.82). (60) We use here the convention of summation over repeated indices. (61) See Chap. lA, §2 and 3.


We shall now make some simplifications(62) to take into account the one-dimensional character (in IRx) of the wire problem. In the case where f = 0, we finally obtain the following mathematical model: the displacement U satisfies

(3.81) 1 i)~a2u _~~(a2U)_a2U =0(62) XEIR, r>O c2 ar2 0)0 ar ax 2 ax 2 '

ii) U(X,O) = UO(X) , au (X, 0) = UI(X), ar

the positive constant 0)0 which has the dimension of a frequency and the positive constant c which has the dimensions of a speed, can be expressed in terms of constants previously introduced. We suppose that the wire is infinitely long and that the initial displacement and velocity UO and U I are known at each point x.

Statement of problem. We pass from problem (3.81) to a dimensionless problem by the change(63) of variables

(3.82) 0)0

X = - X, t = 0)0 r c

and we set:

(3.83) def

u(x, t) U(X,r) = u(~x,_t ) 0)0 0)0

(3.81) becomes:

(3.84)

f i) ~:~ - 2 :t (::~) - ::~ = ° , l ii) u(x, 0) = UO(x) = UO(;O x),

xEIR, t>O

au 1 (c) -a (x, 0) = ul(x) = _U l -x . t 0)0 0)0

We may also make the change of variables:

(3.82)' cr = t, X = x (simple change of notation) .

By then settmg:

U(X,r) = U( x,~) = u(x,t) (3.83)'

0)0 and I: = -, (3.81) becomes:

c

{i) a2u _ ~ ~ a2u _ a2u = ° (64)

at 2 I: at ax 2 ax 2 (3.85) iJ 1

ii) u(x,O) = UO(x) = UO(x) , a~ (x, 0) = ul(x) = ~ UI(X) .

It is this problem (3.85) which we propose to study at present.

(62) This equation is analogous to IA (3.6), but for the one-dimensional case. (63) The interest in this change is essentially to simplify the formulae. (64) The study of (3.85) is often preferred to that of (3.84) in applications, in order to study the influence of the parameter E.

50 Chapter XIV. Cauchy Problems in ~n

Study of problem (3.85)

1) Study of problem (3.85) in g"(\R). Suppose that Uo and u 1 E g"(\R). We shall try to solve problem (3.85) in g"(\R). Perform, for this the Fourier transform in x. We denote by u = u(y, t) the Fourier transform of u(x, t); (3.85) becomes:

(3.86)

Let

{

. 02 U 2 0 2 A 2 A

1) ~ + - -;- y u + y u = 0 ut e ut

ii) u(y, 0) = UO(y) , ~(y,O) = u1(y). (65)

w == w(y) = 1 ~ IYIJy2 - e2 , Iyl > e,

1 -IYIJe2 - y2, Iyl < e. e

The differential equation (3.86)i) has for its general solution:

(3.87) Ii) Iyl > e, u(y, t) = e-t~(voCoshwt + v1 sinwt) y2

ii) Iyl < e, u(y, t) = e-t'(wocoswt + w1 sinwt).

The quantities Vo, V1, Wo, W 1 (which are functions of y) are determined by the initial conditions (3.86)ii). We find easily:

(3.88) {i) Iyl > e,

ii) Iyl < e ,

Thus the solution of (3.86) is written:

(3.89)

A() t~ [AO (h y2 shwt) A1 shwtJ u y,t = e-' u (y) c wt + -S~ + u (y)~ {

i) Iyl > e,

A ~ [A ( y2 sinwt) A sin wt ] ii) Iyl < e, u(y,t) = e- t • UO(y) coswt + -S--W + u1(y)--W .

We shall clarify in which sense (3.89) is the ~olution of (3.86). We first of all define the functions U ° and U 1 by:

{

- ~ [ y2 sh wt ] Iyl > e, Uo(y, t) = e- t • chwt + -S~

- ~ [ y2 sin wt ] Iyl > e, Uo(y, t) = e- t • coswt + -S--W

(3.90)

(65) With the notation ao and a' denoting the Fourier transforms of Uo and u" as in (3.16).

§3. Wave Equations

(3.91) { Iyl > e, iTdy, t) = e-ti/.-Sh:t

_ y2 sin rot Iyl < e, U1(y,t) = e- t£ __ .

ro

51

We remark that iTo and iT 1 are the respective solutions of (3.86) for the initial conditions:

(3.92) uO(y) = 1, u1(y) = 0

UO(y) = 0, u1(y) = 1 .

These solutions 00 and 01 may again be called (by analogy with the case of wave propagation) the fundamental solutions of (3.86). With the help of these two functions, 00 and 0 1 , the solution of (3.86) given by (3.89) becomes:

(3.93) u(y,t) = UO(y)Oo(y,t) + U1(y)01(y,t).

Remark 15. If t < 0, we see with the help of (3.87)i) that u(., t) does not

necessarily belong to 9'''(~) since e _t(y2 + lylJy2 - 1) ¢ 9'''(~) for t < 0, therefore

u(. , t) ¢ 9"'(~): there is not, in general, a solution of problem (3.85) for t < 0 .

This expresses the irreversibility of problem (3.85): the solution u of problem (3.85) may again be put in the form (3.32)', but {G(t)} is an evolution semigroup and not an evolution group(66). 0

Examination of the properties of the solution u thusfound: For all t > 0, iT 0(.' t) and 0 1 (. , t) are multipliers: 0 0(. , t) and 0 d. , t) E (9 M(~Y) (67).

We verify easily that the derivatives in y of Oo(Y, t) and 0 1 (y, t) are bounded by polynomials in y (for all given t).

We could show that the mappings t -+ u(., t) and t -+ ~~ (., t) are continuous

from ~ to 9"'(~) and take for their values at the origin UO and u 1 respectively. Consequently, problem (3.85) is well-posed in 9"'(~) for all t > 0 and (3.85) has a unique solution u(x, t) whose Fourier transform is given by (3.89) (or (3.93».

2) Study of problem (3.85) in L 2(~). Suppose for the moment that U O and u 1 E L2(~). Observe that the fundamental functions 0 0 and 0 1 of (3.86) are bounded for all t > 0, therefore:

0o(.,t) and Ol(.,t)ELOO(~).

Thus the mapping of L2(~) x L2(~) -+ L2(~)

(uo,ud -+ u(x,t) = §";I U(y,t) = §";l(UO(y)Oo(y,t) + U1 (y)U 1(y,t»

is continuous (by composition of continuous mappings).

(66) See Chap. XVII. (67) See Appendix "Distributions".

52 Chapter XIVo Cauchy Problems in [hln

Further, for t > 0, a~o (y, t) is given by:

(3094) {

avo ~ shwt i) Iyl > e, Tt(y,t) = e- ' , ----;-(_y2) ,

1010) Iyl avo () -I~ sinwt ( 2) 0 < e, Tt y,t = e '~ -y ,

avo therefore, Tt(y, t) E LOO(lRy) for all t > O.

Likewise we would show that: a~l (y, t) E LOO(lRy) for all t > o. We deduce that the mapping:

(3.95) (UO,U1)H ~~(.,t) = g-;l(uO(y)a~O (y,t) + Ul(y)a~l (y,t))

is continuous from L2(1R) x L2(1R) for all t > o. We could show that the mappings

t -+ u(., t) E L2(1R), t -+ ~~ (., t) E U(IR) are continuous from IR+ into L2(1R) and

take the values UO and u 1 respectively for their values at the origin. Consequently, problem (3.85) is well-posed in L 2(1R), t > O. We remark that this is not the case for the wave equation (see §3.1).

Remark 16. The mathematical model (3.81). The study of the resolution of problem (3.85) has shown us some profound differences between 0 the wave equation and equation (3.81)i).

The viscosity term [ :02 :r (:~~) ] in (3.81 )i) implies that problem (3.81) has a

solution in Y"(IR) (and in U(IR)) only for t > 0, whereas the Cauchy problem for the wave equation has a solution in 9"'(IR) for all t. Formula (3.89) allows us to understand the behaviour of the solution u of (3.85) for e -+ 00; we note that for each y, with Iyl < I; for e -+ 00, we have w -+ Iyl and

A( AO 1 sinl Ylt u y, t) -+ u (y)coslylt + a (y) -Iy-I-

which is the solution of the wave equation. The method which consists of approaching the wave problem via problem (3.81) is called the method of parabolic regularisation (see Chap. XVIII). We shall remark again on the difference between problem (3.81) and the problem:

{

~ a2 U + ~ au _ a2 U = 0 c2 ar2 wo ar ax 2

(3.96)

U(X,O) = UO(X) , ~~ (X, 0) = U1(X)

(see Example 5, §3 of Chap. XVII B). In 9"'(IR), or in a Hilbert space such that

§4. The Schriidinger Equation 53

U(., r) E Hl(IR), ~~ (., r) E L2(1R), the solution U of (3.96) is again given by a

formula of the form (3.32)' where {G(t)} is an evolution group. (3.96) conserves the hyperbolic character of the wave equation, contrary to (3.81)i) which is parabolic by nature. 0

We may verify that the solution u of (3.85) is, for all given U O and u 1 E L 2(1R), regular in t (~OO and even analytic), but is not necessarily regular in x for given t, t > 0, as opposed to the solution of the heat flow problem (see §2 and also §3, the end of Sect. 1.3.1).

§4. The Cauchy Problem for the Schrodinger Equation, Introduction

The reader has found an introduction to the Schrodinger equation in Chap. lA, §6. We shall only treat elementary cases in this Chap. XIV. We shall return to more 'realistic' cases in Chaps. XVII and XVIII.

1. Model Problem 1. The Case of Zero Potential

We look for u(x, t) = u satisfying(68)

(4.1) {i) i ~~ + Ltu = 0 In IR~ x IRt+ ,

ii) u(O) = Uo .

Looking for solutions of (4.1) in 9"(lRn ), we obtain after taking the transform in x, the equivalent system:

(4.2) {au 2' 0

i,at - I~I u(t) =

u(O) = UO •

Fourier

Then by continuing as in §2 and 3 ofthis Chap. XIV, we obtain the unique solution u(t) given by

(4.3)

The Cauchy problem for (4.2) is well-posed in .@'(IR~) and in 9"(IR~) and is reversible, since the function R(. ,t):y --+ e-ilyl2t is in (DM(IR~) for all fixed t. Again denoting by R(t) the operator of multiplication in 9"(lRn) by R(. ,t) we establish that the family {R(t)}tE~ is a group in 9"(lRn).lfinstead of taking Uo in

(68) The system of units adopted in order to write the Schriidinger equation in the form (4.l)i) is such that ii = 1 and m = 1/2; see Chap. lA, §6.


9"(W), we take it in L 2(lRn)(or, which is equivalent, if we take Uo in L 2(lRn», we see that (4.3) defines u(t) E L 2(W) (where 12(. , t) = u(t». Further

(4.4) II u(t) IIL2(~') = 11120 IIL2(w) .

The group {R(t)}t E ~ is then a unitary group in L 2(lRn) (see Chap. XVII) and problem (4.2) is again well-posed in L 2 (IRn). By returning to the space 9"(lRn) by taking the inverse Fourier transform in y, we obtain a unique solution to problem (4.1) in 9"(lRn), given by

(4.5) u(x, t) = ff; l(R(., t» * Uo = R(., t) * Uo . (x) (x)

From the calculation performed in Appendix "Distributions", §3, of Vol. 2

( 1 i)n (iIXI2) (4.6) R(.,t) = ff;l(R(.,t» = (4nt)-nI2 ~ exp 4t

( nn) (iIXI2) (69) = (4nt)- nI2 exp -i 4 exp 4t

If we set for all t E IR

(4.7) G(t) = R(., t) * x

the family of convolution operators {G(t)}t E ~ is a group in 9'(lRn) and 9"(lRn), and a unitary group in L 2(W).

To summarise, we have

Theorem 1. i) The Cauchy problem (4.1) is well-posed in the spaces 9'(lRn), 9"(lRn) and U(lRn). ii) The problem is reversible. iii) The solution is given by:

(4.8) u(t) = G(t)uo

where {G(t)}tE~ is a group in 9'(W), 9"(lRn) and a unitary group in L2(lRn). iv) For Uo E L2(lRn), we have

(4.9)

Remark 1. We note that the action of G(t) over Uo does not have a regularising 6d 0

(69) With the convention, i -n/2 = (exp _ i n;), this formula may again be rewritten:

( IXI2) R(. ,t) = (4n:it)-n/2 exp --4it

which we can compare with (2.29) (changing t to it).

§4. The Schriidinger Equation 55

Remark 2. Elementary solution(70). We look for E(t) with support in IR~ x IRt satisfying in the sense of E&'(IR~ x IRt).

(4.10) oE(t)

i at + AE(t) = - b(x) ® b(t) .

By Fourier transform in x, (4.10) is equivalent in 9"(lRn ) to

(4.11) a ~ -

i at E(t) - lyl2 E(t) = - l(y) ® b(t) .

Then

(4.12) E(t) = iY(t). R(t) ;

from which we have the unique elementary tempered solution of (4.10)(71):

(4.13) E(t) = i Y(t). R(t)

where R(t) is given by (4.6). Some consideration analogous to those developed in §2 and 3 for the nonhomogeneous Cauchy problem may be developed here. D

Remark 3. Calculation of the elementary solution by self-similar solutions (Compare with Remark 4 of §2 and with Remark 8 of §3). We look for E(x, t) (in 1 space dimension), the solution of

(4.14)

with

(4.15) E(x,O) = ib(x),

and E(x, t) tempered in x. Then IkIE(kx, k2 t) satisfies the same conditions for all t E IR, therefore

IkIE(kx, k2 t) = E(x, t) ;

by taking k = l/jt, and setting ~ = xt- 1/2 :

(4.16) 1

E(x, t) = jtf(~), f(~) = E(~, 1) ,

and f(~) is even in ~ (making k = - 1). Equation (4.14) is satisfied if and only if

(4.17) - i[f(~) + ~f'(~)] + 2f"(~) = 0 .

As in Remark 4, §2, we see that there is only one even solution (up to a

(70) See Chap. V, §2. (71) The elementary solution E defined by (4.13) and (4.6) does not correspond to a function of Lloc(lR~ x IR,) (for n > 1). Indeed, we must use finite parts (Pf) to define E directly from (4.6).

56 Chapter XIV. Cauchy Problems in [Rn

multiplicative constant) of this equation, therefore

From which

(4.18)

i~2

f(~) = Ce4 .

E(x, t) C ix 2

= ~e4r -Ii ' and we must choose C so that E(x, 0) = ib(x). If we take qJ E ~(rR),

> 0

(4.19) f E(x, t)qJ(x)dx = ~ ff(xt-1/Z)qJ(X)dX

and we must have in the limit

(4.20) f + 00 i~2

C e4d~ -00

i.

We may give a sense to this integral; it is 'Fresnel integrai'. Its calculation, see 1 i

Appendix "Distributions", §3, Vol. 2 leads to C = - Ft:: x -1--" Finally, v 2n + I

(4.18)' ( ) ( liZ) (72) (73)

E(x,t) = Y(t)exp i~ .(4nt)-I/Z.exp - :it tErR, xErR.

Remark 4. Here is another procedure (presented in a formal manner but which can be justified) for the calculation of E. Let F(x, t; A) be the solution of

where ). E C. We have:

A of _ OZ F = 0 ot ox2 ' F(x, 0; A) = Ab(x)

F(x, t; 1) = _1_ e-x2/4r

2fo and E(x, t) is the solution of (4.14), (4.15)

E(x, t) = - F(x, t; - i) .

(72) The reader will establish the analogies between the formulae (4.18)' and (2.13). These analogies will be studied in Chap. XVIIA, §6. (73) This expression may equally well be written

( rr) ( IXI2) E(x, t) = iY(t)exp - i 4 (4rrt)-1/2 exp i 4t '

which illustrates the similarity between (4.6) and (4.13).

§4. The Schrodinger Equation 57

But (formally) taking t complex, we see that

G(x, t) = A-I F(x, At; A)

. aG aZG satIsfies at - axz = 0, G(x,O) = c5(x), therefore G(x, t) = F(x, t; 1) and con-

sequently A-I F(x, At; A) = F(x, t; 1)

therefore

F(x t· A) = AF(X ~. 1) = A e-;'x2 /4t , , , A' 2) ntA - 1

Taking A = - i (and with Ji = eilt /4 ) this again gives (4.18). o

2. Model Problem 2. The Case of a Harmonic Oscillator

Following the example of a potential v(x) = X Z cited in Chaps. lA, §6 and IXB, §2, we study the following problem. We look for u satisfying:

{i) i ~Ut + -21 ,1u - -21lxlzu = 0 in IR~ x IRt+ (74)

(4.21) u

ii) u(O) = Uo .

If we take the Fourier transform with respect to x, we obtain the equivalent system in 9"'(lRn)

(4.22) { . au 1 A 1 Z A

I at + 2,1 yu - 21yl u = 0

u(O) = uo ,

and we see that the partial differential equation (4.21) is invariant under the inverse Fourier transform. This means that the Hamiltonian operator Jf

def 1 Ixlz I (75) (4.23) Jf = - 2,1 + 2

which is self-adjoint in L2(lRn), commutes with the Fourier transform in LZ(lRn), which is a bounded unitary operator ~ of L Z(IR") into itself (see Chaps. VI and VIII). It follows that the operators ~ and Jf have a common basis of eigenvectors. We know, for example in dimension n = 1, that the Fourier transform has the Hermite functions as eigenvectors. More precisely, if Hk is the kth Hermite function, we have (see Chap. VIII, §2):

(4.24) ~(Hk) = (i)k Hk , kEN

and the countable system of Hk constitutes a basis of the Hilbert space LZ(IR"). The

(74) Here, we have taken m = 1. (75) For the definition of the domain of this operator in L2([Rn), see Chaps. XV and IXB, §2.

58 Chapter XIV. Cauchy Problems in [J;l"

Hk are also the eigenfunctions of the Hamiltonian Yf. We may then solve problem (4.21) in dimension n = 1 in the space L2(~) (in arbitrary dimension n, we shall take with k = (kl' ... ,kn) E Nn)

Hk(X) = CkHk,(xd· .. HkJXn)

where Ck is a normalisation constant). We now examine the case where Uo is given in L2(~). Then

00

(4.25) Uo = L 1l0kHk with k=O

We look for u(t): x -+ u(t, x) in the form:

00

(4.26) u(t) = L Ilk(t)Hk k=O

the solution of (4.21). By substituting (formally) this expression in (4.21) and subject to justifying the differentiation operations, we obtain a system of ordinary differential equations:

(4.27) { ill~(t) - Pkllk(t) = 0

Ilk(O) = 1l0k' k = 0, 1,2, ...

where Pk denotes the eigenvalue of Yf relative to Hk (Pk = k + 1/2, see Chap. VIII, §2). The Ilk are determined uniquely in the following manner

(4.28)

We note that this defines u(t) E L2(~n), since thanks to (4.25)

00

(4.29) L Illk(tW < + 00 . k=O

It remains to be seen in what sense (4.21) is satisfied. This method of solution or method of diagonalisation is the subject of Chap. XV to which we refer. This problem is treated there in §5.

§5. The Cauchy Problem for Evolution Equations Related to Convolution Products

1. Setting of Problem

We return to the wave problem in dimension n = 1 which we have treated in §3, and set

(5.1)

§5. Equations Related to Convolution Products

Then equation (3.2)i) may be written (for n > 1):

auz aZUl

at axz ' (5.2)

Consequently, if we introduce the following vector U and matrix A:

(5.3)

(3.2) is equivalent to

(5.4) {i) au + AU = 0

at

ii) U(O) = Uo = (UO,u l ).

59

Similarly, the problems posed in §2, §3, §4 are in the sequel more general, classes of problems. Denote by d(DJ a matrix of linear differential operators with constant coeffi-cients:

(5.5) { .91 == d(Dx) = (aij(DJ)i,i= 1. .... N

aij(Dx) polynomials in -aa , ... , -aa with constant coefficients. Xl Xn

We look for a vector U = (u l , . .. , UN) where ui:(x,t) -+ ui(x,t), i = 1,2, ... ,N, satisfying

{ .) au 1 at + dU = 0

ii) U(x,O) = Uo(X) = (UOI "'" UON ) given.

(5.6)

Remark 1. Every linear differential operator with constant coefficients may be represented by a convolution(76), we can again write (5.6)i) in a more general form (that is to say, encircling the case where .91 is no longer a matrix of 'coefficients of linear differential operators' with constant coefficients):

(5.7) au --a + .91* U = O. t x

o

2. The Method of the Fourier Transform

We shall now impose a supplementary condition on U; in this manner, problems (2.1), (3.1), (4.1) will be on an equal footing.

(76) See Appendix "Distributions", Vol. 2, §2.4: P(D)u = (P(D)b) * u.

60 Chapter XIV. Cauchy Problems in IJiln

We suppose that £1 is a vector space (of distributions) in which the Fourier transform !!F operates. We then denote by :it 1 the image of £1 under !!F; therefore

(S.8) ·Yf'1 = !!F £1 ;

for example £1 = 9"(IR") or £1 = L2(1R") and we suppose that

(S.9) { for all t > 0, u(t):x -+ u(x, t) belongs to £1

u(t) a function or distribution of the variable x .

We set £ = £f, :it = :itf (77). The formulation of the Cauchy problem is then:

(S10) {we look for a function t -+ Vet) defined on IR+, with values in £ . satisfying (S.6), Vo being given in £.

We now carry out a Fourier transform in x (dual variable y). For u E £1' we shall have 0 = (01"", ON) E:it and system (S.6) is then equivalent to:

(S.11 ) { ;) ~~,(,) + .,#(;y). U(I) ~ 0

ii) V(O) = Vo ,

where 0 is a function or a distribution (which is vectorial) in the variable Y and where

d(iy) = (aap(iYl,iY2"" 'iY"))a.p:1. .... N;

:it being the space of distributions over IR;, we use the same method for the resolution of (S.11) (called the variation of 'constants') as was used in §2, §3, §4. Thus, we obtain in [.@'(IR~)]N ::::l :it, the unique solution of(S.I1)

(S.12) Oy(t) = exp( - td(iy)). OOy ;

(S.12) is valid for all distributions OOy and not only for OOy E :it since, as the aap are polynomials in y, each coefficient of exp(t d(iy)) is in ct'OO(IR"). We may then distinguish two principal cases: First case. For all t ~ 0 formula (S.12) defines an element of:it which is the same as saying that the function y -+ exp(t d(iy)) is, for t ~ 0, a muitiplier(78) of :it. Then problem (S.10) has a unique solution for all Vo E £. We then say that problem (S.10) is well-posed in £. If £ = £f, we say, by an abuse of language that problem (S.lO) is well-posed in £1' Second case. For certain Vo in £, the right hand side of (S.12) is not in :it and the Cauchy problem is ill-posed in £(79).

(77) Or more generally .Jft' = .Jft'l X ••• x.Jft'N for UOi E .Yfi' i = I to N. Thus in the case of the wave problem (5.4), we may take

.Jft'l = HI(IJil) , .Jft'2 = L2(1Jil) or .Jft'l = L2(1Jil) , .Jft'o = H-1(1Jil).

(78) This generalises the usual notion of a mUltiplier. (Note that for each y, exp(t d(iy)) is a priori a nondiagonal matrix). (79) If .Jft' = .Jft'~, we then also say that the Cauchy problem is ill-posed in .Jft'l'

§s. Equations Related to Convolution Products 61

Remark 2. If problem (5.10) is well-posed (for t ~ 0) in Yf, the solution of(5.10) is given by:

(5.13)

where Uy(t) is given by (5.12). D

Remark 3. 1) If problem (5.10) is assumed well-posed for t ~ 0, the function R(t):y ........ exp(td(iy)), which we call the resolvant of system (5.11), defines a semigroup in if. If we set G(t) = R(t) * = ~; 1 (R(t)) * ,then the solution of (5.10) given by (5.13)

x x

may be written

(5.14) V(t) = G(t) Vo

and {G(t)}, " 0 is a semigroup in Yf. 2) Naturally, it may be that problem (5.10)-(5.11) is also posed for all t ~ O. Then {R(t)} t E ~ defines a group in if and problem (5.11) is reversible. It is then the same for problem (5.10) and {G(t)} is a group which operates in Yf. This is the case in the problems considered in §3 and §4. D

Remark 4. We look for u, the solution of the following problem:

(5.15) t 1 !2~ + L1u = 0

au u(O) = uo , "&(0) = U 1 ;

(5.15) becomes, by Fourier transform in 9"([Rn) (for given U o and U 1 in 9"([Rn)).

(5.16) ld2 U 2 ' - - Iyl u = 0 dt 2

'(0) _' du(O) _ ' u - uo , dt - u.

whose general solution is given by

(5.17) sinh (I y It) ,

u(t) = cosh(IYlt)uo + Iyl U 1 .

. sinh (I y It) . The functIOns y ........ cosh (I y It) and y ........ are not III (I) M for t i= 0, there-

Iyl fore problem (5.15) is ill-posed in Yfl = 9"([Rn)(80), or even, for example, in Yfl = U([Rn). D

a2

(80) The operator - + Ll is elliptic; every Cauchy problem for an elliptic operator is ill-posed in at2

Y"(see Chap. V, §3 and §4; this follows essentially from Corollary 1, §3 and from Proposition 8, §4).

62 Chapter XIV. Cauchy Problems in ~"

Remark 5. Hyperbolic Systems. Suppose that problem (5.10) is well-posed. The solution being given by (5.14), R(t) is a matrix (ri/x, t»i,j = 1, ... ,N. In other words:

(5.18) N {if U(t) = (Ul (t), ... , UN(t»

ui(x, f) = jf:l

rij(x, t) ! uo/x) .

We then say that the system is hyperbolic if the functions (or the distributions) x -+ rij(x, t) have compact support(8l). In this case, note that (5.18) has a sense for arbitrary

UOj E g&'(lRn), j = 1, ... ,N .

Thus (5.18) (that is to say (5.14» defines a solution of problem (5.10) with values in [g&'(lRn)]N when the system is hyperbolic and the given data UOj E g&'(lRn). We also have here the uniqueness of this solution in [g&'(lRn)]N. 0

To summarise, we establish that there are three sorts of Cauchy problems: I) ill-posed problems; 2) well-posed problems if we impose conditions of growth to infinity on the space variables; 3) hyperbolic problems, which are well-posed in g&' (IR") (i.e. upon which we are not obliged to impose conditions of growth to infinity in the space variables.

Remark 6. Non-homogeneous problems. Now let F: t -+ F(t) = (J;, (t), ... ,!n(t» be a function defined for t > 0 with values in Yef and, for example, continuous. We look for t -+ U(t) of class eel in t > 0 and of class ceO for t ~ 0, satisfying

{ dUdt(t) + d U(t) = F(t)

(5.19)

U(O) = Uo .

Then by Fourier transform in x, we obtain the system equivalent to (5.19)

{dU(t) A A T' + ~(iy)U(t) = F(t)

U(O) = Uo .

(5.20)

The unique solution of (5.20) is given in [g&'(IR")]N by

(5.21) U(t) = e-·J(iy)tUo + f>-·"'(iYHt-U)F(Ci)dCi,

and if for t ~ 0, R(t) is a multiplier of ie, the unique solution of (5.19) is given by

1 U(t) = G(t)Uo + f~ G(t - Ci)F(Ci)dCi

(5.22)

(i.e. U(t) = R(t) * Uo + R(t) * Y.F), (x) (X.I)

IHI) See also in Chap. V the notion of hyperbolicity.

§S. Equations Related to Convolution Products 63

a formulae to be compared with formulae (1.7) and (1.8) of §1. o Remark 7. The preceding considerations are applicable to an evolution system of the type:

(5.23)

of type (5.19) where .s;I = ;j * is a matrix of convolution operators in x. These x

problems are always well-posed in 9" (IRn) if the rij(., t) are in (9 M' o

3. The Dirac Equation for a Free Particle

We have already met the Dirac equation in Chap. lA, §6. With a system of units such that c = h = m = 1, the equation for a free particle can be written

(5.24) . at/! 1-=

at - (i L (X.i~ - f3)t/! j=1.2.3 oXj

or again, explicitly (82) :

.0 I at t/!1

.0 I at t/!2

(5.25) a

i at t/!3

.0 I at t/!4

-1 a a a

0 i- i-+-oX3 oX1 oX2 a 0 . a t/!1

0 -1 i--- -1-OX1 OX2 oX3 t/!2

a a a 0 t/!3 i- i-+-oX3 oX1 oX2 t/!4

. a a a 0 1--- -i-

ax! oX2 OX3

(82) Note that there are other possible choices of the matrices [Xi and (3 (see Itzykson-Zuber r 1]).


We have seen in Chap. IX that the Dirac 'Hamiltonian' operator for a free particle

" .0 H = - i L... (Xl - + fJ j=1,2,3 oXj

(5.26)

given explicitly by the first matrix on the right hand side of (5.25), defines a selfadjoint operator in [L 2 ([R3 )]4.

We shall try to solve (5.24) with an initial condition 1/1(0) = 1/10 given in :ff

= L 2([R3)4.

From which we have the Cauchy problem: find 1/1 satisfying:

(5.27) { 01/1 + ( L (Xj~ + ifJ)I/I = 0 ot j=1,2,3 oXj

1/1(0) = 1/10 .

Setting

(5.28)

(5.27) becomes

(5.29)

or again:

(5.30)

with

(5.31)

{ ou " . 0 - + L. (Xl-U = 0 ot j= 1,2,3 OXj

u(O) = Uo

{ ou + ,s;1(~)u = 0 ot ox

u(O) = Uo

" .0 + L. (Xl-

j=!,2,3 OXj

0

0

o OX3

o . 0 ---l-

ox! OX2

0

0

o . 0 --+1-OX 1 OX2

o OX3

0 OX 3

0 0 ---i-

ox! OX 2

o

o

which is a problem of type (5.6).

0 0 --+i-

ox! OX2

0 OX3

o

o

(83) We use here the notation u and Uo instead of V and Vo, we have - in this example - no risk of confusion.

§5. Equations Related to Convolution Products 65

Therefore we apply the Fourier transform with respect to x, which leads to:

(5.32) {au A

~t + d.(iY)U = 0

u(O) = Uo

with

d(iy) = d(iy)

~ -c ;~~;~ y,

o o

From which we have the solution, from (5.12)

(5.33) u(t) = exp( - td(iy))uo .

We now must calculate the matrix exp( - td(iy)) explicitly. Taking into account Chap. lA, §6, we show that

(5.34)

exp( - td(iy)) = exp ( - it. L !X.i yi ) = /coslylt - i. L !X.li~isinIYlt, )=1,2,3 )=1,2,3 Y

where I y I = J yi + y~ + y~ and / is the 4 x 4 unit matrix. We can write this as:

sinlylt exp( - td(iy)) = /coslylt - -IY-I-d(iy).

Note that

d(iy)uo(y) = A(:x )uo(Y) ;

from which

• • sin ( I y It) ( a ) (5.35) u(y, t) = cos(IYlt)uo(Y) - Iyl d ax uo(Y)·

We recover the form of the solution of the wave equation (see (3.31)). We have seen

(in §3) that coslylt and sinlylt are multipliers (e (9M(1R3)) for any fixed t, which Iyl

allows us to envisage more general spaces for Uo than (L 2 ) 4. Thus, if Uo is in (9"(1R 3 )4 then so is u(t), t e IR. The Cauchy problem (5.32) is therefore well-posed in [9"(1R 3 )]4. The family of operators e -ld(iy) = R(t) forms a group in (9")4. The system (5.32) is therefore reversible in (9"(1R 3 ))4. If Uo is in (L2 (1R 3 )\ the norm of u(t) is conserved throughout time

Ilu(t)II(L2)4 = lIuo ll(L2)4, rtt e IR.

66 Chapter XIV. Cauchy Problems in ~n

The group defined by R(t) introduced in Remark 3 is therefore unitary in (L 2 ([R3))4. By the inverse Fourier transform, the unique solution of problem (5.30) is given by

u(t) = ~-l(coslylt)*uo - ~-l(si~~flt)*d(:x)uo.

W h . 3 S h' d sin I y It. d e ave seen In §, ect. 1.3.1., t at Since coslylt an -Iy-I- are In (!)M an are

entire holomorphic of the type exponential, their inverse Fourier transforms have compact support in x, for all fixed t, and the system is hyperbolic from Remark 5. This allows us, from Appendix "Distributions" (Vol. 2), to define convolutions with very large classes of uo: we may take

Uo E [.@'([R3)]4 .

The solution I/I(t) of problem (5.27) is then in [.@'([R3)]4 for all t (returning to the function 1/1 with the help of (5.28)).

Remark 8. We have thus obtained an explicit expression for the solution 1/1 of the Cauchy problem (5.27), (5.28) with the help of the Fourier transform. This method is analogous to the method of solution by expansion in plane waves, in current use in physics (see particularly Itzykson-Zuber [1], pp. 44 to 63). 0

§6. An Abstract Cauchy Problem. Ovsyannikov's Theorem

The abstract theorem which we shall now demonstrate is a generalisation of the classical Cauchy-Kowalewsky theorem to solve the Cauchy problem for partial differential equations in the class of analytic functions. For this we start by giving

Definition 1. Let S = {X p } p ;. 0 be a family of Banach spaces where for all P ~ 0, X p is a vector subspace of X o. We suppose in addition

(6.1) X p' c+ Xp and II· lip :::; II· lip' , P' > P

where II .11 p denotes the norm of X p' Such a family of Banach spaces is called a scale of Banach spaces.

Setting of problem. We are given: 1) two constants > 0, Po, Ao and a function UO having the following properties: for P E [0, Po [, UO is continuous in [0, Ao (Po - p)] with values in X p and satisfies

(6.2) IluO(t)lIp :::; Ro , P E [0, Po[, t E [0, Ao(Po - p)[

where Ro is a real constant > 0;

§6. Ovsyannikov's Theorem

2) a function F: (t, s, u) ~ F (t, s, u) continuous from

{O ~ t ~ T} x {O ~ s ~ T} x {uEXp.;llullp. < R}

with values in X P' where

R > Ro > 0, T > 0, 0 ~ P < P' < Po ,

satisfying, besides:

(6.3) F(t, s, 0) = 0 ,

and

(6.4)

there exists a constant C > 0 independent of (t, s, u, v, p, p') such that

Ilu - v lip' II F(t, s, u) - F(t,s, v) lip ~ C----c,-----'-P - P

for all p, p' with P < P' < Po and for all u, v E X p'

with II u lip· < R, II v lip' < R, t, s E [0, T] .

This set, we look for u in X p satisfying:

(6.5) u(t) = UO(t) + L F (t, s, u(s)) ds for t ~ 0 .

67

We shall demonstrate the following theorem of Ovsyannikov and for the proof we refer to Kano-Nishida [1].

Theorem 1. If UO and F satisfy (6.2), (6.3), (6.4), then (6.5) has a unique solution U and there exists a constant a > 0 such that we have: U is continuous in [0, a(po - p) [ with values in X P' (p < Po), and satisfies

(6.6) II U(t) lip < R, tE[O,a(po - p)[.

Remark 1. Condition (6.3) is not restrictive. In effect, if

. K F(t, s, 0) #- 0 wIth II F(t, s, 0) lip ~ ,

Po - P then

II UO(t) + t F (t, s, 0) ds lip ~ Ro + KA o ,

so that we are reduced to the case of a function F satisfying (6.3) by changing

i) UO(t) into UO(t) + t F(t, s, O)ds

ii) F(t, s, u) into F(t, s, u) - F(t, s, 0) . D

Proof of Theorem 1. Denoting by X the space of functions u which for all P E [0, Po [ and t E [0, a(po - p) [ are continuous functions of t with values in X p

68 Chapter XIV. Cauchy Problems in 1Rl"

and such that

(6.7) M[u] sup {IIU(t)ll p (l- t )}<+oo; pe[O,po[ a(po - p)

le[O,a(po-p)[

M[u] is a norm over X, and X equipped with this norm is a Banach space. We look for the solution of the integral equation (6.5) whose norm M[u] is finite for a suitable constant a > 0.

1. Demonstration of existence

1st stage. To show the existence, we use a method of successive approximations: we define by recurrence a sequence {Uk} kEN with

{i) uo(t) = UO(t)

~~ r ii) Un + 1 (t) = UO(t) + Jo F(t, S, un(s))ds ,

where

(6.9) II udt) lip < R for ° ~ t < adpo - p), ° ~ p < Po

where ak satisfies

(6.10) ao

ak + 1 = ak - 2k + 2 '

ao . lim ak = -2 = a, ao to be chosen later. Then If we set

k --+ + 00

so that

(6.11 )

Vk is for all p E [0, Po [ a continuous function of t with values in X p over [0, ak(po - p)[; further we have:

(6.12) sup {ll vdt)ll p (l _ t )} < pe[O,po[ ak(po - p)

00 .

Ie [0, adpo - p)[

Suppose that the ui, i ~ k are determined and satisfy II ui(t) lip ~ R for t E

[0, ai(p - Po)[, P E [0, Po [; if we set

(6.l3) Ak = Mk[Vk] < + 00 ,

then

we deduce for t E [0, ak + 1 (Po - p) [

§6. Ovsyannikov's Theorem 69

from which we have by recurrence:

(6.14) II uk+ dt) lip ~ jto Aj j (1 - aja: 1 ) + II Uo(t) lip.

Suppose now that it is possible to find the Ai, ao such that we have:

(6.15) and ~ Aj R - Ro L.., < 2

j=O 1 _ _ aj_+_1

aj

Then for all t E [0, ak + 1 (Po - p)[

R - Ro (6.16) II Uk+ 1 (t)llp < Ro + 2 '

and F(t, S, Uk + 1 (s)) is well-defined, for all k, if we choose ao small enough. From (6.13) we have:

lIuk+1(t) - udt)lIp ~ Akj(1 _ (t )) < Akj(1 _ ( t )) ak Po - P a Po - P

for

from which

M [Uk+ 1 - Uk] ~ Ak .

Since the series L Ak < + 00, it follows that the sequence {Uk} converges to U E X which satisfies, from (6.16):

R - Ro Ilu(t) lip ~ Ro + 2 ' 0 ~ t ~ a(po - p).

Further, this function U satisfies (6.5). In effect, for t E [0, a(po - p)[, p < p' < Po, we have:

II uo(t) + I F(t, s, u(s)ds - u(t) lip

from (6.4). Thus

~ f~ II F(t, s, u(s)) - F(t, s, uds)) lip ds + II u(t) - Uk + 1 (t)llp

~ _,_c_ rt II u(s) - uk(s) lip' ds + II u(t) - Uk+ dt) lip ~ Ek p - PJo

II uo(t) + f~ F(t, s, u(s))ds - u(t) lip ~ Ek for all k

with limEk = 0 from which we have (6.5). The existence in Theorem 1 is therefore shown, subject to verifying (6.15).

70 Chapter XIV. Cauchy Problems in [f;l"

2nd stage. Verification 0/(6.15). From the definition of Uk and from (6.1S)ii) we have:

Uk + 1 (t) = f~ [F(t, S, Uk + 1 (s)) - F(t, s, uds))] ds .

From (6.4) we obtain

il II Uk (s) II II Uk + 1 (t) lip ~ C ( ) p(s) ds

o P s - P

for t E [0, ak + 1 (Po - p)[ and a certain choice of p(s) with p < p(s) < Po _ _ s_ ak + 1

(for example p(s) = (Po - (s/ak+ d + p )/2.) From (6.13)

II Uk + dt) lip ~ C f~ -l-_--A-k -S-_(p(S)d~ p)

adpo - p(s))

'" CA, f ( d') o 1 _ s (p(s) _ p)

ak + 1 (Po - p(s))

~ C 1 il 2ak+ dak+ dpo - p) + s)d . '" Jl.k { } 2 S , o ak + 1 (Po - p) - s

from which we deduce:

Ilvk + 1 (t)ll p "; 2cak + 1 Ak (1 + 1 ) t ak + I(PO - p) ak + dpo - p) 1 _____ _

ak+l(PO - p)

and Ak + 1 = M k + 1 [Uk + 1 ]

~ 2Cak + 1 Ak sup (1 + t ) t pE[O,PO[ ak+dpo-p) ak+dpo-p)

IE[O.ak+! (po - p)[

from which:

(6.17)

Note that

Ao = MO[I F(t,s,UO(S))dS]

~ ~~~ CRo(f~ (1 _ s/ao(po _d:(S)))(p(S) _ p) (1 - aO(Pot _ P))) (84)

~ 4aoCRo ·

(84) With p(s) = ~ [po + p - .~J. 2 ao

§6. Ovsyannikov's Theorem 71

Choosing ao such that 4Cao < ~; then

(6.18)

and

We thus deduce (6.15) and (6.16), from which we have the existence in Theorem 1.

Demonstration of uniqueness

Let U 1 and U 2 be two solutions of (6.5) satisfying (6.6);

w(t) = U 1 (t) - U2 (t)

must satisfy:

(6.19) w(t) = f~ {F(t, s, U1 (s» - F(t, s, U2 (s»} ds .

Then for all P1 < Po we see that

From (6.19) and (6.4) we obtain:

II w(t) II ::::;; C rt II w(s) IIp(s) ds p Jo p(s) - P

where p(s) satisfies P < p(s) < P1 - s/a. The same reasoning which results in (6.17) leads here to:

t )-1 a(p1 - p)

from which

M1 [w] ::::;; 4CaM1 [w] .

It follows that M1 [w] = 0 if 4Ca < 1 which is possible by choosing a small enough. We deduce

IIw(t) lip =0 forall 0::::;;t<a(P1-p)

and for all P1 < Po. From which w = 0; which proves the Theorem. D


Review of Chapter XIV

The study of the problems of this chapter, where the spatial variable x ranges over the whole of [Rn, has brought up problems with very different properties. At this stage of the exposition we have approached essentially three different categories of problems. In all of these problems, there exists a semigroup {G(t)} of class reo which allows us to proceed from the given data to the (unique) solution via a formula of the form:

u(t) = G(t)uo + (G * f)(t), t ~ O.

In the case of diffusion problems, we have seen that the Cauchy problem may not, in general, be resolved starting from initial data at t = 0 towards t < O. The problem is called irreversible, and the semigroup {G(t)} may not be extended into a group. We have further established a large degree of regularity of the solution u in x, even though its initial value Uo is very irregular. This regularisation is due to the dissipative character of the semigroup {G(t)} (a notion which will be studied in Chap. XVII). We shall see in the following chapters that these characteristics of diffusion problems are linked. In the case of wave problems, the Cauchy problem is reversible and the semigroup {G(t)} may be extended into a group. The solution u of the problem is not, in general, more regular in x than the initial values U O and u1 .

We have even observed a propagation of singularities. The characteristic curves have illustrated this propagation of solutions and notably of their supports. The possibility of having compact support in x for these solutions, illustrated by the method of elementary solutions, has then characterised hyperbolic systems (to which the wave equation belongs). The elementary problems relating to the Schrodinger equation have shown the possibility of extending {G(t)} into a unitary group in L 2 ([Rn). We shall see later (particularly in Chap. XVII) that this property exists under very general conditions relating to the potential. In the cases treated, this group does not have any regularising action on the initial data. We shall also see in the following that this property is equally general. Finally, these three types of problem in [Rn have been seen as particular cases of a more general problem, that is the evolution problem connected with the convolution product. These problems are invariant under translation in [Rn and therefore may be studied by the use of the Fourier transform over the space variable.

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