/2,
As a consequence, there is no later that operator
for all
t
>
O.
-
T(t)
A
E
L(E)
such that
T(t) ~ etA. We will
see
can be considered as the exponential of the unbounded
t~. ax
97
3. Linear (unbounded) operators on a Banach space Definition 3.1. pair
(D,A)
map: D
Let
where
E be a E-Banach space. A linear operator on D is a ]K-vector subspace of
and
E
E
is a
A is a lK-linear
-+ E.
A
is the domain of the operator. We will often denote by the operator itself and the domain by D(A),
- The subspace
D
- The graph of
A
G(A)
~
is defined by
{(u,f) E E
It is clear that
The range of R(A)
G(A)
A(D(A»
E, there exist
E
f
~
Au}' EX E.
is an(-vector subspace of
is the set
A
{f
~
u E D,
E,
x
u
E
R(A) , We have
and is denoted by
D(A) , Au
~
f),
In the sequel, we shall say that a linear operator
(D ,A)
on E is unbounded
if we have
Sup Ilull
UED,
IIAul1
0, and for all
Definition 3.5.
u
E
D(A), IIAu + Aull:c. A Ilull
A linear accretive operator
A
if, in addition to the property above, for all DCA),
U E
Au + Au =
on
E
fEE
is called m-accretive the equation
f
has at least one solution for all
A > O.
, Remark 3.6.
Since
A
is accretive, such a solution is always unique. More
it !:i
,
precisely, we have the following Proposition 3.7.
Let
'I
be the identity operator on
I
in the obvious way. A linear accretive operator
s EJR
A > a
II (A + AI)-lll
Proof.
any
A'
the solution of
Ilfll:c. Aa Ilull·
A>
for all
< -
-
It is clear that it
fEE
A + AI
the operator
A
Au + AOti = f
f
All we have to shaw is that in fact We shall consider the case where
on
E
is
a.
is accretive and
Hence the operator
A
A + sI,
E. In this case, Aa > 0, R(A + Aa I ) has an inverse in L(E) and
m-accretive if, and only if for some for all
and define
E
fr U
RCA + AOI) = E, Lhen for
is unique and such that
belongs to
R(A + AI) = E
1
with nann .:::. -:\ . a for all A > O. L(E)
A E ]0, 2A [. Then the equation
O
Au + AU"" f
can be rewritten under the form
f + (Aa-A)u u
CA + A I)-l [f + (Aa-A)U] a T A (u) •
Now it is obvious that
T
: E A
+ E
is Lipschitz-continuous with Lipschitz
constant
From Banach fixed point theorem we deduce that The case of an arbitrary
A E JO,+
0 [
RCA + AI) = E
for A E JO,2A [· O
follows by induction.
99
II
The following property will appear fundamental in the study of evolution equations Ccf. § 5). Theorem 3.8. Proof.
Any m-accretive operator in a Banach space is closed. CA + 1)-1
Since the graph of
so is the graph of Remark.
is closed by the closed graph theorem,
A + I, hence also the graph of
A
is closed.
An important property of m-accretive densely defined operators is
that they can be approximated by bounded operators. This property is the object of Proposition 3.10 below and will be very useful in the construction of linear semi-groups'. In the sequel, we set for all
A
>
0
We note that for a]l
IIJ(A,A) II
O.
x E DCA), we have
J(',A)x - x ~ J(',A)x - J(',A)lx + AAxi ~
-AJ (A ,A)Ax
.. IIJ("A)x - xii
Since
for all
100
as
A + O.
!!JcA,A)11 < 1, it is clear that the result then extends to any
Proposition 3.10.
with
,::,'"Axll +0
A,
~
x
If E
DCA)
~
E, then
D(A), lim !!A,x - Ax II ~ 0 ,+0
A(I + ,A)-l
E
L(E)
and
IIA,II."-
i,
for all
,>
O.
xE DCA).
n--
AA Ax
Since
Proof.
x c D(A)
and
=
1
III - J(A,A)] we have Y E E. Then as a consequence of Proposition 3.9:
A)..x '" J"CA,A)y -+ Y
Remark 3.11.
in
E
The property
on
E
A -+ O.
DCA)
m-accretive. For example, let A
as
is not automatically satisfied if
E
c ([ 0,1])
E
A is
and define an unbounded opera tor
by
D(A) = 10 c C1 ([0,1]), 0(0) = u(I)} Au
u',
=
for all
x E DCA).
The solution of U E
DCA),
u l
is given for any
+
AU =
f
C([O,1]) by
f E
';'
e
a(x) =
-A
1
~
l-e
It follows easi.ly that
A
Je
,,
A Yf(y)dyl.
:i
,
I',I
o
I
is m-accretive in
D(A) = 10 E E, 0(0) = u(I)}
Ii
E. On the other hand, we have
11q"I
which is not dense in E.
In this example, even the result of proposition 3.10 is not satisfied. Indeed, for any
x
E
E
and
proposition 3.10 would imply
if
¢
E
D(A)
is such that
A > 0
we have
A\x
E
D(A) , hence the result of
Ax E DCA) for any X E D(A). On the other hand, t'(O) I ¢'(1), then M ¢ D(A).
4. Linear (unbounded) operators on a Hilbert space In this section, we denote by
{
11011
H
the inner product of =
1/2
the norm of
a fixed real Hilbert space, and by
(u,V) E H x H u E H.
4.1. The adjoint of an unbounded linear operator Let
A
be a (possibly unbounded) operator on
H
with dense domain. We set
101
"
D(A*)
{v E H,
l1
Sup
< +oo}.
uED (A) u ,::1
II II Then u
~
D(A*)
is a vector subspace of
H. For any
is linear and uniformly continuous on
defined by the norm of
v E D(A*) , the mapping DCA)
for the topology
H. As a consequence of the unique extension principle
for uniformly continuous map defined on a dense subset, we obtain the e:x:istence of for all
~ E L(H,
JR)
U E D(A),
such that ~(u)
= .
From the Riesz representation theorem we conclude that there exists
cP
E H
such that
for all since
cP
U E
DCA), = .
is uniquely determined when
definition
U E
.
A*
DCA), and for all
{(v ,~) E H x H, for all
}.
Definition 4.1.1.
A*
E
G(A),
is often the most convenient in the applications. A*
defined just above
A.
The operator
A*
is always closed. Also if
A is closed,
(A*)* = A.
Proof.
We define the Hilbert structure on
use the orthogonality theory in we have 102
(u,£)
The (possibly unbounded) operator
is called the adjoint of Proposition 4.1.2.
v E D(A*),
by its graph, we find
G(A*)
This definition of
then
is given, we can set as a
A*v. In other terms, we have
for all
If we define
v
H x H in the obvious way and we
H x H. It is clear from the definition that
Hence
G(A*)
G(A). I f
A
is always closed. In addition, we find is closed, then
G(A)
= G(A)
(A*)'
O.
for all
Proposition 4.2.2.
H is monotone if, and only if it
A linear operator on
is accretive. Proof.
- If
A is monotone, then we have
IIAu + ;,u11 > ;,2 IIul1
Hence
2
2
=
IIAul1
2
, for all
A is accretive (in
- Conversely, if for all
u
E
+ z)' + ;,211u11
2
u E D(A), and for all
1> O.
H).
II Au 112
A is accretive, then we have
D(A), and for all
::\ > O. On dividing by
+ 2::\ . :':. 0, ::\ >
a
and letting
A -+ + 00 we find:
Hence
~
0,
for all
U E
D(A).
A is mcnotone.
Corollary 4.2.3.
In a Hilbert space, a sum of two accretive operators is
accretive. Proof.
Obvious since the sume of two monotone operators is monotone.
Definition 4.2.4.
A monotone operator
A on
H
is maximal monotone if we
have R(I + A) = H.
Hence a maximal monotone operator is exactly an m-accretive operator in
H. 103
The two following important results are proved in [2J and we state them without proof.
Proposition 4.2.5.
H. (Hence Tille
A is maximal monotone in
If
converges to
Alex
Ax
in
H as
Ie
-7
H, then 0
+
D(A)
for any
is dense in
x E D(A» .
recall that this result is not valid for m-accretive linear operators
in an arbitrary Banach space, cf. Remark 3.11. Theorem 4.2.6.
If
A
is linear monotone (with
maximal monotone if, and only if
G(A)
D (A) = H), then
is closed and
A*
A
is
is monotone.
The two following consequences of Theorem 4.2.6 are important in the applications.
Corollary 4.2.7. adj oint, i.e. Then
A
Let
A*
A.
CorollarY_.i·2.:8.
A
be linear monotone with a dense domain and self-
is maximal monotone.
domain, i.e. Then
A
Let
A
A
be any skew-adjoint linear. operator with a dense
satisfies
A*
-A.
is maximal monotone.
Remarks 4.2.9. theorems (cf.
a) Corollaries 4.2.7 and 4.2.8 joined to perturbation [8J) are quite efficient in praetiee to construct important
classes of maximal monotone operators. b) In proving that
to establish that
A* = A
or
D(A*) c D(A)
M~ = -A, the most difficult part is usually
[or equivalently that
that it is not sufficient to check that
A*u '" Au
on
G(A*) c G(A)I. Note D(A): to forget this
R(I + A) = H) can be a source
point (or equivalently to omit the proof that of important mistakes in the applications. c) The sum of two maximal monotone operators in
H
is not always maximal
monotone despite the fact that the two domains must be dense.
As an example, let
.,'.
104
1
H
~
D(A)
{u E= H (0,1),
D(B)
(u
E
2 L (0,1) and
u(O)
O} ,
Au
1 H (0,1), u(l )
O} ,
Bu
du dx du dx
One remarks that
B
A*
and
(A,B)
H. By
are both maximal monotone in
definition, we have 1
D(A+B)
H (0,1), (A+B)u O (A+B)u + u
The equation Hence
A+B
f
=
°
=
on
D(A+B).
has no solution in
D(A+B)
if
f
0, T(t)
E
u Ct) :=: TA (t)u converge in C([O,T] ,E) A O to a function uCt) = T(t)u EO C([O,+o:>[ ,E) and
°
A+
fol1o~ing
iii) for all
E
E, the functions
> 0, as
we have the
TA (t) = e
A > 0, we set E
-tA,
E, T(O)u
O
~
u '
O
and IIT(t) II::. 1, s> 0, T(t+s) = T(t)T(s).
L(E)
E
t > 0, and for all
b) If in addition
U
oE
T(t)u
D(A), then
O
E
1 C ([O,+oo[)
and is the unique
solution of the problem 1
(u(t) E C ([0,+oo[,E) and u(t)
j:~ + Au(t) lu(o)
=
t
D(A), for all
>
t
°
> 0
(5.1)
uO•
Finally for all all
= 0, for all
E
t
>
° we have
du dt
for
-T(t)Au ' hence O
t > O.
Proof of Theorem 5.1.1.
This proof will be carried out in several steps.
Step 1.
Estimate of the solution when 1 the formula: AA = "I(1 -
A is changed to
AA' A > O. We recall
(I + AA)-l E L(E)
and
II
J, II::.
1.
1
t
-);",;\tJ,
As a consequence, for all
t
>
0
we have
e
e
hence 105
-tA II e
--"< e A e
A II
This implies:
Step 2.
1 xt
for all
t> 0, and for all
Convergence of
u). (t)
From the definition of
when
A > 0, Ilu (t) 11.
S
ElR, and
"J...>
0, and
]J
> 0
-stA - (1 +s)tA -tA s (tA -tA ) A A e \1 = e \1 e \1
As a consequence, for all
~(e
[0,1]
S E
-stA - (1-s}tA
~)
A
ds
te
-stA - (1-s)tA A U (Au -AA)
C(O,l ;L(E». We now remark that
the right-hand side being in -tA II u A(t) -
U
u
(t) II
=
II e
"cuo) - e
-tA u (u ) II O
1
-stA - (1-s)tA Ilbd~{(e A ~)uo}dsll
°
0
and
t > O. We also have
v, (t) " -T, (t)A,u ' hence O
and
Sup Ilv, (t) - T(t)Auoll->o tE[O,T] By writing
as
,-+
a
t
0
for any
T > O.
t
f
+
a and letting
vA (s)ds, for all
>
i'
"
A + 0, we therefore obtain
Ii II
t
u(t)
Hence
U
U
o
-
f
a
T(s)AuOds, for all
E
:~"
and
, -> 0:
u(t) E D(A)
du -Au(t) "" lim [dt'(t)]
-T(t)Au
,+0
Finally, let
1
u E C ([O,Tr ,E)
[O,T[. We set 1 VEe (O,t)
:~ for all
vCt)
=
for all
IIAuoll,
t
>
j
I:
O.
i
, ,
E
and
O
du " dt(t), for all
t
>
O.
be a solution of this problem, and let
T(Q.-t)u(t)
for
t E [O,£.]. It is immediate to see
with
" T(£-t)Au(t) + t
"II:~II ~
O
hence by letting
that
-T(t)Au O
'l
" -A(J,T, (t)u ) and
Finally we
Q, E
t > o.
]0,£.[. Hence
T(£-t):~ v
"
a
is constant on
[O,t]
and we have 107
T(O)u(~)
~
Since
"v(O)
is arbitrary, we obtain that for all
u(t) "T(t)u ' O
t
E
[O,TI.
Hence the proof of Theorem 5.1.1 is completed.
5.2. Application to some typical examples Example 5.2.1.
Let
E = H, a real Hilbert space and
unbounded) operator with
DCA)::: Hand
A
a monotone (possibly
A* = A. Then for any
DO E DCA),
there exists one and only one solution of (5.1). The simplest possible example in the theory of partial differential equations is when
Z H " L (0,1) Z
D(A) " H (0,1) n H~(0,1) Zu l'f u(x) C D(A), Au -- - 3 Z
ox
Then the semi-group associated to
-
A
in
H
corresponds to the resolution
of the problem:
l
~~ :Z ~ " 0 o
u(t,O)x" u(t,1) u(O,x)
uO(x),
Now it is well-known that the orthogonal family (sin
+=
If we set
n
(t) " e
-n
Z"If Zt
u
n
n
(0), for all
IN
is total in H. u
n
nElN'{O}.
From this formula, it is immediate that, even if
108
ne
u (t) sin n~x, then the Fourier components
n=l
are given by u
I
u(t,x)
ll1fX)
u(O) ¢ DCA), we have
(t)
This property is in fact general and can be proved by' induction starting from the following. Theorem 5.2.2.
Let
A be as in example 5.2.1. Then, for any
U
o
E
H, we
have t > 0, T(t)u
for all
Proof.
O
E
D(A)
It is straightforward to check that
for all
)..
>
O. Let dU
II u (t) II "nd A
A
Ildt ll
ul-.. (t)
A)...
is monotone and
A~:=
Al-..'
be as in the proof of Theorem 5.1.1. Then
are non-increasing with respect to
t > 0, From
:1' "
we deduce that for all
T
>
0:
iii i'
I
il II
On the other hand, we have the formula:
'I
I As a consequence: T
-f
o
Thus we finally obtain: T
J
o Since
dU A
2
t Ildt(t) II
du II d:(t) II
~
1
2
dt.:: ililuoll
du II dtA(T) lion
[O,T]
we deduce
109
2
du
~11_\T)112 2
dt
du, II-(T) dt
Hence:
11
O.
On the other hand, an immediate computation shows that for
A~
J(~-A,AA) 0
for all
Z E
AA'
D(A),
hence
IIA~z II
H,
X E
::.liA).z II
~ >
if
A.
From the inequality above we deduce for all By letting
T > 0, and for all
A ~ 0, we first obtain ~
for all
By letting then for all
~ +
Let
to a group, and
E
b), for all
A be such that
D(A)
Hand
T(t)
constructed in Theorem 5.1.1 extends for
T(t): H ~ H is a surjective isometry for all T(t)D(A) c D(A)
t> 0, on changing
A
by
(-A)
t
0, II A u (T) II
0, u(T) E D(A)
Example 5.2.3.
A*
> 0, and for all
t
°
>
unless
U
o
E
any "smoothing property" as the one described in Theorem 5.2.2. The simplest possible example in the framework of P.D.E. is when
H "" L2( JR)
du D(A ) ~ H (lR), Au ~ - dx' for all U E D(L). Then the group 2 defined on H = L (JR) by the formula 1
110
and
T(t)
is
r
~
(T(t)u)(x)
Example 5.2.4.
u(x+t),for all
Let
E
= {u
E
(t,x) EJR.
C([O,ll), u(O)
= u(1)l.
And define
1 D(A) = {u E C ([O,ll), u(O) = u(1), u'(O) = u'(l)} du Au = dx'
for all
u E DCA).
By a straightforward comparison with the example given in Remark 3.11, we check that
A
is m-accretive in
this case, the semi-group ~
(T(t)u) (x)
wi th
T(t)
u(x-t),
E
't.Jith
also extends for
for all
t
< 0
to a group given by
(t ,x) E JR x [o,ll
the 1-perioclic extension of
u
DCA) "" E. It turns out that in
u
on JR. Hence we can see clearly that
the difficulty pointed out in Remark 3.11 was only due to the boundary e([O, 11).
condition and the space
6. CO-serni-groups and their generators 6.1. General properties Defini tion 6.1.1.
{T(t)}O
is bounded on
0, we set
t
=n
[0,11. Let
M
Sup
IIT(t) II
and
w = Log M > O.
O O)}
D(L).
E
h->O
Proposition 6.1.5. a) for all
X
Let
and
L
be as above
T(s)xds E D(L)
E, we have
E
for all
t
~
0, and
t
T(t)x - x b)
for all
x
o
D(L), we have
E
T(t)x
E
D(L)
1
T(t)x E C ([O,+oo[,E)
and
t > 0, :t (T(t)x) = LT(t)x
for all Proof.
L(f T (s)xds)
=
with
T(t)Lx.
a) By the semi-group property ii),
1 t+h 1 h T (h) - I t T(s)xds ~ T(s)xds - -1 T(s)xds, h 1 0 o t
f
for any
f
f
x E E and
t
y-f
T(t)x-x, thus
h
>
O. As
h + 0+
T(s)xds E D(L)
the right-hand side converges to
and
Ly - T(t)x - x.
0
xED (L)
b) Let
and
t
~
0, h > O. Then
T(h~ - I T(t)x _ T(t){T(~)-I xl as
h
+
O. Hence
for all
t
T(t)x E D(L) >
t > 0
T(t)x - T(t-h)x _ T(t)Lx
112
T(t)Lx
LT(t)x - T(t)(Lx). We also find:
d+ 0, dt (T(t)x) - T(t)Lx.
On the other hand, if
h
and
+
and
0 < h < t, then
T(t_h)[T(h)x-x h
Lx] + T(t-h) [Lx·· T(h)Lx].
As
h -+ 0+, both terms on the right-hand side converge to
0
in
E. We
conclude that
~t (T(t)x) ~ T(t)Lx. Finally, T(t)x E C1 (lo,+oo[ ,E) d
Corollary 6.1.6.
L
- For all
Proof.
x
n
in
x
D(L) , x
E
D(L)
is a closed operator and x
x c= E, let
and converges to - Let
T(t)Lx.
LT(t)x
dt (T(t)x)
and
->
n
x
as
E
and
~ ~
t
f
t 0 t -+ O. t
LX
n
E.
T (s )xds in
-+ y
for
t
>
O. Then
xED (L) t
E. As a consequence of b) of
Proposition 6.1.5, t
T(t)x By letting
- x
n
n
+
S
t > O.
T(s)Lx dx, for all
o
n
n
+00, we obtain t
f
T(t)x - x ~
T(s)yds, for all
t
>
o Hence
t
T(t)x-x
f
t
t
Remarks 6.1.7.
T(s)yds
->
y
as
t -+
o. a+ •
Thus
x E D(L)
and
Lx.
y
0
a) In fact it is known that
n
u
P(L )
is dense in
E. For a
proof, cf. [8]. b) Proposition 6.1.5, b) can be considered as a generalization of'Theorern 5.1.1,b).
6.2. The theorem of Hille-Yosida-Phillips In the applications, what is generally given is a linear unbounded operator L
on
E, and the question is to decide whether
a CO-semi-group on
L
can be the generator of
E. A general answer to this question is given by the
following result. Theorem 6.2.1 (Hille-Yosida-Phillips). generator
0
0
•
f a C -seml-group
T (t)
on
A linear operator
E with
L
II T(t) II.:: Me
wt
on
E
is the
(M > 1,
W
> 0)
113
i f and only if the following conditions are satisfied i)
D(1)
ii)
For all
~
E and for all
A
n
Remark 6.2.2.
E
w, L - AI
~ (1- l1)-1
l > w, R(l,1)
iii) for all
>
lN, II [R(l,1) Inll.::
a) When
E
is one-to-one. L(E)
and we have
M n (l-w)
M > 1, the property iii) is not so convenient to
check directly. In practice the best way is to change the norm in order to reduce the situation to the case
of Theorem 6.2.1 when
M> 1
E
in
1. In fact (cf.[sl) the proof
M
can be reduced to the case
M
1
=
by a
suitable "abstract" renorming. b) When
IIR(A,L)
M = 1, property iii) is in fact equivalent to
c) For simplicity, we shall only prove Theorem 6.2.1 when on replacing
L
L- wI
by
we can assume
M
II.:: A~W' 1. In addition
=
w = O. Finally the basic step in
the proof of Theorem 6.2.1 can be stated as follows.
Theorem 6.2.2.
A linear operator
contraction semi-group on E
Proof.
a) Let
T(t)
L
on
E
is the generator, of a
if, and only if
-L is rn-accretive with
be a contraction semi-group on
generator. We already know (Corollary 6.1.6) that A> 0
h > 0,
and
U
E
D(L), from
IIT(h) II.:: 1
E
D(L)
and
L
D('L) "" E.
its
E. Also for any
we deduce
;;i;1
: '11'
By letting
h
r
is rn-accretive, let
-l-
0, we obtain that
(-L)
is accretive. To check that
(-L)
I
,
J(x) "
I
il
r:,
J
e
-t
T(t)xdt,
x E E.
for all
o
J E L(E)
It is clear that
T(~)-I J(x)
ij,
~~ h
I"
1'1",11'
+00
J
e
-It-h)
h
+00 h e -1
I" :j ,
"~h-
f
0
114
e
IIJII.:: 1. Also, for any
with -t
(T(t+h)x - T(t)x)dt +00
T(t)xdt
h
e e -t T(t)xdt - h
e
S
0 h h
S
0
e
-t
-t
T(t)xdt
T (t)xdt.
h > 0
and
X E E:
As
h
~
0, we therefore obtain
l'
em
h+O
Hence
T(h)-I h
J(x)
J(x) E D(L) ~
LJ(x) = J(x) - x
and
J(x) - LJ(x)
We conclude that
J(x) - x.
=
x,
=
for all
R(I-L)
b) Conversely, if
x
E. Hence
(-L)
E
E.
(-L)
is m-accretive.
is m-accretive and
D(L)
=
E, we deduce from
Theorem 5.1.1 that there exists a contraction semi-group
T(t)
on
E
such
that for all Let
L
x E D(L), lim
h~O+
be the generator of
T(h)x- x h
T(t)
and
Lx. y E D(L). Since
(-1)
is m-accretive,
we can solve the equation
Ii
I
-Ly + y, x E D(L).
-Lx + x
I,
Clearly
G(L) c G(L).
Since deduce G(L)
=
(-L) y
is accretive, from the relation Ly = Lx = Ly (and of course
x, hence
G(L)
and
L
-L(y-x) + y - x
=
"I'I
0 we
II
y E D(L». Finally
is the infinitesimal generator of
T(t).
7. Analytic semi-groups Let
E
be a complex Banach space, and
operator on
L
a (possibly unbounded)
~-linear
E.
Definition 7.1.
The resolvent set
p(L)
is the set of all complex numbers A
such that i)
for all
ii)
R(L- AI)
iii) (L- AI)-1
E E, Lu
U
AU => u
o
E E
L(E).
Proposition 7.2.
peL)
holomorphic map from
is open and the map p (L)
into
L (E)
AI--+
(L- ;\.1)-1 E L(E)
is an
considered as a [:-Banach space. More 115
precisely, if
Iz I
i
~
0, T(t) maps E into 2 D(L) and we have
is analytic and
In addition, for any
Proof.
21T
ex
n n - 0
sufficiently large, we have I A - A I < dist (A ,C) < n
n-
II (L-A I) 111· n
As a consequence of Proposition 7.2, we find Finally
~
c peL)
A E peL)
since
and the proof of step 1 is completed.
117
·,11' I.
Let
Step 2.
be such that
8
0 < 8
O. Also an argument
in Proposition 2.2 shows that
has the semi -group property on the open ~ convex cone D.
!lEep 4.
The estimate of
IILT(t)ull
We have shown in step 2. that for
for
t
E
]0,1].
tE ]0,+00[:
LT(t)u
It immeditately follows that we have
Hence
f rS M
< - t'
for all
E
]0,1]
M > O. Hence the proof of Theorem 7.5 is completed.
for some finite constant
Remark 7.6.
t
a) The result of Theorem 7.5 remains true with obvious
modifications if instead of R '- (O) '- (z
i
we assume for some
,I "
120
E
~,
1 Arg
\) > 0:
z - n
1 .::.
a),
\ I
R
c {v}
(z
U
~
E
{v},
"
I Arg(z-v)
-
nl
< a}.
b) The concluding estimate in Theorem 7.5 generalizes the conclusion of Theorem 5.2.2. In fact, by using the remark a) just above we see that the result of Theorem 5.2.2 is still valid (with another constant than
general) if
A
is assumed to be self-adjoint in
H
~
in
12 of a and the generator
(1,u).
CO-serni-group of type
c) The generalized version of Theorem 7.5 above includes the case where
L(E). It is established in [8] that if the final estimate holds with
L E
1 C < -,then in fact e
LEL(E).
8. Some convergence and approximation results . The results of this section are valid in the framework of Co -sem1-groups, but
for simplicity we give the statements and the proofs only for contraction semi-groups. For the general case, cf.[B J.
B.1. The Trotter approximation theorem Let
~A
and
(A)
IN
be m-accretive in a Banach space
nnE D(A ) ~ E. We set R(A,A)
~ (A+H)-1
n
and
T (t) n
r~spectively
for
A
>
E
with
D(A)
0, and we denote by
the contraction semi-groups generated by
T (t)
(-A)
and
(-A) . n
Theorem 8.1.1.
The following statements are equivalent:
A>
a) There exists for all b) for all
°
such that
x E E, lim IIR()"A )x - R(A,A)xll n++= n
A > 0, we have
for all
x
E
E, lim
n++ oo c) for all
o
x E E
II R(), ,A )x - R(A ,A)x II n
and for all
o
t > 0, we have
lim II T(t)x - T (t)xll ~ O. n n-++ oo Proof. c) ~b) It is not difficult to check (cf. proof of Theorem 6.2.2) that for all
A> 0
and for all
x E E,
121
,
1 +00
f e-AtT(t)xdt
R(A,A)x
o hence for all
x E E
- T(t)xll dt •
The result therefore follows as an immediate consequence of Lebesque's dominated convergence theorem. b)
=;.
a) Obvious.
a)
=;.
c) The proof of this part relies on a computational lemma.
Lemma 8.1.2.
Let
B be m-accretive in
semi-group generated by
E
and
(-B). Then for any
S(t)
denote the contraction
x E E, we have, for all
t
t
R(A,B)[T(t) - S(t)]R(A,A)x =
f
S(t-S)[R(A,A) -R(A,B)]T(s)xds
o Proof of Lemma 8.1.2.
A simple computation yields
dd [S(t-S)R(A ,B)T(s)R(A ,A)x] S BS(t-s)R(A,B)T(s)R(A,A)x - S(t-S)R(A,B)AT(s)R(A,A)x S(t-s)[I - AR(A,B)]T(s)R(A,A)x - S(t-S)R(A,B)T(s)[I - AR(A,A)]x S(t-S)[R(A,A) - R(A,B)]T(s)x.
The result of lemma 8.1.2 follows by integrating in Proof of a)
=;.
c) of Theorem 8.1.1.
We fix
x
E
E
s and
on t
[O,t]. >
O. Then
II (T (t) - T (t) )R(A ,A)x II n
< liT (t)(R(A,A)x - R(A,A )x)11 n n
->
+ IIR(A,A )(T (t) - T(t»xll n
n
II [R(A,A ) - R(A,A)] T(t)xll n
.: 0
r I \
The two first terms obviously tend to Since
II R(A ,An) (Tn (t) - T (t) II .::.
that the left-hand side tends to
0
i
as
and
0
n
7
+00,
A > 0
is fixed, in order to show
x E E, it is sufficient to
for any
check that for all
We set
y
x
~ Ax
8.1.2 with
B
D(A), lim IIR(A,A )(T (t) - T(t»xll ~ n++ OO n n
E
o.
x = R(A,A)y, and we apply the formula of len~a
+ AX, hence
A . We obtain:
=
n
IIR(A,A )(T (t) - T(t»R(A,A)Y11 n n t < IIIR(A,A )T(s)y - 0 n
Since
{T(s)y}
u
O O.
is the contraction semi-group with generator
1 D(A)=UeE,fEC (lR)
1·1.
E dnd
II (I+A)-1 1I .:'O 1,11(1+~)-11I.:'O 1Example 8.2.2.
,
-D(A)
k-m
ill
and
f'eEl.
f E D(A).
-I f(x+h~-f(X) 1
Ckf(x+mh)
L
= -(lI
h
£) (x). Then
(lIh)kf(x)
and from Theorem 8.2.1 we deduce
-A
such that
+00 f (x+t)
lim
L
h+O h:=:O uniformly on IR
for any
tEE.. This formula is a generalization of Taylor's
expansion of entire functions in the framework of continuous functions. 8.3. The exponential formula
Let
A
be m-accretive in
E
with
D(A)
E. We already saw that for any
A > 0, we have
J
R(A ,A)x
e
-At
T(t)x dt
o for all
Tet)
x E 3:, where
is the contraction semi-group generated by
L = -A. Conversely, the semi-group resolvent
can be computed in terms of the
R(A,A). For example the following result is established in [8].
Theorem 8.3.1.
As
lim
'l'(t)x
n++oo
x
n-++oo,wchave t -n (1+- A) x n
(~) nR(~ A) nx
lim n-++ oo
for all
T(t)
and
E :l
t'
t
t
> O.
Theorem 8.3.1 is in fact also valid (first formula) in the
Remark 8.3.2.
nonlinear framework (cf.
[5]). For the proof in the linear case, we refer
to [4 J or [8].
Remark 8.3.3. uniform on
a) The convergence of
[O,TJ
T
for any
>
t n
(I +-A)
-n
x
to
T(t)x
as
n -r +00
is
O.
b) The result of Theorem 8.3.1 can be viewed as the convergence of an implicit difference scheme to the "solution" of the problem
du + Au(t)
dt
{ u(O)
0,
t
> 0
x.
Indeed, we have
t n
(1+-A)
-n
x
u
ll,TI
where
{u}
U,p 02p~n
is defined by
125
1
j
~(u
l
t
u
U,p
-u
n,p-l
n,p
for all
x
n,D
) + Au
= 0,
< p < n
n E IN.
9. The inhomogeneous equation and the "variation of parameters" formula
In this section we consider the inhomogeneous initial value problem
du dt
Lu(t) + f(t),
u(O)
= x.
We assume that
>
°
}
(9.1)
is the infinitesimal generator of a CO-semi-group
E
on U
L
t
such that the problem above with 1 + C (JR ,E) for any initial value x
E
A 1
C (]O,+ooi,E), u(t) E D(L)
Proposition 9.1.
has a unique solution
(cf. Proposition 6.1.5). + is called a strong solution of (9.1) if uE CelR ,E) n
u(t)
function
f = 0
T(t)
u(t)
If
for
°
>
t
E
DeL).
and (9.1) is satisfied.
is a strong solution 0'£ (9.1) with
+
f E C( JR ,E),
then we have t
for all
Proof. and
Let
f 0
h
t
t
>
> 0
0, uCt)
i
T(t)x +
°
T(t-s)f(8)d8.
be fixed and consider
g(s)
(9.2)
°
T(t-8)U(S). For
O.
is given by the
formula t
u(t) ~
a+
J T(t-s)T(s)ds
tT(t)x,
for all
t > D.
o 127
Hence
u(t)
is not in
D(L)
for any
t > O. Also
differentiable (as an E-valued function) for
tT(t)x
is nowhere
t > O.
- It turns out that the weak solutions still have very interesting properties, like the continuous dependence property shown in Proposition 9.5 below and the "weak differentiability" established in section 10.
.
1 + 2 (f, f) E [L (ID. ,E) J • Let u 1 oc and u be the respective weak solution of -du = Lu(t) + f(t) and du dt dt Lu(t) + f(t) such that u(O) = x and ~(O) "" x. Then, for all t > 0: Proposition 9.5.
Let
(x,x)
E
E
x
E and
Ilu(t) - ~(t) II:: Me'tit{ Ilx - ;;;11 + Proof.
t
! Ilf(s)
o
•
- £(s) lids).
An obvious consequence of the linearity of
liT (t) II
O.
is a strong
solution of (9.1). Corollary 9.7.
If
1
+
fEe (ID. ,E)
and
x
E
strong solution.
Proof.
We have for
vet)
128
t
>
0
t
t
o
o
J T(t-s)f(s)ds
J T(s)f(t-s)ds
D(L)
the solution of (9.1) is a
ft
.. v(t+h) - v(t) h
a
T(s) f(t+h-s) - £(t-s) ds h
1 t+h
f
h
+
T(s)f(t+h-s)ds,
for all
h"
a
small.
t
By letting
h
~
0, we obtain
with
ve
v' (t) = T(t)f(a) +
t
f
T(t-s)fj(s)ds. Hence the result is a consequence of Proposition 9.6.
a
Remark
9.s. -
The above proof also works if
[O,T], (For example if
every bounded interval enough to assume that
f
f
is absolutely continuous on
E
is reflexive, it is
is Lipschitz continuous.)
- If additional assumptions are made on
L, it may happen that all solutions
of (9.1) are strong solutions. In this direction the following result is
established in [8]. Theorem 9.9.
f(t)
E
'.;Ie assume that
+
T > a there exists a measurable + w : ]O,T[ +JR which satisfies the conditions is such that for all
C(JR ,E)
function
satisfies the hypothesis of Theorem 7.5 and
L
(s,t) e ia,T) x ia,T), !!Ht) - £(s)!! < w(!t-s!) h { there exist hela,Ti, fW(T)dT
a
{ u(a) = x.
We have shown in section 9 that the "variation of parameters formula" permits to define a unique "weak solution" of (9.1) which depends continuously on the data
(x,f)
and is differentiable for a rich set of data (say
x E D(L)
and 129
1 + fEe (lR ,E».
We will now show the following:
fEe l (nt ,E)
- the condition
1
can be replaced by
f E L1
00
+ (lR,D (L» •
- If we take any weak solution of (9.1), this solution is in fact differentiable in a weak sense, which pennits to say that the equation is
"really" satisfied in some larger Banach space.
10.1. Strong solutions for D(L)-valued fCt) Proposition 10.1.1.
all
t
0
>
1
and
f
Assume that
LfCt) E L1
E
C( JR+ ,E)
with
+
00
(JR ,E). Then for all
f(t)
E
x E D(L)
D(L)
for almost
the solution of
(9.1) is a strong solution. Proof.
For almost all
s E [O,t), t> 0
T(h~ - I T(t-s)f(s) Hence
.
1lll
T(t -s) T(h)-I f(s), h
T(h)-I 1 T(t-s)f(s) ~ T(t-s)Lf(s)
h+O Also for any
II
we have
T(Il~ -
f'or a 11
for a.a. s
E
h > 0.
[O,t].
1
S
E
D(L)
I SII
and any
~ 11* f dds
h > 0:
(T(s)S)dsll.". MewlhlllLSII.
o
On applying this inequality with
S
=
T(t-s)f(s)
we obtain that
is bounded a.e. by
IIT(h;) - I T(t-s)f(s) - T(t-s)Lf(s) II
\Ii
·1:,1
,.
By Lebesgue's theorem it follows that
il':I1'
for all
Lv(t)
t
>
0,
vCt)
E DeL)
. T(h)-I() 1 lffi+ h v t h+O
and t
f
T(t-s)Lf(s)ds.
o
1 + + LfCt) E L ( 1R ,E) we have in addition Lv(t) co: C( JR ,E). We now 1oc show that VEC 1 (lR+,E). For any t> 0 and h> 0, we have
Since
v(t+h) h- vet)
~
T(Il); I vet) +
*
t y T(t+h-s)E(s)ds. t
Hence
o
v(t+h) - vet) h
lim
h+O h>O < h < t,
LV(t) +f(t). On the other hand if
t
>
a
and
vet) - vet-h) h
vet-h) - vet)
-h T(h)-I
."c-'=h~-= v (t-h) +
1
h
t
i
T(t-s)f(s)ds •
t-h
Also we have the inequality
vet-h»~ 112 MewlhIIILv(t)
IIT(h) h- I (v(t) -
1 + vEC(m.,E)
Finally we conclude that dv dt
~
for all
Lv(t) + f(t),
- Lv (t-h) II
and
t > O.
Hence the proof of Proposition 10.1.1 is completed. 10.2. The differentiability of "weakll solutions We have seen that even if
f _ 0, for
xi D(L)
the solution of (9.1) can be
nowhere differentiable as an E-valued function. However, in the applications to partial differential equations, it always happens (even in the lIworst" case where
L* "" -L
with
E
Hilbert and
D(L) f E) that the weak solutions are differentiable in some larger space, generally a space of generalized functions (or distributions of finite order). The following result explains why this will be, in fact, always the case. Theorem 10.2.1. CO-serni-group
There exists a Banach space T(t)
on
X such that
X with infinitesimal generator
E c+X L
and a
such that we
have D (i) ~ E {
Proof.
for all Let
x E E, and for all
M ~ 1, w
for all
t
> 0,
>
a
t > 0, T(t)x
T (t )x.
be such that
IIT(t)11 2 Me
wt
. 131
A
We pick any p
a
>
w and let
for the semi-norm
E
defined by
for all Since and
f a
p(u)
U E E, T(t)u
IIT(t) IlL
ret)
check that
~
t
ex) < Me
wt
-1
u. For all
X. We call
s,
>
t
0
T(t)u - u D(L)
consider the sequence
f
(AOl - L)
Then
u
E. Since
G(L)
E E
x
E
{ du
de
~
E
D(i) f
U E
the generator of
E
and
n
t + 0, hence
E c D(L) .
and
AOx - Lx = f E X. We in X as n -)- +00, and
+ f
(and L
lim
(u ) n
in the sense
uED(L». is the generator of a CO-serni-group of type
x E E
x
and
=
for all
t
u
~ X
+
> O.
(M,w)
E E.
f E C( 1R ,E), the solution
1 C (JR+,X) n C(JR+,E)
Lu(t) + f(t),
u
X, we have
is one-to-one. Hence
For any
E, we set
(9.1) satisfies
u(t)
as
X
such that
is closed in
AO > wand
X, AOI - L
Corollary 10.2.2.
132
L
f · n
f
Now since
E, let
is a Cauchy sequence in
n
such that
T(s)v - vII.
- v
converges in
t
n
LeX)
p(u).
s
Finally, to show that -1
tilt
we have
s
t
Me
and by density it is not difficult to
is a CO-serni-group on
It follows that
on
X,
T(t)u.
(T(t)U - u _ T(s)u - u) P
of
T(t) E
X. Then obviously, G(L) c G(L). Also, let
in
(AOI - L)
=
~
0, p(T(t)u)
~ 0, there exists a unique
t
Moreover, we have
un
.
can be identified with a dense subspace of
E
u E E, and for all
for all
=
ufO,
if
- L)-1 u ll
E erX. Also we obviously have
Hence for all
ret)
II (AOl
u E E, p(u)
for all
v
be the completion of
X
u(t)
of
Proof:
Apply Proposition 10.1.1 and Theorem 10.2.1.
Remark 10.2.3.
a) It follows from the closed graph theorem that
L
is a bounded operator, so
is also the extension of
L
L: E -r X
in the sense of
L (E ,X) •
b) The following alternative construction, generalizing an idea of [7], seems to be more convenient than Theorem 10.2.1 to determine concretely First of all we define the adjoint G(L*)
{(v,g) c: E' x E', for all
:=
-
IIAO~
introduce
L*~II*
Y = F'
Theorem 10.2.4.
II II *
where
E G(L), = }.
(u,f) ~0I
- L*
has an inverse in
denotes the norm in
The extended space E
above. Moreover, i f we identify
Proof.
by its graph:
L
E'. We finally
and we have the following.
isometric to the closure of
for all
L.
D(L*) = F is a Banach space for the norm defined by
L(E'). The vector space =
of
AD > w ,
It is readily verified that for any
II~IIF
L*
X and
u E E,
in X
cons t ructed at Theorem 10.2.1 is
X
y
with
and for all
¢
II IIF
for the norm dual to
E
cly(E) , then
F,
.
By definition, we have
Now for any
U E
E, we have
11
and the equality occurs if we choose J
is the duality map: E
subspace of
Ilully
-r
AOt/> - L*¢
E'. It follows that
E
E
J«~OI
- L)-l u )
where
can be identified with a
F', and under this identification we have: =
p(u),
for all
uE
E. 133
Clearly, then, the completion of closure of
E
in
E
Y fot' the norm
for the norm
II II Y'
belongs to have
L(E,Y)
E
A : E
+
Y defined by
E, (Au)(¢) = .
and
Au
=
Lu
for all
u E D(L). Hence
1\(E) c X and we
1\ = L.
Remark 10,2,5. also
u
is isometric to the
The last statement becomes obvious
by using remark 10.2.3, a): indeed the operator for all
p
If
E
is a Hilbert space, then
Y = X, Both of these properties can fail if
D(L*) E
E'
is dense in
and
is a non reflexive
Banach space. Remark 10.2.6.
Although the idea of "extrapolation space" is quite natural
and almost implicit in the treatment of linear partial differential equations of evolution type, from the abstract point of view it has beeL studied only recently. In [9J, F. Weissler outlined Theorem 10.2.1 and applied this idea to the study of the semi-linear heat equation in LP • In [7], the properties of
X and
L
have been studied for
E
a real
Hilbert space with applications to the asymptotic behavior of damped hyperbolic systems with oscillating forcing term. In the case of general Banach spaces, the duality method used in [7] is still directly applicable but technical. Finally, Theorem 10.2.1 appeared in a slightly different ferm in the paper [6J of G. Da Prato and P. Grisvard, where more general extrapolation properties are also studied. Also our Theorem 10.2.4 seems to be new, although it is in fact the precise form in which Theorem 10.2.1 will be applicable in practice in the theory of partial differential equations.
References [1 J Balakrishnan, A. V., Applied functional analysis, Springer-Verlag, Applications of Mathematics, No.3 (1976). [2]
Brezis, H., Analyse fonctionelle appliquee, Masson, Paris (1982).
[3]
Brezis, H., A. Pazy, Convergence and approximation of semi-groups of nonlinear operators in Banach spaces, J. Funct. Anal. 9 (1972), 63-74.
134
[4] Chernoff, P.R., Product formulas, Nonlinear Semigroups, and Addition of
Unbounded Operators, Mem. A.M.S. 140 (1974), 1-121 . • [5] Crandall, M.G., T. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Am. J. Math. ~ (1971), 265-298:
[6] Da Prato, G., P. Grisvard, Maximal Regularity for Evolution Equations by Interpolation and Extrapolation, J. FUllet. Anal. ~ 2(1984), 107-124. [7J Haruux, A., Damping out of transient states for some semi-linear, quasiautonomous systems of hyperbolic type, Rend. Ace. Naz. Sci. dei XL, (Memorie di Matematica) 101 (1983), VII, Fase. 7, 89-136.
[8] Pazy, A., Semi-groups of linear operators and applications to partial differential equations, Springer, ,Berlin (1983).
[9] Weissler, F.B., Semilinear evolution equations in Banach spaces, J. Funct. Anal. }2, 3 (1979), 277-296. [10]Yosida, K., Functional Analysis, Springer (1965). A. HARAUX
Analyse Numerique Universite Pierre et Marie Curie 4, place Jussieu F-75230 Paris Cedex 05
France
135