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0
denote by
Em;
since
IIS(:)vnil
with
t
Given
i s t h e supremum of t h e sequence
m(Z)
a dense sequence i n t h e unit sphere of Moreover,
S ( t ) E m E Em for
t h e norm of t h e r e s t r i c t i o n of
i s separable,
with
i s t h e closed subspace generated by t h e
i s separable and
complement of t h e n u l l s e t
En
and a separable subspace
hence i f Em
n’
en
to
El, {vn]
Em and is i t s e l f measurable.
em we have
m(s
+ t)
=
u E Em, hdl 5 11 5 SUP f lb(s>vli; v E Em, b l l < _ m ( t > l Accordingly, a c o n t r a d i c t i o n r e s u l t s from:
1/s(s)S(t)uli;
< - m(s)m(t).
LEW 1.2.
t
defined i n
Let
1 0
m( 0, t ,d e ) ,
m(s)m(t) (0,
c a< B
_ we have eU d U ( a - d) 2 [ O , a ] s o t h a t Id] + ] a - dl >_ a , where 1.1 indic a t e s Lebesgue measure. But Id1 = la - d l , hence Id1 >_ a/2. Assume now t h a t m($) is unbounded i n [ a , f31, so that t h e r e e x i s t s Proof.
Let
a sequence
[anl
a > 0.
If
t h e r e with
s
= a, t
m(an) +
m.
.
Applying t h e argument above we
5
FIRST ORDER EQUATIONS
deduce t h e existence of a measurable set
dn
[0, p ]
in
with
.Jm(an)
i n dn, which c o n t r a d i c t s t h e a/2 and m ( t ) 2 /dnl 1 an/2 f a c t t h a t m(t) i s everywhere f i n i t e . This completes t h e proof of
Lemma 1.2. End of proof of Theorem 1.1. Let 0
I h / 0.
bargained f o r i n t h e d e f i n i t i o n .
I n t h e uniform
Co exp ( & )
un(0)
t 20.
exp ( - u t ) u ( t )
4
u(0)
so then
When t h e Cauchy +
exp ( - & ) u ( t )
This is considerably more t h a n we
6
FIRST ORDER EQUATIONS
The Cauchy problem i n
$1.2
my.
(-m,
We d e c l a r e t h e Cauchy problem f o r (1.1)w e l l posed i n
-a
< t
a ) ) .
(-m,
i s s t r o n g l y continuous f o r a l l t. t ? 0.
The proof of (2.1) is t h e same a s that f o r t h e case
To show
(iii)we note that t h e assumptions imply that t h e Cauchy problems f o r
both (1.1)and U ' ( t ) = -Au(t) a r e w e l l posed i n ' t
2
0,
(2.3)
u(t^)
s i n c e t h e correspondence
~ ( - 2 ) maps
-t
s o l u t i o n s of (1.1) i n t o s o l u t i o n s of (2.3) and v i c e versa. be t h e propagator of (2.3) i n
t 2 0.
s (t)= Applying Theorem l . l ( i i i ) t o h
we deduce t h a t
S(t)
S(t^)
kt
~-(t)
We v e r i f y e a s i l y t h a t
S(-t)
and t o
(t 2 0)
.
is s t r o n g l y continuous i n t < 0
t > 0
in
S ( - c ) = S-(t^)
and
t > 0.
However,
S(h)u = S ( t + h)S(-t)u,
follows.
In p a r t i c u l a r , t h e Cauchy problems f o r (1.1)and ( 2 . 3 ) a r e
thus s t r o n g c o n t i n u i t y a t
t
= 0
FIRST ORDER EQUATIONS
7
t L 0 which y i e l d s ( 2 . 2 ) by i n t e r v e n t i o n o f Obviously, t h i s i n e q u a l i t y implies t h a t t h e Cauchy
uniformly w e l l posed i n Theorem l . l ( i i ) .
problem i s w e l l posed i n
(-m,
thus completing t h e proof of
m),
Theorem 2.1.
$1.3
The Hille-Yosida theorem. Assume that t h e Cauchy problem f o r (1.1) i s w e l l posed i n
that
is closed.
A
number with
Re
X>
Let
w
R(X)u =
(1.9), X
be t h e constant i n
E
Define an operator i n
w.
[
e
-Xt
S(t)u d t
and
t 2 0
a complex
by
(u
E
.
E)
(3.1)
I
Since t h e norm of t h e integrand is bounded by
ColIu/I exp ( w
is a bounded operator i n
u
Assume now t h a t
E.
s o l u t i o n of (l.l), s o t h a t
Then
D.
E
-
Re X)t, R ( X ) h
S(t)u
is a
s l ( t ) u = AS(t)u. (l) Using a w e l l known r e s u l t
on i n t r o d u c t i o n of closed operators under t h e i n t e g r a l s i g n (HILLE-PHILLIPS
[1957:1,p. 831) we obtain A
LT
-
e-‘%(t)u
dt =
L~
.,
= e -XT S(T)u
for
T > 0.
R(X)u
E
ment of
Letting
D(A)
E,
and
T
-t
00
e-XtAS(t)u d t = JOT e-XtSt ( t ) u d t
-
u
+ X L T
e-XtS(t)u d t
and using closedness of
AR(X)u = XR(X)u
choose a sequence
- u.
{unj
-
-
A ) R ( X )= I
I n p a r t i c u l a r , (3.2) shows t h a t
w e l l one-to-one. Then
(XI
-
-u
(XI
-
X
E
p(A),
A)-l = R(X), The f a c t t h a t
p(A)
A):D(A)
t h e resolvent s e t of p(A)
u. Then
-t
i s onto.
E
u
A
E D(A)
It is a s with
Au = Xu.
/lu(t)II =
u = 0.
W e have then
and t h a t
contains t h e half plane
R(X;A) = Re
X>
w.
i s nonempty allows f o r t h e complete i d e n t i f i c a -
t i o n of t h e subspace Do of “admissible i n i t i a l d a t a ” c o n s i s t i n g of a l l uo such t h a t X€p(A),
S($)uo
and
R(X)E C_ D ( A )
(3.2)
i s a s o l u t i o n of (1.1)with
s o that
-t
.
(exp (Re X)t)(/ull which c o n t r a d i c t s (1.9) u n l e s s proved t h a t
un
so that
To s e e t h i s assume t h e r e e x i s t s
u ( t ) = ( e m (Xt))u
i s an a r b i t r a r y e l e -
such t h a t
D
R(X)un -+ R(X)u, AR(X)un= XR(X)un u n + XR(X)u (XI
u
Now, i f
in
we deduce that
A
is a s o l u t i o n of (1.1). This is done a s follows.
E
E
If
R(X;A)S(t)u is a s o l u t i o n of (l.l), h e n c e i t f o l l o w s f r o m ( 1 . 6 ) t h a t
a
FIRST ORDER EQUATIONS
(t > 0)
R(X;A)S(t)u = S(t)R(X;A)u
u
This e q u a l i t y extends t o a l l
E by denseness of
E
(3.3) f o r u
Write
(XI
=
-
v E D(A)
A)v,
D
(3.3)
and implies t h a t
.
( t 2 0)
S(t)D(A) C_ D(A)
.
(3.4) (XI
and then apply
-
A)
to
The r e s u l t is
both s i d e s ,
AS(t)u = S(t)Au
u
valid f o r
E
Jb
S(t u - u =
for
u
E
D.
Apply
(3.5 we obtain
Making use of
D(A).
R(X;A)
t
t S'(s)u
"
t o both s i d e s and u s e
R(X;A)(S(t)u
u
E;
E
(3.3):
the r e s u l t i s
t
-
U)
S(S)AR(X;A)U
=
Since both s i d e s a r e bounded operators of to all
S(s)Au ds
ds =
u
.
ds
t h e e q u a l i t y can be extended
using (3.3) again and applying
(XI
-
t o both s i d e s ,
A)
we obtain
s(t)u
-
u
=
rt
AS(s)u ds
(3.6)
JO
for a l l
u
E
D(A).
(1.1)f o r any
u
Obviously, t h i s implies t h a t E
A,
D(, = D(A) We can obtain estimates for R ( X ; A ) formula
= (-l)nn!R(X;A)nfl
R(X;A)(II)
S(;)u
i s a s o l u t i o n of
hence
.
(3.7)
and i t s powers using t h e well known and d i f f e r e n t i a t i n g
(3.1) under
t h e i n t e g r a l s i g n ( t h i s can be e a s i l y j u s t i f i e d on t h e b a s i s of t h e dominated convergence theorem).
results for
where ities
u
E
E
and
Re
X>
The formula
0.
Using (1.9) we obtain
Co and w a r e t h e constants i n (1.9). We show next t h a t inequal(3.9) a r e as well s u f f i c i e n t f o r uniform well posedness of t h e
9
FIRST ORDER EQUATIONS
Cauchy problem for (1.1).
3A
THEOREM 3 . 1 (Hille-Yosida).
f o r (1.1)i s 5
and
w
The Cauchy problem
t 2 0 with propagator S ( t ) u(A) i s contained i n t h e half-space
uniformly w e l l posed i n
s a t i s f y i n g ( 1 . 9 ) i f and only i f Re X
b e closed.
R(X;A)
(3.9).
s a t i s f i e s inequalities
The proof of t h e other h a l f below
We have a l r e a d y proved one h a l f .
i s perhags not t h e s h o r t e s t a v a i l a b l e but can be adapted with minor changes t o equations o t h e r than (1.1) ( s e e Chapter 11). We begin by constructing s o l u t i o n s of (1.1)given s u f f i c i e n t l y "smooth" i n i t i a l conditions.
u
E
D(A 3 ),
> w, 0.
w'
Let
Define ( 2 )
N
u(t;u) = u
+
tAu
t2 2 +A u 2 w'+im
+
(3.10)
eXt
Ju'-im
&i
3u ax - R(X;A)A
( t 5 0)
,3
.
A deformation of contour shows t h a t t h e i n t e g r a l i n (3.10) vanishes for
t
5 .O,
thus N
u(0,u) = u ~ ~ ~ ( t=; O u ()e q ~ (~w ' t ) )
Obviously,
as
t
.
(3.11)
+
m.
An e a s i l y j u s t i f i a b l e
d i f f e r e n t i a t i o n under t h e i n t e g r a l s i g n shows that continuous d e r i v a t i v e in
t > 0;
;(E;u)
on t h e o t h e r hand
A
has a can be introduced
under t h e i n t e g r a l s i g n with convergence of t h e r e s u l t i n g i n t e g r a l , s o that
_ 0
E-valued continuous f u n c t i o n defined i n
f ( t ) = O ( e q (at))
such that
(Sf)(:)
f(?)
b e t h e Laplace transform of
as
t
+
f o r some
w
CY
and l e t
t,
(Sf)(X) =
e-Xtf(t)
dt
.
Then
t > 0.
uniformly on compacts of
The proof i n t h e s c a l a r c a s e can b e found i n WIDDER [1946:1, Chapter V I I ] .
The extension t o v e c t o r valued functions i s immediate.
End of proof of Theorem 3.1.
We apply t h e inversion formula (3.14)
in (3.10); i n view of (3.13) we obtain
t o t h e function
(3.15)
s o t h a t , u s i n g i n e q u a l i t i e s (3.9) w e o b t a i n IllY(t;u)ll
5 Collu/I lim
n
Define N
N
S(t)u = u(t;u) for
u
E
ment of
D ( A 3 ),
u(t)
s(%) we
operator such t h a t
(t
0)
t h e solution i n (3.10).
can extend each E ( t )
g($)
.
Arguing as i n t h e t r e a t -
t o a bounded everywhere defined
i s s t r o n g l y continuous i n
t 2 0
and
C l e a r l y , t h e proof of Theorem 3 . 1 w i l l be complete i f we show t h a t any solution
u(t)
of (1.1)admits t h e r e p r e s e n t a t i o n U(t) = iqt)U(O)
.
(3.18)
11
FIRST ORDER EQUATIONS
This is done as follows.
It r e s u l t s from t h e d e f i n i t i o n of for all
R(X;A)S(t)u = g ( t ) R ( X ; A ) u
u
thus f o r a l l
D(A 3 )
E
g(t) u
E
that
E.
Ace ord i n g l y ,
-S ( t ) A u = A i ( t ) u m(t^)
On t h e o t h e r hand,
(u
t 2 0)
D(A),
E
.
(3.19)
i s a continuously d i f f e r e n t i a b l e
= S(i)R(h;A)3
W(t)
o p e r a t o r valued f u n c t i o n t h a t s a t i s f i e s
Accordingly, i f
= Am(t).
i s a n a r b i t r a r y s o l u t i o n of (1.1)we have
u(^t)
d as m ( t
-
+ m ( t - s)Au(s)
(3.19), hence R(X;A)3u(t)
after
-
(XI
Applying
3.3.
= 0
(0
5 s 5 t)
(3.20)
= m(O)u(t) = h ( t ) u ( O ) = R(X;A)3S(t)u(0).
u(A)
(-m,
be c l o s e d .
A
uniformly w e l l posed i n i f and only i f
s)u(s)
t o b o t h s i d e s (3.18) r e s u l t s .
A)3
Corresponding t o t h e c a s e
THEORFM
-
s ) u ( s ) = -Am(t
(-m,
m)
we have
a)
The Cauchy problem f o r (1.1)&
w i t h propagator
i s contained i n t h e s t r i p
satisfying (2.2)
S(%) /Re
XI 5 w and R(X;A)
w,
n
satisfies
5
IIR(X;A)n(l Proof.
Co( /Re
XI -
( IRe XI
=
0,1,. . . )
.
(3.21)
We have a l r e a d y observed t h a t i f t h e Cauchy problem f o r (1.1)
i s w e l l posed i n
t h e n t h e Cauchy problem f o r ( 2 . 3 ) ( o f course
m)
(-m,
a l s o f o r (1.1))i s w e l l posed i n
t 2 0.
i n e q u a l i t i e s (3.21) f o l l o w from Theorem
Since
3.1.
R(X;-A)
= -R(-X;A),
Conversely, we deduce from
t h e same theorem t h a t i f (3.21) holds t h e n t h e Cauchy problem f o r (1.1) and (3.2) i s w e l l posed i n
t 2 0.
t 2 0 with
S
and
S- s a t i s f y i n g (1.9) i n
T h i s i s e a s i l y s e e n t o imply t h a t t h e Cauchy problem f o r (1.1)i s
w e l l posed i n
(-a,
m),
t h u s completing t h e proof of Theorem 2.1.
For t h e s a k e of b r e v i t y i n f u t u r e s t a t e m e n t s , w e i n t r o d u c e t h e A closed, densely defined operator A i s s a i d t o
following n o t a t i o n .
1
belong t o & + ( C 0 , w ) posed i n
t 2 0
i f t h e Cauchy problem f o r (1.1)i s uniformly w e l l
and t h e s o l u t i o n o p e r a t o r
S(c)
satisfies
(1.9).
We
12
F I S T ORDER EQUATIONS
1 1 &+(w) = U {&+(Co,w); C > 13 ( n o t e t h a t Co < 1 i s h p o s s i 01 1 The n o t a t i o n s f o r t h e case b l e ) and &+ = U {Q+(w); -m c w < a].
a l s o write
-m
c t
(Re
e-XtS(t)u d t = R(X;A)u
Then -
contains t h e
an operator valued f u n c t i o n s t r o n g l y con-
S(t)
wt lls(t)lI
W,
n
s
0,1,...)
.
(3.25)
Although t h i s form i s l e s s p r a c t i c a l , s i m i l a r i t y with generation theorems f o r other equations becomes more apparent ( s e e Section 11.2).
13
FIRST ORDER EQ,UATIONS
91.4
Semigroup t h e o r y . Given two Banach spaces
a l l l i n e a r bounded o p e r a t o r s from (uniform o p e r a t o r ) topology.
A
E-valued f u n c t i o n
F
and
E
we denote by
into
E
F
(E;E)
t 2 0
defined i n
t h e space of
endowed with i t s usual
We u s u a l l y a b b r e v i a t e
S(^t)
(E;F)
to
(E).
i s c a l l e d a semigroup
i f (1.7) h o l d s , that i s i f
s(s +
s(0) = I ,
(s, t 2 0)
t ) = S(S)S(t)
(4.1)
*
Equations ( 4 . 1 ) a r e o f t e n c a l l e d t h e e x p o n e n t i a l ( f u n c t i o n a l ) equations. We have s e e n i n t h e preceding s e c t i o n s t h a t t h e s o l u t i o n o p e r a t o r of a w e l l posed Cauchy problem i s a s t r o n g l y continuous semigroup.
As t h e
following r e s u l t shows, t h e converse is as w e l l t r u e . THEORFM 4.1.
t L 0.
S(-)
b e a semigroup s t r o n g l y continuous i n
Then t h e r e e x i s t s a unique c l o s e d , densely d e f i n e d o p e r a t o r
such t h a t
E
1
E+
i s t h e e v o l u t i o n o p e r a t o r of
S(t)
Proof.
A
We d e f i n e t h e i n f i n i t e s i m a l g e n e r a t o r
A
of
S(t)
by t h e
formula 1 AU = lim I;(S(h) kt0+
The domain of
A
a r b i t r a r y and
a > 0
u
c o n s i s t s of a l l
E
E
-
such t h a t
r a S(S)U
=
ds
a Jo
The second e q u a t i o n ( 4 . 1 ) implies t h a t
hence
ua
E
a+ h
-
I)ua =
D(A)
S ( S ) U ds
ua -+ u
as
(4.3) e x i s t s .
For u
-
.
+
(4.4)
h
S(s)u ds),
(with AU& =
But
(4.3)
define ua
g 1( S ( h )
.
I)u
a
( a f o r t i o r i , D(A))
-+
0,
1 g(s(a) - I)U)
.
t h u s t h e s e t of a l l elements of t h e form
is dense i n
E.
(4.5) ua
FIRST ORDER EQUATIONS
14 We prove next t h a t
t 2 0
for h, 0 ,
i s closed.
A
thus if
u
E
D(A)
u
If
then
u
after (4.5). Au -+ v. n
E
Let
E
as w e l l and
D(A)
.
(4.6)
E,
{u n
D(A)
be a sequence i n
un + u,
such t h a t
Then 1 j-(S(h)
1 1 ) u = l i m h(S(h) n+
-
h + O+
Taking limits a s
Let now
u
E
-
we see t h a t
shows t h e closedness of
ht
E we have
S(t)u
AS(t)u = S(t)Au Hence, for any
E
h
1 ) u = l i m (Au,) n n+m
u
E
D(A)
.
h
= v
and t h a t
Au = v,
which
A.
D(A).
Integrating
(4.6) i n
0
ds
S(s)u
S(s)Au ds =
5
s
5t
we o b t a i n
tAut = S ( t ) u
=
-
u
h
using
(4.6) i n t h e l a s t e q u a l i t y .
differentiable i n
t
>0
This shows t h a t
with
,
S ' ( t ) u = S(t)Au = AS(t)u
so that
t >_ 0.
S(^t)u i s a s o l u t i o n of (4.2) i n
This proves ( a ) i n t h e
d e f i n i t i o n of uniformly w e l l posed Cauchy problem. dence p r o p e r t y ( b ) i s checked a s f o l l o w s . for a l l
u,
llS(t)ull
t 2 0.
0
A~(Z)
D(A),
E
( 0
(r)
S(i)
More g e n e r a l l y , we can c o n s t r u c t
lib+,
i n such a way t h a t
where
h >0
i s a r b i t r a r i l y preassigned.
See ZAECZYK
[1975:11;
e a r l i e r (although l e s s elementary) example can be found i n HILLEPHILLIPS
[1957:1, p . 6651 where
EXERCISE
14.
n(A)
3.:?
Prove Lemma
i s i n f a c t empty ( 1 )
so t h a t
u s e the f a c t that
(Hint:
with
(6.18) EXERCISE 15.
Prove t h a t t h e “ r e d ” i n e q u a l i t i e s (3.24) imply t h e i r
complex c o u n t e r p a r t s ( 3 . U ) .
(Hint:
if
Re h >
f~
express
R(A;A)
by
means of i t s Taylor s e r i e s
with
p
real.
EXERCISE
R(A;A)
=
Then l e t
p
16. Let
i n f i n i t e s i m a l generator, Show t h a t i f
u
E
D(A)
be a s t r o n g l y continuous semigroup,
S(t)
t h e niimber defined by
(11
(6.15),
its
A
w’ >w,
0.
then
(6.20) t h e l i m i t being uniform on compact s u b s e t s of
l i m i t is
EXERCISE 17.
Let
E
such t h a t
h(h ; A ) / / 5 u
E
D(A)
For
t = 0
the
be a ( n o t n e c e s s a r i l y densely defined) l i n e a r
A
operator i n a Banach space
Show t h a t i f
t > 0.
(6.3)).
(compare with
$u
C/h
R(X;A) exists for
( h > w)
.
X >
w
(6.21)
then
l i m
A-
+m
hR(h;A)u = u
.
(6.22)
and
FIRST ORDER EQUATIONS
23
FOOTNOTES TO CHAPTER I (1) We d e n o t e h e r e by
not t h e d e r i v a t i v e of
( E ) ) applied t o valued funct i on s .
u.
S'(;)u S(;)
t h e d e r i v a t i v e of t h e f u n c t i o n
S(t^)u
(which may f a i l t o e x i s t i n t h e norm of
The same o b s e r v a t i o n a p p l i e s t o o t h e r o p e r a t o r The c o r r e c t n o t a t i o n
(S(i)u)'
becomes cumbersome
later. ('2)
Roughly speaking,
R( i;A)u.
u(t)
i s t h e i n v e r s e Laplace t r a n s f o r m of
However, t h e c o r r e s p o n d i n g i n t e g r a l may n o t be c o n v e r g e n t
-1
-2
...
t h u s we u s e t h e w e l l known formula R(h;A)u = h u + A Au + + A- mAm -1 u + hmmR(h;A)Amu f o r m = 3, which a l l o w s d i f f e r e n t i a t i o n w i t h respect t o
t.
24
CXIIPTER I1
THL CAUCHY PROELEM FOR SECOND ORDER EQUATIONS COSINE FUIVCTION THEORY
The Cauchy problem f o r second o r d e r e q u a t i o n s .
$11.1
A t h e o r y f o r t h e equation
u"(t)
(1-11
Au(t)
=
t h a t p a r a l l e l s c l o s e l y t h a t of (1.1.1)can be developed without undue difficulty.
We prove t h e fundamental r e s u l t s i n t h i s chapter.
s o l u t i o n of (1.1)in t E-valued f u n c t i o n u(:)
30
A
i s a twice c o n t i n u o u s l y d i f f e r e n t i a b l e
such t h a t
u(t)
E
D(A)
and (1.1)h o l d s t h e r e ;
s o l u t i o n s i n d i f f e r e n t i n t e r v a l s a1.e defined accordingly.
The Cauchy
problem i s now t h a t of f i n d i n g s o l l t i o n s of (1.1)t h a t s a t i s f y t h e i n i t i a l conditions u(0)
=
u0
, u'(0)
=
u1
(1.2)
The Cauchy problem i s w e l l posed o r p r o p e r l y posed i n
2
t
i f and only
0
if (a)
uo,ul
E
D
There e x i s t s a dense subspace there e x i s t s a solution
D
of
u ( 7 ) of
E
such t h a t for any
(1.1)2
t
2
satisfying
0
(1.2). (b)
t
2
0
There e x i s t s a nonnegative, f i n i t e f u n c t i o n
C(ll>
The d e f i n i t i o n s of well posed problem in posed problem i n
[0,m)
and
(-my=)
(-m,m)
(t
2
(1.3)
0)
and uniformly well
a r e obvious analogues of t h e f i r s t
o r d e r case and we omit t h e d e t a i l s . We d e f i n e now two p r o p a g a t o r s o r s o l u t i o n o p e r a t o r s
c(
L
0)
- c .
The checking t h a t
w,O.
(1.1) i s done j u s t a s i n Theorem
t h e same way t h a t
dh
A5
llL(t;u)ll
= O(exp
-u ( ~ , u )
= U,
(w't))
-
u'(O,U)
u(t;u)
is actually a
1.3.1; a l s o , we prove i n
as = 0
t
-
m,
(2.13)
and
L m e - h t L ( t ; u ) d t = hR(h2 ;A)u
for
Re7 >
W'.
It follows from t h i s e q u a l i t y and from Lemma 1.3.2 t h a t
31
SECOND (\WF:Y FCUATIONS
(2.14) hence, using i n e q u a l i t i e s
(2.1),
l l h t ; ~ ) l l5 C o I I ~ I / l i m n-
p-y,1
-(n+l)
wt
wt
= cOllulle
ft 1 0 )
.
(2.15)
m
Condition ( a ) i n the d e f i n i t i o n of uniformly well posed problem f o r (1.1) i s v e r i f i e d a s follows:
if u
u
0’ 1
E
D(A3),
i s given by
(1.2)
u(t)
G(t;u )
=
0
-1
Lt-
@(t)U
(E)
u
E
D(A3)
- valued
We s h a l l show t h a t i f 0
we define
(2.16)
= i(t;u)
E,
ohtaining a
s t r o n g l y continuous function with
-S ( t ) u 2
(b)
a n d extend it by c o n t i n u i t y t o all of
A second operator valued function
t
.
u ( s ; u ~ )d S
To check t h e continuous dependence statement
for
a solution satisfying
u(;)
-8 ( t )
=Lt
i s defined by
.
“@s)u d s
i s an a r b i t r a r y s o l u t i o n of
(1.1) i n
we must have
u(t)
=
E(t)u(O) + Z(t)u’(O).
(2.18)
This i s done a s i n the f i r s t order case and we only sketch t h e d e t a i l s . The f i r s t step i s t o show t h a t and consequently with functions
h(t)
A;
c(t)
= “@t)R(h;A)3 and
=
“(t)R(A;AP
commute with
h ( t ) = i(t)R(A;A)3
continuously d i f f e r e n t i a b l e and s a t i s f y h’(t)
z(t)
and
R(X;A)
t h e second i s t o note t h a t t h e operator valued
h’(t)
= AZ(t)R(A;Af’=A h ( t ) .
=
a r e (twice)
h ( t ) , and the equality
Accordingly
32
SECOND OFTIER EQUATIONS
and
(2.18)
h ( 0 ) = R(A;A)3,
follows noting t h a t
h(0)
=
This com-
0.
p l e t e s t h e proof of Theorem 2.1.
REMARK 2.2. Theorem 2.1 shows i n p a r t i c u l a r t h a t we have t h e r e l a t i o n 2 2 t h e region t o t h e l e f t o(A) 5 ;Reb 5 w } = {A;Reh 5 b) - (Imh)2/4m23, 2 I n p a r t i c u l a r , if o f a p a r a b o l a p a s s i n g through t h e p o i n t s U2, 2 2iw
{w
.
a(A)
o=O,
i s contained i n t h e n e g a t i v e r e a l axis.
A n o t a t i o n s i m i l a r t o t h a t f o r t h e f i r s t o r d e r c a s e w i l l be u s e f u l here.
@(t) s a t i s f i e s
g2(Co,o) w
2
for
The following analogue of Theorem
&
THEOREM 2.3. t h e h a l f plane continuous i n
t
2
km
u
E
2 Q
(Co,w)
wt Cge
@(
=
for
< 0 by Theorem 2 . 1 ) .
1.3.4 holds:
(t
L
:1 1 )
hR(h2 ;A)u
2 R( A ; A )
exists i n
(2.20)
0).
(Reh
> ho).
(2.a)
i s t h e s o l u t i o n o p e r a t o r of
The proof i m i t a t e s t h a t o f Theorem
REMARK 2.4.
8(u)
E
eht@(t)u d t
F
05
and such t h a t
0
Assume t h a t , f o r each
A
if
-
I)U
a
0
it r e s u l t s t h a t
i s closed we n o t e t h a t
s,t;
=
(3.1)
D(A)
i s dense i n
implies t h a t
C(s)C(t)
E. =
T o show
C(t)C(s)
34
SECOND ORDER EQUATIONS 2 2 ( @ ( h ) -I)@(t)ll = @ ( t7 ) ( @ ( h ) -I)U
h
h2 for
h
#
Accordingly,
0.
u
if
D(A)
E
@(t)u
then
D(A)
E
and
A@(t)II = @(t)Au
f o r any
u
E.
E
Hence
after (3.6). Let Aun v. Then
{u }
-
2
(1.7)
n
( @ ( h )- 1)u
be a sequence i n
= l i m
so t h a t t a k i n g l i m i t s as
h
2 ( @ ( h )2
h
n-m
h
+
D(A)
h
I ) u n = l i m (Au ) n n- m
u
we deduce t h a t
0
D(A)
E
-
un
such t h a t
=
h
v
u,
,
and
Au = v
a s claimed.
If u
E
D(A)
we have
r t ( t- s ) @ ( s ) A u d s
=
' 0 = A
rt(t-
s ) @ ( s ) u ds =
Lt t2
( t - s ) A C ( s ) u ds
Au
t
=
C(t)u
- u.
(3.9)
0
Hence
@(G)u i s twice continuously d i f f e r e n t i a b l e i n @"(t)u
so t h a t
=
@(:, = 7 t2n+1hn/(2n + l)!). Show t h a t (6.4) and (6.5) imply
40
SECOND ORDER EQUATIONS
; A
0 = SuprRe A’/‘
(i)
Alternately,
E
.
~ ( A ) I
(6.8)
i s t h e l e a s t p o s i t i v e number such t h a t
(,lo
a(A)
i s con-
t a i n e d i n t h e closed r e g i o n t o t h e l e f t of t h e parabola
5
(d
>0
(with
$/4‘,’2
2
p a s s i n g through t h e p o i n t s with
(? -
=
, -+ S i O i2 .
(0
,
(6.9)
I n p a r t i c u l a r , (6.6)and(6.7)hold
n a t u r a l l y depending on
C
( 6 . 6 ) and (6.7) do not n e c e s s a r i l y hold w i t h 8(;)
5;
0’
A
s(5)
and
s(t^)i s
@(;)
Show t h a t
(1)
holomorphic
@(;)
i n particular,
continuous i n t h e norm of (E)(of course, t h e norm of ( E )
w =
@(i)
can be extended t o f u n c t i o n s
(as (E)-valued f u n c t i o n s ) f o r all
is
Produce a n example t o show t h a t
contained i n t h e negative r e a l a x i s . and
a(A)
if
(11)
is
always continuous i n
l/@(;)ll
due t o formula ( 1 . 5 ) and boundedness of
on
compact s u b s e t s ) . n
EXERCISE 2. topology of
@ ( t )be a c o s i n e f u n c t i o n continuous i n t h e
Let
(E)
t
(continuity a t
suffices).
= 0
A
use formula ( 2 . 1 1 )
n = 1 and c o n t i n u i t y of
for
- I I/
Ilh2R( h2;A)
show t h a t 2 2 h R(h ; A )
of
< 1 for
h
h a s a bounded i n v e r s e ) .
r e p r e s e n t a t i o n s ( 6 . 2 ) and
EXERCISE 3.
Let
-
1x1 -,
as
0
t
at
=
0
to
@(;)
admits the
(6.4).
E = C
m
@(;)
(Hint:
s u f f i c i e n t l y l a r g e so t h a t
Show t h a t
be t h e Banach space of all (complex-
(-m,m)
0 valued) continuous f u n c t i o n s u ( x ) u(x)
Show t h a t t h e
@(o
A
-
D1
moment's c o n s i d e r a t i o n shows i h a t
(1.8)
i s a Eanach space equipped
11. 1l0.
with
THEORFM 1 . 3 .
The space
i s a phase space for
C?
5
x E 1 with = E 0 < t < ffi; moreover,
= E
m
(1.1)
-m
-
-
and
Eo = D1
(1.9)
I11 = D1'
6(
/l[u,vl\\,
\/U!le =
Em we have
5 (co +
IlG(t)l!
I f , say, t h e norm
0 .
Since
C/lAl
Ih- h0 1 < %/Cl
0.
Then
a r c s i n (l/C1)
=
e x i s t s a constant
lIR(h;A)(/
replaced by (cp-).
or
Cp ( r e s p .
c+(cp-)w i t h
a, t h e r e
we w r i t e
T,
5
i s d e f i n e d i n t h e same way w i t h
exists i n the sector
f o r every
Cp,
b e l o n g s t o any s e c t o r
THEORD4 3.1. Assume t h a t
R(A;A)
Given
+ i n d i c a t e s e x c l u s i o n of
subindex
(2.1)
i n a sector containing t h e p o s i t i v e r e a l axis.
We i n t r o d u c e some n o t a t i o n s .
R(h;A)
is the
A'
C
=
C
Z+h' 1 ) .
Cp'
such t h a t
(3.1)
CLpO
]/R(hO;A)l] 5 5 l/llR(hO;A)ll
it f o l l o w s t h a t
and c a n be e x p r e s s e d
there by t h e power series m
R(A;A)
=
T j=0
Since
(pO
- A)~R(A@;A)~+'.
(3.2)
57
PHASE SPACES
with 0
0. I n a
arg p =
p l a n e minus t h e l i n e
R(A;-(-A)1'2)
(3.13) y i e l d s a n a n a l y t i c e x t e n s i o n t o t h e h a l f p l a n e 1 t h u s R(k;-(-A)1/2) exists i n (q' + T I ) ) .
symmetric f a s h i o n ,
> 0,
R e ei"'2p
It remains t o estimate t h e r e s o l v e n t . so t h a t
with
For
c > 0,
p E
$
1 55 (n +
+ T ) ) , Im p
5
0
0
p
w e use (3.13) i n s t e a d o b t a i n i n g t h e
Gf
t h e mere f a c t t h a t
and
-p
p
b e a complex number s u c h t h a t
belong t o
p(
-( -A)1/2)
w e have
= ~ ( ~ ; - ( - A ) " ~ ) R ( - ~ ; - ( - A )l/2)E = ~ ( -2p; A ) E = D(A).
that
p((-A)'"
improvement of a ( p a r t i c u l a r c a s e ) of Theorem
Proof: L e t both
( ( -A)1'2)2
Im p 20
Then
T h i s completes t h e proof of Theorem 3.2.
An i m p o r t a n t consequence fGllOWing
IJ. E
TI).
so t h a t
x+( 2-1 (ql
same e s t i m a t e .
Let
5
p = (p\eiJi w i t h
c+( c+(12(ql + n)),
C
-A,
-p2
D( ( (
f
#
is the
2.5:
E
p(A).
Since =
Since (2.22) i m p l i e s
(3.14) s t a n d s proved.
5111.4 T r a n s l a t i o n of g e n e r a t o r s o f c o s i n e f u n c t i o n s . If A i s t h e i n f i n i t e s i m a l g e n e r a t o r of a s t r o n g l y c o n t i n u o u s
60
PIIASE SPACES
semigroup
then
S(:)
A-bI
any complex number) i s th:
(b
infinitesimal
The S ( G ) = e-btS(f). b corresponding r e s u l t for cosine functions i s somewhat harder to prove.
generator of t h e s t r o n g l y continuous semigroup
LEMMA 4.1.
be t h e i n f i n i t e s i m a l generator of a strongly
A
~ ( ts a t i s f y i n g
continuous cosine f u n c t i o n
Il@(t111 5 coeb It1 and l e t
b
(-m
be a n a r b i t r a r y complex number.
0,
+
a)
[Rehl -h))-(n+m)) (5.25)
69
PHASE SPACES
Proof:
Using
(5.22)
i n formula
(1.3.8) a l l
result instantly.
The corresponding formula f o r
estimates
when
(5.25)
is a group.
S(t)
a r e a consequence of Lemma 1.3.2. =
( - l ) n n ! R(h;A)n
-A
(5.23)
t a k e s care of
The opposite i m p l i c a t i o n s
In f a c t , since
we o b t a i n from formula
inequalities
R(A;A)(~)
=
(1.3.14) ( s e e a l s o
(1.3.15))
that
Ils(t>llIc0 Colimtmn-n(n + 1)
... (n + m > ( l - - wt )-(n+m+l) n 5
t
The corresponding e s t i m a t e i n way.
w t -(n+l) + l i m (1- F )
assumed for
h >
co(l + tm)ewt. (t 2
0)
for groups follows i n t h e same and
(5.25)
need o n l y be
real.
End of proof of Theorem
%(.)
0
(5.23)
We observe i n passing t h a t
=
satisfying
(5.20).
IIR(~;%)~II(~)
5.5.
g e n e r a t e s a group
5.6 we o b t a i n
5 c ( ( I h l - r ~ - b ) -+~n ( l h I (111
%
The o p e r a t o r
Applying Lemma
- w - b)-(”+l))
,...
+ b , n = 0, 1
).
(5.26)
Consider t h e s e r i e s
.. . R( h;\)(R(
R( h ; $ ) ( R ( A;\)b’B+
for
kl,k
*,...
=
t h u s each term of
where
of
k =
(5.26)
O,l,Z
,..., Ihl
(5.27)
>
W
h;%)bp?”
(5.27)
+ 2b. It i s e a s y t o see t h a t
can be w r i t t e n i n t h e form
and p + q = k + n. We make use of t h i s r e l a t i o n and j t o deduce t h a t t h e g e n e r i c term i n t h e series (5.27) i s k
bounded i n norm by a n expression of t h e form
PHASE SPACES
70
Cbk
( Ih( - fIi
1 - b)k+n
+
Cb
k
k + n (/Al-fIl-b)
(k + n ) ( k + n + 1)
+ mk
( [ A ] - 0 1 - b)k+n+2
k+n+l
' (5.30)
*
We observe next t h a t
111 >
for
U
..
+
where it must be remembered t h a t
2b,
k =
..
cki
and t h a t
.,k assume independently all t h e v a l u e s 0,1,. We d i f f e r e n t i a t e n next (5.31) r e p e a t e d l y w i t h r e s p e c t t o Ihl o b t a i n i n g t h e e q u a l i t i e s
kl,
1
+
k
n
n)b ( , h / - u - b ) k+n+l
1 Accordingly,
k
k
+
n)(k
+n +
( / h i - u - 2 b ) n+l
-
l)b
k
- (
( ( ] A / -td-b)k+n+2
the series
(5.27)
n(n
(5.32)
'
+ 1)
I A l - u - 2b)n+2
(5.33) '
i s convergent i n t h e norm of t h e space
( 3 ) and we can e s t i m a t e t h e norm o f t h e sum by ( a constant t i m e s ) t h e
sum of
(5.31), (5.32) and (5.33).
We observe f i n a l l y t h a t
(2.57)
is
nothing b u t
and check (by d i r e c t a p p l i c a t i o n of t h e d e f i n i t i o n ) t h a t R(A;%)
(R(h;\)bp)j
=
R(A;%
(5.34)
+ bB) = R(X;%).
The end r e s u l t i s t h e sequence o f i n e q u a l i t i e s
l ] ~ ( X ; % ) ~ l / ( ~y) C ( l h l - ~ - 2 b ) -+ ~Cn(1Al - u - 2 b ) - ( n + 1 )
+
Cn(n
+ 1)( I A / -
U s i n g (a s l i g h t Il'$(t)ll
5 c(1
+
- 2b)-(n+2)
( / A 1 > w + 2b, n
=
... )
0,1,
(5.35)
71
PHASE SPACES
-00
C
t
C
(5.17).
which completes t h e proof of
m,
W e a t t e n d f i n a l l y t o t h e l a s t statement i n t h e proof o f Theorem
5.5. with u(0)
u(
0,
1< p
0.
Lp( (-T,T);E)
t h a t i s , converges i n
hp(E),
f o r every
Passing if necessary t o a subsequence we can t h e n i n s u r e t h a t
0. that
Applying t h e n r e p e a t e d l y e =
(-co,~),
graph theorem
e
- e 5 e,
t h u s by a
c o n t a i n s an i n t e r v a l
t h e second r e l a t i o n
S b ( t ) E _C D(A;l2)
so t h a t
$12Sb(t)
e
for a l l
(6.20)
t;
(-a,a),
we deduce
by t h e closed
i s a bounded o p e r a t o r .
Consider now t h e group
l$(ll 5 co Here
c(:)
(-02
Proof:
If
b
Im
5
0
(ii)
0
(iii)
0.
2W
a d m i t s a n extension
such t h a t
Lj(t)
(i)
i s an
l+(t)
to
(E)-valued a n a l y t i c
There e x i s t a constant
then t h e operator
%(i)
i s strongly C > 0
satisfies
-Ab
such t h a t
(2.1)
It follows t h e n from Theorem 3.2 t h a t t h e r e e x i s t s (see (4.11)). $ > 0 such t h a t R ( h ; (-%) 1/ 2 ) e x i s t s i n larg A \ < ~i + ~ r / 2 and satisfies
-
Thus Theorem 7.1 a p p l i e s t o show t h a t
-(-%)1/2
generator of a s t r o n g l y continuous semigroup
is the infinitesimal
bb(S)
analytic i n
larg 51 < q, 5 f 0. Since Theorem 3.2 does not provide d i r e c t i n f o r mation on t h e growth of b b ( t ) f o r t r e a l we s h a l l o b t a i n t h i s information by means o f an e x p l i c i t r e p r e s e n t a t i o n for b b ( t ) .
Define
To show t h a t t h e l i m i t e x i s t s we perform a n i n t e g r a t i o n by p a r t s , obtaining t h e equivalent express ion
rm
Tb(t) =
/
'- 0
h(t,h)R(h;-%)2
dh
,
(7.19)
84
PHASE SPACES
there e x i s t s a constant
-< c,h1/2
for t
2
(7.20)
C,
5
(t
+
2
may i n p r i n c i p l e depend on
C'
need only use
(7.20)
t >0
continuous i n
h> 0
in
b > w),
i s c e r t a i n l y the case i f
E
(7.20)
O),
(7.19)
h
at
= 1
(7.U) we o b t a i n t h e estimate
and
ll'b(t)ll where
2
Dividing t h e domain of i n t e g r a t i o n i n
6.
and using
(A
such t h a t
C6
' > '1, %'
(if
6.
so t h a t
(7.22) e x i s t s we
in
C' = 0
(7.22):
xb(t)
W e prove e a s i l y t h a t
i n t h e norm of
(E).
this
is
On t h e o t h e r hand, i f
D ( A ) we have Tb(t)u-u
=
f a s i n thlp(R(A;-%)u-$
l i m
a + ~0
and t h i s expression tends t o zero when
t
+
0.
Although
a c t u a l l y strongly continuous a t t h e o r i g i n ( i . e .
for every
u
E
E)
u ) dh
TJO
xb(t)u
xb(i) +
u
as
is
t
+
r e s u l t w i l l be obtained below.
If we t a k e
u
E
D(A)
then t h e previous
s t e p s show t h a t t h e following computation i s j u s t i f i e d :
A1/2
R(h;-%)u
=
d h = R(p;-(-%)1/2)~.
Since t h e same Laplace transform r e l a t i o n must of needs hold for bb(;)u,
-
where
0
we need not prove t h i s d i r e c t l y , a s a f a r stronger
bb(;)
we have
i s t h e a n a l y t i c semigroup generated by 3,(t)u
=
Irb(t)u
(by uniqueness of Laplace
(7.24)
PHASE SPACES
u
transforms) f o r We extend
u
a f o r t i o r i for
D(A),
E
85
E
l+( t oit) h e upper h a l f plane
E.
z2
by mea.ns of
0
t h e formula
%(C)
=
% ( t+
\(;)
Since
i'I) = % ( t ) b b ( z )
i s s t r o n g l y continuous i n
s t r o n g l y continuous i n upper h a l f plane
2
0,
l+(c)
group and
m
...
and t
S(c)
S(t)E
5 D(Am)
(7.26)
complex a s w e l l ) .
t > 0;
be an a r b i t r a r y element of
Lrb('c)u
E
D(-(-%)'12)
=
D(i(-%)'/*)
i n f i n i t e s i m a l g e n e r a t o r of with r e s p e c t t o
t
E.
t
-
- ( - A b ) 1/2*
T > 0,
Then, i f
is the
Since
=
\(:),
obviously,
W e apply t h i s obser-
v a t i o n (for m = 1) t o t h e a n a l y t i c semigroup generated by
u
for
R(A;A) dh
e
( 7 . 2 6 ) can be extended t o t Let
is
i s a n a n a l y t i c semi-
AmS(t) i s (E)-continuous i n
(so t h a t , i n c i d e n t a l l y ,
bb(q)
0 and
1
AmS(t) =
(7.25)
i s closed, t h e r e s o l v e n t e q u a t i o n
A
i t s i n f i n i t e s i m a l generator t h e n
1,2,
and
m
0).
i s s t r o n g l y continuous i n t h e
and Cauchy's formula t h a t i f A
=
m Em,k
( t ) - m j
as
j
h
m
.
(8.28)
F i s t h e E l b e r t sum of a l l t h e f i n i t e dimensional spaces 2 t h u s i s a r e f l e x i v e Banach space; s i n c e t h e
F. = E J m ( j ,m(j+1)' norm i n 1x1 5 TT
i s dominated by
6
L
times t h e supremum norm i n t h e
same i n t e r v a l , F i s a subspace of t h e space Lo (-,a) of a l l odd, 2lr 2 ~ - p e r i o d i c f u n c t i o n s which a r e square i n t e g r a b l e i n 1x1 5 TT endowed The cosine f u n c t i o n @(;) i s defined by 2 i t s i n f i n i t e s i m a l g e n e r a t o r A i s d /ax2 with maximal domain.
with t h e corresponding norm.
(8.1);
Assume t h a t f o r some for
%(;).
shows t h a t
LL,(;)
i s bounded i n , say, defined b y (8.28)
b
we can f i n d a group decomposition
(8.9)
must obey (8.9) and t h a t i n case Ill.+(t)/lF It1 51 t h e same must be t r u e of ~ ~ ~ o (b,(t) t ) ~ ~ ,
with
b = 0.
( t . ) i s t h e sequence i n
But i f
J
then
.sup
3 21ym( j),m(
thus
(8.U)
An argument very s i m i l a r t o t h a t p e r t a i n i n g t o t h e space
j+l)(tk)
~ ~ b o ( t .+ k )m~ ~by v i r t u e of
ym(k),m(k+l)(tk)J
(8.28),
(8.29)
and a c o n t r a d i c t i o n i s
obtained. The following r e s u l t shows t h a t t h e problem of f i n d i n g a group
94
PHASE SPACES
decomposition of a cosine function becomes m d i c a l l y simpler
if one
i s allowed t o enlarge t h e underlying space.
THEOREM 8.4.
Let c ( ; )
i n t h e Banach space_ E
be a s t r o n g l y continuous cosine function
satisf'ying
wltl Then t h e r e e x i s t s a Banach space /Iu/IE5 (C
(u
E
E)
(8.30)
m).
5 F,
E
F such t h a t
l I ~ l5l ~Cllu/lE
(8.90))
t h e constant i n
0) i s
< m ) )
(b21
@(t^)
1
i n v e r t i b l e and
KV denotes t h e Macdonald f u n c t i o n defined by
where
for
v
#
..
+3,~2,.
and extended lyi c o n t i n u i t y t o a l l values of
(WATSON [1944:1, p. 781).
of a well known i n t e g r a l formula (GRADSTEIN-RLDZYK [l963 :1, p.
EXFRCISE
3.
Let
A,
v
We note t h a t (9.4) i s a vector-valued analogue S(t)
b e as i n Exercise 1.
Given
u
763 ] ). E
E
we say
97
PHASE SPACES
that
S($)u
L
i s c o n t i n u o u s l y d i f f e r e n t i a b l e of o r d e r
and o n l y i f t h e r e e x i s t s
p >
sal/fs(s)(l i n t e g r a b l e i n
s >_ 0
fB(g)
and a f u n c t i o n
w
t
2
0
if
continuous, w i t h
and such t h a t
i m
(t > - 0).
‘-1 f p ( s ) d s
=
e-%(t)u
&
0
(9.6)
h
The f u n c t i o n
of
c1
i s c o n t i n u o u s l y d i f f e r e n t i a b l e of o r d e r
S(c)u
that
i s t h e d e r i v a t i v e of o r d e r
f,(:)
-
u E Ea = D ( ( b 1 ( n o t e t h a t b y Lemma 4.2,
D((b1
a
- A) )
e-@u. c1
Show
i f and o n l y i f
(9.7)
A)‘)
does n o t depend on
b).
The
r e s u l t shows, i n p a r t i c u l a r t h a t t h e d e f i n i t i o n of c o n t i n u o u s d i f f e r e n t i a b i l i t y of o r d e r EXERCISE 4.
2.
Given
E
a &
order
function at
u
t m
E -X
fe($)
does not depend on t h e
~1
[1966:1 3 or
KOMATSU
( t h e author,
[1983:3]).
we say t h a t
@(;)
i f and o n l y i f there e x i s t s
continuous i n
0 and u
0, 6
i s c o n t i n u o u s l y d i f f e r e n t i a b l e of o r d e r
EXERCISE
with
‘= D((b21 - A ) a ) .
For t h e e x c e p t i o n a l v a l u e s of EXERCISE
2 ‘
Show
then u
E x e r c i s e 2.
e-p%(t^)u.
[1983:3]).
A, @(;)
b e as i n
E E
then
‘+by
@(t)u
201
Let
E
b e a space s a t i s f y i n g
98
EVSE SPACES
@(z)
(6.1) w i t h 1 < r < m, A,
i's i n E x e r c i s e 2.
c o n t i n u o u s l y d i f f e r e n t i a b l e of order
2a w i t h
a
Then
>
0
@(t^)u i s i f and o n l y i f
(9.9) h o l d s . A
EXERCISE 7.
0
- 0.
a in
Let
EXERCISE 0
_
t
Assume t h a t 0.
i s c o n t i n u o u s l y d i f f e r e n t i a b l e of o r d e r
S($)u
Show t h a t
on compact s u b s e t s of
b e a s t r o n g l y c o n t i n u o u s semigroup,
c(t^)u
i s Kdlder c o n t i n u o u s w i t h exponent c1
< t < m.
( t h e author,
[1369:2]).
Let
@($) a s t r o n g l y c o n t i n u o u s c o s i n e f u n c t i o n i n 2.
Banach s p a c e ,
t h a t Assumption 5 . 1 h o l d s i f and o n l y i f s t r o n g l y continuous f u n c t i o n i n
i
> - 0,
V(Z)E C - D(A)
=
/ ii
space,
~ ( 2 )a
AV(F)
Show
is a
-
log s ( S ( s + t ) - S(s
t))ds.
(9.10)
0
use t h e e x p r e s s i o n for
EXERCISE 10.
and
where
r l
V(t)
(Hint:
b e an a r b i t r a r y
E
(b21
-
o b t a i n e d i n E x e r c i s e 2).
( t h e a u t h o r [1169:1]).
Let
E
b e a n a r b i t r a r y Banach
strongly cosine function satisfying
Using f o r m u l a (6.18) show t h a t
@ ( s ) u ds
TI
_
E.
v.p.
i n d i c a t e s l i m i t as
Formula ( 9 . B ) i s a n o p e r a t o r
a n a l o g u e o f t h e scalar f o r m u l a
v a l i d for
a
>_
0
(9.Q)
(GWSTEIN-RIDZYK [1963:1, p. 4211)
99
PIUSE SPACES
EXERCISE 11.
Under t h e assumptions i n E x e r c i s e 10, show u s i n g
formula (6.23) t h a t
u
for e v e r y
D(A).
E
Formula
(9.14) i s a n o p e r a t o r a n a l o g u e of t h e
s c a l a r formula
valid for
a
>
0.
(GRADSTEIN-RIDZYK [I963:1,p. 4201).
EXERCISE l2. Using E x e r c i s e s 5 and 8 show t h a t f o r m u l a (?.l2), as < t < M for
w e l l as i t s more g e n e r a l v e r s i o n (6.18) h o l d i n
ucE,y>O. Y EXERCISE
13. Using E x e r c i s e s 5 and 8 show t h a t formula (9.14), as < t < m f o r u c: E (6.23), h o l d i n
w e l l as i t s more g e n e r a l v e r s i o n f o r any
y
> 1/2.
EXERCISE 14. n o t bounded i n
F?(t^)
equals
Y
Show t h a t t h e s i n g u l a r i n t e g r a l o p e r a t o r (8.3) i s
C271(-m,m).
EXERCISE 15. of
-M
Prove Theorem
R(h;U)
1.3 showing t h a t t h e Laplace t r a n s f o r m
and a p p l y i n g Theorem
1.3.4.
FOOTNOTES TO CHAPTER I11
(1) Elements of
Eo x El
and similar p r o d u c t s p a c e s w i l l b e d e n o t e d
as "row v e c t o r s " or "column v e c t o r s " a c c o r d i n g t o convenience. (2)
3
T h i s e s t i m a t e c a n b e c o n s i d e r a b l y improved (see Chapter VI, E x e r c i s e s
t o 8).
(3) (4) (5) (6) (7)
See f o o t n o t e ( 2 ) . See f o o t n o t e ( 2 ) . See f o o t n o t e ( 2 ) . See f o o t n o t e ( 2 ) . See f o o t n o t e ( 2 ) .
100
CHAFTER I V APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS
Wave equations:
5IV.l
t h e D i r i c l l e t boundary condition.
W e consider i n t h e f i r s t s i x s e c t i o n s of t h i s chapter t h e equation U"(t) = A ( @ ) u ( t ) .
(1.1)
Here
m
m
Au =
D j ( a . (x)Dju) Jk
with
x = (xl, . . . , x m ) , D ' = a/ax
j
~ ( x )a r e defined i n a domain 0 A(B)
denotes t h e r e s t r i c t i o n of
condition
@
r
a t t h e boundary
or
bj(x)Dju
-e
+
c(x)u
j =1
j=1 k = l
and t h e c o e f f i c i e n t s a . ( x ) , b . ( x ) , Jk J of m-dimensional Euclidean space Rm A
obtained by means of a boundary
of t h e form
D " u ( ~ ) = y ( ~ ) ~ ( ~( X)
-
E
r),
(1.4)
D" denotes t h e conormal d e r i v a t i v e t o be defined below ( s e e (4.1)). 1 Since t o r e p l a c e a j k ( x ) by ( a j k ( x ) + a k j ( x ) ) does not change t h e
where
a c t i o n of
A
on smooth f u n c t i o n s we s h a l l assume from now on t h a t
a
jk
We r e q u i r e t h e valued. A
If t h e
a
jk a
(x)
= a
kj
(x).
t o be real-valued; t h e
b. J
and
c
can b e complex-
have f i r s t order p a r t i a l d e r i v a t i v e s , we can w r i t e
jk
i n t h e more n a t u r a l form
m
Au =
m
m A
a . (x)DjDku + r b . ( x ) D j u j=1 k = l j=1 J
+
c(x)u
,
(1.5)
101
PARTIAL DIFFEREWIAL EQUATIONS
where
The passage from ( 1 . 2 ) t o (1.5) and v i c e v e r s a i s no longer p o s s i b l e i f
the
a
assume
(1.5) represent
a r e not d i f f e r e n t i a b l e ; i n t h i s case ( 1 . 2 ) and
jk
[1974:1]).
q u i t e d i f f e r e n t e n t i t i e s ( s e e PUCCI-TALENT1
We s h a l l always
i s w r i t t e n i n t h e form ( l . 2 ) , c a l l e d t h e divergence o r
A
v a r i a t i o n a l form.
The c o e f f i c i e n t s
a
b c w i l l be required t o be jk’ j’ merely measurable and bounded; we p o s t u l a t e i n a d d i t i o n t h a t A be uniformly e l l i p t i c i n t h e sense t h a t
(1.6) f o r some
ic
> 0.
Our f i r s t r e s u l t concerns t h e D i r i c h l e t boundary condition (1.3). No assumptions whatsoever w i l l be placed on t h e domain boundary
r.
or the
0
In t h i s high l e v e l of g e n e r a l i t y , it i s obvious t h a t
(as well as t h e boundary condition
u
=
0)
Au
will have t o be understood
i n a s u i t a b l y generalized sense; for instance, i n view of t h e l a c k of smoothness o f t h e
a
jk’
i t i s not c l e a r whether
can be applied t o
A
any nonzero function. The b a s i c space i n our treatment i s supporting r o l e w i l l be played by
a(n)
( o f Schwartz t e s t f b n c t i o n s )
+(O), in
H
2 = L (0).
An important
t h e c l o s u r e of t h e space
$(n);
m(X)
c o n s i s t s of all i n f i n i t e l y d i f f e r e n t i a b l e functions in
0, and t h a t t h e space
a;LI
functions
u
$(n)
(k
an i n t e g e r
and
+(n)
2
L2(n).
with support
1) c o n s i s t s o f
having p a r t i a l d e r i v a t i v e s of order
understood i n t h e sense of d i s t r i b u t i o n s ) i n
8(n)
we r e c a l l t h a t
5k
(derivatives
The spaces
( f o r all necessary f a c t s on t h e Sobolev spaces
ADAMS b975 :1I )
Hk(n), #(n)
consult
.
The f i r s t stage of o u r argument w i l l be t h e c o n s t r u c t i o n of operator
d(0)
a r e f i l b e r t spaces equipped with t h e s c a l a r product
Ao(B),
where
A0
i s the
self a d j o i n t p a r t ol
A,
the
102
PARTIAL DIFFEREmIAL EQUATIONS
With t h i s i n mind, we introduce a new s c a l a r product i n
#,(fi)
by t h e
formula
(u,v),
=
[(a- c ) c v dx
ifi
where
dx
=
... dxm
dxl
(u,~),
(and conjugate l i n e a r i n C > c
constants
h ( y y ajk&Dkv)
dx
(1.9)
and
cY>v It i s obvious t h a t
+
=
e s s . sup c
-
and t h a t
= (V,U)~
u).
.
(1.10) (U,V)~
i s linear i n
v
Moreover, we see e a s i l y t h a t t h e r e e x i s t
such t h a t
0
( w e u s e t h e uniform e l l i p t i c i t y assumption for t h e f i r s t i n e q u a l i t y ) . Accordingly, t h e norm
(1.12) corresponding t o t h e s c a l a r product norm of
4
defined by
(1.7);
(1.9) i s
equivdent t o the original
t h u s we s h a l l assume from now on
&(n)
endowed with (1.12) ( o f cotirse, t h e same arguments apply t o t h e
space
$(",
a f a c t t h a t w i l l be used i n 61V.4).
A function
u
E
$(n)
D(AO(f3)) i f and only i f
belongs t o
(1-13 1
i s continuous i n t h e norm of
L
2
(i)):
i f t h i s i s t h e case, w e extend
( s i n c e €$(a) i s dense L2(Q) 2 i n t h e topology of L (n) t h i s extension i s unique). L e t
t h e l i n e a r f u n c t i o n a l (1.1)) t o a l l of in v
L E
2
(Q)
2
L (0) be such t h a t
(1.14) Define A0(B)" (Motivation i s obvious:
= azI
-v .
i f the coefficients a
(1.15) jk
and t h e boundary
r
103
P A R T I A L DIFFEREIWIAL EQUATIONS
u
are smooth and
u
and
w
=
r,
on
0
=
r).
on
0
-A u = v in n 0 f o l l o w s f o r any smooth w such t h a t
i s a smooth f u n c t i o n such t h a t then
(1.14)
We check e a s i l y t h a t t h e d e f i n i t i o n of A (f3)
above does
0
a.
not depend on
We wish t o show t h a t t h e o p e r a t o r
j u s t defined i s s e l f a d j o i n t .
AO(B)
We b e g i n by proving t h a t
( U - AO(B))D(AO(f3)) h > v.
f o r any
In fact, l e t
(1.16)
= L2P)
be a n a y b i t r a r y element o f
v
L2(Q)>.
Define a l i n e a r f u n c t i o n a l by
w
(1.17) i s
Since
4(0),
L * ( ~ I it i s a s well continuous i n
continuous i n
u
thus there exists
E
$(n)
(1.16)
such t h a t =
( U , d A
hence
(1.17)
*
(V,.>X
-+
(1.18)
(v,w>, u
note t h a t o u r c o n s t r u c t i o n of
follows:
yields the
estimate
Rewriting ( 1 . 1 4 ) i n t h e form
w = u
and t a k i n g
we s e e t h a t
(1.19)we deduce t h a t from
(1.18) t h a t i f
R(h;Ao(p))
u, v
( h- AO(f3)~,v)= so t h a t
Ao(B)
defined.
A1
E
D(A,(B))
(U,V),
i s symmetric.
u
E
4(Q)i n
D(AO(@)).
X-D(A0(B))
To prove t h a t
result.
h > v.
combining with
It a l s o f o l l o w s
then =
=
( u , hv
- A(B)v) i s densely
AO(B)
We f i n a l l y prove t h a t
In o r d e r t o d o t h i s it i s sufficient t o show that
i s dense i n all
i s one-to-one;
exists for
t h e topology of
c a s e , t h e r e would e x i s t an element
to
- AO(B)
w
E
d(". $(."I
I f t h i s were not t h e with
( U , W ) ~=
I n view of (1.20) t h i s i m p l i e s t h a t which, d u e t o
A (B) 0
(1.16),
D(AO(B))
shows t h a t
w
0
for
i s orthogonal
w = 0.
i s s e l f a d j o i n t we make u s e of t h e following
104
PARTIAL DIFFERENTIAL EQUATIONS
Let
LEMMA 1.1. X l b e r t space
number
A.
Proof: -
Assume t h a t t h e resolvent
H.
Then
Let
be a. densely defined symmetric operator i n t h e
A
u,v
be two a r b i t r a r y elements of
( R( A;A)u,v) =
so t h a t
R()\;A)
( ( AI
p(A)
contains a r e a l
i s s e l f adjoint.
A
=
(R( A ; A ) U ,( AI
- A ) R (A;A)u,
R( A;A)v)
H.
Then
- A ) R ( A;A)V) =
(u, R( A, A ) V )
i s symmetric; t h u s
where t h e interchange of i n v e r s e s a n d a d j o i n t s i s e a s i l y j u s t i f i e d (see
RIESZ-SZ.-NAGY [1955:1I ) .
This ends t h e proof.
i s s e l f a d j o i n t and bounded above (by Ao(B) generates a strongly continuous cosine f u n c t i o n A0 ( @ ) @( 0 3
b-,
s ~m =; 0 3 ~ .
E
A l i t t l e use w i l l be made i n t h e following l i n e s of t h e Sobolev spaces
&’P(n)
c o n s i s t i n g of a l l f i n c t i o n s
LP(n);
p a r t i a l derivatives i n
t h e space
Also, we s h a l l employ t h e spaces u in
-
continuous i n
n,
C(l)(F)
having f i r s t
i s normed with
c o n s i s t i n g of a l l f u n c t i o n s
having continuous f i r s t p a r t i a l d e r i v a t i v e s
each d e r i v a t i v e admitting a continuous extension t o
THEOREM
1 f p c
u r LP(fi)
&”(n)
m.
4.1
& n
men
(a)
be a bounded domain of c l a s s
if
Dju
a. and l e t
114
PARTIAL DII’FERENTLAL EQUATIONS
t h e r e e x i s t s a constant
u
f o r everx
then
E
(depending only on 0, p , q ) s u c h t h a t
C
C(’)(F).
if
(b)
(4.3) holds f o r everx
q
2
1.
For t h e proof of a c o n s i d e r a b l y more g e n e r a l r e s u l t see ADAMS We n o t e t h a t Theorem 4.1 h o l d s a s w e l l f o r domains
[1975:1, p. 1141.
which a r e “piecewise of c l a s s c y l i n d e r s whose base i s a
(in
C (’”’
such a s , say, p a r a l l e l e p i p e d o n s or
- 1)- d i m e n s i o n d
also, t h e boundedness h y p o t e s i s i s not e s s e n t i a l : r e s u l t holds i f
THEOREM 4.2. 8
( b u t not
Let fl
n)
f o r instance, t h e
i s bounded.
be a domain o f c l a s s
, 15p
C( 0 )
(or, r a t h e r , t h e s e t of r e s t r i c t i o n s of f u n c t i o n s of
dense i n
exists i n
cy.
i s symmetric s o t h a t , using Lemma 1.1 we show t h a t
i s s e l f a d j o i n t and bounded above by
depending not only on t h e cosine f u n c t i o n
OIV.5
(4.12)
L2W,
a, where
v but also on t h e c o e f f i c i e n t
@,(t)
generated by
Ao(@)
ff
y.
i s a constant Accordingly,
i s t h i s time given by
The phase space.
The arguments i n s I V . 2 have an obvious counterpart h e r e . c o n s t r u c t i o n of t h e square r o o t
B of
Ao(B)
The
proceeds i n t h e same way,
as does t h e proof of THEORFM 5.1 D(E) =
d(n).
(5.1)
The phase space f o r t h e equation
u " ( t ) = Ao(B)u(t) i s now
(5.3)
El
=
$(".
(5.4)
117
PARTIAL DIFFEREWIAL EQUATIONS
Again, t h e phase space The group
Go(;)
( 5 . 3 ) i s t h e same one provided by Theorem 111.1.3.
propagating t h e s o l u t i o n s of ( 5 . 2 ) i s given by (2.11)
with i n f i n i t e s i m a l g e n e r a t o r D(210(f3)) = D ( A O ( f 3 ) )
(2.12), i t s domain being i d e n t i f i e d by
x €$(D).
To t a k e c a r e of t h e f i r s t order terms we
use Theorem 2.3 a p p l i e d t o t h e bounded p e r t u r b a t i o n o p e r a t o r (2.13).
I n t h i s way we o b t a i n ;
Let
THEOREM 5.1.
r,
A
0
be a domain of c l a s s
t h e operator ( l . 2 ) ,
(3
measurable and bounded on
I-.
with domain
D(A(f3))
=
CiLi
t h e boundary c o n d i t i o n
w i t h bounded boundary
( 1 . 4 ) with y
Let -
D(AO(f3)).
d(n)
Then t h e space
X L'(0)
is a
phase space f o r t h e e q u a t i o n
u"(t)
Q1v.6
=
.
A(B)u(t)
(5.6)
The Cauchy problem.
A l l t h e r e s u l t s i n S e c t i o n IV.3 have a n immediate c o u n t e r p a r t h e r e ; we
d e f i n e t h e semigroup B O ( i ) given by ( 3 . 1 ) i n t h e product space 2 2 = L (0) X L ( a ) ; again, depends on t h e p a r t i c u l a r square r o o t
z0(t)
of
Ao(B)
chosen.
B
However, we need
This can be achieved by r e p l a c i n g l a r g e i n t h e d e f i n i t i o n of
Ao(f3);
t o have a bounded i n v e r s e .
c ( x ) by
m Pu = C b . ( x ) D J u + j =1 J TmOREM boundary
y
2
6.1.
r, A
fi
t h e o p e r a t o r (1.2), B
D(A(B))
i s w e l l posed i n
-m
= D(Ao(f3)).
< t
’ ;X
a(x)) < m,
(7.8)
N
a(x), t h e spectrum of
where
E
i s e a s i l y i d e n t i f i e d as
A,
As proved i n E x e r c i s e 11.5, (7.8) i s e q u i v a l e n t t o t h e f a c t t h a t
~(x)
i s contained i n a r e g i o n of t h e form Re h < - w2 - ( I m h ) > / 4 3 . LEMMA 7.2.
-
A
2
i f and o n l y i f
E
p
(a)
(7.9) i s even ,(b)
a
is P -
r e a l with (-l)p’zap (c) j
is r e a l i f
aj
i s odd,
j
>
j
i s even
>
p/z,
,
(7.10)
(d) aj i s imaginary i f
p/2.
Assume t h a t ( a ) , ( b ) , ( c ) and ( d ) hold.
Proof:
P(t) = where
j
p,
Since
a n i n e q u a l i t y of t h e type of
51.
not hold i n t h i s c a s e f o r l a r g e
This ends t h e proof
of Lemma 7.1. We note t h e c u r i o u s consequences o f Lemma
belongs t o
2
,
d
-
A =
+
-
=
(-$I8
6
+(-&)5 (d/dx) 5
does not, i n s p i t e of t h e f a c t t h a t (d/dx)8
although t h e o p e r a t o r
t h e operator A
of
(dx)
7.1:
than
(d/dx)
6
i s a “tamer” p e r t u r b a t i o n
.
I n t h e following s e c t i o n we s h a l l attempt a t h e o r y of t h e equation
(7.l),
b u t only i n t h e c a s e where
t h e D i r i c h l e t boundary c o n d i t i o n . c o e f f i c i e n t s of
O1v.8
A
of o r d e r > p/2
B
i s t h e h i g h e r order v e r s i o n of
Lemma
7.1 i n d i c a t e s
that the
w i l l have t o be s u i t a b l y r e s t r i c t e d .
Higher o r d e r e q u a t i o n s ( c o n t i n u a t i o n )
We study here t h e e q u a t i o n
(7.1) w i t h
an operator
A
of t h e form
121
PARTIAL DIFFERFNTIAL EQUATIONS
c
7
Au =
(-l)Ial-'D"(a+(x)D
Bu ) +
I4 5 k
la1 5 k The c o e f f i c i e n t s
101 T
am, ba
k
a r e r e a l and defined i n a bounded domain
Rm.
in-dimensional Euclidean space
of
W e s h a l l assume t h a t t h e c o e f f i c i e n t s
of t h e p r i n c i p a l p a r t of t h e operator
a
(8.1)
bo/(x)Dau.
A,
OB
c
(-l)ial-lDw(a~D')
,
(8.2)
Ictl=k [BI=k
a r e continuous i n
-
n;
t h e r e s t of t h e
simply measurable and bounded i n r e s t r i c t i o n of
A
R.
a
*'
a s well as t h e
A(B)
The operator
obtained by imposition a t t h e boundary
b,
are
denotes t h e
r
of t h e
D i ric h l et b ound a ry cond it ion
... =
u = D"u =
(Dw)k-l~ = 0
(x
E
r)
(8.3)
(8.3) w i l l be s a t i s f i e d only i n a generalized sense t o be
(although
c l a r i f i e d l a t e r ) . We assume t h a t
and t h a t
A
f o r some
K
i s u n i f o r d y e l l i p t i c , which i n t h i s case means t h a t
> 0.
The following r e s u l t (Ggrding's i n e q u a l i t y ) w i l l be b a s i c .
To s t a t e
it w e introduce t h e Sobolev spaces wk'p(fi) (1 5 p < m ) c o n s i s t i n g of u defined i n fl and having p a r t i a l d e r i v a t i v e s of
all f u n c t i o n s
5k
order
For
p
(understood i n t h e sense of d i s t r i b u t i o n s ) i n
LP(R);
the
w k ~ p ( n >i s
norm of
=
2
wky2(n) =
( t h e only case of i n t e r e s t t o u s ) we s h a l l w r i t e
$(Q).
( i n t h e norm of
The space
Hk(n)).
$(n)
The statement t h a t
v e r s i o n o f t h e boundary conditions THEORail
8.1 L e t
L
i s t h e c l o s u r e of u E %(a)
(8.3).
be a d i f f e r e n t i a l operator;
d)(n)
in
$(n).
i s t h e weak
122
PARTIAL DIFFERENTIAL EQUATIONS
i n a bounded domain
7
c
bI5k
lPl5k
Q
5 Rm.
(-l)lN(-lDTY(aOIT;Dpu)
Assume t h a t a l l t h e c o e f f i c i e n t s
a r e measurable and bounded and t h a t
ICY~ =
=
k.
i s continuous i n
a$
Then t h e r e e x i s t constants
x
14 5 k Is I I
0
when
such t h a t
C,CY
JaaM(x)D?DBu
-
dx
2
k
For a proof see FRIEDMAN [1969:1,p.321. W e proceed t o t h e c o n s t r u c t i o n of a phase space f o r t h e equation
where
A.
i s t h e s e l f a d j o i n t p a r t of A, A~ =
7 Ao(B)
The d e f i n i t i o n of renorm t h e space
where
CY
$(n)
Y
(8.8)
(-i)lml-lDm(aOIT;Dpu).
IBl5k
l+k
follows t h a t f o r t h e second order case.
W e
by means of t h e s c a l a r product
i s t h e constant i n (8.6).
We have
(8.10) The second i n e q u a l i t y follows f r o m t h e boundedness of t h e c o e f f i c i e n t s
of A ; t h e first i s a consequence of Theorem 8.1.
u
E
4(.".)
belongs t o
D(AO(B))
i f and only i f t h e l i n e a r f u n c t i o n a l
w -, ( u , ~ ) , i s continuous i n t h e norm of element of
2
L (a)
L
2
(Q),
A,(@).
being t h e orily
that satisfies
W e show i n t h e same way as i n t h e case a d j o i n t and t h a t
An element
Ao(B)
k = 2
i s bounded above
g e n e r a t e s t h e cosine f u n c t i o n
(by
that o!),
Ao(B) so that
i s self Ao(B)
123
PARTIAL DIFFERENTIAL EQUATIONS
C(t) and a square r o o t
cash t A o ( B ) 1 / 2
=
=
(8.12)
can be defined as i n gIV.2: we have
B = A,(@)'/' D(B)
,
#(n)
=
D((U h >
t h e l a s t i n e q u a l i t y holding for
- A ~ ( B ) 1) /2 ),
(8.13)
Theorem 111.5.4, combined with
cy.
(8.13) i m p l i e s t h a t Q =
i s a state space for (8.7).
(8-14)
H p ) x L2(Q) To show t h a t
Gf
i s as well a s t a t e space
f o r the f u l l equation
we i n c o r p o r a t e t h e lower order terms i n
(8.1) through p e r t u r b a t i o n
(Theorem 2.3) d e f i n i n g
(8.16) and
:]
? = [ : We o b t a i n i n t h i s way: THEOREM 8.2.
Let A
(8.1), @
be t h e operator
the Dirichlet
boundary c o n d i t i o n (8.3), and l e t
(8.18)
A(B) = Ao(B) + P w i t h domain
Then t h e space
D(A(@)) = D(Ao(@)).
a phase space f o r t h e equation
$(Q)
x L2(n)
(8.15).
The t r e a t m e n t of t h e Cauchy problem f o r (8.15) f o l l o w s word by word t h a t f o r second order e q u a t i o n s i n pIV.3; THEOREM
8.3.
L A A
we only s t a t e t h e f i n a l r e s u l t .
be t h e operator (8.1),
boundary c o n d i t i o n ( 8 . 3 ) , and l e t
B the Dirichlet
124
PARTIAL DIFFERFNTIAL EQUATIONS
A ( B ) = Ao(B) with domain
D(A(B))
( 8 -19)
P
Then t h e Cauchy problem f o r t h e
= D(AO(B)).
equation (8.15) i s well posed i n
9IV.g
+-
0, e
=
5 ~ / 5f o r
T )- f ( t ) l l
such t h a t
6 = F(E/?)
[O,aI
i n t e r v a l s of l e n g t h
56
with
f i n i t e number
R
and p i c k r a t i o n a l s tl,
t h e s e i n t e r v a l s ; once t h i s i s done, determine
'G = ' G ( t )
t
be an a r b i t r a r y r e a l number. e
E
t
and a p o i n t
(-t, -t
i n the interval
It +
such t h a t
j
b(t + kn) - f ( t
T -t
and a l l
5
E/5
if
.I
r
of sub-
i n each of
such t h a t
j
=
1,2
,..., r.
Pick a t r a n s l a t i o n number
+ a)
J
. ..,t r
no
+ km)ll 5 ~ / 5 f o r m,n 3 no,
l\f(t. + kn)-f(tj Let
a11 t
\if(t)-f(t')II
a
( t h a t t h i s i s p o s s i t l e follows from Lemma 3.1 ( a ) ) .
6
Finally, cover t h e i n t e r v a l
J
e(E/5;f)
=
(so t h a t
.c 6 .
+ km)ll 5 / k ( t + k n >
0 < t + T
21%/
i n t e r v a l of l e n g h t of
Let
f(t).
Obviously,
h2
=
h2-hl
( 5 + bl)/2
of
1% 1 +
and d e f i n e
f( 2(
not containing any E - t r a n s l a t i o n number
Ih21) h
3
(a2,b2)
not containing any € - t r a n s l a t i o n number
= (a
2
+ b2)/2.
Then
h
3
-Ll and
h3-h;,
n - yh n - h 2 , . . . , h n - h n - l
{hn)
f(t).
which shows t h a t t h e sequence
Then, i f
(f(G
sequence uniformly convergent i n
+
-a
hn))
f(s)
ds
2
0
if
f
2
0
(-m
0
o+, [1958:1, p . 5671),
a s claimed.
This ends t h e proof
147
I N HILBERT SPACE
of Theorem
6.1.
The next r e s u l t i s a n exact c o u n t e r p a r t of Theoyem 2.2 for c o s i n e However, t h e method of proof i s somewhat d i f f e r e n t .
functions.
THEORZM 6.2. @(s
+ t) +
@(s
Let c(
=
e(E,T,u)
0, E
=
5
Ct;o 5 t 5
t h e c h a r a c t e r i s t i c f u n c t i o n of
t
Proof: Set equation
=
s
= u/2
cr
e
E
E
H,
(6 -12)
ll@(t)uII < ~ l l u 1 1 3 .
e.
i n t h e (second) cosine f u n c t i o n a l
0/2
=
@(u) + I
-
we have
Accordingly, i f
so t h a t
hence
(11.1.9). m e r e s u l t i s 2qu/2)2
Hence, i f
T,
(v);
using
(6.11).
+ l), u
1/(2C
m
&
f! e.
5 1/(2C + 1)
we deduce t h a t
It follows t h a t i f
shows t h a t t h e f u n c t i o n s
x (i)
and
u
x
E
f2;)
e
then
2a
#
e,
which
have d i s j o i n t support.
Hence
by
(i)
and
t i o n we o b t a i n
(v).
Taking t h e change-of-variable property i n considera-
(6.13),
thus ending t h e proof of Lemma 6.4.
149
I N HILBERT SPACE
Proof of Theorem 6.2.
m e operator
i s t h i s time defined by
P
ds
E y v i r t u e of Lemma
6.4
with
E =
1/(2C m
t h u s it f o l l o w s from t h e d e f i n i t i o n of
+ 1)
P
.
(6.14)
we have
that
On t h e o t h e r hand, it i s obvious t h a t
Accordirgly, i f inequalities Let now
i s t h e p o s i t i v e , s e l f a d j o i n t square r o o t of P,
Q
(6.8) h o l d . t
be a real number,
u,v
elements of
H.
Using t h e
c o s i n e f u n c t i o n a l e q u a t i o n s and Theorem 5.2 we deduce t h a t ( P @ ( t ) u , v ) = LIM
L T ( @ ( s ) C ( t ) u , @ ( s ) v d) s
T-‘M T L =
1LIM 2 T-m
$lo -T
( @ ( s + t)u,C(s)v) ds
$k
T
+ 1_
*
LIM T-m
+ 1_
LIM T-m
(@(s - t)u,@(s)v) d s
PT
‘
for
u,v
E
H.
$ j o ( @ ( s ) u , @ ( s+ t ) v ) d s
Accordingly,
(6.18) P r e - and p o s t - m u l t i p l y i n g by QC(t)Q-l
Q
-1
= Q-’@(t)*Q
we obta.in =
(Q@(t)Q-’)*.
(6.19)
150
I N HILBERT SPACE
This completes t h e proof of Theorem
6.2.
The coriiments following Theorem 2.2 apply h e r e a s w e l l :
replacing
t h e o r i g i n a l s c a l a r product by ttie ( t o p o l o g i c a l l y e q u i v a l e d c ) s c a l a r product
(2.12)
-
@ ( t )s e l f a d j o i n t .
r e n d e r s each
COROLLARY 6.5.
Assume i n a d d i t i o n t h a t
C(;)
Q
B
u
B
2
t
E.
Then t h e r e e x i s t s a s e l f ad-
tnd a bounded s e l f a d j o i n t o p e r a t o r
0
(6.8)
s a t i s f y i n g i n e q u a l i t i e s of t h e form
@ ( t=) Q - l c o s (tB1/')Q @(;)
Conversely, e v e r y
(-a
E 2 EI f o r some
(6.7).
and such t h a t
u , v )d s
m
(&( ; ) U , V ) and
so t h a t both almost p e r i o d i c f u n c t i o n s
By Theorem
have t h e same Fourier s e r i e s .
(@(;),u,v)
3.8 we must have
& ( s ) = @ ( s ) by a r b i t r a r i n e s s of u,v.
( & ( s ) ~ , v= ) ( @ ( s ) u , v ) , thus
This conclludes t h e p r o o f 3f Theorem 7.1. I n t h e language of a b s t r a c t d i f f e r e n t i a l equations, t h i s r e s u l t can be f o r n u l a t e d a s follows.
Let
THEOREM 7.2. t h e f i l b e r t space
H
A
be a closed, densely defined o p e r a t o r i n
such t h a t t h e Cauchy problem for
u”(t) i s w e l l posed i n
-m
0 ; cf.
holds i n Cgsaro mean,
o r i n Abel mean
u EXERCISE
=
-
m
l i m (1 A) F A ~ M 2u h-. 1k=O -(kp)
.
15. S t a t e and prove a s u i t a b l e converse
(8.22) of Exercise
14.
165
CHAPTER V I
THE PARAEOLIC SINGULAR PERTUREATION PROELEM
Vibrations of a membrane
4VI.l
i n a viscous medium.
r
Consider a uniform membrane fixed t o t h e boundary
of a two
R and immersed i n a viscous medium.
dimensional domain
The small
o s c i l a t i o n s of t h e membrane a r e described by t h e equation pvtt where brane
+ yt
UAV
=
(X
E
a) ,
(1.1)
i s t h e v e r t i c a l movement of t h e mem-
v = v ( x , t ) = v(xl,x2,t) and t h e c o n s t a n t s p,y,a
are, r e s p e c t i v e l y , t h e mass d e n s i t y
p e r u n i t a r e a of t h e membrane, t h e c o e f f i c i e n t of v i s c o s i t y of t h e medium and t h e t e n s i o n of t h e membrane.
v satisfies
The displacement
t h e boundary c o n d i t i o n
o
v ( x , t )=
r) ,
(.
(1.2)
and t h e i n i t i a l c o n d i t i o n s v(x,O) If we d e f i n e
=
v ( x , t ) = u(x, (u/y)t)
and t h e boundary c o n d i t i o n ( 1 . 2 ) ; u(x,0)
(~p)'/~/y
Setting
=
0
u
R)
.
(1.3)
s a t i s f i e s t h e equation
=
-Y 0u
JX)
(x
E
0)
.
(1.5)
w e can w r i t e (1.5) i n t h e form
= F
E U E
then
E
the i n i t i a l conditions are
u ( x ) , ut(x,O)
2
where
(x
u 0 (x),vt(x,O) = u,(x)
tt
+ u
t
= a u
(.
E
n>
9
w i l l be small i f the medium i s h i g h l y viscous
can write t h e second i n i t i a l c o n d i t i o n i n t h e form ut(x,O) = c-1(;)1k1(x)
.
(1.6) ( y >> 1). W e
166
PARABOLIC SINGULAR PERTWATION
This suggests t h e problem of studying t h e behavior of t h e s o l u t i o n of
(1.6) on
as
-
E
allowing f o r dependence of t h e i n i t i a l c o n d i t i o n s
0,
This w i l l be done i n t h e r e s t of t h e c h a p t e r for a n a b s t r a c t
E.
model encompassing e q u a t i o n s l i k e
E x p l i c i t s o l u t i o n of t h e perturbed
Singular perturbation.
sV1.2
(1.6).
equation.
w i l l be t h e i n f i n i t e s i m a l g e n e r a t o r
Throughout t h i s c h a p t e r A
c(:)
of a s t r o n g l y continuous c o s i n e f u n c t i o n space
(that is,
E
A
E
8;
i n t h e complex Banach
see: Exercise 2 ) .
We c o n s i d e r t h e
Cauchy problems
2
E U"(t;E)
+
=
U(0;E)
+
= AU(t;E)
U'(t;E)
f(t;E)
3 o),
(t
(2.1)
u
0
( t ) ,
U'(0,E)
= U1(E),
and
u'(t)
-
=
Au(t)
+
(t
f(t)
-
3 0),
~ ( 0 =) u 0'
-
(2.2)
Roughly speaking, t h e r e s u l t s i n t h e f o l l o w i n g s e c t i o n s e s t a b l i s h that
if
u(t;E)
u(t)
as
E
0
if
uO(E)
i s suitably restricted.
ul(E)
u,
f(t;E)
f(t)
and
R e s u l t s o f t h e same t y p e hold
f o r derivatives.
A s a f i r s t s t e p , we compute t h e e x p l i c i t s o l u t i o n of ( 2 . 1 ) using simple changes of dependent and independent v a r i a b l e . Writing -t/2&2 t/2E U(t;E) = e v(t/E;E) ( o r , equivalently, v(t;E) = e u(Et;E)) we e a s i l y show that
i s a s o l u t i o n of t h e i n i t i a l v d u e
v(t;E)
problem v(t;&) + e
V(O,E)
= UO(E),
Conversely, e v e r y s o l u t i o n u(;;E)
=
V'(0,E)
t/2E
1 U (E) 2E 0
f(&t;&),
(2.3)
+
EU1(E).
v ( t ^ ; ~ ) of ( 2 . 3 ) g i v e s r i s e t o a s o l u t i o n
of t h e i n i t i a l v a l u e proklem ( 2 . 1 ) .
The ( g e n e r a l i z e d )
s o l u t i o n of t h e i n i t i d value problem (2.3) i s g i v e n by V(t;E)
=
C(t;E)V(O;E)
+
s(t;E)v'(O;&)
+,JtS(t
-
s ; E)es/2E f ( E s ; E )
ds,
(2.4)
PARAEOLIC SINGULAR PERTURBATION
c(:p
where
S(t;E)u =
i s t h e cosine f u n c t i o n generated by
167
A + ( 2 ~ ) - ~and 1
W e know from Lemma 111.4.1 t h a t t h e
C(s;E)u d s .
Cauchy prtbpem f o r
However, t h e s e r i e s r e p r e s e n t a t i o n (111.4.7) i s not
i s well posed.
convenient here and we provide a d i f f e r e n t r e p r e s e n t a t i o n below which, A
i n c i d e n t a l l y , proves a f r e s h t h a t LEMMA 2.1.
A
kt
8, t h a t
E
be well posed, and l e t
b
(2E)-21
E
8.
i s , l e t t h e Cauchy problem f o r =
U"(t)
+
Au(t)
(2.6)
be an a r b i t r a r y complex number.
Then
t h e Cauchy problem f o r U"(t)
i s well posed; t h e propagators
=
(A
2
+
b I)u(t)
cb(;),
S b ( t ) of (2.6)
by Cb(t)u =
c(t)U
d t Il(b(t2
+
(2.7)
bt
(t2
-
-
('1
a r e given
s~)'/~) C(s)u d s , ( 2 . 8 )
s2)1/2
and
where
Io,
5
a r e t h e Bessel f u n c t i o n s defined by (2.10)
The proof can be c a r r i e d out i n ( a t l e a s t ) t h r e e ways, all of which we sketch b r i e f l y below.
In t h e f i r s t , one t a k e s advantage of
t h e d i f f e r e n t i a l equation
(2.11) s a t i s f i e d by if
u, v
then
E
u(f)
Iv(x)
D(A)
and of a l i t t l e i n t e g r a t i o n by p a r t s t o show t h a t
(so that
= cb(;)u
c(t*)u, S(t*)u a r e genuine s o l u t i o n s of ( 2 . 6 ) ) i s a genuine s o l u t i o n of (2.7): uniqueness
+ %(t*)u
of s o l u t i o n s of ( 2 . 7 ) i s proved transforming them back i n t o s o l u t i o n s of
(2.6) by means of formulas (2.8) and (2.9) with b replaced by i b .
168
PARAEOLIC SINGULAR PERTUREATION
For d e t a i l s on t h i s l i n e of approzch (&though t h e context t h e r e i s somewhat d i f f e r e n t ) see SOVA
[1970:4].
The second uses Theorem 11.2.3; w e
show t h a t t h e Laplace transform of C b ( c ) u given by ( 2 . 7 ) equdls 2 2 2 The t h i r d i s based on Lemma 111.4.1, h R ( A ; A + b I ) u = AR(A2 - b ;A)u.
en i n (111.4.7) combined
p l i c i t c o n s t r u c t i o n of each of t h e terms s e r i e s r e p r e s e n t a t i o n (2.10) f o r
I o ( x ) , I1(x).
ex-
with the
We omit t h e d e t a i l s .
(2.8), (2.9) t o t h e equation
Applying t h e t r a n s l a t i o n formulas (2.5) we o b t a i n C(tjE)U
+
= C(t)U
and
=L t
S(t;E)u
-
l o ( ( t 2 s')'/~/~E)C(S)U d s
both v a l i d f o r a r b i t r a r y
u
and
t.
c ( s ) u ds
(2.12)
,
(2.13)
Using t h e s e formulas i n ( 2 . 4 ) and
making a n obvious change of v a r i a b l e i n t h e i t e r a t e d i n t e g r a l r e s u l t i n g from t h e i n t e g r a l i n ( 2 . 4 ) we o b t a i n t h e f i n a l formula for u ( c ; & ) , we have used u(t;&) = e
+
(11.4.3) f o r t h e s o l l i t i o n of t h e nonhomogeneous equation:
-t/2E
2 ( t /& )uo ( E 1
-
t e - t / 2 E 2 ~ t / &T ( ( ( t / E ) 2 E2
((t/E)*
2 = e-t/2E
where
C(t/&)u,,(E)
-
s)ll2/2E)
c(s)(, u O ( E ) ) d s S2)1/2
1
+
cp(t,s;&)C(s)(~ uO(E)) d s ' 0
2 = e-t/2E
C(t/E)Uo(&)
+
R(t;E)($
U,(E))
+
qt;E)($
U O(E)
+
E2U,(&))
(2.14)
169
PARABOLIC SINGULAR PERTURBATION
where t h e d e f i n i t i o n s o f t h e s c a l a r f u n c t i o n s
Cp(t,s;&),
qt;&)a r e
R(t;&),
t h e o p e r a t o r valued f u n c t i o n s t h e formula (3).
$(t,s;E)
and
e a s i l y read off
The f o l l o w i n g r e s u l t i s a n immediate consequence of formula (2.14) and t h e correspondence between s o l u t i o n s of ( 2 . 1 ) and t h o s e of (2.3):
LEMMA 2.2.
Every g e n e r a l i z e d s o l u t i o n
t 10.
continuous i n
f u
0
then
(E) = O
of
u(t;E)
(1.1)
u ( t ; & ) i s continuously
differentiable. precisely, i t s
We examine now t h e i n i t i a l value problem (2.2),
r e l a t i o n with t h e second order i n i t i a l value problem f o r (2.6).
Let A
THEOREM 2.3.
(2.6) be w e l l posed.
8,
E
t h a t is, l e t t h e Cauchy problem f o r
Then t h e i n i t i a l value problem f o r
u ' ( t ) = Au(t)
i s w e l l posed i n t
2
(2.15)
t h e propagator of (2.15)
0,
g i v e n by t h e
a b s t r a c t W e i e r s t r a s s formula
S(t)u = The propagator in -
if
5 >
-
Re A
0;
m
2
e-'
/4tC(s)u
more p r e c i s e l y ,
then
A
E
G(rp)
(t > 0 ) .
ds
c a n be extended t o a ( E )
S(?(E)
-
In
h-(E) =
(1 + 4 u ~ ~ & ~ )>~1/ /~ ~/ ~ &W .e~ o b t a i n i n t h i s way t h e following r e s u l t :
where
Co,w
a r e t h e c o n s t a n t s i n (2.17).
To study convergence of
u(t;E)
we s h a l l use t h e asymptotic
s e r i e s f o r t h e Bessel f u n c t i o n s ,
where
( s e e WATSON [1948:1, pp. 203 and 1981). The asymptotic s e r i e s ( 3 . 5 ) w i l l be used with
x =
2E
((:)*
-
both t o c a l c u l a t e limits (as i n (3.10)-(3.11),
(3.25)-(3.26),
(4.14)-
(4.15)) and f o r e s t i m a t i o n purposes (as i n (3.14)-(3.15), (3.27)-(3.28), (4.16)). The second a p p l i c a t i o n deserves some comment. Since t h e functions x = 0 , (3.5) w i l l not provide good bounds
t o be estimated are r e g u l a r a t
near zero. However, t h i s i s not very s i g n i f i c a n t , s i n c e t h e s e functions converge t o limits t h a t do have s i n g u l a r i t i e s a t
x = 0. To improve t h e
estimations, we s h a l l r e p l a c e remainders of t h e form
O(x%)
by
o ( ( x + a)%) which a r e r e g u l a r a t t h e o r i g i n . It i s p l a i n t h a t t h e observations above can be a p p l i e d a s w e l l t o t h e asymptotic s e r i e s obtained from t h e s e r i e s (3.5) f o r functions of t h e form
173
PARAEOLIC SINGULAR PERTUREATION
x-'Iu(x),
where
B
and
a r e a r b i t r a r y r e a l numbers. Likewise, we s h a l l
u
( 3 . 5 ) term by term t o provide asymptotic s e r i e s for d e r i v a -
differentiate
t i v e s of a r b i t r a r y o r d e r . This can be easily j u s t i f i e d . To t h i s end we We proceed t o t h e e s t i m a t i o n of u ( t ; & ) - u ( t ) . d i v i d e t h e i n t e r v a l of i n t e g r a t i o n i n ( 2 . 1 4 ) i n
0
5
5
s
s(E),
where t h e asymptotic development ( 2 . 5 ) c a n be u s e d , and t h e "small" interval
s(&)
5 t 5 t / & where
rougher bounds w i l l s u f f i c e .
The
d i v i s o r y p o i n t between t h e " i n n e r " and " o u t e r " i n t e g r a l s i s defined by
where
q
< 1/2 w i l l be s p e c i f i e d l a t e r .
The l e n g t h of t h e second
interval is
We use t h e asymptotic development ( 3 . 5 ) t o o r d e r
0
5
s
5
S(E):
* X
IJX)
=
(1 +
o($))
m
=
in
0
.
(3.9)
(2lrx)l
A f t e r d i v i d i n g by
x
and performing a few c a n c e l l a t i o n s we o b t a i n
where =
X(t,S;E)
To o b t a i n (3.11) from
Obviously,
(1.10)
-
- q ( F )2 ) -3/4(1 + o($)).
(3.9) we
T(x),
-1
m
=
0.
e x i s t s a constant
have used t h e i n e q u a l i t y
(3.11) i s uniform i n t h e i n t e r v a l
To o b t a i n a n e s t i m a t e f o r
x
Since C
(3.11)
(lrt)
cp
we use formula
xmlI (x) i s r e g u l a r f o r 1
such t h a t
0
5
s
5
s(E).
(3.5) f o r t h e f u n c t i o n x
=
0,
there
PARABOLIC SINGULAR PERTURBATION
174
where
x; t h e choice of 2 i n t h e denominator of increasing, since ( 2 + x) -3/2 ex
does n o t depend on
C
(3.13) makes t h e f b n c t i o n (a
+
x)"
ex
is increasing f o r
l a t i o n s leading t o (3.10)-(3.11)
holds i n
0
5 s 5 t,
and t h e c o n s t a n t
C
a>
CY. Essentially
t h e same manipu-
reveal that t h e e s t i m a t e
where
does not depend on
E,t.
I n e q u a l i t y (3.14)
l e a d s t o t h e estimates below.
holds, where
i s a constant t h a t does not depend on
C
s,t,E.
We make use of (3.14)-(3.15) observing t h a t 2 1/2 2 (Es/t) ) 5 1 ( E s / t ) /2 i n t h e exponent and t h a t (Es/t) 2 ) 1/2 >_ (1 (Es(E)/t)')1/2 = 2~ i n t h e denominator of
Proof.
(1 (1
-
-
(2.14);
t h e term
-
2 4E /t
i n s i d e t h e p a r e n t h e s i s i s p o s i t i v e and can
be dropped.
where t h e c o n s t a n t
C
does not depend of
Proof. We use a g a i n (3.14)-(:,.15)
hand s i d e of t h e i n e q u a l i t y
s,t,E.
keeping i n mind t h a t t h e r i g h t
(3.13) is a n increasing f i n c t i o n of
x.
Accordingly, we can e s t i m a t e t h e r i g h t hand s i d e of (3.14) by t h e value obtained i n s e r t i n g t h e highest p o s s i b l e value of (which i s t h e summand
(1
-
2
(Es(E)/t)2)1/2
4E /t
= 2q).
(1
-
(Es/t
2 1/2
)
Once t h i s is done we d i s c a r d
i n t h e o u t e r p a r e n t h e s i s o f (3.15).
The r e s u l t
i s (3.17). A s a n immediate consequence of (3.17) and of the estimation (3.8)
PARABOLIC SINGULAR PERTURBATION f o r the length of the interval
s(E)
t h e r e and i n o t h e r i n e q u a l i t i e s
C
s ,t ,&
0 . We s a y t h a t a family of f'unctions converges uniformly i n t > - t ( E ) t o a f u n c t i o n g(:) if and t(E)
> 0
f o r each
E
only if sup
1 h
-
llg(t;E)
g(t)lI = 0 .
Ed0 t)t(E)
I f t h e supremum i s t a k e n i n s a y that
t
2
t ( E ) 0
for
a r b i t r a r y we
uniformly on compacts of
g(:)
t(E). We prove below that f o r every
on compacts of
t ,t(E)
s u b s e t s of
as long as
E,
u
E
E, R(t;E)u-r S(t)u
uniformly w i t h r e s p e c t t o
t(E)/E2
4
(E
m
.+ 0 )
uniformly
u on bounded
.
(3.20)
I n f a c t , assume t h i s i s f a l s e . Then t h e r e e x i s t s a bounded sequence
[u,]
sequence
{t,]
For each
n
C
E,
a sequence 2
such that
we choose
tn/En
'n
-21-
{En]
*
with and
m
End
0
and a bounded
lIR(tn;En)un
- S(t,)unll
2 6 > 0.
such that 'in
-
WE
n
n --
(3.21)
n (note t h a t zero,
'n
-
n 1/2
< 1/2: moreover, s i n c e both as
n
+ m).
En
and
E n t n-1/2
tend t o
We d i v i d e t h e i n t e r v a l of i n t e g r a t i o n i n
(2.14) according t o t h e e q u a l i t y (3.7) with
q = q n'
We have
176
PARABOLIC SINGUMR PERTURBATION
The f i r s t i n t e g r a l tends t o zero as convergence theorem: that
(note that
1
-
(Es/t)2
uniform estimate
due t o the dominated
m
+
9
-
cp(t,,s;E,)
n
i n f a c t , t e asymptotic r e l a t i o n (3.10) shows e-'
/4tn-+ 0
2 211, -,
1. hence
(3.16).
as
-
n-,
Es/t
m
0)
for
s
fixed
and we have t h e
The second i n t e g r a l tends t o zero by (3.18).
A s f o r t h e t h i r d it i s e a s i l y seen t o telescope making the change of variable
ti1/'s
=
(J
tn a r e bounded. I n f a c t ,
and r e c a l l i n g t h a t t h e
Now, it follows from (3.8) and (3.20) that S(E
n
)
=
tn 2 1/2 (1 - 4q ) n n
t
2
s(En) >_ 2 t 1/4&-1/2 n
Thus
8n
-
m
(1
-
,3/4
2'in)l/2 2 2
as
n
-
m
n -
-
(3.24) El/2 n s o t h a t (3.23) tends
t o zero (we note t h a t i f
w = 0 t h e i n t e g r a l (3.23) tends t o zero 2 under the s o l e assumption that t,,/En -,m , where t h e tn may be
unbounded; t h i s f a c t bears on a resuLt below). a contradiction and j u s t i f i e d
OUT
claim about
We have then obtained sf.
We prove next t h e corresponding statement f o r
5(t;E).
The
estimates a r e obtained i n a s i m i l a r fashion, thus we only s t a t e the final results.
Formula (3,10)-(3.11) has t h e following counterpart:
with X(t,S;E) = (77t)-1/2(1 The estimate is uniform i n The inequality
holds i n
0 5 s 5 t, where
-
2 (y))-1/4(l +
0 5 s 5 s(E).
.
(I(%))(3.26)
PARABOLIC SINGULAR PERTURBATION
177
p(t,S;E) = t and t h e constant
(3.28)
does not depend on
C
we use t h e asymptotic formula (3.5) f o r
m = 0
E,t.
To o b t a i n (3.25)-(3.26)
m = 1; t h e same formula with
yields the inequality
Using t h e i n e q u a l i t y (3.26)-(3.27) we e a s i l y o b t a i n t h e following counterparts of L e m 3.2 and Lemma 3.3 :
holds, where the constant
where t h e constant Using
C
does not depend on
does not depend on
C
s,t,E.
s,t,E.
(3.31) and (3.8) we obtain
We prove t h a t
6(t;E)u uniformly i n of
t >_
S( t ) u
uniformly with respect t o
t(E)
i n e x a c t l y t h e same way used f o r
E
(3.33)
R;
u
i n bounded s e t s
d e t a i l s a r e omitted.
A f t e r a n elementary estimation of t h e f i r s t term i n (2.5) t h e proof of t h e following r e s u l t i s complete:
Let
THEOREM 3.6. UO(E)
and l e t
u(t;E)
-b
v,
uO(E),ul(E) 2
E UJE)
E
E
-. uo
be such that
- v
(E
+
0)
be the generalized s o l u t i o n of ( 3 . l ) ,
number such t h a t (3.20) holds.
,
(3.34) t(E) > 0
Then
U(itjE) -. u(Z)
(3.35)
178
PARABOLIC S INGUMR PERTURBATION
unifwmly i n compacts of s o l u t i o n of ( 3 . 2 ) w i t h respect t o
uo,v
REMARK 3.7.
if.
2
t
t(E)
u(%)
where
u(0) = uo.
i s t h e generalized
The convergence i s uniform with
(Iuo(I, /(v(I a r e bounded.
does not converge t o
uO(E)
thus t h e r e i s a "boundary l a y e r " near zero where approximation t o
t >_ 0 u
Obviously, uniform convergence i n
expected s i n c e i n g e n e r a l
u(^t)
cannot be
as
0
-
E
0,
i s not a good
u(;;&)
[1981:1] f o r a thorough
( s e e KEVORKIAN-COLE
treatment of t h e one dimensional case). Note a l s o t h a t m i f o r m -opt -w2t e u(t;E) t o e u ( t ) i n t >_ t ( E ) cannot be
convergence of
assured even i n t h e s c a l a r case.
To s e e t h i s , l e t
s o l u t i o n of t h e i n i t i a l value problem
-w
< w2
since ?(&),A*(&) e-',2tew2t = 1,
we have
e
L3t -2 1 with
0 l e t t ( E ) be such that 2 -, a , where 0 < a < m . Then we seE from (3.3) that
follows. 0 , 0 c E c 1/&)
independent of
u)
> 0
and of
5
116"(t;E)(E-1R(&-1;A))ll
C(w4
f
2 w 2 / t + l/t 2 ) eu) t
The proof i s straightforward b u t tedious. that
U E
D(A),
B"(t;E)u
s o that
.
(4.26)
Assume f o r t h e moment
e x i s t s ; a n e x p l i c i t formula f o r it
can be obtained from (4.1): -t/2&2 G,"(t;E)u =
E3
-t/2E2 C'(t/E)u
-
4
-t/2E C(t/E)U
2E
+
te
2 C(t/E)U
8E6
C(s)u ds
te
2 -t/2E f
t e 4E7
C(s)u ds 2
-t/E
Jo
Ii(((t/E)2
-
2 1/2
s )
/2&) C(s)u ds
Il(((t/E)2
-
S2)l/'/2E) C(s)u ds
PARABOLIC SINGULAR PERTURBATION
187
e 4E5
where terms a r e grouped t o g e t h e r as t h e y appear i n d i f f e r e n t i a t i n g
(4.1).
Note a l s o that t h e t h i r d and f o u r t h i n t e g r a l s a r e i n d i v i d u a l l y
divergent and must be combined i n t o one.
We t a k e a look f i r s t a t t h e
terms that l a y o u t s i d e of i n t e g r a l s . For t h e second we have
and t h e same estimate o b t a i n s for t h e t h i r d and t h e f o u r t h ,
SO
that
t h e y s a t i s f y (4.26) even without t h e i n t e r c e s s i o n of t h e mollifying operator then
E-1R(E-1;A).(4)
For t h e f i r s t term we note that i f
C(^t)v i s continuously d i f f e r e n t i a b l e w i t h
hence
C' ( t ) v = d(t)Av
Since
w
7
v
E
D(A)
C " ( t ) v = C(t)Av,
and we have
0, l/S(t)ii 5 C e x p ( ~ t ) ( ~ and ) t h e r i g h t hand s i d e of (4.29)
can be estimated i n t h e same way as (4.28). To e s t i m a t e t h e six i n t e g r a l s i n (4.27) we d i v i d e t h e domain of integration a t specified l a t e r .
s = s(E)
given by
(3.7), with q < 1/2 t o be
For t h e f i r s t o u t e r i n t e g r a l we t a k e advantage of
(3.17) f o r cp(t,s;E), divided by t E 2 ; f o r t h e i n t e r v a l of i n t e g r a t i o n we use (3.8). The r e s u l t is a bound of t h e form t h e estimate
The s e c o n d , f i f i h and s i x t h i n t e g r a l s a r e t r e a t e d i n t h e sane way: i n a l l c a s e s , due t o t h e a d d i t i o n a l f a c t o r e s t i m a t e of t h e form
t/E2
we end up with a n
188
PARABOLIC SINGULAR PERTURBATION
A s pointed out a f t e r (4.27) the t h i r d and f o u r t h i n t e g r a l s must be
combined i n t o one t o a m i d divergence a t
s = t/E
w r i t t e n s e p a r a t e l y only f o r typographical reasons).
( i n f a c t , they a r e The basis of the
r e s u l t i n g estimation w i l l be t h e asymptotic s e r i e s f o r the h n c t i o n obtained from (3.6): Q(x) = X-~(X-~I~(X))'
we deduce from it that
The combined integrand of the f o u r t h and f i f t h i n t e g r a l (including f a c t o r s outside of the i n t e g r a l ) i s
-
2 -t/2&2 t e Q ( ((t/E)2
16E~
-
~~)'/~/2E)C(s)u.
I n view of (3.30) we have
where p(t,S;E) = t
(3
(4.34)
is increasing w e can bound the r i g h t hand side of (4.34) by i t s value a t s = s ( E ) subsequently deleting the f a c t o r 6$/t from t h e outer parenthesis. The r e s u l t i s an upper bound f o r
Since
-t
x)-5/2ex
t h e combined integrand of the form
Therefore, the i n t e g r a l can be bounded by the following expression:
This completes the consideration of the outer i n t e g r a l s .
We look a t t h e inner i n t e g r a l s . t h e i n t e g r a l belaw:
We begin by grouping them i n t o
PARABOLIC SINGUL4R PERTURBATION
189
O(t,s;E)C(s)u ds.
Using t h e asymptotic developments (3.5) f o r
Io, I1 and Ii of
m = 1 i n t h e f i r s t and f o u r t h i n t e g r a l s and of order
order
i n t h e r e s t we o b t a i n f o r
B
m = 2
a n expression of t h e form
a l i n e a r combination of terms of t h e form
with X
with
(4.36)
j = 2,1,0,C
f o r each t e r m
expression f o r
J
>
We then use T a y l o r ' s formula of order 2 ) 'j, ending up w i t h t h e following
U . , ~ .0.
J
(1 - ( E s / t ) X:
o(&)) 2
where each
i s independent of
X,(t,s)
t > 0
cosine f u n c t i o n
and apply formula C(^s) = cos og,
(in fact,
E
(4.27) i n
where
t h e space
(4.39)
is a f i n i t e
X,
>_ 0 ) . We
s@t-* w i t h ct:B
l i n e a r c o m b i d t i o n of terms of t h e form then f i x
2j
E = C
t o the
i s a r e a l parameter.
0
Naturally, t h e r e s u l t must be t h e second d e r i v a t i v e of t h e s o l u t i o n of E 2t " ( t j E )
4-
< ' ( t ; E ) = w 2< ( t j E )
,
(4.40)
w i t h i n i t i a l conditions 0
t6'(t;E)
be such t h a t (3.20) holds.
-
tS'(t),
t*E'l(t;E) (E-$(E-l;A>)
uniformly on compacts o f
t >_
t(E)
4
Then (4.52)
t2S"( t )
,
(4.53)
i n t h e topology o f
(E).
Obviously, r e s u l t s of t h e type o f Lemma 4.4 and Theorem 4.6 can be obtained f o r d e r i v a t i v e s of any o r d e r o f
6.
We omit t h e d e t a i l s .
I n a s e n s e , t h e r e s u l t s above do not t e l l t h e whole s t o r y about
5" s i n c e t h e smoothing o p e r a t o r
E-lR(E-l;A)
only plays a r o l e i n
t h e f i r s t term on t h e r i g h t hand s i d e of (4.27); b e s i d e s , t h e f i r s t
t e r m a l s o makes it necessary to s e p a r a t e t h e c a s e t i o n purposes.
is :
A s t a t e m e n t on
@'(t;E)
u)
= 0
f o r estima-
t h a t avoid t h e s e inconvenients
192
PARABOLIC SINGULAR PERTURBATION
THEOREM 4.7 w
2
and of
0
IlG"(t;E)U
(a)
-
E
.There e x i s t s a constant
( t > 0, 0
0,
*
m(A) = ( E A Then m' (A)
(5.24)
+
A)
q2
(A,O)
is alternating.
The proof i s l e f t t o t h e reader (Exercise 1).
.
(5.25)
196
PARABOLIC SINGULAR PERTURBATION LEMMA 5.4.
t
+
Aza.
Let
f ( % ) be continuous i n
Assume t h e Laplace transform
m.
t >_ 0 , f ( t )
=
O(exp a t )
Pf(^A) i s a l t e r n a t i n g i n
Then f ( t ) >_ 0
( t >_ 0 ) .
(5.26)
The proof is an immediate consequence of Lemma 1.3.2 ( s e e (1.3.14)). End of proof of Theorem 5.1. We go back t o (5.18). The d e f i n i t i o n (5.15) of t h e flmction h ( t , s , A ; t ) , Lemma 5.3, Lemma 5.2 and the is comments preceding it show t h a t h, a s a function of A, a l t e r n a t i n g f o r any s , t 2 0 , E > 0. Since t h e f i n c t i o n k ( s ) defined i n (5.19) i s nonnegative, it follows from (5.18) t h a t the Laplace i s a l t e r n a t i n g . Thus, by Lemma 5.4, transform of r(ht;E) r ( t ; E ) >_ 0 ( t 2 0 , E > 0 ) . Taking i n t o account t h e a r b i t r a r i n e s s of
*
u and u
,
(5.7) follows, completing t h e proof of Theorem 5.1.
I n a l l of t h e r e s u l t s t h a t follow u(:;&)
u(z))
(resp.
is the
s o l u t i o n of t h e homogeneous i n i t i a l value problem (2.1) (resp. ( 2 . 2 ) ) .
and applying (5.6) and (5.7) to t h e f i r s t term on t h e r i g h t hand s i d e t o estimate t h e other summands we use (3.4) which implies
of (5.28): (taking u
0
(E)
= 0
or
Ilc(t;E)ll _ 0,
E
’0)
*
(5.29)
W e obtain a simpler but l e s s p r e c i s e bound noting t h a t @,(t;&), u?t (Lemma 3.1) and i n t e g r a t i n g ( 5 . 7 ) by p a r t s ; it r e s u l t s YW(t,E) 5 e t h a t Ow(t;E) 5 (1+ w2t)eat so t h a t (5.27) becomes 2 lju(t;E) u(t>li 5 c O2 (1 ~ + w 2t ) e w2t / l ~ u +o ~coew ~ t ~ l u o ( ~ )~ , I I +
-
-
197
PARABOLIC SINGULAR PERTURBATION 2 wLt e I/ul(~)I(
( t 2 0,
+ c0E
Theorem 5.5 implies t h a t when
t
D(A)
E
0
-
/lU(t;E)
uniformly on compacts of
u
u(t)ll
E
> 0)
.
(5.30)
we have 2
= O(E
1
(5.31)
if
0
I1uO(~)- uoIl = O(E
2
and
llu,(~>Il =
o(1).
(5.32)
Estimates of t h e same s o r t can be e a s i l y obtained f o r t h e d e r i v a t i v e u'(t;E)
if
u
0
E
2
D(A )
and
uO(E)
In f a c t ,
D(A).
E
v(%;E) = u ' ( % j E )
i s t h e s o l u t i o n of t h e i n i t i a l value problem (2.1) w i t h v(O;E)=U'(O;E)=U~(E),
~ ( 2 =)
On t h e o t h e r hand,
= u"(0;E) = E
v'(O,E)
-2
(AuO(E)
-
ul(E)).
(5.33)
i s t h e s o l u t i o n of (2.2) w i t h
u'(t)
(5.34)
~ ( 0= ) ~ ' ( 0 =) AuO. Accordingly, we have
THEOREM 5.6.
Assume t h a t
u
2
0
E
and
D(A )
uO(E) E
2 2 l ( u ' ( t ; E ) - u ' ( t ) l l 5 COE @ w ( t ; E ) l ( A uolI + CO+w(t;E)lIU1(E)
-
+ COYW(t;E)/lU1(E)
2
D(A).
-
Then
AuolI
AU,(.)ll
2
2 w t 2 (1 + w t ) e IIA uolI
5
COE
+
c0 ew t(I/U1(E)
2
- AUoll
.
( t 2 0) It follows from t h i s r e s u l t that i f
- Auo(E)/I)
IlU,(E)
+
uo
(5.35) 2
D(A )
E
uO(E) E D(A)
and
then llu'(t;E) uniformly on compacts of Ilu,(E)
-
t 2 0 2
- u'(t)l/ =
2 O(E
(5.36)
)
if
- Au~(E)II =
2
)
and
IIu~(E)
= Of& )
and
/ I A u O ( ~ ) AuoII = O ( E ).
AuoI/ =
O(E
O(E
> , (5.37)
or,e q u i v a l e n t l y , i f Ilu,(E)
- Auoll
Theorems 5.5 and
2
-
2
(5.38)
5.6 a r e e a s i l y s e e n t o i n p l y convergence r e s u l t s
v a l i d f o r a r b i t r a r y i n i t i a l conditions.
PARABOLIC SINGULAR PZRTURBATION
198
Let
5.7.
THEOREM
(resp. ( 2 . 2 ) w i t h u
t i o n of (2.1)
E E
0
uO(E)
~ ( 2 ) ) be
(resp.
u(^tjE)
-. uo,
arbitrary).
-. 0 &s
E'u1(E)
t h e generalized solu-
E
+
Assume t h a t
(5.39)
0.
Then U(tjE)
uniformly on compacts o f Proof.
u.
U(0) =
6> 0
Pick
r(^t)be
Let
u(t)
+
E
(5.40)
0
+
t >_ 0 .
u
and choose
E
D(A)
with
;1
- uoI/ 5 & .
t h e s o l u t i o n of t h e i n i t i a l value problem (2.2) with
Applying Theorem 5.5 and i n e q u a l i t i e s
(5.29), ( 5 . 3 0 ) , we
obtain
-
IIU(tF)
5
COE
5
U(t>ll
2 (1+
2 (u
-
IIU(tF)
+
t ) e w t/lAiiI
5
COE
2
2 w t
e
l l ~ ~ l +l
i ~-
uOl/
2
coew t l l u o ( ~ ) 2 w t
11u1(@)11+ C06 e
-
uo1l
.
(5.41)
> 0
s u f f i c i e n t l y small we c a n obviously make t h e r i g h t hand 2 2C0 & ew a i n 0 5 t 5 a , a > 0. This ends the proof.
E
side e wt
+ Taking
c0E 2ew
U(t)lI
-
C o e " tlluo(E)
2
t
-
II3t)
:(t>lI +
2
2
Concerning d e r i v a t i v e s , we have
Let
THEOREM 5.8.
that
u(t)
u(tjE),
u~,u~(E E )D(A) AuO(E) a Au, u l ( f ~ )
-
b e its i n Theorem 5.7.
Au0 -as
E
+
Assume
0.
(5.42)
-
Then
U'(tjE) uniformly on compacts of
U'(t)
4
as
E
(5.43)
0
+
t >_ 0 .
The proof follows t h e l i n e s of t h a t of t h e previous r e s u l t . 6 > 0,
,(^t)
and choose
u
E
2
D(A )
such that
llAu
- AuoI/ 5 6 .
Let
Then, i f
is a g a i n t h e s o l u t i o n of t h e i n i t i a l value problem (2.2) w i t h
U(0) = uo we a p p l y (5.29) and (5.35),
obtaining
199
PARABOLIC SINGULAR PERTURBATION
2
+ c 08 e w t
(t
0,
E
(5.44)
> 0).
T h i s completes t h e proof.
5.9.
Convergence i n (5.31) and (5.36) is uniform i n t ? 0 ( r a t h e r t h a n j u s t uniform on compacts of t ? 0 ) i f w = 0 . O f REMARK
course, t h e same observation applies t o a l l t h e other r e s u l t s i n t h i s section. For easy reference l a t e r w e c o l l e c t t h e s e p a r t i c u l a r cases of Theorems 5.5, 5.6 5.7 and 5.8 under a single heading. THEOREM 5.10.
cosine m c t i o n
Assume that A
c(Z>
with I M t ) II 5
Let
u(ht;E)
(2.1)
,
u(%)
generates a s t r o n g l y continuous
co
e w Let
u
f
Ha.
(t
2
0)
.
(5.53)
Define
IuI,
= ~~u~~ + sup ( w 2
+
l/t)a-le-W
2 tl/AS(t)ul/.
(5.54)
t>_o Obviously,
I *ICYi s
a norm i n
31
CY
(which, i n c i d e n t a l l y , makes
Ha
a
Banach space). The following two r e s u l t s a r e formal counterparts of Theorem 5.5 and 5.6.
The proof of both r e s u l t s i s based on a d i f f e r e n t estimation of t h e operator
a.
commute with A
Assume f i r s t t h a t
u
E
D(A);
since
S
and
6'
and with each other we can w r i t e
=L t
B(t;&)u
It follows from (4.1) that
V(t
-
s;E)AS(S)U d s .
(5.57)
202
PARABOLIC SINGULAR PERTURBATION
-
Take the
(1 cu)-th power of both s i d e s , take the
a-th
power of both
sides of (4.20) and multiply the i n e q u a l i t i e s thus obtained term by term.
The r e s u l t is ll5'(tF)Il
Hence, i f
u
(u)
2
+
2
l/tlffew
(t
2 01.
(5.59)
Za,
E
/b(tF)UII 5
w
t
2
(w
(t;E) 0
7.7.
such t h a t
t h e constant i n (4.20) and (4.26)) and define
=
o
for
t c 0.
There e x i s t constants
E
0'
B > 0
independent of
E
215
PARABOLIC SINGULAR PERTURBATION
-
(7 19)
Proof:
Write 6"(sjE)U
for
u
E
D(h),
of (4.27) and
where
5
0
=
X0 ( S ; E ) U +
is t h e f i r s t term on t h e r i g h t hand side
Xl i s t h e sum of t h e r e s t .
r a t h e r , Theorem 4.7)
(7.20)
Zl(S;E)U
Using Theorem 4.4 ( o r ,
we deduce 2
~ ~ ~ ( s ;5&c(w2 ) ~ -t ~ w2s-'
kt
t > 0, u
Kp(S;E)U-Kp(S
By Theorem
E
D(A).
-
tjE)U
+
s-2)eW
5
C's-2e ps
( s > 0)
.
(7.21)
We have =
4.1 we have
Putting together (7.21) and (7.23) we can estimate t h e integrand of t h e
f i r s t i n t e g r a l i n (7.22) by
Cb-2, thus t h e i n t e g r a l i t s e l f is
bounded by a constant times
1 s - t
- -s1 '
(7.24)
The second i n t e g r a l i n (7.22), a f t e r i n t e g r a t i o n by p a r t s , becomes
(7.25) A look at t h e integrand i n (7.25) makes p l a i n that it can be estimated
by a constant times
216
PARABOLIC SINGULAR PERTURBATION
thus t h e i n t e g r a l c o n t r i b u t e s another serving of (7.24).
Putting
t o g e t h e r a l l estimations and taking advantage of t h e f a c t t h a t
D(A)
is dense i n E we deduce t h a t
-p
c
+
~
2 -t),-( S-t)/2E eW( S-t)/E e
(5
-
,
E
(7 27)
t h e last two summands o r i g i n a t i n g from estimation of t h e boundary terms On t h i s basis, we proceed t o estimate t h e i n t e g r a l
i n (7.25).
(7.28) The i n t e g r a l of t h e f i r s t t e r m i n (7.27) i s computed as i n Lemma 6.3. The i n t e g r a l of t h e second term i n (7.27) i s
To compute t h e i n t e g r a l of t h e last term we make t h e change of variables
s
-
t =
t h e domain of i n t e g r a t i o n i s then
0;
< s +
(P
- 4
- b))1/2 - w ) I E 2 ( p -id2i-
A/
1x1
V/ 11-11
=
-
+ l)1’2(P/lUl)1’2)
(Re((U
2
UlAl-1’2}\U
0 < Re 1-1
i s t h e unique multiple of
h
- kdl
by
i s bounded away from zero i n t h e s t r i p t l y small, where
(p
(7.32) 1/2
]A\-’, setting u = E A we see t h a t it i s enough t o show t h a t
Multiplying numerator and denominator noting t h a t
217
(7.33)
+ 111’2 2 with
1.
sufficien-
E~
111 = P.
on t h e l i n e
p
and
We check e a s i l y t h a t (7.33) never vanishes, thus we only have t o show t h a t it i s bounded away from zero for 1 ~ +1 m. Note t h a t , f o r 1U1 = r a t t a i n s i t s minimum a t Re((u + 1)1’2(~/1~])1’2)
u
=
+ir, thus
(7.34) On t h e o t h e r
I
EOlUl -
1x1
hand,
q u + 111’2,
>
lhl-1/21U
so t h a t
Ei21~1
thus our claim holds f o r
Proof of Theorem 7.6.
T h a t the kernel
independent of
< 1, 1/2w.
K (t;E) P
satisfies
(a)
in
was shown i n Lemma 7.7, while (6.3), with B
E
likewise independent of
was t h e s u b j e c t of Lemma 7.8.
E
t h e operator
-Lt
f(;) i s bounded i n
-
_ 0 .
The function w(;;E)
= u(;jE)
- vo(G;E)
i s a s o l u t i o n of t h e i n i t i a l value problem ~
+
E2w"(tjE)
~ ( 0 ; s )=
1
w = w
U (E)
0
-
w'(t;E) = A w ( t ; E )
+ vo,
2 e-t/E Avo,
~ ' ( 0 j E ) = Ill(&)
- E -2
(8.10) To.
i s t h e s o l u t i o n of t h e homogeneous equation with t h e assigned i n i t i a l conditions and w2 is t h e s o l u t i o n
Write
-k
w2
where
w1
of the inhomogeneous equation with zero i n i t i a l conditions and 2 f ( t ; E ) = -e-t/E
We apply t o w1 while
w2
(8.W
AV0 '
Theorem 5.5 (with t h e simplified estimate (5.30)),
i s handled by means of Theorem 7.2 ( s p e c i f i c a l l y , t h e f i r s t
i n e q u a l i t y (7.4)). The final estimate i s
with t h e obvious modification i n t h e last term i f
w = 0.
This ends
t h e proof. For
N = 1 an a d d i t i o n a l c o r r e c t o r must be used, namely
2 vl(tjE) = -e -t/@
THEOREM 8.2.
1'
Assume t h a t ( 8 . 3 ) holds f o r
N = 1, t h a t i s
221
PARABOLIC SINGULAR PERTURBATION
u
0
u
=
(E)
= E - 2v 0
1 u , u 1, vo, v1
-~ and t h a t
!I
U(tjE) = u ( t ) +
D(A
U
2
0
t
u n i f o r m l y on compacts o f
O(E ),
f
+ d v l i-
o(1)
(8-131
1. Then
V0(tj&) f
0
2
t Eul
(E)
E(ul(t) +
(resp.
uo(i)
where
0,
+ O(E 2
Vl(tjE))
1
(8.14)
&
u,(t))
the s o l u t i o n o f (8.2) w i t h
u0 ( 0 ) = uo If -
i.
vo
(resp.
w = 0 , (8.14) holds uniformly i n
u,(o)
= u1 +
(8.15)
t >_ 0 .
We c o n s i d e r t h i s t i m e t h e f u n c t i o n
Proof.
w(^tjE) = U(ntjE)
-
-
vo(i;E)
= Aw(tjE)
E2w"(t;E) i w ' ( t ; E ) =
U
As i n Theorem
0
(E)
+
vo
C EV1,
2
-
W'(0,E) =
(8.16)
Evl(t;E)
t h a t s o l v e s t h e i n i t i a l v a l u e problem
w(O,E)
q.
e-t/E
U (E)
1
-
Av
0
-
2 @e-t/E Avl,
E -2V
(8.17)
-1
- E
y
1'
0
8.1, we write w as t h e sum o f a s o l u t i o n w1
(8.18)
of t h e
homogeneous e q u a t i o n t a k i n g t h e a s s i g n e d i n i t i a l c o n d i t i o n s and a solution
w2
of t h e nonhomogeneous e q u a t i o n w i t h z e r o i n i t i a l c o n d i t i o n s .
We a p p l y a g a i n t h e s e c o n d i n e q u a l i t y (7.4) t o
and
w2
(5.30) t o
W1,
obtaining
Obviously, a d i f f e r e n t t a c k must b e a d o p t e d f o r N >_ 2 , s i n c e t h e f i r s t term o n t h e r i g h t hand s i d e s of ( 8 . 1 2 ) and (8.19) c a n n o t b e squeezed smaller t h a n
level.
O(E2).
We p r o c e e d at first o n a p u r e l y f o r m a l
The a p p r o x i m a t i n g h n c t i o n w i l l b e of t h e form
u
N
(tjE)
= u (t) + EUl(t)
0
N
f
*-.
-t E UJt)
,
(8.20)
222
PAMBOLIC S I N G L U R PERTURBATION
a r e defined a s before and t h e where u0 (t), u1(^t) s a t i s f y t h e d i f f e r e n t i a l equations un' ( t ) = Aun(t)
-
~:-~(t)
u
( t >_ 0 )
(t), n 2 2, .
(8.21)
Noting t h a t t h e c o r r e c t o r s 2 vo, v1
used i n t h e cases N = 0,l a r e of 2 t h e form v O ( t j E ) = v (t/E ), vl(t;E) = v (t/E ), we s h a l l use a 0 1 combination of c o r r e c t o r s of t h e form II ( t ; E ) = v (t/E N 0
The
v
n'
n >_ 2
2
)
+
tVl(t/E
2
+
) +
N 2 VN(t/E ).
E
(8.22)
w i l l s a t i s f y t h e d i f f e r e n t i a l equations vn" ( t )
+
vA(t) = Avn-,(t)
(t
2
0)
,
(8.23)
and t h e decay condition vn(t)
-, o
as
t
4
m
and
(8.24)
n = 1,2
Note that t h e equation s a t i s f i e d by , ) : ( u U'n (
. is
(8.25)
t ) = Aun(t),
vn, n = 1 , 2 s a t i s f i e s v p )
+ vA(t)
= 0
.
(8.26)
Consider now t h e f'unction
N
I1=0 p )
= ( E 2U
N
+ &-2
N
c &"Vi(t/E2) + E-2 c E"V;l(t/E2)
+ E-2
17;O
+ up)) +
E ( E 2 u;l(t)
c E"(yll(t/&2) t VA(t/E2)) -2
+ up))
223
PARABOLIC SINGULAR PERTURBATION
+
N-2
c
EnAvn(t/E2)
n=0 N
=
c
EnA(un(t)
+
2
vn(t/E ) )
ti=O
The i n i t i a l conditions on u
Il’
u0 ( 0 ) = uo
-
E N - 1 AvNml(t/E2)
n = 0,l
+ vo,
ENAvN(t/E 2 ).
(8.28)
a r e those i n Theorem 8.1: = u1
u,(o)
+ v1
.
(8.29)
On t h e other hand, t h e i n i t i a l conditions on v n = 0 , 1 must be n’ 2 2 those t h a t insure t h a t v (t;E) = v (t/E ), v (t;E) = v,(t/E ), 0 0 1
v0 ( t ; E ) , v1(t;E)
where
a r e t h e correctors used i n Theorem 8.2.
Accordingly, $0)
= vo, vi(0) = vl,
hence, taking (8.26) and (8.24) i n t o account,
v ( t ) = -e 0
For
n
2
2,
-t vo, v,(t)
t h e i n i t i a l conditions f o r
= -e
un(i)
-t
v
1’
and
vn(t)
are,
respectively un(o) = un
thus for
un(E) tU(t;E)
-
must be constructed a f t e r
(8.32)
vn(o)
vn(t).
The i n i t i a l conditions
are obtained from (8.29) (8.30) ( 8 . 3 2 ) and (8.33):
PARABOLIC SINGULAR PERTUBBATION
224 lo ( 0 ; E ) = N
cN Enun(O) + cN Envn(o) = =O
Il=O
= u
0
- v
+
v
-
0
-
-
vn(o>>
G 2
+
EV
cN Envn(o) = cN n=2
0
=
c
+ N E n(un
“(9 + vl)
f
E
nun
(8.34)
=O
N
N
n=0
n=0
c Enu’n( 0 ) 4- c Enm2vn
N-2
c
E=O
EnU’(0) = n
c
n - 2 v n + E N-1 U&l(O)
E
I1;O
N
f
E
up)
(8.35)
Hence, i n view of (8.3), IlU(0F)
- mN(o;E)I/
= O(EPst1)
-
=
(8.36)
and IlU’(0;E)
lo$OjE)I/
O(EN-l)
.
(8.37)
We face now t h e problem of making a l l t h e s e computations valid. Roughly speaking, t h i s amounts t o :
(a) showing t h a t every d e r i v a t i v e w r i t t e n ( a s i n (8.171, (8.21), (8.24), e t c . ) a c t u a l l y e x i s t s . (b)
etc.),
showing t h a t every time we w r i t e
(as i n (8.17), (8.21),
Au
u a c t u a l l y belongs t o t h e domain of
A.
This w i l l be done by r e q u i r i n g “smoothness” conditions of varying degree on t h e c o e f f i c i e n t s
un’ vn
u o ( t > = S(t>(U0 + v&
i n (8.3). u,W
We begin with
= S(t)(U1+
a r e made e x p l i c i t i n (8.31).
“J
(8.38)
while
vo(t), vl(t)
v,(t),
v3(t) we solve (8.23) with t h e i n i t i a l condition (8.33) at and t h e decay condition (8.24) as t -. m:
t = 0
To construct
225
PARABOLIC S TNGULAR PERTURBATION
v;(o)
v2
=
-
v"(t) 3 v ~ ( o= ) v3
3
v2(t) -,o
u$o), f
-
vl(t) 3
= -e -tA v ~
-, o
u~(o), v 3 ( t )
t
as
=,
4
( t 2 0),
-
t
as
(8.40) m
.
Solving e x p l i c i t l y these e q u a t i o n s , v2(t)
= te
-t Avo
-
-t v ( t ) = t e Av
3
-t
(v2
- e -t (v3
1
- Au0 -
2Avo)
9
(8.41)
- AU1
avo)
9
(8.42)
-
A(uo + v,), U i ( 0 ) = A(U1 + vl). u s i n g t h e equation (8.21) and t h e
where w e have used t h e fact t h a t We compute next
e
u 2 ( t ) , u3(t)
U'
0
(0) =
i n i t i a l c o n d i t i o n (8.32) :
u;(t)
2
AU2(t) - S(t)A (u0 + v0)
=
( t >_ 0 ) , (8.43)
u2(o) =
?(t)
Au3(t)
=
-
U2
+ To
7
2
( t 2 0),
S(t)A (ul + vl)
(8.44) u ( 0 ) = u + v1 3 3
9
2 ) where we have used t h e f a c t s t h a t u " ( t ) = S(t)A (uo t v,), ~ " ( t = 0 1 2 (see (8.29) and (8.37)). = S(t)A (ul -t v,), v;)(O) = vo, v i ( 0 ) = v1
Hence
U,(t)
=
S(t)(U2 + v0)
=
S(t)(U2
f
v0)
-
u ( t ) = S(t)(u3
3
With
S(t
Lt
-
s)S(S)A
2
(u0 + v0) dS
tS(t)A 2 (u0 + v0) ,
-k
2
vl) - tS(t)A (ul
(8.45) -k
v~).
(8.46)
up(%),u 3 ( t ) ,
see that
v4(i),
y2(i), v3(t) already manufactured, we can e a s i l y v 5 ( t ) w i l l have t h e form v4(t)
=
e-tP4(t), v5 ( t )
=
e-tP5(t)
,
(8.47)
PARABOLIC SINGULAR PERTURBATION
226
where
is a polynomial of degree 2 whose c o e f f i c i e n t s a r e l i n e a r
P4(%)
combinations of
AJu
0’
AJvo ( j 5 j), Au2
and
Av2
uo, vo 7u2yv2 replaced by On t h e other hand, we have
t h e same polynomial with respectively.
U4(t)
- P4(0))
2tS(t)A3(uo
u (t) = S(t)(u
5
+
u
(i)
(resp. u5($))
-
vo)
- P5(0))
5
42
is
2
-2tS(t)A3(ul + vl) thus
P (t)
+ tS(t)A (u2 + v0) -
= S(t)(~4
-
and
5 u1,v1,u3,v3
t2 S(t)A4 (uo + v,),
+ tS(t)A2 (u3 i- vl)
(8.48)
-
- t2 S(t)A4(ul + v,),
(8.49)
can be constructed i f uo, vo E D(A 4 ), 4 2 D(A ), u3 E D(A ). However, i f we wish (8.47)
u E D(A ) (resp. ul,vl E 2 t o be a genuine solutions of (8.23) we a c t u a l l y need that
vo E D(A 5 ) and u E D(A 3 ), u4, P4(0) E D ( A ) ; i n view of our 0’ 2 previous comments about P4, it is s u f f i c i e n t f o r t h i s t h a t 4 3 2 uo, vo E D(A ), u2 E D(A ) v2 E D(A ) and u,, E D(A). Likewise, i f u
we wish (8.48) t o be a genuine s o l u t i o n of (8.21) we must a s k t h a t
3
2
E D(A5), u3 E D(A ) v3 E D(A ), u5 E D(A). It w i l l be of ul’ “1 i n t e r e s t l a t e r t o a s c e r t a i n t h a t u 4 ( t ) , u ( t ) a r e twice continuously 5 6 d i f f e r e n t i a b l e . This w i l l be t h e case i f u0’ V0Y U1’ v1 E D(A 1, u2, u3 E D(A 4), v2, v3 E D(A 3) and u4”-15 E D(A). From t h e s e observations we surmise t h e following r u l e s , v a l i d f o r
arbitrary m
2
1. I n t h e f i r s t place, we have
v,(t> where
(resp. Pml(%))
P,(t)
a r e l i n e a r combinations of (j
5
2m.-
A L ~A , J
3),
= e -tP*,(t)
= e-tP&),v;w,(t)
-
(8.50)
i s a polynomial whose c o e f f i c i e n t s
AJu ,AJ, 0
... Ajua-4,A3uh-4.(j
(j ~ 5~an
,
0
(j Y ~-2’~-ZjYvN-2’vN-3 u ~ , u ~ , vE ~ D(A , ~ ~).
THEOm8.3. Odd’
-4-
D(A )
3
N
,...,
N
N
n=O
n=O
(8.55) uniformly on compacts of
t 2
o
(uniformly i n
t
o
if
U)
= 0).
g
PARABOLIC SINGULAR PERTURBATION
228
i s even t h e same r e s u l t obtains under t h e assumption t h a t 2 Nt2
N>_ 2
uN>vN E D(A
1,
Proof. N = 2m
+
%-17%-29vN-19v~q-2
f
We consider f i r s t t h e case
1 and apply r u l e ( a ) .
N
odd
Avo(t),
...,AvN(t)
>_ 3;
we s e t here
Since conditions (8.51) a r e s a t i s f i e d
...
(with something t o spare) we deduce t h a t with
1.
D ( A ~ ) , . * * > U ~ > U O0, V E ~D(A >V
continuous.
vo(t), , v N ( t ) E D(A) Taking (8.50) i n t o account we
deduce t h a t
(8.56) This w i l l be used t o estimate the l a s t two terms on t h e r i g h t hand s i d e of (8.28):
f o r t h e f i r s t two terms we simply use t h e f a c t ,
u~-~(t)and
assured by (b), t h a t differentiable.
%(t)
a r e twice continuously
Using t h e f i r s t inequality (7.10) i n (8.28) we
obtain
where, i n v i e w of (8.56), the contribution of t h e l a s t two terms i s O(EW1),
This ends the proof.
The case
N
even >_ 2
i s handled
much i n t h e same way and we omit t h e d e t a i l s .
@?I. 9 E l l i p t i c d i f f e r e n t i a l equations. We apply the theory i n t h e lust eight sections t o t h e d i f f e r e n t i a l operator
m
m
i n a bounded domain R
.
m
of m-dimensional space w i t h boundary
T; here
229
PARABOLIC SINGULAR PERTURBATION
A(p)
d e n o t e s t h e r e s t r i c t i o n of
o b t a i n e d by means of t h e D i r i c h l e t
A
boundary c o n d i t i o n
o
=
U(X)
r),
(X E
(9.2)
or b y means of t h e v a r i a t i o n a l boundary c o n d i t i o n N
D ~ ~ ( X= )
The c o n s t r u c t i o n of
(x
y(x)u(x)
E
r).
(9.3)
w a s c a r r i e d o u t i n Chapter IV i n c o n s i d e r a b l e
A(@)
9IV.3 ( f o r t h e D i r i c h l e t boundary c o n d i t i o n ) and i n SN.6 (for t h e boundary c o n d i t i o n ( 9 . 3 ) ) t h a t A ( B )
d e t a i l ; i n p a r t i c u l a r , it w a s shown i n
g e n e r a t e s a s t r o n g l y continuous c o s i n e f u n c t i o n , t h u s a l l t h e r e s u l t s i n t h i s chapter apply automatically.
SVI. and u 0
5 , for 5.13.
i n s t a n c e t h e “ i n t e r m e d i a t e ” e s t i m a t i o n s i n Theorems 5.12 Combining Theorem
D ( ( b 2 1 - A(@))‘)
E
u n i f o r m l y on compacts of D((b21
-
5.u w i t h Lemma 5.15 w e deduce t h a t
-
u ( t ) I I = 0(E2’)
t > - 0.
p
Hl(n).
E
-o
(9.4)
The most i n t e r e s t i n g c a s e i s
D((b21 - A ( B ) ) if
as
can b e identified.
A(f3))‘)
if
l a r g e enough) t h e n
(b
jlu(t;E)
where
Of s p e c i a l i n t e r e s t a r e t h o s e i n
c1 =
1/2,
I n f a c t , w e s h a l l show t h a t
(9.5)
= Hi(Cl)
i s t h e D i r i c h l e t boundary c o n d i t i o n and
bl,
...,bm
belong t o
To show ( 9 . 5 ) we n o t e t h a t it h a s a l r e a d y b e e n proved t h a t
D ( ( b 2 1 - AO(@))1/2) especially
=
(see ( N . 2 . k ) ) a n d r e c a l l Theorem IV.2.2, HO(R) 1
(IV.2.6)). Thus, D((b21
We s k e t c h t h e p r o o f of
w e o n l y h a v e t o shod t h a t
- Ao(p))lb2) (9.6).
cosine function generated by used t o c o n s t r u c t
Let
- A(p))”I2)
.
(9.6)
C ( t ) = c o s h t Ao(@)1’2
Ao(p).
b e the 0 It f o l l o w s from t h e p e r t u r b a t i o n
e0(t) (or
6(t) from
cosine function generated by
= D((b21
A(p)lb
d i r e c t l y ) that
C(t),
the
c a n b e e x p r e s s e d b y means of t h e
perturbation series C ( t ) U = C0(t)U where domain
+
gTJF*Co(t)u
+ qTJF*qTJF*Co(t)u +
d e n o t e s t h e ( o n l y ) bounded e x t e n s i o n o f
O1 H (Q))
t o a l l of
* * *
,
So(t)P
(9.7) (with
L2( Q ) ; t h a t t h i s e x t e n s i o n e x i s t s follows
230
PARABOLIC SINGULAR PER’IURBATION
S ( t ) P i s bounded ( i n t h e norm o f 0
from t h e f a c t t h a t
1
L2(n))
in
~ ~ ( n )s,i n c e s o ( t ) P = (sinh t Ao(B)1/2)Ac(B)-1/2P, and
Using (1.5) and t h e “ r e c i p r o c a l ” series Co(t)u = we show t h a t
@(t)U
-
qqF*C(t)u
+
f40P*rn*C(t)u +
(9.8)
@ ( t ) u i s continuou.;ly d i f f e r e n t i a b l e i f and o n l y i f (9.6) follows f r o m Theorem
@,(ti) i s c o n t i n u o u s l y d i f f e r e n t i i b l e , t h u s
111.6.4. However, i n t h e p r e s e n t s i t u i t i o n , estimates on rates o f convergence l i k e (1.4) c a n b e o b t a i n e d under weaker assumptions b y more e l e m e n t a r y methods.
We s k e t c h below t h i s theeory i n a s u i t a b l y ” a b s t r a c t ” v e r s i o n .
Let
E = H
be a H i l b e r t s p a - e and
A.
a s e l f adjoint operator
such t h a t
with
K
>
0.
We c o n s i d e r t h e o p e r a t o r A = A.
where
P
+ P,
(9.10)
i s such that
m-l
(9.11)
i s bounded, where B = ( - A )1/2 d e f i n e d as i n srV.3. Using essentially 0 t h e same methods i n srV.4 we show t h a t A g e n e r a t e s a s t r o n g l y
c o n t i n u o u s c o s i n e f u n c t i o n , t h u s ill r e s u l t s i n t h i s c h a p t e r apply, i n particular those i n
sVI.5. We e x p l o i t t h e s e below.
Using t h e f u n c t i o n a l c a l c u l u s f o r s e l f a d j o i n t o p e r a t o r s we can d e f i n e f r a c t i o n a l powers
( -Ac)‘
of
-Ao
where
E = C1, A u = ku w i t h
A
E
= H
i s a H i l b e r t space and
It f o l l o w s from Fxercise 11.5 -that A
E
A
8(C,w)
g e n e r a t e s a s t r o n g l y continuous c o s i n e f u n c t i o n
3.1,
is a
(that
C(t)
satisfying
Ilc(t)ll 5 i f and only i f
cr(A),
C e U J l tI
t h e spectrum of
(-m
A,
- u0ll
THEOREM 5.6.
-m
= o ( E CY
Let
E,A
< t
II
if = ~ ( E c y - ~ )a s
E
4
o
be as i n Theorem 5.5, and l e t
.
(5.37) u(t;&)
be a s o l u t i o n of t h e homogeneous problem (2.1) with uO(&) E D @ ) , u ( t ) a s o l u t i o n of t h e homogeneous problem (2.2) with uo E D((-AO)1- ). Then, i f (5.24) holds t h e r e e x i s t s a constant
C(U)
such t h a t
~ l u l ( t ; E )- u * ( t > l l sc b > E a ( i + I t l ) a / 2 e " l t l ~ ~ ( - ~ o ) 1 ~ u o ~ ~ (5.38)
258
OTHER PROELEMS
The proof c o n s i s t s i n applying Theorem t h e proof of Theorem
u0
4.2).
u0 (E)
D((-AO)lw),
E
E
llul(E)
-
iAuOII
~ ' ( t )( s e e
u'(t;E),
A s a consequence, we o b t a i n e a s i l y t h a t , i f D(A)
then
- u'(t)ll
IlUYtjE)
uniformly on compacts of
5.5 t o
(Hint:
0
t h i s f o l l o w s from a s t a n d a r d
theorem on p e r t u r b a t i o n of g e n e r a t o r s of c o s i n e f u n c t i o n s . a u t h o r [ 1971:11, SHIMIZU-MIYADNU
See t h e
[1978:I], TAKENAKA-OKAZAWA [1978:11
o r TMVIS-WEBB [1981:1].
4.
EXEBCISE
(Hint:
i s a s o l u t i o n of
(8.5) w e have
u(t^)
If
i s a s o l u t i o n of
6.
u(t)
and i n t e g r a t e ) .
(8.5) w e have
m u l t i p l y t h e e q u a t i o n (8.5) scalarly by
EXFXCISE
5;
u(t^)
m u l t i p l y t h e e q u a t i o n (8.5) scalarly b y
EXERCISE 5.
(Hint:
If
u' ( t ) and i n t e g r a t e ) .
The assumptions a r e t h e same i n E x e r c i s e s
3, 4, and
we r e q u i r e i n a d d i t i o n t h a t
(Hint:
m u l t i p l y (8.7) b y
and add t o (8.6).
Take r e a l p a r t s .
Combine t h e f i r s t t h r e e i n t e g r a l terms i n t o one and use t h e f a c t t h a t
for a l l
u
E
D(Q), v
EXERCISE
7.
E
H,
consequence of (8.8)).
Assume t h a t
268
OTHW PROBLEMS
Re(Bu,u)
_ 0 .
f o r any s o l u t i o n of (1.1) I f the function
t >_ 0
(1.3)
C(t)(llu(O)lI + l l U ~ ( 0 ) l l )
C(t)
in
(1.3) can be chosen nondecreasing i n
( o r , more g e n e r a l l y , bounded on compacts of
t
0)
then we
say t h a t t h e Cauchy problem f o r (1.1) i s uniformly w e l l posed ( o r
t >_ 0 .
uniformly properly posed) i n
The propagators o r s o l u t i o n operators of (1.1)a r e defined by
u(2)
where
(resp.
u ( 0 ) = u, u ' ( 0 ) = 0
C ( t ) (resp. of
D
Since both C(t)
and
0
a(t)
v(%))
i s t h e s o l u t i o n of (1.1)with v(0) = 0, v ' ( 0 ) = u).
(resp.
The d e f i n i t i o n of
s(t)) makes and
sense f o r u E D ( r e s p . f o r u E D1). 0 Dl a r e dense i n E we can extend (using (1.3))
t o all of
E
a s bounded operators; t h e s e operator-
valued functions r e s u l t s t r o n g l y continuous i n Iic(t)li
5 C ( t > , ils(t>iI 5 C ( t >
Moreover, by d e f i n i t i o n ,
C ( 0 ) = I,
S ( 0 ) = 0.
t >_ 0
U(t) =
c(t)u(o) +
and s a t i s f y
( t >_ 0 ) .
(1.5)
F i n a l l y , we prove e a s i l y
u(%) i s a n a r b i t r a r y s o l u t i o n of (1.1)i n
t h a t if
(1-4)
S ( t ) u = v(t),
@ ( t ) u =u ( t ) ,
t 2 0
then
S(t)Ul(O).
(1.6)
The proof i s t h e same a s t h a t of (11.1.6). We s h a U assume from now on that t h e operators
A
and
B
are
closed. $vIII.2
Growth of s o l u t i o n s and existence of phase spaces.
The d e f i n i t i o n of phase space i s , except f o r small modifications, t h e same i n $111.1. A phase space i n
t
0
f o r t h e equation (1.1)
equipped with any of i t s product n o r m , @ = Eo x E 1' El a r e Banach spaces s a t i s f y i n g t h e following assumptions:
i s a product space where
E 0 (a)
(-.
D1
dense i n (b)
and
E ,E
6 E with bounded inclusion; moreover, 1Do Eo El) is dense i n E i n t h e topology of Eo ( r e s p . i s
0
n
0
El i n t h e topology of
El).
There e x i s t s a s t r o n g l y continuous semigroup G ( t )
272
THE COMPLETE EQUATION
E = E
in
0
X
t 2
E
o
1
such that
with
for any s o l u t i o n u ( i )
u(0)
E
E ~ u , l(0)
E
E ~ .
The comments a f t e r t h e d e f i n i t i o n of phase space i n $111.1apply here:
we omit the d e t a i l s . We examine i n the rest of t h e s e c t i o n t h e r e l a t i o n of t h i s notion
w i t h t h a t of w e l l posed Cauchy problem i n the case where
E = Q2 is
t h e set of a l l sequences with
2 ~ ~ { u n= ] ~ Iu ~n
c
l2
u = [un jn >_ 1)= {un] of complex numbers and A , B a r e the operators c
ACunI = lanun),
B{ un] = Fbnun],
(2.2)
rb ) sequences of complex numbers t o be determined l a t e r : n t h e domain of A c o n s i s t s of a l l {u ) E E w i t h {a,.,) E E. The n domain of B i s s i m i l a r l y defined; observe that both A and B a r e
{an]
and
normal operators commuting w i t h each other.
u(%) = [un(%)] i s a s o l u t i o n of (1.1)then each u (%) s a t i s f i e s the s c a l a r equation n u''(t> + b n u ' ( t ) + a u ( t > = 0. On t h i s b a s i s , we deduce that the n?? propagators C(%), b ( t ) of (1.1) must be given by
where
+
hn,A,
If
a r e the r o o t s of t h e c h a r a c t e r i s t i c equation h2 + b A + a n = O ,
(2.5)
h = h- (a case t h a t we w i l l n n Obviously, a necessary condition f o r t h e Cauchy problem
w i t h the modifications de rigueur when
avoid here).
f
f o r (1.1) t o be w e l l posed i n
a(t) =
Ilc(t)II
t
2
= SUP
n>_l
0
i s that
-
A+ e n
n
A$
THE COMPLETE EQUATION
273
and
be bounded on compacts of
0
5 t
1 for a l l
Also, a,
m(n> >_
t >_ 0 ,
i s increasing i n
m(^t)
accordingly t h e f u n c t i o n on compacts.
n
thus bounded
n = wn
'n
(n
2 1)
.
(2.17)
Define yn = l o g I n view of t h e i n e q a l i t y
Cyn
+ Lwl/". n n
= log w;/n
log x i x
5
2-1/2ex,
valid for
(2.18) x
5.
0, we
h.ave yn We s e l e c t now
a n,bn i n
n
-
n'
(2.19)
(2.5) i n such a way that (2.20)
We have i
( h I = p > e . n n -
(2.21)
On t h e o t h e r hand, i n view of (2.19),
2
(8, thus the sequence
A =
{A'-]n /A/
-
2 1/2
VJ
>_ Yn>
i s contained i n t h e region e,
Accordingly t h e r e e x i s t c o n s t a n t s
0 C_
Re h
0> 8
7
5 I m A. 0
independent of
fl
such that (2.22)
275
THE COMPLETE EQUATION
I n view of t h e d e f i n i t i o n (2.14) of
m(t)
we obtain from (2.23) and
(2.24) that
e(m(t> -et> 5 a(t>5
5
e(m(t) - e t > in
t 2 0.
T(t)
o(m(t>
+ e
t
1,
(2.25)
5 o ( m ( t > + et >,
(2.26)
The i n e q u a l i t i e s on t h e right-hand s i d e s of (2.25), (2.26)
imply that t h e Cauchy problem f o r (1.1) i s w e l l posed i n only remains t o choose t h e sequence i n e q u a l i t i e s (2.10) a r e s a t i s f i e d .
0
To do t h i s , we assume ( a s we
Keeping i n mind that t h e
constant i n (2.22) i s independent of t h e choice of
Both conditions (2.U) a r e obvious. the greatest integer
5 t.
It
0.
i n such a way t h a t t h e
u(%) i s nondecreasing.
obviously may) t h a t
t
we s e t
fl,
On the other hand l e t
t
2
1, n = [ t ] ,
Then, taking (2.16) and (2.27) i n t o account,
we o b t a i n
whence t h e f i r s t i n e q u a l i t y (2.10) r e s u l t s from (2.25); t h e second follows i n a similar way from (2.26).
EXAMPLE 2.2.
Let
a > 0.
Then t h e r e e x i s t A , B of t h e form
(2.2)such t h a t t h e Cauchy problem f o r (1.1)i s w e l l posed i n 0
b u t not w e l l posed i n any i n - t e r v a l 0
2
0
0.
SUP
s20
e
-us
IFr( s ) u l l ,
(4.5)
Eo
=
(4.6)
D ( A ) _C E o .
i s a Banach space i s much t h e same as t h a t f o r t h e
(111.2.1) and w e omit i t .
THEORE3l 4.2.
t
2
W e obviously have Do
equation
c o n s i s t s of all
so l a r g e t h a t ( 3 . 2 ) , t h e f i r s t i n e q u a l i t y (3.57)
W'
and ( 4 . 2 ) h o l d .
The proof t h a t
Eo
is
Eo
W',
The space
i s continuously d i f f e r e n t i a b l e i n t
l I ~ 1 1=~ IIuII + where
(4.4)
Let t h e Cauchy problem f o r
and l e t Assumption 3.1 be s a t i s f i e d .
f o r (4.1). Proof:
We must show t h a t
( 4 . 1 ) be well posed i n Then
Em i s a phase space
29 1
THE COMPLETE EQUATION
We prove f i r s t t h a t each Em. q t ) i s a bounded operator i n Em. I n order t o do t h i s we t a k e u E D0 and f i x t > 0 . Due t o time invariance of ( 4 . 1 ) t h e f u n c t i o n i s a s t r o n g l y continuous semigroup i n
u(i)
=
C ( t + g)u
i s a s o l u t i o n of ( 4 . 1 ) t h u s it follows from formula
(1.5) that C(S
+
= C(s)C(t)u
t)U
This e q u a l i t y i s extended t o
+ S(s)C'(t)u
u
aJ_1
E
Eo
(s,t
as follows:
2
(4.8)
0).
integrate i n
0 5rzt,
-rote(. +
T)u dT = c ( s )
Lt
C(7)U dT
+
and extend (4.9) t o a r b i t r a r y
u E E by denseness of we d i f f e r e n t i a t e and o b t a i n (4.8). The analogue of ( 4 . 8 ) f o r S ( t ) i s S(S
+ t)u
= C(s)S(t)u
+
u(s) = 8(s
+
Do;
u
for
E
2
0),
(4.10)
t)u, u
E
D1;
since all
u
operators i n (4.10) are bounded we can extend t h e e q u a l i t y t o all We note i n passing t h a t (4.8) i t s e l f can be extended t o all
a( s ) C ' ( t )
modified form observing t h a t
Eo
(s,t
S(s)S'(t)u *I
and i s shown by applying (1.5) t o
(4.9)
S(s)(c(t)u-u),
u
f
E
E
E.
in a
must have a bounded extension.
We s h a l l not make use of t h i s i n what follows. We prove t h a t each
qt)
i s a bounded o p e r a t o r i n
Em. To do
t h i s , we m u s t show t h a t t h e o p e r a t o r s C ( t ) :Eo
-
c ' ( t ) : Eo
E 0 E
are bounded i n t h e spaces i n d i c a t e d .
8 ( t ) :E
-
8 ' ( t ) :E
Eo
-.
(4.11) E
This i s r a t h e r obvious f o r
c'(t)
E ) and f o r a t ( t ) (from Assumption 3.1). 0 Note a l s o t h a t it follows from Corollary 3.7 and Lemma 4.1 t h a t
(from t h e d e f i n i t i o n of
292
THE COMPLETE EQUATION
( h e r e and i n o t h e r i n e q u a l i t i e s
C
denotes an a r b i t r a r y constant, not
n e c e s s a r i l y t h e same i n d i f f e r e n t p l a c e s ) . Continuity of
C(t)
C(s)C(t)u = C ( s + t ) u - S ( s ) C ' ( t ) u j
form
and d i f f e r e n t i a t e with respect t o
apply t o an element
u
+ t ) u - 8'(s)CI(t)u.
i s a bounded operator from
C(t)
i n the of
E~
We obtain
s.
Cl(s)C(t)u = C l ( s If follows t h a t
Write ( 4 . 8 )
i s proved a s follows.
Eo
(4.13) into
and
Eo
(4.14) f o r some constant
Finally, boundedness of
C.
Write (4.10) i n t h e form
It follows t h a t
' 0
i s continuous i n
+ t ) u - 8t(s)Sl(t)u.
)
5
wt
(t
Ce
(4.15)
L
(4.16)
0)
W e have t h e n completed t h e proof t h a t each
C.
Em: moreover, t h e r e e x i s t s a constant
wt
l l ~ t I l l ~ 5~ Cme ) f o r some constant
W e then
The r e s u l t i s
i s a bounded operator and
8(t)
lls(t)ll(E.E f o r some constant
i s shown a s follows.
+ t)u-S(s)S'(t)u.
C(s>S(t)u = 8 ( s
d i f f e r e n t i a t e t h i s e q u a l i t y term by term. C f ( s ) S ( t ) u= S l ( s
S(t)
C,
t h e constant
(t
C
10 )
q t )
such t h a t (4.17)
being t h e same i n Corollary 3.7
w
and Lemma 4.1. The semigroup equation
follows from (4.8) and (4.10) and t h e i r d i f f e r e n t i a t e d versions (4.13) and (4.15). n
The next step i s t o show t h a t
q t )
i s s t r o n g l y continuous.
It
i s enough t o prove t h a t Ilqh)u as
h
qtk
-
O+.
- uII( Em)
+
(4.19)
0
However, we s h a l l skip t h i s step since we show below t h a t
has a derivative at t h e o r i g i n ( i n t h e norm of
s)
for u
in
293
THE COMPLETE EQUATION
Gm; t h i s , combined w i t h t h e uniform bound (4.28)
a dense subset of obviously i m plies
THEOREM 4.3.
(4.19),
q;)
since
i s a s t r o n g l y c o n t i n u o u s semigroup w i t h
8 given by
infinitesimal generator
8=
=
c l o s u r e of
71,
(4.20)
where
w i t h domain D(%) = D ( A )
The f u n c t i o n
i s a s o l u t i o n of
u(;)
u(t)
(4.22)
( D ( A ) fI D ( E ) ) .
X
(4.1) o n l y i f
= [u(t),u'(t)l
(4.2?)
i s a s o l u t i o n of
u'(t)
=
%u(t).
Proof: We b e g i n b y showing t h a t t o p o l o g y of
Eo.
i s dense i n
D(A)
To d o t h i s we s e l e c t a
(4.24)
"6-sequence"
Eo
{@,I
i n the
of scalar
f u n c t i o n s l i k e that used i n t h e proof o f C o r o l l a r y 3.5 ( b ) , and show that
u as
n
-
( f o r any
f o r each
m
u
E
E)
(4.13) we s e e that
u
n
= J$ , ( t ) c ( t ) u
E
E
0' i s obvious.
dt
That ( 4 . 2 6 )
-
(4.26)
u
h o l d s i n t h e t o p o l o g y of
Assume now t h a t
u
E
EO.
Then, using
E
294
THE COMPLETE EQUATION
and we check e a s i l y t h a t
e-WSC'(s)un converges uniformly i n
to
un
emWSCt(s)u, s o t h a t
-
u
in
20
t
EO'
W e show next t h a t
u
Em f o r each
in
l i m i t r e l a t i o n s as
E
h
-
D(3).
This i s equivalent t o t h e following f o u r
0+:
-
h'l(C(h)u for
for
u
u
u
for
E
E
E
u
E
h-lS(h)u
-
u
(4.28)
h-lC'(h)u
-
-Au
in
Eo
(4.29)
D(A) n D(B),
D(A),
D(A)
n D(B).
- u)
+
--. -Bu
in
0
(4.31)
E
To prove (14.28) we use (4.13) i n t h e form
s,
i s bounded i n norm by a constant
t h e constant described a f t e r
e W I S , m'
(4.32) as h
(4.30)
E
in
and
This expression, as a f u n c t i o n of
times
in Eo
D(A),
h-'(S1(h)u for
u ) -, 0
(4.5).
The l i m i t of
is
C"(~)U-S'(S)C"((~)U= C"(S)U + S'(S)AU = 0 after (3.11). C1(s)(h"S(h)u
To show (4.29) we w r i t e (4.10) in t h e form
- u ) = h''(St(s
+ h)u
- S t ( s ) u ) - g'(s)h''(8'(h)u - u ) - c ' ( s ) u , (4.33)
which i s bounded i n norm as well by a constant times
as
h
-
O+
eWts; i t s l i m i t
is
S"(S)U-S'(S)S"(O)-C'(S)U
=
S"(S)U + S'(S)BU
+
S(S)AU = 0
(4.34)
295
THE COMPLETE EQUATION
i n view of Corollary 3.5(d).
F i n a l l y , t h e two l i m i t r e l a t i o n s (4.30)
and (4.31) a r e obvious, since and
u
E
h-l(W'(h)u
- u) n D(E)
-D(A) 1
D
-
-
h-lC'(h)u
( s e e Corollary
3.5
%
=
=
-AC(O)u- EC'(0)u
= -Au
-Eu f o r
(c)).
q;)
Having proved (4.27), we know t h a t semigroup and t h a t , i f
C"(0)u
- BS'(0)u
S"(0)u = -A8(0)u
i s a s t r o n g l y continuous
i s i t s i n f i n i t e s i m a l generator, t h e n
(4.35)
U C B . T o improve (4.35) t o (4.20) it w i l l be s u f f i c i e n t t o prove t h a t Uh
f o r all u
E
D(9J)
s e l e c t a sequence
In f a c t , i f
{un]
(t)u d t
= kJhE
5 D(Z)
E
D(U)
(4.36)
(4.36) i s t r u e and
-
un
with
u
in
u Qm
E
(g, we may
(that
D(8)
is
dense i n ( u )h + 11
follows from (4.26) and f r o u Corollary 3.5 ( b ) ) . Then whereas u(un) h = 8 ( u n ) h = h -1(F(h)un- un) -+ h -1( S ( h ) u - u);
uhn€ D(E)
f o r any
u
E
Gh = Assume that tends t o 8 u
u
that
E
h > 0
Qm and any
and
-
h-'(5(h)u
.
u)
(4.37)
u E D(%). Taking i n t o account t h a t t h e r i g h t s i d e of (4.37) as h+O+ it follows from t h e fact t h a t i s closed
D(@
u
and
uu
= %u, which completes t h e proof of (4.20).
The i n c l u s i o n r e l a t i o n (4.36) c a n be reduced t o t h e f o u r r e l a t i o n s
(4.38)
(4.39) / g h C l ( t ) u d t = C(h)u
L h S t ( t ) u d t = S(h)u-u If
u
E
D(A)
E
-
u
D(A)
we have
E
D ( A ) fl D ( B )
n D(B)
(U E
(u
D(A)
=L
E
D(A)),
(4.40) (4.41)
D(E)).
h
AJOhC(t)u d t
so t h a t (4.38) h o l d s .
8( a + p l o g (1 + I h l ) I n Exercises ASSUMPTION
u
for all
of
E
{u
(5.8)
5 t o 11 we r e q u i r e p a r t ( a ) of Assumption 3.1, t h a t i s
5.1.
i s continuously d i f f e r e n t i a b l e i n t
S(i)u
E
5.
U s i n g Exercise 2 show t h a t t h e operator
0
Do Tl D(B); Bu
D1]
E
8(t)B
(with
h a s a bounded extension
t o all
given by = C(t)
EXERCISE 6.
Define
with
m(F)
-
Sf(t)
.
(5.9)
R(h;c~) as i n (3.19),
R( h;fn)u
Jbwe-htm(t)s(t)u
=
,
dt
a t e s t f u n c t i o n i d e n t i c d l y equal t o
(5.10)
1 near zero.
(a s l i g h t l y extended v e r s i o n o f ) (5.6) show t h a t if u such t h a t
Au, Bu
E
where
N(t;rp) = 2 v f ( t ) 3 ' ( t ) EXERCISE
Show t h a t for
7.
E
Do
Using
n D(B)
is
D1 t h e n (3.37) holds, that i s ,
+ $h;a)u
R(h;a)P(h)u = u
Define
-t
p"(t)s(t)
R(h)
as i n
+
,
(5.11)
a ' ( t ) W .
(3.41)
for
Reh
3 W,
w l a r g e enough.
u as i n Exercise 6 we have R(h)P(h)u = u
EXERCISE 8.
h(:;rp)
2
E E.
EXERCISE domain
.
For Reh
given by (3.46).
R( A). EXERCISE 9 .
Define
2
w,
W
.
l a r g e enough, d e f i n e
S(;)
Prove that t h e Laplace transform of
(5.12)
- (3.48),
by
8 equals
301
THE COMPLETE EQUATION
jta-'/r(a) ya ( c o n v o l u t i o n by
;lo
0)
(t < 0 )
Ya produces t h e " a n t i d e r i v a t i v e of order a " ) . Show
t h a t , m u l t i p l y i n g (5.12) by o b t a i n , u s i n g Exercise
for
2
(t
and i n v e r t i n g Laplace t r a n s f o r m s we
8,
as i n Exercise 6.
u
EXERCISE 10. Assume t h a t t h e s e t of d1 u
Au, Eu
E
D1 i s dense i n t h e space X
Do
E
n D(B)
= D(A)
n D(B)
such t h a t
endowed with t h e
norm
EXERCISE 11. Snow t h a t , i f
(Y1
€3 I
+ Y2
Combining (5.13) and
@ E
u
E
D1,
+ Y @A) * 3
SU = Y
€3 u
3
(t
2
(5.14)
0).
( 5 . 1 4 ) , prove t h a t , under t h e c o n d i t i o n s of
Exercise 10,
qt) = E(t). EXERCISE 12.
Under t h e c o n d i t i o n s of Exercise 1 0 , show t h a t
(3.53) ( r e s p . (3.54)) h o l d s for st(;)
formula
t h a t there exist constants
EXERCISE
u
E
E
SL(t)B).
c(t)u
Show
i s continuously d i f f e r e n t i a b l e i n
if
t 10
(4.3) h o l d s , so t h a t Ilcl(t)UII
C
(resp. f o r
such t h a t
C,W
13. Under t h e c o n d i t i o n s of Exercise 8 show t h a t ,
i s such t h a t
t h e n formula
with
(5.15)
( b u t not
w)
5
Ce
wt
may depend on
under t h e p r e s e n t hypotheses.
(t
u.
L
0)
-
(5.17)
Show t h a t Theorem 4 . 2 i s v a l i d
THE COMPLETE EQUATION
302
EXERCISE w e l l posed i n
14. 0
5t 5
E
D(A)
n D(B)
i s well posed i n
2
0
u E.
i s dense i n
t
t h a t Assumption
a (a > 0),
t h e r e and t h a t t h e s e t of all
Bu
(5.3) i s 3.1 i s s a t i s f i e d
We suppose h e r e t h a t t h e Cauchy problem f o r
f
D ( A ) I- D(E)
such t h a t
Then t h e Cauchy problem for
(5.3)
and Assumption 3.1 i s s a t i s f i e d . Note t h a t 611
t h e assumptions i n t h i s Exercise a r e s a t i s f i e d f o r t h e incomplete equation
u"(t) + Au(t) = 0
(5.18)
under t h e only assumption t h a t t h e Cauchy problem for
(5.18) i s w e l l
posed; of course, t h e r e s u l t for (5.18) can be proved i n a more elementary way by ad hoc methods. FOGTNOTES TO CHAPTER VIII
(1) We note t h e i n c o n s i s t e n c y of n o t a t i o n involved i n w r i t i n g t h e incomplete e q u a t i o n
u" + Au
=
0,
and not
u"
=
Au
as i n Chapters I1
and 111. (2)
Although t h e argument could be completed using (3.25), t h e " l e f t -
(3.41) s i m p l i f i e s some of t h e arguments. ( 3 ) We might s e t h e r e W1 = min(U,wl): f o r i f W ' < U, R(A) c a n be ana'Lytically continued t o Reh > W ' by means of Q ( h ) . ( 4 ) Convolution by Y i s employzd h e r e t o avoid using convolution of 3
handed" r e p r e s e n t a t i o n
distributions.
303
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