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Chapter 2 Evolution Operators
Chapter 2
:
Evolution Operators
Orientation. The reader mostly interested in Probabilistic Methods (Chapter 3) should read Section 1 and Sections 2.1 to 2.5, and then proceed with Chapter 3. The reader mostly interested in High Frequency wave propagation should read Sections 3.1 to 3.5 and Section 3 . 8 ,
and then proceed
with Chapter 4 . Sections 1 and 2 treat the case of parabolic operators with coefficients rapidly varying in the space and in the time variables, with different scales. The method of asymptotic expansion shows that there are three cases to be considered. Proofs of convergence are given in Section 2; Sections 2 . 6
and 2 . 7 give the proof under the
weakest possibly hypothesis on the coefficients and are somewhat complicated. Section 3 studies the operators of hyperbolic type, or of Petrowsky or of Schrodinger type. Some non linear problems are considered in Sections 2.12 and 3.7. Section 3.9 gives an example (somewhat connected with Chapter 1, Section 15)
of a situation where
local operators,of evolution admit for homogenized operator a non local evolution operator. 233
234
EVOLUTION OPERATORS Many other situations can be studied by using ideas and
techniques of Chapters 1 and 2. Some of them are briefly indicated in Section 4 (comments). Another situation will be studied by these techniques in Chapter 3, Section 5 (since the motivation for such a study appears in the Probabilistic context).
1.
Parabolic operators.
1.1
Asymptotic expansions.
Notations and orientation. Let 8 be a bounded (to fix ideas!) open set of Rn, with boundary
r
(which will be assumed smooth or
not, depending on the situation).
Let T > 0 be a given finite number.
We set
We will consider first, equations of the following type:
with initial condition (1.3)
UE(X,0)
=
u 0 (x)
,
x E 8
,
and with suitable boundary conditions, defined in Section 1.2 below. On the functions aij we shall always assume that . ! a
17
remains in a bounded set of Lm( S 1
(1.4)
aTj(x,t)cicj 2 a
~
in , ~ a.e. c ~
s ,
, a >
o ,
si E
R
.
In order to obtain constructive formulas, we shall assume much more on the structure of functions aE namely, ij;
PARABOLIC OPERATORS: (1.5)
ASYMPTOTIC EXPANSIONS
aE (x,t) = aij(x/E.t/E k ) ij
235
,
where k > 0 is for the moment unspecified, and where: aij(y,T) is Y-r0 periodic
aij(y,T)ciEj
clcici
,
, t
a.e. in y and
We want to study the behavior of uE
E
-+
T
.
0.
We begin by giving a precise definition of uE (Section 1.2) and we study next the asymptotic expansion of uE.
As we shall see the
formulas obtained depend on k (which appears in (1.5)).
Remark 1.1. We shall explain in the comments on Section 4 the results one of the form
obtains when considering functions a:j (1.7)
aij (x,t) = aij (x,x/E,~,~/E k)
and more generally aij(x,t) = aij(x,-,x x
(1.8
E
1.2
,...,t,-t
t EkF1--*) *
E2
Variational formulation. We need some notation.
We shall use, as in Chapter 1, spaces
V such that
'That
is aij(y,r) is Y periodic as a function of y and aij(y,r
+
T
0
)
=
a. .(y,r) , 17
.
a.e. in y , ~
EVOLUTION OPERATORS
236
Given a H i l b e r t s p a c e L ~ ( O , T ; 8 ) t h e s p a c e of (0,T)
-t
( o r a Banach space!) w e d e n o t e by
(classes o f ) functions t
-t
v ( t ) from
which a r e measurable and which s a t i s f y
8
= ess.supl l u ( t )
II
8
Provided w i t h t h e norm g i v e n i n ( l . l O ) ,
<
m
,
Lp(0,T;8
if p = )
m
.
is a
Banach s p a c e (and it i s a H i l b e r t s p a c e i f p = 2 and i f M i s a Hilbert space). W e s h a l l set
(1.11)
2 H = L ( 0 )
,
V C H C V '
and w e o b s e r v e t h a t
1 For u , v E H ( C ? ) , w e d e f i n e : (1.12)
a'(u,v)
=
au av dx (x,t) ax. 3 axi
.
W e have t o be c a r e f u l s i n c e a E( u , v ) depends on t and s i n c e , even
i f t h e f u n c t i o n s a . 's a r e smooth ( f o r e v e r y E ) t h e t - d e r i v a t i v e s of lj
t o t h e s t r u c t u r e of a . given i n (1.5) 1j t h a t , i n what f o l l o w s , t h e a p r i o r i e s t i m a t e s s h o u l d n e v e r u s e t h e a i j ' s are unbounded-due
d a E( u , v ) . dt For u
E
V w e d e f i n e A'u
E
V'
by
PARABOLIC OPERATORS:
I n f a c t A'
ASYMPTOTIC EXPANSIONS
231
depends on t :
(1.14)
AE = A E ( t )
.
T h e " a b s t r a c t " f o r m u l a t i o n of t h e p r o b l e m w e w a n t t o s t u d y i s
l e t f a n d u o be g i v e n , s a t i s f y i n g
now: (1.15)
f E L2(0,T;V')
(1.16)
uo
Then
E
H
,
.
u E i s t h e s o l u t i o n of 2 L (0,T:V)
(1.17)
u
(1.19)
UE(0) = u
E
0
,
.
Remark 1.2. 2 I f v E L (0,T;V)
a bounded s e t of L m ( S
then AEv )
E
2 L (0,T;V')
and s i n c e a .
11
remains in
,
w h e r e here and i n w h a t f o l l o w s , t h e c ' s d e n o t e v a r i o u s c o n s t a n t s w h i c h do n o t depend on
E.
T h e r e f o r e (1.18) gives:
238
EVOLUTION OPERATORS
I c[ I
(1.22)
IUE
II
+ 11
.
L2 (0,T;V) But it is known (cf. for instance Lions-Magenes [l]) that: (1.23)
if v
6
2 L (0,T;V) and
2
L (O,T;V'), then v
is a.e. equal to a function, still denoted by v, such that t
+
v(t) is continuous from [O,Tl
+
H.
Therefore we see that (1.17), (1.21) imply that (1.19) makes sense.
Remark 1.3. It is proven in Lions [l] that the problem (1.171, (1.18), (1.19) admits a unique solution. One has the estimate
The verification of ( 1 . 2 4 ) is easy. (1.18) by uE.
We multiply both sides of
We denote by (f,g) the scalar product in H and we set
But we observe thatt
'This @ E
2
is obvious if
L (0,T;V) and
2
E
@
is smooth and it can be justified if
L'(0,T;V').
PARABOLIC OPERATORS:
2 39
ASYMPTOTIC EXPANSIONS
so t h a t (1.26) g i v e s ( w e w r i t e a'($)
for
Hence t
b But d u e t o (1.4), w e c a n f i n d A t (1.28)
aE
V)
2 a1
W e d e n o t e by
1\71
l2 -
I 1 I I*
A v/
2
such t h a t
,
v
t h e norm i n V'
.
E
V
.
W e o b t a i n from ( .27) 2
+;
/
IluEl
2do
'
0
1 +One c a n t a k e X = 0 i f V = H O ( 0 ) or i f V = s p a c e w h i c h d o e s not contain the constants.
240
EVOLUTION OPERATORS
Using G r o n w a l l ' s lemma,
(1.24)
follows.
Remark 1 . 4 . One c a n r e p l a c e ( 1 . 1 8 ) by t h e e q u i v a l e n t f o r m u l a :
1
V = Ho(B').
Example 1.1:
Then ( 1 . 1 7 ) i m p l i e s (1.31)
u
= 0 on C
.
I t i s t h e D i r i c h l e t ' s boundary v a l u e problem.
Example 1 . 2 :
V
1 = H (8).
Then one deduces from (1.30) t h a t , i n a "weak" s e n s e , (1.32)
au
2 = 0 on C
where
a
-=
av
(1.33)
a
a?.(x,t)v. 11 7 axi
,
AE
v = { v l , . . . , ~ n } = normal t o
r
d i r e c t e d toward t h e e x t e r i o r
PARABOLIC OPERATORS:
ASYMPTOTIC EXPANSIONS
241
I t i s t h e Neumann's b o u n d a r y v a l u e p r o b l e m .
Remark 1 . 5 . ( 1 . 2 4 ) w e see t h a t w e c a n e x t r a c t a s u b s e q u e n c e ,
By v i r t u e o f
s t i l l d e n o t e d by u E , s u c h t h a t (1.34)
u
2 u i n L (0,T;V) w e a k l y
-+
au
$ is
(and ( 1 . 2 4 ) ) ,
Using ( 1 . 2 2
.
2 bounded i n L ( 0 , T ; V ' ) and t h e r e f o r e
w e c a n assume t h a t +
I t i s known t h a t ,
.
i n L2 ( 0 , T ; V ' ) weakly
when
V
-+
H i s compact, it f o l l o w s from ( 1 . 3 4 ) ,
(1.35) t h a t (1.36)
uE
+
.
2 u i n L (0,T;H) s t r o n g l y
I f we set
w e c a n a l s o assume t h a t (1.38)
6;
+
ci
2 i n L (0,T;H) w e a k l y
.
One v e r i f i e s t h a t , i f a ? . i s g i v e n by ( 1 . 5 ) , w i t h ( 1 . 6 ) , t h e n 13 (1.39)
a!.
13
+
where i n g e n e r a l
74 ( a i j )
&
L m ( S ) weak s t a r
,
EVOLUTION OPERATORS
242
topology).
k).
ci
m
( a1. 3. ) &.a x . ( i n any 3 W e s h a l l o b t a i n below t h e l i m i t of ST (which depends on
But i t d o e s n o t f o l l o w t h a t
-+
The o n l y t h i n g w e c a n s a y f o r t h e t i m e b e i n g i s t h a t
(1.41)
at -
Remark 1 . 6 .
ac. axi
.
= f
The case of c o e f f i c i e n t s a:. i n d e p e n d e n t of t . 13
L e t u s assume t h a t
(1.42)
,
a ? . = a . . (x/E) 13 =I
where t h e a i j ' s
a r e Y-periodic
( i . e . w i t h t h e hypotheses of
Chapter 1 ) . Then t h e answer i s immediate.
Let
g e n i z e d o p e r a t o r of A E , c o n s t r u c t e d i n Chapter 1.
(1.44)
% at +
Q
be t h e homo-
=
,
4 u = f
d e f i n e s t h e l i m i t of u E . m
I n d e e d , l e t $ be g i v e n in?,,
(10,T[). For any f u n c t i o n
v E L ~ ( O , T ; li ) , w e d e f i n e
W e remark t h a t (1.47)
au
a t (0) =
-u
E
d0 (-)
dt
s o t h a t w e deduce from ( 1 . 1 8 ) t h a t
PARABOLIC OPERATORS:
But since
AE
ASYMPTOTIC EXPANSIONS
does not depend on t, (A'u,)
(@) =
AE (uE( $ ) ) and
therefore
(we use (1.36)).
We also know that
(1.50)
u($) in V weakly
uE($)
+
.
hence (1.44) and the result follow.
1.3
Asymptotic expansions.
Preliminary formulas.
We shall consider the three most interesting cases for k appearing in (1.7). P E r k=
In order to avoid confusion, we shall set:
a +
A
(x)
E t E
(1.51) E
and we shall consider the values (1.52)
k = 1,2,3
We shall s e t
'I
. = t/Ek and
243
244
EVOLUTION OPERATORS
.5
all these operators depend on (1.54)
PErl =
Q1+E
E - ~
It follows that
parametrically.
T
-1 0 Q2+EQ3r
where (1.55)
Q1 = A1
(1.56)
PCr2 =
,
Q2
=
-
R1 ~ +
E
E
a?
-lR2 +
E
a+ A 3 Q 3 = -at
,
a + A2 0
;
R3 ,
where
a +
A1
,
,
(1.57)
R1 =
R2 = A2
(1.58)
P E r 3 = E - ~ +S E~-2 s2 + €-Is3
R3 =
+
E
0
a +
A3 ;
s4 ,
where
a ,
S1 = aT
(1.59)
S2 =
A1
,
S3 =
,
A2
S4 =
a +
.
A3
We use now these formulas to find an asymptotic expansion of uE.
We distinguish the three cases k = 1,2,3. ' We look for uE in
the form (1.60)
u
€
=
u
0
+
EU1
+
E2U2
+
. ..
,
U. =
u . being Y periodic in y and 3
'Another
3
T~
U.(X,Y,t,T)
,
periodic in
T
3
.
ansatz will be necessary when k is not an integer.
PARABOLIC OPERATORS: 1.4
Asymptotic expansions:
245
ASYMPTOTIC EXPANSIONS
The case k = 1.
By identifying powers of
in the expansion of PEV1u = f , we
E
obtain:
,
(1.61)
QluO = 0
(1.62)
QlUl + Q2UO = 0
(1.63)
Q1u2
t
+ Q2u1 + Q3uo
=
f ,
... .
In (1.61), x , ~ , Tare parameters, so that (1.61) is equivalent to (1.64)
uo does not depend on y
We denote by tively,
T ) so
m
Y
.
(respectively,
that ??I
=
h
T O
the average in y (respec-
T)
Y
=
.
By taking
n
of
(1.62), we obtain
i.e., since we have (1.641, (1.65)
uo = u(x,t)
auO
=
0 and therefore
.
Then (1.62) reduces to: (1.66)
Alul = -A2u =
au a , .( y , ~ )+ aYi a11 ax j
(x,t)
.
We can write the general solution of (1.66). We introduce
XJ
=
XI ( y , ~ )as the Y-periodic solution, defined up to an additive
constant, of
and we choose the constants such that (1.68)
xJ(Y,T)
+ T~
periodic
;
246
EVOLUTION OPERATORS
(for instance we can take ?!4
Y
( ~ J ( Y , T )=) 0, v
T ) .
Then (1.66) gives (1.69)
.
ax
au u1 = - ~ J ( Y , T ) (x,t) + Gl(x,t,~) j
We can solve (1.63) for u2 iff,
?4 y(Q2u1 + Q 3 u )
= f
,
i.e., (1.70)
aGl
ar +
?4 Y Q 2 (-XI &) 3
74 y(Q3u)
+
We can solve (1.70) forG1 (with a
T~
=
f
.
periodic function of
T)
i.e.,
We observe that
so that we obtain
(1.71)
au
-
9 [aij - aik
$1
2
axiax. a u = f . 3
This is the homogenized operator (as we shall justify below).
1.5
Asymptotic expansions:
The case k = 2.
We use now (1.56), (1.57).
,
(1.72)
RluO = 0
(1.73)
Rlul + R2uO = 0
,
We obtain
iff
PARABOLIC OPERATORS: (1.74)
R1u2
+ R2ul + R3u0
=
f
ASYMPTOTIC EXPANSIONS
247
.
We multiply (1.72) by uo and we integrate over Y.
Let us set
We obtain
Since uo is
au, aYi
T~
periodic,
'0
0
= 0,
(1.77)
i.
au [g,u0],
d-r = 0, so that (1.76) implies
- + AIUO
auO Then AluO = 0 and a?
:yo
= - - 0,
so that
.
uo = u(x,t)
Then (1.73) reduces to
We introduce 8j
( y , ~ ),
Y - T ~periodic
, solution of
(1.78)
(0'
is defined up to an additive constant).
Then
We can solve (1.74) for u2 ifft
tR1@ = F admits a Y-T,, periodic solution iff m ( F ) = 0
(observe that RZji = 0 admits the constants as only Y - T ~per odic solution).
248
EVOLUTION OPERATORS (R2u1
+ R3uO)
,
= f
i.e., au (1.80)
-
n [ a1.3. -
aej] a ik
ayk
_ .
~
a 2u ax.ax. 1
3
This is the homogenized operator of below).
in
(1.71)
P E P 2 (as
for P E r l ,
but coefficients are different.
The case k = 3 .
We now use (1.58), (1.59).
We obtain:
,
(1.81)
S u = 0 1 0
(1.82)
Slul + S2uo = 0 ,
(1.83)
S1U2 + S2U1 + S3Uo
(1.84)
S1u3 + S2U2 + S3U1 + S4Uo
=
0
t
= f
,
... .
It follows from (1.81) that uo does not depend on a
T
0
we shall justify
It has of course the same structure as the one obtained
Asymptotic expansions:
1.6
.
T:
(1.82) admits
-periodic solution in u1 iff
I
Ts2uo
=
0
,
i.e.,
We define (1.86)
6,
=
a
It is of course.an elliptic operator in Y, so that (1.85) is au equivalent to 2 = 0, V j and therefore uo = u(x,t)'. Then (1.82) aYj
PARABOLIC OPERATORS:
ASYMPTOTIC EXPANSIONS
reduces to __ a-r = 0, i.e. (1.87)
u1 does not depend on
T
.
We can solve (1.83) for u2 iff RT(S2U1 + S3Uo)
=
0
$ 1 , ~ )is Y-periodic
,
,
i.e.
If we introduce @ J ( y )by:
,
(1.88) J
- y . )
= o ,
(which defines tJj up to an additive constant), we see that (1.89)
u1
'
= -@'(y)
au ax. +
u,(x,~)
3
.
Then (1.83) can be solved for u2; if 7
we have (1.91)
0 u2 = u2 + u2(x,y,t)
.
Equation (1.84) can be solved for U3 iff (1.92)
R T ( s2u 2 + s3u1 + s4u 0)
and (1.92) can be solved for (1.93)
R
1.r r(S2U2
+
G2
s3u1
+
= f
iff S4U0) = f
.
249
250
But
EVOLUTION OPERATORS
m'Y
r
(S2u2) =
RT m
Y (Szu2) = 0, so that (1.93) reduces to
We have again the same structure for the homogenized operator, with again different formulas.
1.7
Other form of homogenization formulas. Let us denote by q k. . = homogenized coefficient of 17
k = 1,2,3.
Let us check the following formulas:
where al is defined in (1.75) and where
For (1.961, we will actually prove that
This is sufficient since
a2 u
(-----ax.ax.
1
7
I
for
PARABOLIC OPERATORS:
251
ASYMPTOTIC EXPANSIONS
taking into account that
= o . Therefore, by comparing the expressions (1.71), (1.80), (1.94) with (1.95), (1.96)I , (1.97), we have to verify that
(1.101)
;+$’ - Yj&)
= 0
This is straightforward.
al(xj
-
.
Indeed (1.67) implies that
yj,x i) =
o
for every
T
,
hence (1.99) follows. If we multiply (1.78) by 0
i
and integrate in y and in T, we
obtain (1.100) and (1.101) follows from (1.88). Formulas (1.95), (1.96), (1.97) imply the ellipticity I Of k a’ f f k = -qij axiax j
(1.1021
~
-
Indeed, if we set:
+ we have
=
cj(xj
-
yj)
,
J, =
cj(ej
-
yj)
,
P =
c j @j
252
EVOLUTION OPERATORS
Hence the result follows.
1.8
The role of k. Let us consider now k > 0 not necessarily an integer. We set
(1.106)
ui = solution of (1.171, (1.181, (1.19) relative to AE =
A(t/E
k)
,
and we denote by uk , k = 1,2,3, the solution of (1.107)
*+ k at
,
qkuk = f
(1.108)
Uk E
(1.109)
uk (0) = u0
L2(0,T;V)
,
.
Then we shall see that the limit of uz when k < 2 (respectively, k > 2 ) does not depend on k, i.e. that :u
-f
u1 (in the appropriate
topology) when k < 2 (respectively, -+ u* when k
> 2).
But the
asymptotic expansion depends on k, and actually new "ansatz" will be necessary for the cases when k is not an integer.
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS 2. 2.1
253
Convergence of the homogenization of parabolic equations. Statement of the homogenization result. We shall prove in several steps the
Theorem 2.1. We denote by :u k solution of (1.17), (1.18), (1.19) relative to AE = A(t/E ) We assume that (1.5), (1.6) hold true.
the solution of (1.107), (1.108), (1.1091, k = 1,2,3, where (respectively, 0 ’, respectively,
a
3
)
the k u
a1
is given by (1.71)
(respectively, (1.80), respectively, (1.94)). Also assume that there exists p > 2 such that f E Lp(0,T;W-1rp(C3) ) and that uo or at least that f E
-+
and
E
uo belong locally to these spaces.+
WitP(b 1 , Then, as
0, one has
(2.1)
:u
(2.2)
u :
(2.3)
:u
+
-+
-+
u1
in L2 (0,T;V) weakly if k < 2
,
u2
2 in L (0,T;V) weakly if k = 2
,
u3
2 in L (0,T;V) weakly if k > 2
.
We begin by the case (2.2) which is the simplest; we shall next proceed with (2.1) and (2.3). 2.2
Proof of the homogenization when k = 2. Given P(y) = homogeneous polynomial in y of first degree, we
define w(y,.r) as the solution (defined up to an additive constant) of
t1.e. if ~3
1
c
5
I
c
0 , f E LP(O,T;W-~’P(O I ) ) ,
This hypothesis is not necessary if k = 2 or if k # 1 (cf. (2.22) below).
uo E W’~P(S aij E C
0
1 ) .
254
EVOLUTION OPERATORS
- -:y +
(2.4)
,
AIw = 0
-
w
P i s Y-r0 p e r i o d i c
.
We write:
(2.5)
AE(t/E
2
) = AE
,
u2 = u E
E
.
I f w e set
*
(2.6)
w - P = - ~
then
* * * - -aare + A1(B -
(2.7)
,
P) = 0
b e i n g Y-ro p e r i o d i c
O*
.
W e now d e f i n e wE by
w (x,t) =
(2.8)
2
EW(X/E,~/E
.
)
W e observe t h a t
- -aa wt E + AE*wE =
(2.9)
For any (0 E C ? z ( S ) , and ( 2 . 9 ) by $ u E .
s
.
0 =
8
x
] O , T [ , w e m u l t i p l y (1.18) by $wE
W e i n t e g r a t e over
s
;
w e o b t a i n (compare w i t h
C h a p t e r 1, S e c t i o n 2 ) :
+
[[>]@w,
(2.10)
s +
Is
[c:
[?]$u,]dxdt
wE
-
aij
2$
uE]dxdt =
j
f$wEdxdt
.
0
The f i r s t term i n ( 2 . 1 0 ) e q u a l s (2.11)
Since w
+.
P in L
2
( s ) s t r o n g l y and
t o t h e 1 m i t i n a l l terms of
s i n c e w e have (1.36
(2.10), ( 2 . 1 1 ) .
W e obtain
I
w e c a n pass
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
(2.12)
I
-
UP
Q =
i
a+
dxdt
‘
+
s f$Pdxdt
K )* axj u l d x d t
( a i j aw
.
n
Q
Using (1.41) w e see t h a t
J s
dxdt
+
s
sJ
5,
so t h a t (2.12) r e d u c e s t o
Therefore
(2.13)
k)au.
6. ap = 1
311 ( a k j ayk
ax,
ax j
L e t us take
W e define 8
(2.14)
i* by
- -aei* ar
+
A;(ei*
ei* b e i n g Y - r o Then w = -(Bi*
or
-
-
yi) = 0
periodic
,
.
yi) and (2.13) g i v e s :
a ax. l
($P)dxdt
8
255
256
EVOLUTION OPERATORS
(2.15)
The variational formulation of the problem is:
[>,v]
+
[ti&]
= (f,v)
,
v v
E
v ,
,
\J
v
E
v
and passing to the limit we find (2.16)
au (=,V)
av + (5.,-) 1 axi
= (f,v)
This proves the Theorem (for k
=
.
2) if we verify that we obtain the
same formula than in (1.80), i.e. that
This is equivalent to proving that
or that
We multiply (2.14) by 8 1 ,
(1.78) by Bi*.
This shows that the
left hand side of (2.19) equals
and that the right hand side of (2.19) equals
and the two expressions ( 2 . 2 0 ) ,
(2.21)
are indeed equal.
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
257
Orientation. We consider now the cases when k # 2.
We begin with the study
of this question under the extra hypothesis (2.22)
2.3
,
a . E Co(F x [ O , - r 0 1 ) ij
.
periodic
Reduction to the smooth case. We are going to show
Lemma 2.1. Under the supplementary hypothesis (2.22), it suffices to prove Theorem 2.1 under the hypothesis (2.23)
aij
E
C"(Y
,
x [O,.rO1)
periodic together with all its derivatives
.
Proof.
of (1.30), k fixed arbitrarily. [ O , T ~ ] ) by j!a E C"(Y x [ O , T ~ ~ ) aij ,B
We denote by uE the solution !u
We approximate aij in C"(Y x being periodic together with all its derivatives and satisfying (2.24)
agjcitj 5 a l ~ i t iI
al > 0
,
y,T
b!
.
B k We denote by uE (not to be confused with uE) the solution in
L'(o,T;v) (2.25) where
of aUE
IK,v]
+ aEB(u!,v)
=
(f,v)
,
tl
v
E
v
EVOLUTION OPERATORS
250
We shall check below that
B
(2.27)
HUE
-
UEII
L2 (O',T;V)
< -
c
B supllaij i,j
-
a
iJ
11
co(iX
Let us consider now the case k = 1 to fix ideas.
[o,Tol)
.
We define
x j r B by A1(~JfB y.) = 0 7
where A!
=
- a aYi
(aBj(y,T)
"1,
,
aYj
xJtB
Y - T ~
One .,bows easily that (normalizing x
n Y (XI)
= 0,
nY
,
and we define
-
for instance
periodic
=
o
yi)d.r
.
J , ~ 1 in ' the ~ same manner,
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
259
It follows from (2.27), (2.28), (2.29) that it is enough to 2
prove that u l P B + u l t B in L (0,T;V) weakly when E
E
+
0, 6
fixed.
The same remarks apply to all values of k, so that it remains only to prove (2.27), which is standard. m
E
=u'-u E
Indeed if we set
E
we have
where
so that
and therefore
I IfE Thus (2.27) fol ows from (1.24).
2.4
Proof of the homogenization when 0 < k < 2 . We now prove (2.1) under the assumption (2.22), by Lemma 2.1.
We can therefore assume that ( 2 . 2 3 ) holds true.
260
EVOLUTION OPERATORS
2.4.1
The case 0 < k
5 1.
Let P(y) be given as in Section 2.2.
Let w be the solution
(defined up to an additive constant) of (2.30)
*
A1w = 0
,
w
-
P(y)
is Y - T ~periodic
k We use notation (2.5) and we set uE
-
w
=
uE.
.
If we set
p = -x*
then (2.31)
*
A1(X*
-
P)
=
0
,
x* being Y - T ~periodic
.
We now define w E by
.
x -1t w (x,t) = w(-
E 'Ek
We observe that AE*wE = 0
.
The same procedure as for (2.10) leads to
f@wEdxdt
=
.
n
l4
The first term in (2.32) equals (2.33)
-
I
s
uEwE
2 dxdt -
aw
s
Since we have assumed the coefficients aij to be smooth, it follows from (2.30) that
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
261
so that we have in particular
11
uE@
at awE
s
awE If k = 1, =
aT
dxdtl 5 CE'-~
[ElEand aw
+
0
,
aw
if k < 1
(=)=
+
.
0 in Lm(
s)
weak star
so that
I
uE@ at dxdt awE
+
0
s
.
In all cases, the limit of (2.33) is
-
UP
s SO
2 dxdt ,
that (2.41) gives in the limit
-
I
UP
s
2 dxdt + I f@Pdxdt
=
.
[ci
P
-
s
s This is the same formula as (2.12), with a different w. i* by obtain (2.13) and if we define x (2.341 then
A:(xi*
-
y.)
=
0
,
xi* being Y-periodic
,
We
262
EVOLUTION OPERATORS
(2.35)
ci
=
7/1 [aij
-
a
The Theorem is proven, provided we verify that (2.35) furnishes the same formula as in (1.71), i.e. that
The verification is made along similar lines as for (2.17).
The case 1 < k < 3/2.
2.4.2
We prove now the result for 1 < k < 3/2 and we shall indicate next how to proceed when k
+
2.
We begin by general considerations on the asymptotic expansions. With the notation (1.51) we have (2.36)
P E r k = €-'A1
+
E - ~ A+ ~E O ( A ~+ =) a +
E -k
a
We are looking for an asymptotic expansion of uE where PErkuE= f. The usual expansion using terms
E ~ U (x . ,y ,t,T)
7
is impossible
with-
out additional terms, since there is otherwise no term to "compensate" - ~ would appear.t the powers ~ j which Therefore it is natural to look for u UE =
uo + EU1
+
E2u2
+
... +
where all €unctions depend on x, y, t, T ~ .
in the form:
P v 0 T
+ E3-kvl +
...
,
and are periodic in y and in
But if we use the above "ansatz" for uE, we do not have T deriva-
tives coming in the identification for uo, ul, u2 and one easily
tExcept if functions u fication is impossible.
j
do not depend on
T,
but then the identi-
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
263
checks that the computations are impossible. Therefore one should add k to the above "ansatz",terms of the type E wo + ~ ~ + + so ~ that w
...,
finally we look for uE in the form UE
=
(2.37)
+
uo
+
Ell1
€kWO
+
+ E 2u2 +
... +
+
... .
Ek+1Wl
Identifying the coefficients of
0
+
E-',
E - ~ ,
E3-kvl +
E'
...
gives
,
(2.38)
AluO = 0
(2.39)
Alul + A2u0 = 0
(2.40)
A1U2
+
€2-kv
A2U1
+
,
A3u0
au, + aw, = +at a-r
Identifying next the coefficients of
E
f
.
-k
,
E
1-k
I
E
-k-2
, E -k-1,
we obtain
(2.43)
AlwO = 0
(2.44)
Alwl + A2w0 = 0
I
.
It follows from (2.38) that uo does not depend on y; then, by integrating (2.41) in y, we obtain
au,/aT
= 0 and therefore
Then (2.41) reduces to AlvO = 0 and v 0 does not depend on y. If we use the functions x j ( y , ~ )satisfying (1.67) , (1.681, we have from (2.39)
~
264
EVOLUTION OPERATORS
I n o r d e r t o be a b l e t o s o l v e ( 2 . 4 2 ) f o r v1 w e need
Y
[?+
A2vO]dy = 0
.
But s i n c e v o d o e s n o t depend on y ,
J
AZvOdy = 0 , so t h a t t h e o n l y way
Y t o s a t i s f y t h i s c o n d i t i o n i s t o choose x j s u c h t h a t
I
XJ(y,T)dy
f o r every T
0
=
,
Y
and G1 i n d e p e n d e n t o f T. Then we can compute v1 no m a t t e r how w e choose v o .
Therefore w e
choose vo = 0 and v1 i s d e f i n e d (up t o t h e a d d i t i o n of a f u n c t i o n of x , t , T) by
AIVl
-
a
aT
j
au ax = 0 . j
W e see from (2.43) t h a t wo i s i n d e p e n d e n t o f y and t h e n w1
can
be computed by ( 2 . 4 4 ) . I t only remains t o s a t i s f y t o (2.40)
(observe t h a t t h i s i d e n t i t y
i s i m p o s s i b l e t o s a t i s f y i f t h e r e i s no w,,!).
By i n t e g r a t i o n
and i n T w e f i n d t h a t u s a t i s f i e s t o ( 1 . 7 1 ) . "
Summing u p :
&
y
with t h e
a n sa t z " u
(2.45)
= U + E U
+
Ek+1Wl
+ +
EZU2
...
+
... +
E3-kvl
+
... +
E kwo
,
where t h e c o e f f i c i e n t s are computed as above, one h a s
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
PErkuE= f
+
terms of order 2 k
-
1 in
E
265
.
Remark 2.1. 1) We have introduced the above ansatz on a rather "algebraic" basis: another motivation will be obtained in Chapter 3 (probabilistic methods). 2)
One sees from the above computation that the limit of uE does
not depend on k (when 1 < k < 3/2)+ but that the expansion itself depends on k. 3)
For k satisfying
the "ansatz" for the asymptotic expansion will be: = u + E U 1 + E 2u 2 + E 2p+l-pkvo
u
+
... +
EkWO +
... .
We can now proceed with the proof of the Theorem. We construct a function w E such that (2.46)
(-
a
at
+
AE)*w
=
E
k- 1 gE '
2 where g E is bounded in L ( Q ) as 2
L (Y
x
( 0 , ~ ~ )as )
E
-+
E
-+
0 and where wE
+ .
yi in
0.
One looks for wE in the formtt (2.47)
wC = E(CX(Y,T)- yi
+
EZB(Y,T)
+
E3-kA(y,T)
.
tOf course once the above (formal) proof is justified. ttWe do not need in this expansion terms in E~ or in
E
k
.
266
EVOLUTION OPERATORS
One finds: (2.48)
A;(a
-
- -a~ + ar
One defines xi* by (2.34) take
ci =
x i*
*
,
yi) = 0
*
AIB + A2(" A;A
= 0
-
yi) = 0
,
.
j Xi*(y,T)dy
V T.
= 0,
Then one can
Y
and one can compute X from the third equation (2.48).
One computes 6 using the second equation (2.48) and one obtains the result.
2.5
The proof is then completed along the usual lines.
Proof of the homogenization when k > 2.
(Preliminary) Remark 2.2. 1) We are going to prove the theorem only for k = 3.
The proof
for the general case proceeds along the lines of Section 2.4.2. 2) We assume again that (2.22) holds true.
See Section 2.6 for
the general case. According to Lemma 2.1 we assume (2.23). We are going to construct a function wE satisfying
such that as
E -+
0
(2.50)
g,
is bounded in L2 ( S
(2.51)
wE
+
)
I
2 yi = P(y) in L ( S
We use asymptotic expansions
Y = c1
being Y-periodic and B and
T
X/E
being
I
Y-To
periodic
.
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS Using the notations ( 1 . 5 8 1 ,
Since a does not depend on
+
-
(2.54)
S*B
(2.55)
s y + s 2 5 + s
S;(CY
*
Then we find ( 2 . 4 9 )
the coefficient of
E - ~
is zero.
* 3
I
(a-yi) = o .
(assuming we can solve ( 2 . 5 4 ) ,
+
g, = h
(2.56)
Yi) = 0
*
1
T,
we have
and of EO are zero iff
The coefficients of 1
(1.59)
267
EL
*
+
with
,
E2m
*
(2.55))
*
h = S2 y + S3 B + S 4 ( c Y - Y ~ )
L
=
s*y 3
+
*
* S4B
m = S y . 4
I
*
We can solve ( 2 . 5 4 ) for B iff (S1 =
a : - E)
(2.57)
*
*
But S2 = A1 and since a does not depend on
(a
74 A
-
-*
A1(a
We denote by
-
is equivalent to
y.) = 0
, this means
.
(compare to ( 1 . 8 8 ) ) the solution which is Y-periodic
(defined up to an additive constant) of ( 2 . 5 8 ) . (2.59)
(2.57)
yi) = 0
Using the notation ( 1 . 8 6 ) (2.58)
T,
i* a = @
.
Now we can solve ( 2 . 5 4 ) ; if we set
Then
EVOLUTION OPERATORS
268
then (2.61)
B
.
~
=
Bo + B(x,y,t)
We can solve (2.55) for y iff
i.e., (2.62)
i *1g +
mT(S;Bo
+
Sl(a
-
yi)) = 0
,
V
iY-periodic ,
which defines (up to an additive constant) the function All the functions introduced are Cm Then we can solve (2.55) for y.
& I
or in ?
Functions
a,B,y
X
i(y). [o,To]. m
are C
so that
in particular
and
We now use (2.49).
We take $
E
C?z(s )
and, as before, we
multiply the equation for uE by $wE and (2.49) by @uE.
=
\
s
f$wEdxdt
-
E
i s
gE$uEdxdt
We can pass to the limit and we find
.
We find
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
-
I
dxdt +
uw
I
Ci
269
Pdxdt
s
s
and finally (2.64)
Si
=
[
74 aij
-
akj
a$i*
au
T]
It remains only to verify that this formula coincides with what we have obtained in (1.94), which is left to the reader
2.6
Proof of the homogenization formulas when a E Lm ij using LP estimates. for
We show now that the assumption (2.22) Theorem 2.1 to be true.
The hypothesis aij E Lm(Rn x R ) is Y T sufficient. We assume f o r a moment the following estimate: (2.65)
for the Dirichlet's boundary conditions there exists p > 2 ,
independent of
E
,
such that
The proof of this result is given in Section 2.7 below. We introduce q such that
We can find a sequence of functions aB such that ij
2 70
EVOLUTION OPERATORS
(2.67)
B a . .E C " ( Y
x
13
[O,rO1)
,
B is periodic together with aij
all its derivatives , sup
l r 7
B Ilaij
-
aij
I I Lq(y
[o,rol)
= p(B)
*
0 as B
and
* "
.
With the notations of Section 2.3 we show that
Then Lemma 2.1 is still valid and the Theorem is proven. With the notations of the end of Section 2 . 3 we observe that
I (fE,V)I
5
CP(B
so that (2
By virtue of ( 2 65) it follows that (2.70)
I Ifc
so that in part cular
and (2.68) follows. For other boundary conditions than Dirich-st, the analogous of (2.65) is valid locally, i.e. for
S'
C
-
8 ' x lO,T[, 0 ' C
and the analogous of (2.68) is valid locally, i.e. in
8
,
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS 2.7
271
The Lp estimates. We now prove the Lp estimates used in Section 2 . 6 above.
are of independent interest.
They
The method of proof is analogous to the
one used in Chapter 1, Section 4, for elliptic problems.
The
notations are those of this Section 4, Chapter 1; in particular, we use norms ( 4 . 5 ) ,
(4.7),
(4.81,
(4.11)
of that section. We define
provided with the norm
We introduce
and we observe that the mapping v Lp(O,T;W-l'p(O (2.74)
)).
Ilfl l y
p
divx v maps (Lp( (j
-+
We then provide Y
=
inf divx g = f
P
) )
onto
with the norm
I 191 I (LP((j ))n
We consider a family of functions a ij such that
We denote by [a. (x,t)] the n 11 and we set
x
n matrix with entrices aij(x,t)
EVOLUTION OPERATORS
272
We set
Theorem 2 . 2 .
Let f and u o be given such that f uo
H = L
E
2
( 0 ) .
u
(2.80)
u(0) = u0
Then assuming only on a , %
L2 (O,T;H-'(S))
and
Let u be the solution of
(2.79)
E
E
L 2 (0,T;H0(8)) 1
I
.
r to be smooth enough there exists 8 , such that if
p > 2,
depending
(2.82)
then
u
when f
E
Lp(O,T;WkrP(O )
and remains in a bounded set of this space
)
& uo remain bounded in LP(0,T;W-1rP(8)) and in WirP(O).
Remark 2 . 3 . We can reduce the problem to the case when uo = 0. can find I$ E Lp(O,T;W;'p(O
)),
$
E
Lp(0,T;Wi'p(8
)),
@
Indeed we depending
continously on uo in the corresponding topology, so that @(O) = u 0 and then we consider u
'This
-
@
instead of u.
hypothesis can be very much improved.
CONVERGENCE O F THE HOMOGENIZATION OF PARABOLIC EQUATIONS
273
P r o o f of T h e o r e m 2 . 2 .
of C h a p t e r 1, w e c a n r e d u c e t h e p r o b l e m t o
A s i n S e c t i o n 4.2
where
&i [ a l j ( x , t )
-
(2.84)
Ak =
(2.87)
v < p .
,
k = 1,2
,
By c h a n g i n g t h e scale o f t i m e , w e see t h a t w e a r e r e d u c e d t o proving t h e theorem f o r
(2.88)
a t + A 1u + u = 0 on C
and w i t h (2.841,.
A u = f
2
r
=
x
,
u(0) = 0
]O,T[
,
f s a t i s f y i n g (2.81)
,
. .,( 2 . 8 7 ) .
W e set
(2.89)
a
P = - - A .
at
I t i s known t h a t , g i v e n F
E
Yp,
( w h e r e p i s a r b i t r a r y f o r the
t i m e being), t h e r e e x i s t s a unique u such t h a t P u = F i n (2.90)
U E X
P '
q , u(0) = 0
.
,
274
EVOLUTION OPERATORS Equation (2.88) is equivalent to
(2.91)
Pu
f
(A1 + A ) u - + A2u = f
,
and if the solution u of (2.90) is denoted by (2.92)
u = P-1F
then (2.91) is equivalent to (2.93)
u
+
(P-l(A 1
+
A ) + P-lA2)u = P-’f
.
We introduce
and the Theorem will be proven if we verify that (2.95)
one can find p > 2 such that k ( p ) < 1
.
We have
But exactly as in Chapter 1, Section 4,3 (t plays the role of a parameter) we have
so that
(2.97)
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS It is straightforward to verify that that, since 1 (2.98)
-
p
+
1 IP-ll 1
v < 1, we shall have ( 2 . 9 5 )
2 ( Y 2 ;x2)
275
= 1 so
if we can prove that
there exists a function p ( p ) which is continuous
and p ( 2 ) = 1
.
Indeed we have then k(p) < p ( p ) and we can find p0 > 2 such that k(po) < 1.
Proof of ( 2 . 9 8 ) . We take F = div g, g
E
(Lp(
in ( 2 . 9 0 ) and we consider the
mapping (2.99)
TT
g
+
gradx u
,
Using Riesz-Thorin's theorem it follows that
Therefore, according to ( 2 . 7 2 ) and to ( 2 . 7 4 ) , we have
276
EVOLUTION OPERATORS
so that we can take p(p) = w:-'(~)
and the result follows.
Remark 2.4. We have, for arbitrary boundary conditions, a local result (as there exists p > 2 (and not too large:
in Theorem 4.3, Chapter 1): p
5
%)
such that if f and uo satisfy locally (2.81) and (2.82),
then (2.103)
$u
E
L p ( O , T ; W ~ ' p () ~)
.
Remark 2.5. .It follows of course from these estimates that, if f and uo satisfy conditions of the type (2.81), (2.82), then (2.104)
au
E -f
1
2.8
au ax i
in Lp(
s)
weakly or in Lp(O,T;Lp( 8
I ) )
weakly
,
The adjoint expansion. Let us remark that convergence can be proven for Dirichlet's
boundary conditions and when coefficients a
are smooth enough ij (cf. Section 2.3) using the method of the adjoint expansion as in
Chapter 1, Section 3.3.
We present this briefly, since the idea is
exactly analogous to the one in the elliptic case.
We can always
assume that uo = 0, and we write the equation in the equivalent form
where here (f,v) =
/
J
s
fvdxdt.
CONVERGENCE OF THE HOMOGENIZATION 6 F PARABOLIC EQUATIONS
Given v E
J
(S
),
asymptotic e x p a n s i o n s - a
we construct-using
sequence of f u n c t i o n s v
such t h a t vE
+
277
v i n L L ( )~ w e a k l y ( s a y ) , a n d
such t h a t
[- ata +
(2.106) where
i.e.
ff
*
AE*]vE
[- 2 + at
+
a * ] v i n L2 (
= a d j o i n t of t h e operator
ff
S)
weakly
,
c o n s t r u c t e d i n S e c t i o n 1,
o r f f 3 a c c o r d i n g t o t h e case k < 1, k = 2 , k > 3.
0 = Ol, ff
Then
and t h e r e s u l t f o l l o w s
U s e o f t h e maximum p r i n c i p l e .
2.9
Assuming a l l d a t a t o
be v e r y s m o o t h a n d f o r t h e D i r i c h l e t ' s
boundary c o n d i t i o n s , we have
Indeed let u s w r i t e (2.108) valid
PE =
a a t + AE
method a n d a d d a n
E
€-'Pa
+
,
E 0 P3
for k = 1 and 2.
(with d i f f e r e n t Pi's) 3
+
= €-'Pl
(For k = 3 use t h e same
u3 t e r m i n t h e proof b e l o w . )
We introduce (with
n o t a t i o n s of S e c t i o n 1 , (2.109)
ZE
-
= u
(u
+
EU1
+
E
2
u,)
and we o b s e r v e t h a t PEZE = -E(P2U2
(2.110)
I f all data m
the L (
S
)
norm.
+
P u )
3 1
-
E
2
P3U2
.
are Cm, t h e r i g h t hand s i d e of But
(2.110) i s O ( E )
in
EVOLUTION OPERATORS
and
so that, by using the maximum principle
hence (2.107) follows.
2.10
Higher order equations and systems. Except in Section 2.9 above, we never used the maximum
principle.
All results obtained sofar extend to higher order
equations and to systems, i.e. to the "parabolic analogue" of the problems considered in Chapter 1, Sections 9 and 10. Section 6, Chapter 1 also extend easily.
The results of
Let us consider an equation
of the type (2.111)
auE a at - axi
[a..(x,-,t,-)
Ek
11
aUE1
-
=
axj
f
,
where we suppose that
m
where Lrn(Rnx RT) = space of L P Y periodic. We also assume that (2.113)
a . .(x,y,t,r)C.C.> aEiCi 1 7 -
13
Then we denote by (2.114)
functions of y and
k
,
7
a > O ,
a k (x,t) the homogenized operator
(x,t) =
a k a - axi sij(x,t) ax
j
which are Y
SiER.
x T~
CONVERGENCE OF THE HOMOGENIZATION W PARABOLIC EQUATIONS k where the coefficients qij(x,t) are computed as the qij's in Section 1, with x and t playing the role of parameters.
If u
k
denotes the solution of uk
L~(o,T;v) ,
E
k
k a u + a (x,t)uk = at
(2.115)
k
u (0)
=
u
0
f
,
,
then (2.116)
u
+
2
uk in L (0,T;V) weakly as
E
+
0
.
One can also extend, always with the same type of method, the results of Chapter 1, Section 8. Let us consider now briefly the "parabolic analogue" of the situation of Section 11, Chapter 1. We assume that the space dimension equals 3 , and we denote by aE(x,t) a 3
x
We
3 matrix.
assume that
a . .( y , ~ )having the same properties than in (1.6) 13
.
I
We consider, with the notations of Chapter 1, Section 11:
and we observe that there exists a unique function uE such that
(2.119)
u
E
L~(o,T;v) ,
2 where f E L ( O , T ; V ' ) ,
uo
E
uE(0) = u 0 3
H = (L2(0))
.
,
279
EVOLUTION OPERATORS
280
Asymptotic expansions. We give only the results of the asymptotic expansion; one has to consider three cases.
Case k = 1. We define jiP(y,r) as a Y-r0 periodic solution of
-
roty a(y,r) (rot jiP Y
(2.120)
div jip = 0 Y
e
P
)
=
o ,
.
Then
U 1 (u,v) =
(2.121)
(
-
R(a
a rot ?)rot u,rot v) Y
and the homogenized equation is (%v) at
(2.122)
Case k
+ Q 1 (u,v) = (f,v)
,
tJ v E
v
-
= 2.
We define BP(y,r) as a Y - r 0
aa.tirp + (2.123)
periodic solution of
rot a(y,r) (rot GP Y Y
div BP(y,r) = 0 Y
-
e ) P
=
o ,
.
Then (2.124)
U
2 (u,v) = ( 7;1 (a
-
-
a rot 8)rot u,rot v) Y
and the homogenized equation is the analogous of ( 2 . 1 2 2 ) instead of
with 0 2
CONVERGENCE OF THE HOMOGENIZATION QF PARABOLIC EQUATIONS Case k = 3 . We define
and we define ip(y) as a Y-periodic solution of rot Z(rot Y Y (2.126) div '$I
Y
= 0
5' -
e
P
=
)
o ,
.
Then (2.127)
u
3 (u,v) =
( n (a - a 17
rot $)rot u,rot v) Y
and the homogenized equation is the analogous of (2.122) with U 3 instead of
a'.
The justification of these formulas is simple if we assume: (2.128)
,
divx f E L 2 ( S )
div uo
E
L2( 8 )
.
Indeed, we define z o as the solution of (2.129)
Azo = div u 0
,
and assuming the boundary (2.130)
zo E ~
0
z
r
E
1
HO(8)
smooth enough, we have
.
~ ( n 8H ~) ( D )
We then define z as the solution of (2.131)
a Az
=
divx f
,
(This is an elliptic problem:
Z ( O ) = zo
Az =
1 0
div
X
,
z =
f(x,u)do
o
on
c
.
+ Az 0 ) ; we have
281
EVOLUTION OPERATORS
282
ii
= UE
-
vz
we see that
aiiE
+ (a rot iiE,rot v)
[K,v] where ? = f =
div uo
-
a - at P,z0
= (f
-
=
( ~ , v ),
Since div fiE ( 0 )
Vz, so that divx f = 0.
=
a - at vz,v)
0, we see that div GE = 0.
Therefore we can assume (if (2.118) holds true) that divx f = 0 ,
divx uo = 0
,
divx u
= 0
.
Then one can prove the above formulas by arguments similar to those of Sections 2.2, 2.4, 2.5.
Remark 2.6. proof along the lines of Chapter 1, Sections 11.4, 11.5, does
A
not seem to work in the present situation.
The homogenization
problem seems to be open if we do not assume (2.128).
2.11
Correctors.
Orientation. We now return to the situation of (1.17), (1.18), (1.19). want to introduce first order correctors '8 (2.132)
u
-
(u +
OE)
2
-+
1
0 in L (0,T;H ( 8)
We
such that )
strongly
.
Remark 2.7. By using cut-off functions m E ( x ) , as in Chapter 1, Section 5.1, we can have correctors giving an approximation in LL(O,T;V) strongly.
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS Remark 2.8. We did not make any attempt to find the analogous results to those of Chapter 1, Section 18.
Notation. We shall denote by u1 the function: 1
(2.133)
&L
=
ax
u1 = - e l
if k = 1
,
if k = 2
,
j
ax j
We shall also denote by the ~ a m enotation u2,u3,..., the next terms of the asymptotic expansion, for k = 1,2,3. We shall use (2.134)
GE
=
u +
=
u
+
if k = 1,2
EU 1
+ 2u2
E U ~ E
,
if k = 3
Theorem 2.3. We assume the hypothesis of Theorem 2.1 to hold true and we assume all data (2.135)
uE
-
smooth enough.' (u
+ EU~) -f
0
&
Then L2 ( 0 , T ; V )
strongly
Proof. We introduce
'This
is made more precise in the proof below.
.
283
284
EVOLUTION OPERATORS
(2.136)
- 6E
uE
<,=
and we consider (2.137)
XE =
(T
-
t)[(
.
0
We observe that m
Since 1<,(0)l2
= 6 2 lu1(0)
we prove that X E replaced by T
-
0 as
+
or c 2 lul(0)
E -+
+ eu2(0] ', we see that, if
0 , then we shall have (2.135) with T
but since T does not play any role, this
q > 0,
q,
m
gives the general result. We can write XE =
1
(T
-
+ aE(uE)ldt
t) [ (u:,uE)
0
+
(T
-
+ aE(GE)]dt
-
+ aE(uE,GE)ldt
,
t) [(G;,GE)
YE
-
ZE
,
0
T
)
The first term in X
1
(T
-
.
equals
t) f,uE)dt
+
T
(T
0
0
+ aE(GE,uE)ldt
=
-
1
t ) (f,u)dt
(T
-
t) [ (u' ui
+ a ( ~ , ~ ) i d .tt
0
tWe set
=
a k , k = 1,2,3 and c7Xu,v
= (
a u,v).
CONVERGENCE OF THE HOMOGENIZATION QF PARABOLIC EQUATIONS
The second term in X
equals
One verifies, exactly as in Chapter 1, Section 5, T
T (T
-
t)aE(GE)dt
I
-f
n
0
tends to
-
5
1u0l2
+
285
(T
T 0
-
(T
-
(5.15), that
t) Q(u,u)dt, so that the second term
t) Q(u,u)dt
=
T 0
(T - t)[(u',u) +Q(u,u)ldt.
Therefore, if we show that (2.138)
YE
(2.139)
ZE
-t
-f
T T 0
0
(T - t) [(u',U) + Q(utu)ldt
(T
-
it will follow that X E
t) [(u',u)
+
+
Q ( ~ , ~ ) l d ,t
0, hence the Theorem follows.
Let $ be a given function in & g ( S ) , (near
r)
r
$ =
1 except on a set E
of measure /El. We define (compare to Chapter 1, (5.21),
(5.22)) T
T
We see exactly as in Chapter 1, Section 5,3, that (2.138), (2.139) will be proven if we verify that, for fixed $, Y
E$
(2.140)
+
1
(T
-
t) [(u',u$) + 0 $(u,u)ldt
(T
-
t) [(u',u$) + Q $ (u,u)ldt
0
zE$ 0
,
EVOLUTION OPERATORS
286
But
s
If we denote by q .
Ij
the coefficients of ftk, k = 1,2,3, we have
hence (2.140) follows for Y
EtJ.
We observe next that
But from the definition of GE it follows that +
au
2
+ C7u in L ( Q ) weakly,
hence the result
aii
$+
A Gc
so that (2.141) gives
follows.
Remark 2 . 9 . We can also write the equation as a first order system and develop arguments along the lines of Chapter 1, Sections 5.5 and 5.6.
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
287
For Dirichlet's boundary conditions we obtain in this manner error estimates which do not rely on the maximum principle.
2.12
Non-linear problems.
Orientation. We confine ourselves to
one aspect of
the homogenization of
non-linear parabolic equations (or inequalities), namely to Variational Inequalities (V.1.) of evolution.
Many other cases can
be treated by using the methods introduced below.
Setting of the problem. We consider operators AE as in Section 1.1; A E = A
X/E,t/E k 1 .
Formally we want to consider V.I. of the following type 3at+ A E u E (2.143) E
u
+
-
AeuE
,
f 0 '
-
UE(0
fluE = o
I
Q ,
in
being subject to usual boundary and initial conditions.
To
simplify a little, we shall take: (2.144)
uE(x,O) = 0
.
The variational form of (2.143) is as follows: 2 (O,T;H), then we are looking for uE such that: in L u (2.145)
and
E L 2 (0,T;V)
auE E at
let f be given
,
2 L (0,T;V')
,
uE 5 0
,
a.e. in Q
,
EVOLUTION OPERATORS
200 m
2 V V E L (0,T;V)
,
in S ,
v 2 0
and such that ( 2 . 1 4 4 ) holds true. We shall show below that this problem admits a unique solution. The main result we want to prove is Theorem 2 . 4 . We assume that the hypothesis of Theorem 2 . 1 hold true.
We
denote by U the homogenized operator found in Theorem 2 . 1 (i.e.
U
=
Uk, k
on V. -
and by
= 1,2,3)
U(u,v) the corresponding bilinear form
Let u be the solution of the V.I. of evolution
u
u(x,O) = 0
.
0
(2.149)
Then when E
I
-
[I%,.
(2.148)
-f
u
+
0, +
(2.150) -auE at + -
u in L2 (0,T;V) weakly
,
au . L2 (0,T;V') weakly at in
.
The proof is divided in several steps. We first introduce the penalized equation associated to the V.I. (2.146).
equation
For 11 > 0, we denote by uEn the solution of the nonlinear
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
(2.151)
u
En
(X,O)
= 0
,
u
En
E
289
L2 (0,T;V
(and as usual, the boundary conditions which correspond to the variational formulation).
The equation (2.1 1) is called the It admits a unique solu-
penalized equation associated to ( 2 . 1 4 6 ) .
tion, for which we now obtain a priori estimates. If we multiply ( 2 . 1 5 1 ) =
E
+
+
a (v ,v
=
+ aE(v ) ) :
+
E
+
by uEn, we obtain (since a (v,v
)
hence
We deduce from ( 2 . 1 5 2 ) (2.153)
1 u+ rl En
that
is bounded in L 2 ( Q
It follows from ( 2 . 1 5 3 )
)
,
V
8
and V n
I
that
(and uE satisfies initial and boundary conditions) so that the usual a priori estimates are valid:
If we let n such that
-+
0,
we can extract a subsequence, still denoted by uEn,
290
EVOLUTION OPERATORS
(2.156)
u
Erl
+
u
E
2 in L (0,T;V) weakly
au
,
au
2 3 + 2 in L (0,T;V') weakly , at at (2.157)
-+
0 in L
2
(S
)
.
2 It follows from (2.156) that uEll+ uE in L ( 0 ) strongly, so that u+
El1
-f
+ in . L2(S
u
E
)
and by comparison with (2.157) we obtain
u+ = 0, i.e. uE -< 0. Therefore uE satisfies (2.146) and (2.144). 2 Let v be in L (O,T;V), v 5 0. We multiply (2.151) by v - u we
-
El1'
obtain :
hence
or
Since
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS it follows from (2.158) that uE satisfies (2.145).
291
Therefore we have
shown the existence of uE, satisfying (2.145), (2.1461, (2.144) and such that
The uniqueness of uE is straightforward. We now establish an error estimate of independent interest:
Proof.
We have
UE
-
UE, = UE
+ u-E, - u+E n and by virtue of (2.154) we
have only to show that (2.161)
I
We take v = -uiq in (2.146) and we multiply (2.151) by uE
-
+ uE,.
Adding up, we obtain PUE
+
uEq
J
+ +
+ u,,]
-
aE(uE + ui,,)
i.e.
1 + and integrating over (O,t), we obtain (since (uE n ,-uE) 1. 0 ) r7
292
EVOLUTION OPERATORS
Using (2.154) and (2.159), the right hand side is bounded by C h i , hence (2.161) follows. We have also proved that
It follows from (2.160) that it suffices to study the limit of u
€17
as
E
+
0
for Q
fixed.
But this is immediate.
By virtue of
(2.155), we can suppose, by extracting a subsequence, still denoted by uEn, that U
(2.164)
-f
En
u
in L2 (0,T;V) weakly ,
rl
au nE:' 9 in L2 (0,T;V') weakly . +
Then uErl+ u
n
in L2 ( F
)
strongly, so that f
-
II
u+
En
f
-f
- n u+n
in
L2( Q ) and then
au ].,$I where
+
a (u17 ,v)
=
(f
-
1 + - u / rl
, v v
E
v ,
Q(u,v) is defined as in the statement of the Theorem.
proof is completed.
The
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
293
Remark 2.10. Theorem 2.4 gives, in particular, the homogenization of the Stefan's free boundary problem for a "granular" medium-i.e. coefficients are functions of
when
t aij(x/E).
X/E:
Remark 2.11. The result of Theorem 2.4 extends to the case of coefficients of x t the form aij (x,-,t,x). E
E
Remark 2.12. Theorem 2.4 can be extended to obstacles of the type b1(x,t) when $,
5
v
5 b,(x,t)
and $2 are smooth enough.
, The case of the "most general"
obstacles yi under which the analogous of Theorem 2.4 still holds does not seem to be known.
Remark 2.13. The homogenization of V.1 of evolution for "arbitrary constraints"
v
E
2.13
K(t) is largely open.
Remarks on Averaging.
Setting of the problem. We consider now the case T =
'This
+w:
is because the Stefan's free boundary problem can be
transformed into a V.I. of evolution.
294
EVOLUTION OPERATORS
s
(2.165)
=
8
x
lo,+-[
and we consider the equation
(2.166)
av
For u,v (2.167)
E
H1(S), we set now
aE(u,v) = 8
8
The precise problem we consider is then: u
(2.168)
E
L~(o,T;v),
v
T finite ,
[auc w , v ] + aE(uE,v) = (f,v) UE(0) =
u0
,
V
v E V
,
t > 0
,
.
We assume the following: (2.169)
aij
and of course
Example 2.1. We take :
,
aa! axk
,
b;
remain in a bounded set of I,-( Q
)
295
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS a . .( x , ~ ), (2.172)
a aij(x,T)
b .( x , ~ ),
axk
3
13
m
belong to L ( 8 x R T ) and are almost periodic in
T
.
We extract a subsequence such that
(2.173) in L m ( 6 ) weak star
.
We want to study the behavior of uE
as
E
-t
0.
In the case of Example 2.1 we have (2.1741
iij =
nT(aij)
,
j =,R',(bj) ,
where we have set here (2.175)
m T ( $ )=
x
\
lim
X++m
$(x,a)do
.
0
Theorem 2.5.
t We assume that (2.169), (2.170) hold true and that (2.176)
1 V = Ho(8)
1 V = H ( 8 )
.
We also assume that f , g
(2.177)
i
E
L2( 8
x
lO,T[)
,
V
T finite,
UOEV.
hypothesis is by no means indispensable. I f V consists of 1 the functions of H ( 8 )which are zero on a subset ro of r , the 'This
result is still valid with a slightly more complicated proof.
EVOLUTION OPERATORS
296 Then a s (2.178)
E
-+
u
0 , one h a s +
,
u i n L2(0,-T;V) w e a k l y f o r e v e r y T f i n i t e
where u i s t h e s o l u t i o n o f
(2.179)
I&[
+ i(u,v)
u(0) = u where (2.180)
i(u,v) =
0
= (f,v)
, v
v E
v ,
,
I
aij
ax a u ax av
8
j
i
dx
+
I Gj 0
j
vdx
.
Remark 2 . 1 4 . I n f a c t w e s h a l l p r o v e more: n a m e l y , (2.181)
uE
+
2 u i n L 2 ( 0 , T ; H ( 0 ) )w e a k l y
, v
T <
m
,
Remark 2 . 1 5 . I n case o f Example 2 . 1
(formulas (2.171)) t h e theorem h o l d s
a
( 2 . 1 8 2 ) ) w i t h o u t c o n d i t i o n s on axk a i j ' u s e s t h e s e m e t h o d s a l o n g t h e l i n e s o f t h o s e of S e c t i o n 2.2. t r u e (without (2.181),
P r o o f o f Theorem 2 . 5 . Thewhole problem i s reduced t o p r o v i n g t h a t , VT f i n i t e ,
One
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
297
Indeed, if we admit (2.183), then one can extract a subsequence, still denoted by uE, such that one has (2.178), (2.181) and (2.182). It follows that, V T finite auc
+
1
au ax
in L2 ( 8 x ]O,T[) strongly
j
and therefore
in L2 ( 0
x
]O,T[) weakly
and the result follows. Estimates (2.183) are obtained by using first local mappings, to reduce the problem to the same question when (2.184)
0
=
Ixlx > 0 1 n
.
Then one uses the method of translations parallel to the boundary. t We suppress the index h
=
"E"
to simplify the writing.
Given
{hl,...,hn-l,Oj, we define
We have
, g + ~ u = f i nS ,
tThis is a classical method, introduced by L. Nirenberg 111 for the study of the regularity of elliptic equations.
When V does not
satisfy (2.176) but consists of functions which are zero on
ro
C
r,
then there is a difficulty because V is not "translation invariant". One can then use a compensation method, due to Aronszajn and Smith, and reported in'lions 121.
298
EVOLUTION OPERATORS
and with obvious notations
3 at + AhUh
fh in
=
q
.
Therefore (2.185)
a(uh at
U)
+ Ah(uh - U) + (Ah - A)u
=
f h
-
f
.
It follows from (2. 8 5 ) and from the hypothesis that
1 IUh so that by letting (2.186)
hl
+
0, we see that when
2
a remains axjax
E
-+
0
in a bounded set of L2 ( 8
x
]O,T[)
,
j
where x‘ i = x i ,
i L n - 1 .
We now return to the equation, which we write (2.187)
au
-
ann
2 2
=
a xn
g *
where q remains in a bounded set of L2 ( 8 x lO,T[) (we use ( 2 . 1 8 6 ) ) ; au we obtain (since a > y > 0) multiplying (2.187) by 1 at nn -
au = 0 on (since u = 0 or axn (2.188)
r)
hence it follows that
au remains in a bounded set of L2 ( 8 x lO,T[)
But then ( 2 . 1 8 7 ) ,
which can be written:
.
OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE
299
imp1ies 2 9
E
bounded set of L2( 8
lO,T[)
x
a xn
and (2.183) is proven.
Remark 2.16. In the case of almost periodic coefficients, Theorem 2.5 is related to the classical theory of averaging for ordinary differential equations with almost periodic coefficients.
Cf. Bogoliubov
and Mitropolsky 111 and the Bibliography therein.
Remark 2.17. We confine ourselves to remarks made in the comments of the last section of this chapter for the case of coefficients aij(x/E,t/E k ) , where the aij's are almost periodic in
3.
T
(and periodic in y).
Evolution operators of hyperbolic, Petrowsky or Schrodinger type. I
3.1
Orientation. We present now some very simple remarks connected with the
homogenization problem for hyperbolic operators.
Further results,
using some different ideas and techniques, are given in Chapter 4 of this book.
We also give in this section some results for operators of
Petrowsky or of Schrodinger type, the hyperbolicity not being used in an essential manner.
We give next some examples of "non-local"
operators which appearin the homogenization process.
300
EVOLUTION OPERATORS
3.2
Linear operators with coefficients which are regular in t. Let us consider the operator
where the a . . ' s satisfy: 13
,
a. , (y) E Lm(R;) 17
(3.3)
a. . is Y-periodic 13
aij(y)SiEj 2 clSiSi
a > 0
I
1 We consider V as usual (Ho(Q) _c V
,
,
a.e. in y
5 H 1 ( 6 ) )and
.
with the usual
notations we consider the following problem:
0
where f, u
, u1 are given such that
(3.7)
f
E
L~(o,T:H)
,
uo
E
v ,
u1 E H
.
It is known (cf. for instance Lions [l], Lions-Magenes [11) that problem ( 3 . 4 ) , (3.5), ( 3 . 6 ) admits a unique solution.
A
priori estimates. We shall write $',$I"
for -, at
a2e at2'
Taking v = u~ in (3.5) gives
OPERATORS OF HYPERBOLIC, PETROWSKY'OR SCHRODINGER TYPE
30 1
We emphasize that in order to obtain (3.8) we used in an essential manner the fact that aE(u,v) is symmetric and does not depend on t.
It follows immediately from (3.8) that
The behavior of uE
E
-t
0 is very simple to study.
It follows
from (3.9) that we can extract a subsequence, still denoted by uE, such that uE
+
u
in L~(o,T;v)weak star
,
u'
+.
u'
in Lm(O,T;H) weak star
.
(3.10)
We use next the method of Remark 1.6 which is entirely general. With the notation (1.46) one has, V
@ E
C,"(]O,T[):
in H strongly, one has: uE ( @ ) + u ( @ ) in
V weakly
, where
where Q(u,v) = homogenized form associated to aE(u,v). u is the solution of 2
IF,.]
(3.13)
u
(3.14)
u(0) = u0
E
v v
6
L-(o,T;H)
,
+ Q(U,V) = (f,v)
(3.12)
L~(o,T;v) ,
,
u'
E
,
u' ( 0 ) = u1
Let us consider now the case when
.
v ,
Therefore
EVOLUTION OPERATORS
302 (3.15) We assume that (3.16)
aij(x,y,t) is Y-periodic (3.17)
a1.7.E Co( 5
We set, t/ u,v (3.18)
E
X
[O,T];Lm(Rn)) Y
x,t
,
,
H1 ( 8 ) :
aE(t;u,v) =
J
8 We can
, v
au av aij(x,x,t) E ax axi dx j
.
efine
(3.19)
8 We consider again (3.4), (3.5),
(3.6) (with aE(u,v) replaced by
aE (t;u,v)), and we have existence and uniqueness of the solution. The a priori estimates are now the following.
Equation (3.8)
is replaced by
and by virtue of the last condition in (3.17) we obtain the same a priori estimates (3.9). By the same technique
as
in Chapter 1, Section 6, one proves
that uE satisfies (3.10) where u is the solution of (3.13), (3.14) and
OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE (3.22)
( a (t)u,v) if
u(t;u,v) =
u,v E
303
cr(8) ,
where G(t) is the homogenized operator of AE(t) for fixed t.
Remark 3.1.
a2 + AE is a second order hyperbolic operator. 7
The operator
at
But the results indicated so far really do not depend at all on the hyperbolicity.
All what we sa d remains unchanged (except obvious
modifications) i f one considers (3.23)
aE(t;u,v) =
I
aaB(x,E X
t)DauDBvdx ,
la1 =
161 = m
,
8 a
aB
a
aB
(3.24)
aaB
= a
'
(x,y,t) being Y-periodic in y Co( 5
E
aa at
Ba
E
Lm( 8
[O,T];Lm(Ry))
x
x
(0,T)
x
Rn) Y
,
,
.
The homogenized operator is computed for every fixed t according to the results of Chapter 1, Section 9. We remark that the corresponding operator is not oft hyperbolic type when m > 1; it is an operator of Petrowskytype.
Remark 3.2. One can also add to the operator AE lower order terms. return to that in Section 3.3 which follows.
We
304 3.3
EVOLUTION OPERATORS Linear operators with coefficients which are irregular in t. A natural problem is-the following:
Instead of the situation
( 3 . 1 ) we consider functions a..(y,T) which satisfy ( 3 . 2 ) and 11
(3.25)
aij(y,r)
E
Lm(Rn Y
x
RT)
,
and the usual ellipticity condition.
Y - r 0 periodic
We then consider the operator
AE defined by
If we assume that (3.27)
a
- aij
E
L-(R;
x
R ~ ),
then problem ( 3 . 4 ) , ( 3 . 5 ) , ( 3 . 6 ) admits a unique solution. are nof a priori estimates independent of
But there
Indeed if we define
E.
(3.28)
then ( 3 . 2 0 ) is replaced by
which does not give estimates independent of
as E
-t
E.
0 does not seem to be known in general.
The behavior of uE We shall give in
Section 3 . 4 below some formal computations (which can be justified in
cases). It is possible to consider lower order terms which have
coefficients irregular in t.
Let us consider
+At least there do not seem to be known a priori estimates strong enough to pass to the limit in
E.
OPERATORS OF HYPERBOLIC, PETROWSKY.OR SCHRODINGER TYPE A'
(3.30)
given by (3.1)
305
,
a x t BE = b. (-,-) 7 E E k ax. 3
where (3.31)
b.(y,r) is Y--r0 periodic 3
,
b. E L m ( R y 3
x
RT)
.
there exists a function uE and only one which
We have then:
satisfies (3.4), (3.6) (3.32)
(u2.v) + aE(uE,v) + (B'U',~)
=
(f,v)
The a priori estimates are as follows.
, v v
E
v
Taking v = u
I
. in (3.32),
we obtain
But
so that (3.33) gives:
t
and we obtain (3.9). Therefore one can extract a subsequence, still denoted by uE, such that one has (3.10) and (3.36)
BEuE
+
rl
Then (3.32) gives, V
m
in L (0,T;H) weak star C$
EC;(]O,T[)
.
306
EVOLUTION OPERATORS
and ( B E u E )( @ )
+
i n H-weakly,
Q(@)
follows f r o m (3.37) t h a t ,
t h e r e f o r e i n V' s t r o n g l y a n d it
a ( u , v ) b e i n g t h e homogenized f o r m
a s s o c i a t e d t o aE
Therefore
and t h e p r o b l e m i s now t o compute
We s t u d y t h i s q u e s t i o n i n t h e
Q.
following s e c t i o n s .
3.4
.
A s y m p t o t i c e x p a n s i o n s (I)
a purely formal fashion-the
L e t us consider f i r s t - i n
(3.4),
(3.5),
( 3 . 6 ) f o r A E g i v e n by ( 3 . 2 6 ) .
W e have w i t h t h e u s u a l
notations: (3.39)
a2 + A E
+
= E-2Q1
+
E-lQ2
E 0Q 3
,
at2
Q1 =
a2 2 + aT
Q3 =
at2+ A 3
a
a a 11 . . (y,T) A 1 = - -ayi aYj
A 1 '
I
a2
'
1
Then, i f w e look f o r uE = uo (3.40)
QIUO
=
(3.41)
QlUl
+ Q2UO = 0
0
r
r
+
E
U
+ ~
problem
..., w e
f i n d as u s u a l :
OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE (3.42)
Q1u2 +
+ Q3u0 = f
Q2Ul
307
.
We are thus lead to the following question:
given a function F
such that (3.43)
F
E
L2(Y
( 0 , ~ ~ ) )
x
is it possible to find (3.44)
9 a-r
+ A 1@
@
which is Y--ro periodic and satisfies
,
= F
i.e., @ takes equal values on opposite faces of Y and @(Y,O) = @(YtTo) , A necessary condition for
(3.45)
7?( (F) = 0
@
a@
(Y,O) =
a@
(y,-ro)?
to exist is
.
Let usassume that this condition is sufficient (so that (3.44) defines @ up to an additive constant). "generically" the case.
It can be proven that this is
Then (3.40) is equivalent to uo = u(x,tl and
(3.41) reduces to Q u + A 2 u = 0 . 1 1
If we introduce XJ as some Y--r0 periodic solution of
then
and (3.42) can be solved for u2 iff
+ ,Q ~ u =~ f) ~ ( Q ~ u
,
i.e.
,
308
EVOLUTION OPERATORS
Of course we add the appropriate initial and boundary conditions.
In
which sense u is an “approximation“ of uE is an open question.
3.5
Asymptotic expansions (11). We now consider the situation (3.30) and we take
(3.49)
k = l .
Then
We find
,
(3.51)
RluO = 0
(3.52)
Rlul + R2u0 = 0
(3.53)
R1u2 + R2u1 + R3u0 = f
,
-
We are lead again to (3.44) but this time with
of T.
A1
independent
It follows from de Simon [l] that for almost every T ~ condi,
tion (3.45) is necessary and sufficient for (3.44) to admit a Y-ro periodic solution.
We therefore assume
OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE (3.54)
309
condition (3.45) is necessary and sufficient for the existence of $ which is a Y - T ~ periodic solution of (3.44)
.
Then (3.52) gives u1 by formula (3.47), (3.46); but since A1 does not depend on
T,
being defined by
we have
and
Then (3.53) admits a solution for u2 iff
7/1 (R2u1 + R3UO)
= f
,
i.e. 2
*+
(3.56)
nC
a u +
at2
This can actually be proven: Theorem 3.1. I
We assume that AE is given by (3.1), (3.2), (3.3) and that BE is given by (3.30)
that
bj
E
C1(?
x
with
k = 1.
[O,T~].
solution of (3.38)
with
0
Then
We assume that (3.54) holds true and uE satisfies (3.10) where u is the
given by
Sketch of Proof. We introduce BE* = adjoint of B i.e., BE*v = such that
a (b.(x,t)v). € - fax J j
in the sense of distributions,
We construct a family of functions wE
EVOLUTION OPERATORS
310
+
a2
+ BE*]wE
AE
=
€4,
,
(3.58) w
+ 1
o
-+
in L ' ( Q )
as
E
,
gE bounded in L 2 ( Q )
-+
o
.
For the construction of w, we need all coefficients aij and b smooth-we
j
to be
can always reduce the problem to this case by
approximation.
We look for wE in the form:
(3.59)
-1
W
=
+ E 2@ ( y , T ) ,
ECL( Y,T
4-
a and B being Y--to
periodic
.
*
We find, using (3.50) and since R1 = R1:
*
(3.60)
R a 1
+ R2(-l)
(3.61)
RIB
+
= 0
,
*
*
R2a + R3(-l) = 0
and g,
=
*
.
R2B
We define a as a Y--r0 periodic solution of (3.60), i.e. (3.62)
R1a = -
4 ab
(YtT)
*
Then (3.61) gives (using the fact that R3(-l) = 0):
*
R1B = - R2a which admits a Y - T ~ periodic solution by virtue of (3.54).
Then we
have (3.58). We now use $
E&,;(
Q )
and we take v = Ow, in (3.32).
multiply (3.58) by $uE. We obtain, by subtracting
We
OPERATORS OF HYPERBOLIC, PETROWSKY,OR SCHRODINGER TYPE
The first term in ( 3 . 6 3 ) equals
In ( 3 . 6 4 ) , the first term converges to 0, since u
aw = aa strongly and 2 at aT
+
E
aB + aT
Therefore ( 3 . 6 4 ) converges to If we set E:
=
aij
aa
-
I
(=)
?
2 = 0 in L (
-+
u in L
(u,@")dt.
0
au
$, the
second term in ( 3 . 6 3 ) equals
The third term in ( 3 . 6 3 ) equals b;uEwE
s
$ dxdt 1
(s)
s ) weakly.
j
-
2
+
(bj)
1
s
u'$
j
dxdt
.
311
EVOLUTION OPERATORS
312
The right hand side in (3.63) converges to
(3.38)) =
-
(u",@)dt 0
(citq) lo!u,@), =
f 0
Q(u,@)dt =
T 0
-
(n,@)dt.
T 0
(f,$)dt = (using
Since
it follows that
It remains to show the identity of (3.65) with (3.57), i.e. that
We multiply (3.62) by
xi.
It follows that
hence the result follows.
3.6
Remarks on correctors. We present here only a very preliminary result on the question
of correctors. (3.67)
1
=
We consider the case (3.1).
2~ ax ,
We have
xJ(y) being solution of A1(xJ
-
and being Y-periodic
,
j
y . ) = 0 3
and we use cut-off functions mE(x) as in Chapter 1, Section 5.2. set (3.68)
-
u
= U + E ~ U E
l
'
More generally we suppose that ue is the solution of
We
OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE (3.69)
313
(uE,v) + aE(uE,v) = (fE,v)
where (3.70)
fE,f E L2 (0,T;H) ,
fE
f in L1 (0,T;H) strongly
+
We have again (same proof), (3.10) and (3.12).
.
If we assume
that (3.71)
u E L ~ ( o , T ; H ~ ( o ),)
at
E
L~(o,T;v),
then -
-
= uE - uE
z
(3.72)
I
zE
+
-t
2
0 in L (0,T;V strongly
,
0 in L 2 0,T;H) strongly
The proof of this result relies on the following identity. 1 (T - t)2 $(t) = 2 We have
.
We set
uE,$)]dt
To prove this identity, we can assume that f
.
and f are smooth and witt
values in H (by approximation of these functions).
It is then
obtained by integration by parts. Since T does not play any role here, the result will be proved if we verify that X E
+
0.
314
One verifies that
T
EVOLUTION OPERATORS
0
@'aE(iiE)dt +
T 0
@ ' a(u)dt and one can pass
to the limit directly in all other terms.
3.7
The result follows.
Remarks on nonlinear problems. Let us give a simple example.
We consider the situation of
Section 3 . 6 and we set (3.73)
2
@(v) =
lgrad vI dx
.
8
Let us consider the nonlinear equation (ui,v) + aE(uE,v) + @(uE)(udfv)= (f,v) (3.74)
uE E L-(o,T;v) UE(0) = 0
,
,
u:
E
UE(0) = 0
,
L ~ ( o ~ T ; H,)
.
We assume that (3.75)
af
f,E
E
2
L (0,T;H)
.
Then there exists a unique solution of ( 3 . 7 4 ) which satisfies (3.76)
u :
E
L~(o,T;v),
u" E L-(o,T;H) E
.
The proof of this fact is standard. One obtains the a priori estimates (3.77)
I bElI L- ( 0, T;v)
+
1l~:Il
L~(o,T;v)
OPERATORS OF HYPERBOLIC,
PETROWSKP OR SCHRODINGER TYPE
315
Therefore we can extract a subsequence, still denoted by uE, and such that u
+
u and u'
+
u
(3.78) 1
uE
I,
+
u' in L m ( O , T ; V )
m
in L ( 0 , T ; H )
weak star
,
weak star
.
Since grad uE is bounded in Lm(O,T;(L2 ( 8 ) )"), $(uE) is bounded in L m ( O , T ) and we can assume that (3.79)
since u
$(uE) + 5 in Lm(O,T) weak star I
* u' in L2 ( S
)
;
strongly, we have $(uE
weakly and if we write (3.80)
+ aE(uE,v) = (f
(u:,v)
-
$(uE)u;,v)
gu' in L 2 ( S ) weakly.
we have fE * f
But one can improve this
result; indeed d
u
;iT@
= 2
)
i
I
grad u .grad u dx
8 m
remains in a bounded set of L (0,T) so that we can assume that $(uE) * g in Lm(O,T) strongly and therefore (3.81)
fE
+
f
-
CU' in L 2 ( S
)
strongly
.
Therefore, using (3.72), we see that $(UE)
-
1 $(fiE )+ 0 in L (0,T) for instance
.
But
a 8
(u
+
Em u E
1
)
a (u + Ern u )dx axi E l
,
EVOLUTION OPERATORS
316
so that
.
in L 1 ( 0 , ~ ) Therefore u is the solution of
where $(u) is given in ( 3 . 8 2 ) .
Remark 3 . 3 .
Homogenization of variational inequalities.
The general question of the homogenization in variational inequalities of "hyperbolic type" is largely open. simple result along the following lines. (3.84)
j(v) =
lvldx
g
I
We give only a
We set
g > 0
8
and, always with the conditions of Section 3 . 6 , we consider the solution uE of the V.I.
OPERATORS OF HYPERBOLIC, PETROWSKF OR SCHRODINGER TYPE We assume that f satisfies (3.75).
317
It is known (cf. for instance
Duvaut-Lions [l]) that (3.85) admits a unique solution.
Then one
can prove that (3.78) holds true, with u being the solution of the homogenized V.I. ( u " , -~ u') + U(U,V (3.86)
-
u')
+ j(v) - j(u') L (f,v - u')
u(0) = 0 ,
V V E V ,
u'(0)
=
0
.
Remark 3.4. If in the V.I. of the preceding Remark, we replace j(v) given by (3.84) by (3.87)
j,(v)
=
g
I
lgrad vldx
9
it is likely-but
it is not proved-that
one obtains for the limit a
V.I. of type (3.86) with j replaced by
3.8
Remarks on Schrodinger type equations. We can consider, now with complex valued functions, the equation
(of the Schrodinger type)t
'Of course one has now: (f,v) =
I
8
f Gdx
,
aE(u,v) = 8
318
EVOLUTION OPERATORS
where aE is given as in (3.1)
(3.2), (3.3).
[We can also consider
other cases considered in the preceding sections for aE . I
If we
assume (3.75), we can obtain the following a priori estimate: v = u
I
taking
in (3.89) and taking the real part
1 2 dt
aE(uE) = Re(f,ui)
Hence it follows that (one obtains a preliminary estimate by taking v = u (3.90)
in (3.89))
I bElI L~ ( o ,T ;v) -<
c .
Then we can extract a subsequence, still denoted by uEf such that (3.91)
uE
co
-t
u in L (0,T;V) weak star
and (3.92)
3.9
Nonlocal operators.
Setting of the problem. We consider: (3.93)
A E given with (3.1), (3.3)t
tThe symmetry is now necessary.
OPERATORS OF HYPERBOLICf PETROWSKY OR SCHRODINGER TYPE
bij = bji
(3.94)
m
,
b1.3. E L (R") Y
,
bij Y-periodic
319
,
B > 0
bij(y)S.S. 1 3 2 @Sici and we taket V = H 1o ( b )
(3.95)
.
We set bE (u,v) =
(3.96)
b : j
(x)
au av ax.
dx
3
8
and we consider the equation
(3.97)
UE(0) = 0
.
Remark 3.5. The ellipticity of A E is not necessary in what follows. The problem (3.97) admits a unique solution uE (3.98)
f E L'(O~T;V~)
E L
2 (O,T;V), if
,
and (3.99)
'We
I lUEl I
L~ ( o T;V)
1 can take HO(Q
)
< c .
5V 5
1
H ( 6 )if 1 9 V.
1 If V = H ( a ) , what
we are going to say is valid if one replaces BE by BE + bo (x/E) bo(y) 2 B 0 > O.,
EVOLUTION OPERATORS
320
Indeed taking v = uE in (3.97) we have
hence (3.99) follows.
Therefore one can extract a subsequence, still
denoted by uE, such that uE
(3.100)
+
u in Lm(O,T;V) weak star
.
The problem is now to see what is the "homogenized" equation which characterizes u.
We are going to show it is not a partial
differential equation but an equation with pseudo-differential operator. We take the Laplace transform in t of (3.97).
I
Let us set
W
(3.101)
uE(p) =
e-PtuE(t)dt
,
R e p > O .
0
We do not restrict the generality by assuming that (3.97) is taken for every t > 0 and with f E LL(O,m;H) with compact support. also consider (3.97) as an equation over Rt, with u t < 0.
We can
and f zero for
We then set
and we write (3.97)
This leads to "homogenization with parameter" as in Chapter 1, Sections 8.1 and 15.
We define xJ(p) as a Y-periodic solution of
T
HYPERBOLIC, PETROWSKYiOR SCHRODINGER TYPE
normalized by, say,
I’
x’(p)dy = 0 ; in (3.103) we use the standard
%].
I .
1
notation for A1 and B1 = - -[bij a (3.104)
321
Then we define
qij(p) =
and
Q (p) =
(3.105)
We can take the inverse Laplacetransform of q..(p) and set 17
(3.106)
Q
=
2 - l Q (PI
e - l = inverse Laplace transform
I
Then the homogenized equation is u
E
L~(R;v),
(3.107) Q(t)u = f
u =
o
for t <
o ,
.
The proof proceeds by methods similar to those of Chapter 1, Section 15.
Remark 3.6. The operator Q(;)
is not, in general, a partial differential
operator.
Remark 3.7. Let us consider materials which have a composite periodical structure and with a long memory (cf. for this last point, for instance, Duvaut-Lions [l], Chapter 3 , Section 7). simple model.
We consider a
These general cases will be treated with similar
.
322
EVOLUTION OPERATORS
methods.
We are given functions aij(y) satisfying (3.1), (3.2),
(3.3). We are also given functions bij(y,t) such that
bij - bji
, v
(No ellipticity is assumed). (3.109)
BE(t)(;)v =
.
i,j
Given v
- axi a j
E
Lioc (Rt;V) , we set
-
x bij(;,t
S)
av ax. (x,s)ds 3
0
and we consider the problem:
u
= O f o r t < O ,
where (3.111)
2 f E Lloc(Rt;H)
,
f = 0 for t < 0
This problem admits a unique solution.
.
One has the following
a priori estimates: (3.112)
I luElI
L-(o,T;V)
L- (0,T;H)
< c .
Indeed, if we set b E t;u,v) =
1 8
';1
t;u,v) =
x
bij(F,t)
au av ax 5 dx
(E,t) ax 8
I
j
j
$ dx =
,
OPERATORS OF HYPERBOLICf PETROWSKY OR SCHRODINGER TYPE
323
we obtain
and
so that (3.113) gives:
and therefore, for t 5 T finite (arbitrary)
I t
1 lu,'(t)
l2 +
1
aE (uE(t)) +
bE(t-s;uE(s)fuE(t))ds
0
hence
*
hence (3.112) follows. Therefore we can extract a subsequence, still denoted by uE, which satisfies (3.10).
We show now that u is characterized by the
solution of a pseudo differential operator equation.
EVOLUTION OPERATORS
324 We define, for Re p
J
0
m
2. ,(y;p) = 13 (3.1
-
,
e Ptb. .(y,t)dt
0
13
Then (3.110) can be written (3.115)
(AE
+
gE(p)
+ p 2 I)GE(p) = g(p)
.
(We can always assume that f admits a Laplace transform in t.) We use now "homogenization with parameter".
I
the Y-periodic solution (normalized by
We define
Xl(p)dy = 0) of
Y
we have set
We observe that (3.118)
A1
+ g,(p)
Therefore (3.116
gij (y;p)I
5
& so that the operator
is elliptic for Re p large enough
.
defines XJ (y;p)
(3.119)
and
4 (p) and
Q as in (3.105), (3.106).
We define
.
x3 (p) as
COMMENTS AND PROBLEMS
325
Then the homogenized equation is
(3.120) 2
u E Lloc(Rt;V)
,
u = 0 for t < 0
.
Comments and problems.
4.
Most of the results presented in this chapter are given for the first time.
Preliminary announcements were made in notes by the
authors (cf. Bibliography of Chapter 1). The Lp estimates of Section 2.7 were obtained, in the symmetric case, by Pulvirenti [l], [2].
The method given here extends to
parabolic equations of higher order or to systems. Given a family of elliptic operators, say A E , one can say that A E "G converges" to
c 7 if
V f, (AE)-lf -+
a-lf in a weak Sobolev
space, where the inverse are taken for a given set of homogeneous boundary conditions.
If one considers next a family of elliptic
operators depending on t, say AE (t), one can say that A' converges" to U -1 + A E ) f -t
(where "P" stands for "parabolic") if V f, -1 + a f in a weak Sobolev space (where again the
[A
[&
"P-G
]
inverse are taken for a given set of homogeneous boundary conditions). Then one can compare the "P-G convergence" with the "G-convergence V t " (in general these are different notions!) S.
Spagnolo [l].
Cf. F. Colombini and
Cf. also for this general approach, Spagnolo 111,
121, Sbordone [l]. In case the coefficients a..(y,-r)are Y-periodic in y and almost 13
periodic in
T,
one obtains formulas similar to those of the text,
with a difficulty for k = 2.
In the case k = 1, one considers
xJ(y,-r) as the solution of (1.67) which satisfies, say
EVOLUTION OPERATORS
326
I'
Y
XI (y,T)dy = 0. Then XI is almost periodic in
T
(T
plays the role
of a parameter) and one obtains
where
In case k = 3 , one introduces (compare to (1.86))
and
U
1
is given again as in (1.94) where @ =
lim
A+=
-
lYlA y 0
@(y,o)dody
.
In case k = 2 , one has to consider the T-almost periodic solution of
which is periodic in y, and many problems remain open in this direction.
327
COMMENTS AND PROBLEMS One can of course in the .r-almost periodic case consider correctors, as in the case of periodic coefficients.
Let us give now some indications on possible extensions of the resu ts given in the text. b is Y--r0
(4.5
Let b(y,r) be given satisfying
periodic
b E Lm(Rn Y
,
RT)
x
b(y,r) '> B > 0
,
,
et us consider the operator
and (4.6
a
bE&--
axi(arj
a
k b" = b(x/Elt/E 1
x) , j
aFj = aij (x/E,t/Ek
,
.
(If one considers the case bE(x,t) = b(x/E) when b does not depend on
T,
the homogenization is immediate and is given by
711
Y
(b)
a + ak ,
k = 1,2,3
)
.
The general case (4.6) leads to a number of interesting questions. In case k = 1, the asymptotic expansion, with usual notations, leads to
,
(4.7)
AluO = 0
(4.8)
A U + b -auO +Au = O , 1 1 aT 2 0
(4.9)
A1u2
+
(b
a + A2)u1 + aT
(b
a +
A3)u0 = f
.
It follows from (4.7) that uo does not depend on y . Taking I Y au (b) 2 = 0, so that uo = u(x,t) and with the of (4.8) gives:
Y
aT
notation (1.67), one will have:
328
EVOLUTION OPERATORS We use (4.10) into (4.9).
m
Taking
Y
of the result, we obtain:
1 (b) and we take the average in 2 y
We then multiply by
~
(4.11)
a
1
my(aij
-
aik
T:
el]% 7
One sees the appearance of first order derivatives in x in the homogenization process.
One also sees that one needs neither
regularity on the a. ' s of the type: lj
&
a
to make sense nor regularity of b:=ab
can integrate by parts in
'I
a. . E Lm(Rn 13 Y E
Lm(Ry
x
x
RT)
, in order
R T ) (since one
au in the coefficients of F ). j
The
justification can then be made along the lines of Section 2 .
If
k = 2, one runs into the difficulty of finding Y - T ~periodic solutions of
a question whose general study does not seem to have been made (the same is a fortiori the case for the almost periodic situation). In the case k = 3 , one defines:
and the functions
d
by (1.88) (with this new
El).
Then the
329
COMMENTS AND PROBLEMS homogenized equation is
Singular perturbations and homogenization. Let us consider first the problem
u
being subject, say, to Dirichlet's boundary conditions (with usual
boundary conditions).
Then the homogenized equation is obtained by
a
formulas analogous to those for homogenizing ; i i + AE but where A1 replaced by A1 + A 2 . For instance if k = 2, one considers the Y solution x 3 ( y , ~ ) , which is Y - T ~periodic, of
and
a2= a
2 p A 2 is now given by
(4.17) 2td2 =
qij
[aij
-
d].
aik ayk
One can also consider the problem: (4.18)
au
E
k 2 at + AE(t/E ) u E = f ,
with usual boundary conditions and with uE(0) = 0. If k = 2 , one has (4.19)
uE
+
u in L2 (0,T;V) weakly
&
EVOLUTION OPERATORS
330
where u is the solution of (4.20)
C7 u = f = f (x,t)
,
(t plays the role of a parameter)
(More precisely, (4.21)
u E
If k = 3
v , a 1 (u,v)=
or
(f,v)
,v
v
E
v
)
.
k = 4, one has again (4.19), with u given by
(4.22)
C12(u,v) = (f,v)
,V
E
V
,
if k = 3
,
(4.23)
0 3 (u,v) = (f,v)
, V'v E
V
,
if k = 4
.
v
Actually the proof of (4.22) has been obtained only for the Dirichlet's boundary condition by the method of the adjoint expansion, and assuming all data
to be smooth.
Proofs of (4.21), (4.23)
proceed along the lines of Section 2. There is one more a priori estimate one can obtain when
[One will obtain the same estimates for the equation
Indeed if we multiply (4.18), for k = 1, by AE(t/E)UE, we obtain (we use the symmetry): (4.26) Hence (4.27)
caE [2%~ , u € +] IAE(t/E)u,12 = (f,AE(t/E)uE)
.
.
331
COMMENTS AND PROBLEMS
=
. E (a) (uE,uE). It follows from (4.27) that
2
AE(t/c)uE remains in a bounded set of L ( 0 , T ; H ) .
(4.28)
Using (4.18) and (4.28) we see that (4.29)
E
au 2 in a bounded set of L ( 0 , T ; H ) 2 at remains
.
The behavior of the solution of (4.18) for k = 1 is unsettled.
Reiteration. Let us consider for instance the equation
with the usual boundary conditions and initial conditions, where the aij's satisfy the following conditions: aij(ylz,Tl,T2,T3)is Y-Z periodic in y,z and admits (4.31)
the period T O in the variable j
'lj,
j = 1,2,3 ;
and the usual hypothesis
-
'One
could weaken this hypo~esis but the weakest hypothesis is
not known.
332
EVOLUTION OPERATORS In (4.30) we assume that
(4.33)
0 < kl < k2 < k3
. 2
we have then the "usual" result that uE + u in L (0,T;V) weakly, where u is the solution of
au at + au
=
f subject to appropriate
boundary conditions and where 2 is computed as follows. One considers the operator with coefficients (4.34)
X t a. .(l,I,,v,v,x)
13
E
where k = k3/2.
One homogenizes the corresponding operator according
to the rules of Section 1; therefore we have three cases to consider: k < 2, k = 2 or k > 2,
(4.35)
k3 < 4
,
i.e.
k3 = 4
or
k3 > 4
.
Let us denote by b..(A,iJ,v) the coefficients of the homogenized 17
operator obtained in this way.
We then consider the operator with
coefficients
and we homogenize the corresponding operator: therefore there are again three cases to consider: (4.37)
k2 < 2
,
k2=2
Let us denote by c .
11
k 2 > 2 .
the coefficients of the homogenized operator
(11)
obtained in this way.
or
The coefficients of the operator we are looking
for are then given by 0
I
71 (4.38) 71
o
~
~
~
( = ~dij~
.)
d
7
~
333
COMMENTS AND PROBLEMS The proof is technically long but the ideas are those given in Chapters 1 and 2 . In case of coefficients of the form
one will apply the same rule to aij(x0,y,z,t0,~l,~2,T3) and one will obtain coefficients d . . (xo,to). Then the homogenized operator is 11
d . .(x,t). 11
Homogenization with rapidly oscillating potentials. One can extend the considerations of Chapter 1, Section 12, to the evolution problems, and this for parabolic and hyperbolic equations, and also forschrodinger equations.
For instance let uE
be the solution of
-at(4.40)
AuE + -1 W Eu
uE = 0 on
C
=
f
O
in
E
E
,
uE(xIO) = 0
Q
= Ox
lO,T[
,
.
2 We assume that WE = W(x/E,t/E ) (one can also consider 2) ) is almost W(X/E,~/Ek ) , k # 2 , and a function W (x,~,x/E,~/E periodic. y and in
Then one considers the almost periodic solution T)
and we set
of
=
average in y and A
Then
T
of Wx A
x
(a.p. in
334
EVOLUTION OPERATORS
(4.42
uE
.+
u in L2(0,T;Hi(8)) weakly
,
where u is the solution of
cf. Bensoussan, Lions, Papanicolaou [21 (where one will find analogous formulas also for the hyperbolic case).
Homogenization an8 penalty. Let us consider
(4.44)
where the aijk(y,r) satisfy the usual hypotheses.
We consider the
system : aUEl
at
+
1 (u A h 1 + 2 E
l
-
UE2) =
1'
(4.45)
with boundary conditions, say the Dirichlet's boundary conditions but this is without importance, (4.46)
uEl = uE2 = 0 on E
and (4.47)
UEl(X,0) = Uc2(X,O) = 0
.
This problem admits a unique solution.
Multiplying the first
(respectively, second) equation (4.45) by uEl (respectively, uc2) orLe obtains:
335
COMMENTS AND PROBLEMS
Hence it follows that
and
-
1 (uEl
(4.49)
uE2) is bounded in L2 (0,T;H)
Estimate (4.49) comes from the penalty terms 5 (4.35).
.
1 7
(uEl
-
uE2) in
It follows from (4.38), (4.39) that one can extract a
subsequence still denoted by uEi such that uEi
(4.50)
+
L2 (0,T;V) weakly (where we have the same
u
limit for uEl and for uE2)
.
The limit u is the solution of the following homogenized We give the formulas only for the case k = 2.
problem.
Alj, A2j by
and
{;x
x
;I
A:
= E-2~11
A :
=
E-2
+
E-1~12
+
,
E0
A21 + E - ~ +A E~ 0A23 ~
is the solution which is Y - T ~periodic of
Then the equation for u is (4.52)
au
2 - at -
qij
a 2u ZQT
j
= fl + f2
#
One defines
336
EVOLUTION OPERATORS
where (4.53)
qij =
nm
1
ax;
a i k l - K + aij2
-
aik2
$1
.
The proof proceeds along the lines of Section 2. In (4.52) the boundary condition is (4.54)
u = O o n C .
If in (4.45) one takes boundary conditions which are the same for u
El and for uc2, that is if we take a variational formulation with a space (4.55)
V = W
x
W
,
Hi(0)
5 W 5 H1(S) ,
(with the same W), the result extends:
u will be characterized by
u(t) E W and
In case we take V = V1
x
Vz with different spaces V
and V 12
one obtains (4.57)
u(t)
E
v1
n
v2 ,
and
Problems where one has rapi ly oscillating coefficients in t an a penalty term are studied by Simonenko [l], [21. One can also study the evolution problems with degenerate elliptic part and also with domains with "periodic holes".
Homogenization and regularization. It is known that the solution of a parabolic equation can be obtained as the limit of the solution of elliptic equations, via the
COMMENTS AND PROBLEMS
337
elliptic regularization. Applying this idea to operators of the type
a + A' at
leads to the behavior as
(4.58)
-Em
aLuc
au
at2
at
-- k E + A E u
Ue(O) = 0
E +
0 of equations of the type
= f ,
au EaT( T ) = 0
,
.
The behavior of uE will depend on m and on k, if ' A k cients a. . ( x / ~t/E , )
.
13
is with coeffi-
For instance if we consider
(4.59)
-€
2 3% at2
au au +at > - -axia aij ( E L2 ) E ax ' f , E'
j
then the limit value u of uE is solution of a new parabolic problem obtained in the following manner.
One defines $J(y,.r)as the Y - T ~
periodic solution of
Then (4.61)
& + B u = f at
where B u =
-
,
77i
The results presented for the homogenization of V.1 are sketchy. Other results are announced in the first note of the authors in the Bibliography of this chapter. The problem studied in Section 3.9 corresponds to operators which have been studied (without homogenization) by Showalter and Ting [l]; cf. also other equations of this type, or which present somewhat analogous properties, in Carroll-Showalter 111. results for the homogenization of V . I .
Some
connected with these operators
338
EVOLUTION OPERATORS
are given in the first note of the authors in the Bibliography.
Many
questions arise for these equations. It would be interesting to understand "intrinsically" when the homogenization of partial differential operators leads to non-local operators in the limit (this question could also have a physical interest).
For instance let us
consider the equation
[bij (y) "aYj -1
BE given as in Section 3.9 and the aijvs not necessarily elliptic. We set:
B1 =
a - ayi
and let us assume that there exists
a function ~ j ( ~ , rwhich ) is Y-r0 periodic and which satisfies
"
(i.e., 0
aa.. ay,
(y,-c)dT= 0). Then the homogenized equation (found by
asymptotic expansion) is formally
For the homogenization of first order hyperbolic systems, we refer to Chapter 4.
One can also study (cf. Bensoussan, Lions and Papanicolaou
[ 3 1 ) a case when one has simultaneously "homogenization" and
"penalty":
COMMENTS AND PROBLEMS
339
a u ~ 2 aE a u ~ 2 bE E ax 2 (UEl - UE2) = f2 at (4.64)
(0,l)
x E
,
UEl(O,t) = 0
u (x,O) = 0 Ei
,
,
aE,bE 1. a > 0
1 m - -+ p in L (0,l) weak star aE
,
v in L (0,l) weak star
.
(4.65) -+
afi at E L2((0,1) x Then, assuming that fi, __ m
UEi
-+
UE2(l,t) = 0
,
,
aE,bE E Lm(O,l)
aC
I
,
(O,T)), one has
u in L (0,T;L2(0,1)) weak star
,
(same limit u for i = 1,2) and where u is the solution of the parabolic equation
(4.66)
In ( 4 . 6 4 ) we have only ( 4 . 6 5 ) ,
without periodic, or almost
periodic, structure. For problems of this type for parabolic equations, we refer to Markov and Oleinik [l]. Very many questions arise in connection with the homogenization of second order hyperbolic operators. the text.
Some of them are indicated in
We shall return to this question in Chapter 4 .
For a
340
EVOLUTION OPERATORS
recent work on the regularity of the solution of hyperbolic equations with irregular coefficients we refer to Colombini 1 1 1 . One can also study the homogenization of coupled hyperbolic parabolic systems: cf. Bensoussan, Lions, Papanicolaou [l]. Another problem is the following. Consider the equation
where k E i (4.68)
E
Lm( a ) , k :
-+
Gl
in Lm( 8 ) weak star
.
We assume that (4.69)
kT,B>O
and that (4.70)
kz 1. 0 (but kh can be zero)
.
In (4.67) A is a fixed second order elliptic operator.
Then one
can add to (4.67) standard boundary conditions on uE and for the initial conditions
(4.72)
k9
at
auE where uo and v1 are given: (4.72) gives a condition on ar (X,O) only on the set where ki # 0. This is an equation of "hyperbolic-
parabolic" type.
Cf. V. N. Vragov [l] for the case of fixed
coefficients kl, k2.
One can show the following (cf. Bensoussan,
Lions, Papanicolaou [4]): we do notrestrict the generality in assuming that
341
COMMENTS AND PROBLEMS
@* x
( 4 . 73)
in Lm(B ) weak star
.
Then us * u in Lm(O,T;V) weak star, ui * u' in L2 (0,T;H) weak star, where u is the solution of
+
(4.74)
i2u"
(4.75)
u(0) = u0 ,
(4.76)
u'(0) =
1u'
+
;1L Vl
Au = f
,
.
k2 We remark that in ( 4 . 7 6 ) , domain 0
.
u' (x,O) = u' ( 0 ) is given over the whole
Comparing with ( 4 . 7 2 )
we see that there is, here, some
sort of "increase" in the initial data. Other evolution equations are studied from the new point of homogenization in separated papers by the authors.
For the transport
equations, cf. already the authors' [51 and the Bibliography therein. For other physical aspects of these questions, cf. Sanchez-
.
Palencia [l]
We also refer to the survey of O.A. Oleinik [I], to appear in the Ouspechi Mat. Nauk. We also mention applications of the asymptotic expansion method to problems in turbulence?
.
-t P. Perrier and 0. Pironneau, C.R. Acad. Sc. Paris, 1978.
BIBLIOGRAPHY OF CHAPTER 2 A. Bensoussan, J.L; Lions and G. Papanicolaou [l] Sur quelques problbmes asymptotiques d'dvolution. C.R.A.S. Paris, t. 281 (1975), p. 317-322. [2] Sur la convergence d'opgrateurs diffgrentiels avec potentiel oscillant, C.R.A.S., Paris, February 7, 1977. 131 Remarques sur le comportement asymptotique de systsmes d'6volution. French-Japan Symposium, Tokyo, September 1976. [ E l Perturbations et "augmentation" des conditions initiales. Lyon, December 1976. 151 Boundary layers and homogenization of transport processes. R.I.M.S. Kyoto, 1978. N.N. Bogoliubov and Y.A. Mitropolsky [l] Asymptotic methods in the theory of non linear oscillations. (Translated from the Russian) Hindustan Pub. Corp. Delhi 1961. R.W. Carroll and R.E. Showalter [l] Singular and degenerate Cauchy problems. Math. in Sc. and Eng., Vol. 127, Acad. Press 1976. F. Colombini [l] On the regularity of solutions of hyperbolic equations with discontinuous coefficients variable in time. Scuola Normale Superiore Report, 1976. F. Colombini and S. Spagnolo 111 Sur la convergence de solutions d'gquations paraboliques. To appear. G. Duvaut and J.L. Lions [l] Les Idquations en Mdcanique et en Physique, Paris, Dunod 1972. English translation, Springer, 1975 J.L. Lions [l] Equations differentielles op6rationnelles et problsmes aux limites. Springer, 1961. [2] Lectures on elliptic differential equations. Tata Institute, Bombay, 1957. J.L. Lions and E. Magenes 111 ProblSmes aux limites non homogenes et applications. Vol. 1, 2, Paris, Dunod, 1968. English translation, Springer 1970. P. Marcellini and C. Sbordone [l] An approach to the asymptotic behaviour of elliptic parabolic operators. To appear A. Marino and S. Spagnolo [l] Un tipo di approssimazione dell' operatore Ann. S.N. Sup. Pisa, XXIII (1969), p. 657-673. V . G . Markov and O.A. Qleinik [l] On the heat diffusion in a one dimensional dispersive medium. P.M.M. 39 (19751, p. 1073-1081.
...
342
BIBLIOGRAPHY
343
L. Nirenberq 111 Remarks on strongly elliptic partial differential equations. C.P.A.M. 8 ( 1 9 5 5 ) , p. 6 4 8 - 6 7 6 . O.A. Oleinik [l] Survey on homogenization. Ouspechi Mat. Nauk. to appear. G. Pulvirenti [ l l Sulla sommabilits Lp.. Le Matematiche. 2 2 ( 1 9 7 1 ) , p. 2 5 0 - 2 6 5 . [ 2 1 Ancora sulla sommabilits Lp.. Le Matematiche, 2 3 ( 1 9 6 8 ) , p. 1 6 0 - 1 6 5 . E. Sanchez-Palencia 111 Comportement local et macroscopique d'un type de milieux physiques h6t6rogSne.s. Int. J. Enqrg. Sciences, 1 2 ( 1 9 7 h ) , p. 3 3 1 - 3 5 1 . C. Sbordone [ l l Sulla G-convergenza di equazioni ellittiche e paraboliche. Ric. di Mat. 2 4 ( 1 9 7 5 ) , p. 7 6 - 1 3 6 . R.E.Showalter and T.W. Ting [l] Pseudo parabolic partial dif-
.
.
ferential equations. SIAM J. Math. Anal. 1 ( 1 9 7 0 ) , p. 1 - 2 6 . L. de Simon [l] Sull'equazione delle onde con termine noto periodico. Ren. 1st. di Mat. Univ. Trieste, Vol. 1, fasc. 11, ( 1 9 6 9 ) , p. 1 5 0 - 1 6 2 . I.B. Simonenko [ l ] A justification of the averaging method for abstract parabolic equations. Mat. Sbornik. ( 1 2 3 ) ( 1 9 7 0 ) , p. 5 3 - 6 1 . [ 2 ] A justi€ication of the averaging method ..., Mat. Sbornik, ( 1 2 9 ) ( 1 9 7 2 ) , p. 2 4 5 - 2 6 3 . S. Spaqnolo [l] Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore. Ann. Sc. Norm. Sup. Pisa, 2 1 ( 1 9 6 7 1 , 657-699. [ 2 ] Sulla convergenza di soluzioni di equazioni pakaboliche ed ellittiche. Ann. Sc. Norm. Sup. Pisa, 2 2 ( 1 9 6 8 ) , 5 7 1 - 5 9 7 . V.N. Vragov [ l ] On a mixed problem for a class of hyperbolicparabolic equations. Doklady, 2 2 2 ( 1 9 7 4 ) , Soviet Math. Doklady, 16 (1975),
1179-1183.
:
Evolution Operators
Orientation. The reader mostly interested in Probabilistic Methods (Chapter 3) should read Section 1 and Sections 2.1 to 2.5, and then proceed with Chapter 3. The reader mostly interested in High Frequency wave propagation should read Sections 3.1 to 3.5 and Section 3 . 8 ,
and then proceed
with Chapter 4 . Sections 1 and 2 treat the case of parabolic operators with coefficients rapidly varying in the space and in the time variables, with different scales. The method of asymptotic expansion shows that there are three cases to be considered. Proofs of convergence are given in Section 2; Sections 2 . 6
and 2 . 7 give the proof under the
weakest possibly hypothesis on the coefficients and are somewhat complicated. Section 3 studies the operators of hyperbolic type, or of Petrowsky or of Schrodinger type. Some non linear problems are considered in Sections 2.12 and 3.7. Section 3.9 gives an example (somewhat connected with Chapter 1, Section 15)
of a situation where
local operators,of evolution admit for homogenized operator a non local evolution operator. 233
234
EVOLUTION OPERATORS Many other situations can be studied by using ideas and
techniques of Chapters 1 and 2. Some of them are briefly indicated in Section 4 (comments). Another situation will be studied by these techniques in Chapter 3, Section 5 (since the motivation for such a study appears in the Probabilistic context).
1.
Parabolic operators.
1.1
Asymptotic expansions.
Notations and orientation. Let 8 be a bounded (to fix ideas!) open set of Rn, with boundary
r
(which will be assumed smooth or
not, depending on the situation).
Let T > 0 be a given finite number.
We set
We will consider first, equations of the following type:
with initial condition (1.3)
UE(X,0)
=
u 0 (x)
,
x E 8
,
and with suitable boundary conditions, defined in Section 1.2 below. On the functions aij we shall always assume that . ! a
17
remains in a bounded set of Lm( S 1
(1.4)
aTj(x,t)cicj 2 a
~
in , ~ a.e. c ~
s ,
, a >
o ,
si E
R
.
In order to obtain constructive formulas, we shall assume much more on the structure of functions aE namely, ij;
PARABOLIC OPERATORS: (1.5)
ASYMPTOTIC EXPANSIONS
aE (x,t) = aij(x/E.t/E k ) ij
235
,
where k > 0 is for the moment unspecified, and where: aij(y,T) is Y-r0 periodic
aij(y,T)ciEj
clcici
,
, t
a.e. in y and
We want to study the behavior of uE
E
-+
T
.
0.
We begin by giving a precise definition of uE (Section 1.2) and we study next the asymptotic expansion of uE.
As we shall see the
formulas obtained depend on k (which appears in (1.5)).
Remark 1.1. We shall explain in the comments on Section 4 the results one of the form
obtains when considering functions a:j (1.7)
aij (x,t) = aij (x,x/E,~,~/E k)
and more generally aij(x,t) = aij(x,-,x x
(1.8
E
1.2
,...,t,-t
t EkF1--*) *
E2
Variational formulation. We need some notation.
We shall use, as in Chapter 1, spaces
V such that
'That
is aij(y,r) is Y periodic as a function of y and aij(y,r
+
T
0
)
=
a. .(y,r) , 17
.
a.e. in y , ~
EVOLUTION OPERATORS
236
Given a H i l b e r t s p a c e L ~ ( O , T ; 8 ) t h e s p a c e of (0,T)
-t
( o r a Banach space!) w e d e n o t e by
(classes o f ) functions t
-t
v ( t ) from
which a r e measurable and which s a t i s f y
8
= ess.supl l u ( t )
II
8
Provided w i t h t h e norm g i v e n i n ( l . l O ) ,
<
m
,
Lp(0,T;8
if p = )
m
.
is a
Banach s p a c e (and it i s a H i l b e r t s p a c e i f p = 2 and i f M i s a Hilbert space). W e s h a l l set
(1.11)
2 H = L ( 0 )
,
V C H C V '
and w e o b s e r v e t h a t
1 For u , v E H ( C ? ) , w e d e f i n e : (1.12)
a'(u,v)
=
au av dx (x,t) ax. 3 axi
.
W e have t o be c a r e f u l s i n c e a E( u , v ) depends on t and s i n c e , even
i f t h e f u n c t i o n s a . 's a r e smooth ( f o r e v e r y E ) t h e t - d e r i v a t i v e s of lj
t o t h e s t r u c t u r e of a . given i n (1.5) 1j t h a t , i n what f o l l o w s , t h e a p r i o r i e s t i m a t e s s h o u l d n e v e r u s e t h e a i j ' s are unbounded-due
d a E( u , v ) . dt For u
E
V w e d e f i n e A'u
E
V'
by
PARABOLIC OPERATORS:
I n f a c t A'
ASYMPTOTIC EXPANSIONS
231
depends on t :
(1.14)
AE = A E ( t )
.
T h e " a b s t r a c t " f o r m u l a t i o n of t h e p r o b l e m w e w a n t t o s t u d y i s
l e t f a n d u o be g i v e n , s a t i s f y i n g
now: (1.15)
f E L2(0,T;V')
(1.16)
uo
Then
E
H
,
.
u E i s t h e s o l u t i o n of 2 L (0,T:V)
(1.17)
u
(1.19)
UE(0) = u
E
0
,
.
Remark 1.2. 2 I f v E L (0,T;V)
a bounded s e t of L m ( S
then AEv )
E
2 L (0,T;V')
and s i n c e a .
11
remains in
,
w h e r e here and i n w h a t f o l l o w s , t h e c ' s d e n o t e v a r i o u s c o n s t a n t s w h i c h do n o t depend on
E.
T h e r e f o r e (1.18) gives:
238
EVOLUTION OPERATORS
I c[ I
(1.22)
IUE
II
+ 11
.
L2 (0,T;V) But it is known (cf. for instance Lions-Magenes [l]) that: (1.23)
if v
6
2 L (0,T;V) and
2
L (O,T;V'), then v
is a.e. equal to a function, still denoted by v, such that t
+
v(t) is continuous from [O,Tl
+
H.
Therefore we see that (1.17), (1.21) imply that (1.19) makes sense.
Remark 1.3. It is proven in Lions [l] that the problem (1.171, (1.18), (1.19) admits a unique solution. One has the estimate
The verification of ( 1 . 2 4 ) is easy. (1.18) by uE.
We multiply both sides of
We denote by (f,g) the scalar product in H and we set
But we observe thatt
'This @ E
2
is obvious if
L (0,T;V) and
2
E
@
is smooth and it can be justified if
L'(0,T;V').
PARABOLIC OPERATORS:
2 39
ASYMPTOTIC EXPANSIONS
so t h a t (1.26) g i v e s ( w e w r i t e a'($)
for
Hence t
b But d u e t o (1.4), w e c a n f i n d A t (1.28)
aE
V)
2 a1
W e d e n o t e by
1\71
l2 -
I 1 I I*
A v/
2
such t h a t
,
v
t h e norm i n V'
.
E
V
.
W e o b t a i n from ( .27) 2
+;
/
IluEl
2do
'
0
1 +One c a n t a k e X = 0 i f V = H O ( 0 ) or i f V = s p a c e w h i c h d o e s not contain the constants.
240
EVOLUTION OPERATORS
Using G r o n w a l l ' s lemma,
(1.24)
follows.
Remark 1 . 4 . One c a n r e p l a c e ( 1 . 1 8 ) by t h e e q u i v a l e n t f o r m u l a :
1
V = Ho(B').
Example 1.1:
Then ( 1 . 1 7 ) i m p l i e s (1.31)
u
= 0 on C
.
I t i s t h e D i r i c h l e t ' s boundary v a l u e problem.
Example 1 . 2 :
V
1 = H (8).
Then one deduces from (1.30) t h a t , i n a "weak" s e n s e , (1.32)
au
2 = 0 on C
where
a
-=
av
(1.33)
a
a?.(x,t)v. 11 7 axi
,
AE
v = { v l , . . . , ~ n } = normal t o
r
d i r e c t e d toward t h e e x t e r i o r
PARABOLIC OPERATORS:
ASYMPTOTIC EXPANSIONS
241
I t i s t h e Neumann's b o u n d a r y v a l u e p r o b l e m .
Remark 1 . 5 . ( 1 . 2 4 ) w e see t h a t w e c a n e x t r a c t a s u b s e q u e n c e ,
By v i r t u e o f
s t i l l d e n o t e d by u E , s u c h t h a t (1.34)
u
2 u i n L (0,T;V) w e a k l y
-+
au
$ is
(and ( 1 . 2 4 ) ) ,
Using ( 1 . 2 2
.
2 bounded i n L ( 0 , T ; V ' ) and t h e r e f o r e
w e c a n assume t h a t +
I t i s known t h a t ,
.
i n L2 ( 0 , T ; V ' ) weakly
when
V
-+
H i s compact, it f o l l o w s from ( 1 . 3 4 ) ,
(1.35) t h a t (1.36)
uE
+
.
2 u i n L (0,T;H) s t r o n g l y
I f we set
w e c a n a l s o assume t h a t (1.38)
6;
+
ci
2 i n L (0,T;H) w e a k l y
.
One v e r i f i e s t h a t , i f a ? . i s g i v e n by ( 1 . 5 ) , w i t h ( 1 . 6 ) , t h e n 13 (1.39)
a!.
13
+
where i n g e n e r a l
74 ( a i j )
&
L m ( S ) weak s t a r
,
EVOLUTION OPERATORS
242
topology).
k).
ci
m
( a1. 3. ) &.a x . ( i n any 3 W e s h a l l o b t a i n below t h e l i m i t of ST (which depends on
But i t d o e s n o t f o l l o w t h a t
-+
The o n l y t h i n g w e c a n s a y f o r t h e t i m e b e i n g i s t h a t
(1.41)
at -
Remark 1 . 6 .
ac. axi
.
= f
The case of c o e f f i c i e n t s a:. i n d e p e n d e n t of t . 13
L e t u s assume t h a t
(1.42)
,
a ? . = a . . (x/E) 13 =I
where t h e a i j ' s
a r e Y-periodic
( i . e . w i t h t h e hypotheses of
Chapter 1 ) . Then t h e answer i s immediate.
Let
g e n i z e d o p e r a t o r of A E , c o n s t r u c t e d i n Chapter 1.
(1.44)
% at +
Q
be t h e homo-
=
,
4 u = f
d e f i n e s t h e l i m i t of u E . m
I n d e e d , l e t $ be g i v e n in?,,
(10,T[). For any f u n c t i o n
v E L ~ ( O , T ; li ) , w e d e f i n e
W e remark t h a t (1.47)
au
a t (0) =
-u
E
d0 (-)
dt
s o t h a t w e deduce from ( 1 . 1 8 ) t h a t
PARABOLIC OPERATORS:
But since
AE
ASYMPTOTIC EXPANSIONS
does not depend on t, (A'u,)
(@) =
AE (uE( $ ) ) and
therefore
(we use (1.36)).
We also know that
(1.50)
u($) in V weakly
uE($)
+
.
hence (1.44) and the result follow.
1.3
Asymptotic expansions.
Preliminary formulas.
We shall consider the three most interesting cases for k appearing in (1.7). P E r k=
In order to avoid confusion, we shall set:
a +
A
(x)
E t E
(1.51) E
and we shall consider the values (1.52)
k = 1,2,3
We shall s e t
'I
. = t/Ek and
243
244
EVOLUTION OPERATORS
.5
all these operators depend on (1.54)
PErl =
Q1+E
E - ~
It follows that
parametrically.
T
-1 0 Q2+EQ3r
where (1.55)
Q1 = A1
(1.56)
PCr2 =
,
Q2
=
-
R1 ~ +
E
E
a?
-lR2 +
E
a+ A 3 Q 3 = -at
,
a + A2 0
;
R3 ,
where
a +
A1
,
,
(1.57)
R1 =
R2 = A2
(1.58)
P E r 3 = E - ~ +S E~-2 s2 + €-Is3
R3 =
+
E
0
a +
A3 ;
s4 ,
where
a ,
S1 = aT
(1.59)
S2 =
A1
,
S3 =
,
A2
S4 =
a +
.
A3
We use now these formulas to find an asymptotic expansion of uE.
We distinguish the three cases k = 1,2,3. ' We look for uE in
the form (1.60)
u
€
=
u
0
+
EU1
+
E2U2
+
. ..
,
U. =
u . being Y periodic in y and 3
'Another
3
T~
U.(X,Y,t,T)
,
periodic in
T
3
.
ansatz will be necessary when k is not an integer.
PARABOLIC OPERATORS: 1.4
Asymptotic expansions:
245
ASYMPTOTIC EXPANSIONS
The case k = 1.
By identifying powers of
in the expansion of PEV1u = f , we
E
obtain:
,
(1.61)
QluO = 0
(1.62)
QlUl + Q2UO = 0
(1.63)
Q1u2
t
+ Q2u1 + Q3uo
=
f ,
... .
In (1.61), x , ~ , Tare parameters, so that (1.61) is equivalent to (1.64)
uo does not depend on y
We denote by tively,
T ) so
m
Y
.
(respectively,
that ??I
=
h
T O
the average in y (respec-
T)
Y
=
.
By taking
n
of
(1.62), we obtain
i.e., since we have (1.641, (1.65)
uo = u(x,t)
auO
=
0 and therefore
.
Then (1.62) reduces to: (1.66)
Alul = -A2u =
au a , .( y , ~ )+ aYi a11 ax j
(x,t)
.
We can write the general solution of (1.66). We introduce
XJ
=
XI ( y , ~ )as the Y-periodic solution, defined up to an additive
constant, of
and we choose the constants such that (1.68)
xJ(Y,T)
+ T~
periodic
;
246
EVOLUTION OPERATORS
(for instance we can take ?!4
Y
( ~ J ( Y , T )=) 0, v
T ) .
Then (1.66) gives (1.69)
.
ax
au u1 = - ~ J ( Y , T ) (x,t) + Gl(x,t,~) j
We can solve (1.63) for u2 iff,
?4 y(Q2u1 + Q 3 u )
= f
,
i.e., (1.70)
aGl
ar +
?4 Y Q 2 (-XI &) 3
74 y(Q3u)
+
We can solve (1.70) forG1 (with a
T~
=
f
.
periodic function of
T)
i.e.,
We observe that
so that we obtain
(1.71)
au
-
9 [aij - aik
$1
2
axiax. a u = f . 3
This is the homogenized operator (as we shall justify below).
1.5
Asymptotic expansions:
The case k = 2.
We use now (1.56), (1.57).
,
(1.72)
RluO = 0
(1.73)
Rlul + R2uO = 0
,
We obtain
iff
PARABOLIC OPERATORS: (1.74)
R1u2
+ R2ul + R3u0
=
f
ASYMPTOTIC EXPANSIONS
247
.
We multiply (1.72) by uo and we integrate over Y.
Let us set
We obtain
Since uo is
au, aYi
T~
periodic,
'0
0
= 0,
(1.77)
i.
au [g,u0],
d-r = 0, so that (1.76) implies
- + AIUO
auO Then AluO = 0 and a?
:yo
= - - 0,
so that
.
uo = u(x,t)
Then (1.73) reduces to
We introduce 8j
( y , ~ ),
Y - T ~periodic
, solution of
(1.78)
(0'
is defined up to an additive constant).
Then
We can solve (1.74) for u2 ifft
tR1@ = F admits a Y-T,, periodic solution iff m ( F ) = 0
(observe that RZji = 0 admits the constants as only Y - T ~per odic solution).
248
EVOLUTION OPERATORS (R2u1
+ R3uO)
,
= f
i.e., au (1.80)
-
n [ a1.3. -
aej] a ik
ayk
_ .
~
a 2u ax.ax. 1
3
This is the homogenized operator of below).
in
(1.71)
P E P 2 (as
for P E r l ,
but coefficients are different.
The case k = 3 .
We now use (1.58), (1.59).
We obtain:
,
(1.81)
S u = 0 1 0
(1.82)
Slul + S2uo = 0 ,
(1.83)
S1U2 + S2U1 + S3Uo
(1.84)
S1u3 + S2U2 + S3U1 + S4Uo
=
0
t
= f
,
... .
It follows from (1.81) that uo does not depend on a
T
0
we shall justify
It has of course the same structure as the one obtained
Asymptotic expansions:
1.6
.
T:
(1.82) admits
-periodic solution in u1 iff
I
Ts2uo
=
0
,
i.e.,
We define (1.86)
6,
=
a
It is of course.an elliptic operator in Y, so that (1.85) is au equivalent to 2 = 0, V j and therefore uo = u(x,t)'. Then (1.82) aYj
PARABOLIC OPERATORS:
ASYMPTOTIC EXPANSIONS
reduces to __ a-r = 0, i.e. (1.87)
u1 does not depend on
T
.
We can solve (1.83) for u2 iff RT(S2U1 + S3Uo)
=
0
$ 1 , ~ )is Y-periodic
,
,
i.e.
If we introduce @ J ( y )by:
,
(1.88) J
- y . )
= o ,
(which defines tJj up to an additive constant), we see that (1.89)
u1
'
= -@'(y)
au ax. +
u,(x,~)
3
.
Then (1.83) can be solved for u2; if 7
we have (1.91)
0 u2 = u2 + u2(x,y,t)
.
Equation (1.84) can be solved for U3 iff (1.92)
R T ( s2u 2 + s3u1 + s4u 0)
and (1.92) can be solved for (1.93)
R
1.r r(S2U2
+
G2
s3u1
+
= f
iff S4U0) = f
.
249
250
But
EVOLUTION OPERATORS
m'Y
r
(S2u2) =
RT m
Y (Szu2) = 0, so that (1.93) reduces to
We have again the same structure for the homogenized operator, with again different formulas.
1.7
Other form of homogenization formulas. Let us denote by q k. . = homogenized coefficient of 17
k = 1,2,3.
Let us check the following formulas:
where al is defined in (1.75) and where
For (1.961, we will actually prove that
This is sufficient since
a2 u
(-----ax.ax.
1
7
I
for
PARABOLIC OPERATORS:
251
ASYMPTOTIC EXPANSIONS
taking into account that
= o . Therefore, by comparing the expressions (1.71), (1.80), (1.94) with (1.95), (1.96)I , (1.97), we have to verify that
(1.101)
;+$’ - Yj&)
= 0
This is straightforward.
al(xj
-
.
Indeed (1.67) implies that
yj,x i) =
o
for every
T
,
hence (1.99) follows. If we multiply (1.78) by 0
i
and integrate in y and in T, we
obtain (1.100) and (1.101) follows from (1.88). Formulas (1.95), (1.96), (1.97) imply the ellipticity I Of k a’ f f k = -qij axiax j
(1.1021
~
-
Indeed, if we set:
+ we have
=
cj(xj
-
yj)
,
J, =
cj(ej
-
yj)
,
P =
c j @j
252
EVOLUTION OPERATORS
Hence the result follows.
1.8
The role of k. Let us consider now k > 0 not necessarily an integer. We set
(1.106)
ui = solution of (1.171, (1.181, (1.19) relative to AE =
A(t/E
k)
,
and we denote by uk , k = 1,2,3, the solution of (1.107)
*+ k at
,
qkuk = f
(1.108)
Uk E
(1.109)
uk (0) = u0
L2(0,T;V)
,
.
Then we shall see that the limit of uz when k < 2 (respectively, k > 2 ) does not depend on k, i.e. that :u
-f
u1 (in the appropriate
topology) when k < 2 (respectively, -+ u* when k
> 2).
But the
asymptotic expansion depends on k, and actually new "ansatz" will be necessary for the cases when k is not an integer.
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS 2. 2.1
253
Convergence of the homogenization of parabolic equations. Statement of the homogenization result. We shall prove in several steps the
Theorem 2.1. We denote by :u k solution of (1.17), (1.18), (1.19) relative to AE = A(t/E ) We assume that (1.5), (1.6) hold true.
the solution of (1.107), (1.108), (1.1091, k = 1,2,3, where (respectively, 0 ’, respectively,
a
3
)
the k u
a1
is given by (1.71)
(respectively, (1.80), respectively, (1.94)). Also assume that there exists p > 2 such that f E Lp(0,T;W-1rp(C3) ) and that uo or at least that f E
-+
and
E
uo belong locally to these spaces.+
WitP(b 1 , Then, as
0, one has
(2.1)
:u
(2.2)
u :
(2.3)
:u
+
-+
-+
u1
in L2 (0,T;V) weakly if k < 2
,
u2
2 in L (0,T;V) weakly if k = 2
,
u3
2 in L (0,T;V) weakly if k > 2
.
We begin by the case (2.2) which is the simplest; we shall next proceed with (2.1) and (2.3). 2.2
Proof of the homogenization when k = 2. Given P(y) = homogeneous polynomial in y of first degree, we
define w(y,.r) as the solution (defined up to an additive constant) of
t1.e. if ~3
1
c
5
I
c
0 , f E LP(O,T;W-~’P(O I ) ) ,
This hypothesis is not necessary if k = 2 or if k # 1 (cf. (2.22) below).
uo E W’~P(S aij E C
0
1 ) .
254
EVOLUTION OPERATORS
- -:y +
(2.4)
,
AIw = 0
-
w
P i s Y-r0 p e r i o d i c
.
We write:
(2.5)
AE(t/E
2
) = AE
,
u2 = u E
E
.
I f w e set
*
(2.6)
w - P = - ~
then
* * * - -aare + A1(B -
(2.7)
,
P) = 0
b e i n g Y-ro p e r i o d i c
O*
.
W e now d e f i n e wE by
w (x,t) =
(2.8)
2
EW(X/E,~/E
.
)
W e observe t h a t
- -aa wt E + AE*wE =
(2.9)
For any (0 E C ? z ( S ) , and ( 2 . 9 ) by $ u E .
s
.
0 =
8
x
] O , T [ , w e m u l t i p l y (1.18) by $wE
W e i n t e g r a t e over
s
;
w e o b t a i n (compare w i t h
C h a p t e r 1, S e c t i o n 2 ) :
+
[[>]@w,
(2.10)
s +
Is
[c:
[?]$u,]dxdt
wE
-
aij
2$
uE]dxdt =
j
f$wEdxdt
.
0
The f i r s t term i n ( 2 . 1 0 ) e q u a l s (2.11)
Since w
+.
P in L
2
( s ) s t r o n g l y and
t o t h e 1 m i t i n a l l terms of
s i n c e w e have (1.36
(2.10), ( 2 . 1 1 ) .
W e obtain
I
w e c a n pass
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
(2.12)
I
-
UP
Q =
i
a+
dxdt
‘
+
s f$Pdxdt
K )* axj u l d x d t
( a i j aw
.
n
Q
Using (1.41) w e see t h a t
J s
dxdt
+
s
sJ
5,
so t h a t (2.12) r e d u c e s t o
Therefore
(2.13)
k)au.
6. ap = 1
311 ( a k j ayk
ax,
ax j
L e t us take
W e define 8
(2.14)
i* by
- -aei* ar
+
A;(ei*
ei* b e i n g Y - r o Then w = -(Bi*
or
-
-
yi) = 0
periodic
,
.
yi) and (2.13) g i v e s :
a ax. l
($P)dxdt
8
255
256
EVOLUTION OPERATORS
(2.15)
The variational formulation of the problem is:
[>,v]
+
[ti&]
= (f,v)
,
v v
E
v ,
,
\J
v
E
v
and passing to the limit we find (2.16)
au (=,V)
av + (5.,-) 1 axi
= (f,v)
This proves the Theorem (for k
=
.
2) if we verify that we obtain the
same formula than in (1.80), i.e. that
This is equivalent to proving that
or that
We multiply (2.14) by 8 1 ,
(1.78) by Bi*.
This shows that the
left hand side of (2.19) equals
and that the right hand side of (2.19) equals
and the two expressions ( 2 . 2 0 ) ,
(2.21)
are indeed equal.
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
257
Orientation. We consider now the cases when k # 2.
We begin with the study
of this question under the extra hypothesis (2.22)
2.3
,
a . E Co(F x [ O , - r 0 1 ) ij
.
periodic
Reduction to the smooth case. We are going to show
Lemma 2.1. Under the supplementary hypothesis (2.22), it suffices to prove Theorem 2.1 under the hypothesis (2.23)
aij
E
C"(Y
,
x [O,.rO1)
periodic together with all its derivatives
.
Proof.
of (1.30), k fixed arbitrarily. [ O , T ~ ] ) by j!a E C"(Y x [ O , T ~ ~ ) aij ,B
We denote by uE the solution !u
We approximate aij in C"(Y x being periodic together with all its derivatives and satisfying (2.24)
agjcitj 5 a l ~ i t iI
al > 0
,
y,T
b!
.
B k We denote by uE (not to be confused with uE) the solution in
L'(o,T;v) (2.25) where
of aUE
IK,v]
+ aEB(u!,v)
=
(f,v)
,
tl
v
E
v
EVOLUTION OPERATORS
250
We shall check below that
B
(2.27)
HUE
-
UEII
L2 (O',T;V)
< -
c
B supllaij i,j
-
a
iJ
11
co(iX
Let us consider now the case k = 1 to fix ideas.
[o,Tol)
.
We define
x j r B by A1(~JfB y.) = 0 7
where A!
=
- a aYi
(aBj(y,T)
"1,
,
aYj
xJtB
Y - T ~
One .,bows easily that (normalizing x
n Y (XI)
= 0,
nY
,
and we define
-
for instance
periodic
=
o
yi)d.r
.
J , ~ 1 in ' the ~ same manner,
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
259
It follows from (2.27), (2.28), (2.29) that it is enough to 2
prove that u l P B + u l t B in L (0,T;V) weakly when E
E
+
0, 6
fixed.
The same remarks apply to all values of k, so that it remains only to prove (2.27), which is standard. m
E
=u'-u E
Indeed if we set
E
we have
where
so that
and therefore
I IfE Thus (2.27) fol ows from (1.24).
2.4
Proof of the homogenization when 0 < k < 2 . We now prove (2.1) under the assumption (2.22), by Lemma 2.1.
We can therefore assume that ( 2 . 2 3 ) holds true.
260
EVOLUTION OPERATORS
2.4.1
The case 0 < k
5 1.
Let P(y) be given as in Section 2.2.
Let w be the solution
(defined up to an additive constant) of (2.30)
*
A1w = 0
,
w
-
P(y)
is Y - T ~periodic
k We use notation (2.5) and we set uE
-
w
=
uE.
.
If we set
p = -x*
then (2.31)
*
A1(X*
-
P)
=
0
,
x* being Y - T ~periodic
.
We now define w E by
.
x -1t w (x,t) = w(-
E 'Ek
We observe that AE*wE = 0
.
The same procedure as for (2.10) leads to
f@wEdxdt
=
.
n
l4
The first term in (2.32) equals (2.33)
-
I
s
uEwE
2 dxdt -
aw
s
Since we have assumed the coefficients aij to be smooth, it follows from (2.30) that
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
261
so that we have in particular
11
uE@
at awE
s
awE If k = 1, =
aT
dxdtl 5 CE'-~
[ElEand aw
+
0
,
aw
if k < 1
(=)=
+
.
0 in Lm(
s)
weak star
so that
I
uE@ at dxdt awE
+
0
s
.
In all cases, the limit of (2.33) is
-
UP
s SO
2 dxdt ,
that (2.41) gives in the limit
-
I
UP
s
2 dxdt + I f@Pdxdt
=
.
[ci
P
-
s
s This is the same formula as (2.12), with a different w. i* by obtain (2.13) and if we define x (2.341 then
A:(xi*
-
y.)
=
0
,
xi* being Y-periodic
,
We
262
EVOLUTION OPERATORS
(2.35)
ci
=
7/1 [aij
-
a
The Theorem is proven, provided we verify that (2.35) furnishes the same formula as in (1.71), i.e. that
The verification is made along similar lines as for (2.17).
The case 1 < k < 3/2.
2.4.2
We prove now the result for 1 < k < 3/2 and we shall indicate next how to proceed when k
+
2.
We begin by general considerations on the asymptotic expansions. With the notation (1.51) we have (2.36)
P E r k = €-'A1
+
E - ~ A+ ~E O ( A ~+ =) a +
E -k
a
We are looking for an asymptotic expansion of uE where PErkuE= f. The usual expansion using terms
E ~ U (x . ,y ,t,T)
7
is impossible
with-
out additional terms, since there is otherwise no term to "compensate" - ~ would appear.t the powers ~ j which Therefore it is natural to look for u UE =
uo + EU1
+
E2u2
+
... +
where all €unctions depend on x, y, t, T ~ .
in the form:
P v 0 T
+ E3-kvl +
...
,
and are periodic in y and in
But if we use the above "ansatz" for uE, we do not have T deriva-
tives coming in the identification for uo, ul, u2 and one easily
tExcept if functions u fication is impossible.
j
do not depend on
T,
but then the identi-
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
263
checks that the computations are impossible. Therefore one should add k to the above "ansatz",terms of the type E wo + ~ ~ + + so ~ that w
...,
finally we look for uE in the form UE
=
(2.37)
+
uo
+
Ell1
€kWO
+
+ E 2u2 +
... +
+
... .
Ek+1Wl
Identifying the coefficients of
0
+
E-',
E - ~ ,
E3-kvl +
E'
...
gives
,
(2.38)
AluO = 0
(2.39)
Alul + A2u0 = 0
(2.40)
A1U2
+
€2-kv
A2U1
+
,
A3u0
au, + aw, = +at a-r
Identifying next the coefficients of
E
f
.
-k
,
E
1-k
I
E
-k-2
, E -k-1,
we obtain
(2.43)
AlwO = 0
(2.44)
Alwl + A2w0 = 0
I
.
It follows from (2.38) that uo does not depend on y; then, by integrating (2.41) in y, we obtain
au,/aT
= 0 and therefore
Then (2.41) reduces to AlvO = 0 and v 0 does not depend on y. If we use the functions x j ( y , ~ )satisfying (1.67) , (1.681, we have from (2.39)
~
264
EVOLUTION OPERATORS
I n o r d e r t o be a b l e t o s o l v e ( 2 . 4 2 ) f o r v1 w e need
Y
[?+
A2vO]dy = 0
.
But s i n c e v o d o e s n o t depend on y ,
J
AZvOdy = 0 , so t h a t t h e o n l y way
Y t o s a t i s f y t h i s c o n d i t i o n i s t o choose x j s u c h t h a t
I
XJ(y,T)dy
f o r every T
0
=
,
Y
and G1 i n d e p e n d e n t o f T. Then we can compute v1 no m a t t e r how w e choose v o .
Therefore w e
choose vo = 0 and v1 i s d e f i n e d (up t o t h e a d d i t i o n of a f u n c t i o n of x , t , T) by
AIVl
-
a
aT
j
au ax = 0 . j
W e see from (2.43) t h a t wo i s i n d e p e n d e n t o f y and t h e n w1
can
be computed by ( 2 . 4 4 ) . I t only remains t o s a t i s f y t o (2.40)
(observe t h a t t h i s i d e n t i t y
i s i m p o s s i b l e t o s a t i s f y i f t h e r e i s no w,,!).
By i n t e g r a t i o n
and i n T w e f i n d t h a t u s a t i s f i e s t o ( 1 . 7 1 ) . "
Summing u p :
&
y
with t h e
a n sa t z " u
(2.45)
= U + E U
+
Ek+1Wl
+ +
EZU2
...
+
... +
E3-kvl
+
... +
E kwo
,
where t h e c o e f f i c i e n t s are computed as above, one h a s
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
PErkuE= f
+
terms of order 2 k
-
1 in
E
265
.
Remark 2.1. 1) We have introduced the above ansatz on a rather "algebraic" basis: another motivation will be obtained in Chapter 3 (probabilistic methods). 2)
One sees from the above computation that the limit of uE does
not depend on k (when 1 < k < 3/2)+ but that the expansion itself depends on k. 3)
For k satisfying
the "ansatz" for the asymptotic expansion will be: = u + E U 1 + E 2u 2 + E 2p+l-pkvo
u
+
... +
EkWO +
... .
We can now proceed with the proof of the Theorem. We construct a function w E such that (2.46)
(-
a
at
+
AE)*w
=
E
k- 1 gE '
2 where g E is bounded in L ( Q ) as 2
L (Y
x
( 0 , ~ ~ )as )
E
-+
E
-+
0 and where wE
+ .
yi in
0.
One looks for wE in the formtt (2.47)
wC = E(CX(Y,T)- yi
+
EZB(Y,T)
+
E3-kA(y,T)
.
tOf course once the above (formal) proof is justified. ttWe do not need in this expansion terms in E~ or in
E
k
.
266
EVOLUTION OPERATORS
One finds: (2.48)
A;(a
-
- -a~ + ar
One defines xi* by (2.34) take
ci =
x i*
*
,
yi) = 0
*
AIB + A2(" A;A
= 0
-
yi) = 0
,
.
j Xi*(y,T)dy
V T.
= 0,
Then one can
Y
and one can compute X from the third equation (2.48).
One computes 6 using the second equation (2.48) and one obtains the result.
2.5
The proof is then completed along the usual lines.
Proof of the homogenization when k > 2.
(Preliminary) Remark 2.2. 1) We are going to prove the theorem only for k = 3.
The proof
for the general case proceeds along the lines of Section 2.4.2. 2) We assume again that (2.22) holds true.
See Section 2.6 for
the general case. According to Lemma 2.1 we assume (2.23). We are going to construct a function wE satisfying
such that as
E -+
0
(2.50)
g,
is bounded in L2 ( S
(2.51)
wE
+
)
I
2 yi = P(y) in L ( S
We use asymptotic expansions
Y = c1
being Y-periodic and B and
T
X/E
being
I
Y-To
periodic
.
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS Using the notations ( 1 . 5 8 1 ,
Since a does not depend on
+
-
(2.54)
S*B
(2.55)
s y + s 2 5 + s
S;(CY
*
Then we find ( 2 . 4 9 )
the coefficient of
E - ~
is zero.
* 3
I
(a-yi) = o .
(assuming we can solve ( 2 . 5 4 ) ,
+
g, = h
(2.56)
Yi) = 0
*
1
T,
we have
and of EO are zero iff
The coefficients of 1
(1.59)
267
EL
*
+
with
,
E2m
*
(2.55))
*
h = S2 y + S3 B + S 4 ( c Y - Y ~ )
L
=
s*y 3
+
*
* S4B
m = S y . 4
I
*
We can solve ( 2 . 5 4 ) for B iff (S1 =
a : - E)
(2.57)
*
*
But S2 = A1 and since a does not depend on
(a
74 A
-
-*
A1(a
We denote by
-
is equivalent to
y.) = 0
, this means
.
(compare to ( 1 . 8 8 ) ) the solution which is Y-periodic
(defined up to an additive constant) of ( 2 . 5 8 ) . (2.59)
(2.57)
yi) = 0
Using the notation ( 1 . 8 6 ) (2.58)
T,
i* a = @
.
Now we can solve ( 2 . 5 4 ) ; if we set
Then
EVOLUTION OPERATORS
268
then (2.61)
B
.
~
=
Bo + B(x,y,t)
We can solve (2.55) for y iff
i.e., (2.62)
i *1g +
mT(S;Bo
+
Sl(a
-
yi)) = 0
,
V
iY-periodic ,
which defines (up to an additive constant) the function All the functions introduced are Cm Then we can solve (2.55) for y.
& I
or in ?
Functions
a,B,y
X
i(y). [o,To]. m
are C
so that
in particular
and
We now use (2.49).
We take $
E
C?z(s )
and, as before, we
multiply the equation for uE by $wE and (2.49) by @uE.
=
\
s
f$wEdxdt
-
E
i s
gE$uEdxdt
We can pass to the limit and we find
.
We find
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
-
I
dxdt +
uw
I
Ci
269
Pdxdt
s
s
and finally (2.64)
Si
=
[
74 aij
-
akj
a$i*
au
T]
It remains only to verify that this formula coincides with what we have obtained in (1.94), which is left to the reader
2.6
Proof of the homogenization formulas when a E Lm ij using LP estimates. for
We show now that the assumption (2.22) Theorem 2.1 to be true.
The hypothesis aij E Lm(Rn x R ) is Y T sufficient. We assume f o r a moment the following estimate: (2.65)
for the Dirichlet's boundary conditions there exists p > 2 ,
independent of
E
,
such that
The proof of this result is given in Section 2.7 below. We introduce q such that
We can find a sequence of functions aB such that ij
2 70
EVOLUTION OPERATORS
(2.67)
B a . .E C " ( Y
x
13
[O,rO1)
,
B is periodic together with aij
all its derivatives , sup
l r 7
B Ilaij
-
aij
I I Lq(y
[o,rol)
= p(B)
*
0 as B
and
* "
.
With the notations of Section 2.3 we show that
Then Lemma 2.1 is still valid and the Theorem is proven. With the notations of the end of Section 2 . 3 we observe that
I (fE,V)I
5
CP(B
so that (2
By virtue of ( 2 65) it follows that (2.70)
I Ifc
so that in part cular
and (2.68) follows. For other boundary conditions than Dirich-st, the analogous of (2.65) is valid locally, i.e. for
S'
C
-
8 ' x lO,T[, 0 ' C
and the analogous of (2.68) is valid locally, i.e. in
8
,
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS 2.7
271
The Lp estimates. We now prove the Lp estimates used in Section 2 . 6 above.
are of independent interest.
They
The method of proof is analogous to the
one used in Chapter 1, Section 4, for elliptic problems.
The
notations are those of this Section 4, Chapter 1; in particular, we use norms ( 4 . 5 ) ,
(4.7),
(4.81,
(4.11)
of that section. We define
provided with the norm
We introduce
and we observe that the mapping v Lp(O,T;W-l'p(O (2.74)
)).
Ilfl l y
p
divx v maps (Lp( (j
-+
We then provide Y
=
inf divx g = f
P
) )
onto
with the norm
I 191 I (LP((j ))n
We consider a family of functions a ij such that
We denote by [a. (x,t)] the n 11 and we set
x
n matrix with entrices aij(x,t)
EVOLUTION OPERATORS
272
We set
Theorem 2 . 2 .
Let f and u o be given such that f uo
H = L
E
2
( 0 ) .
u
(2.80)
u(0) = u0
Then assuming only on a , %
L2 (O,T;H-'(S))
and
Let u be the solution of
(2.79)
E
E
L 2 (0,T;H0(8)) 1
I
.
r to be smooth enough there exists 8 , such that if
p > 2,
depending
(2.82)
then
u
when f
E
Lp(O,T;WkrP(O )
and remains in a bounded set of this space
)
& uo remain bounded in LP(0,T;W-1rP(8)) and in WirP(O).
Remark 2 . 3 . We can reduce the problem to the case when uo = 0. can find I$ E Lp(O,T;W;'p(O
)),
$
E
Lp(0,T;Wi'p(8
)),
@
Indeed we depending
continously on uo in the corresponding topology, so that @(O) = u 0 and then we consider u
'This
-
@
instead of u.
hypothesis can be very much improved.
CONVERGENCE O F THE HOMOGENIZATION OF PARABOLIC EQUATIONS
273
P r o o f of T h e o r e m 2 . 2 .
of C h a p t e r 1, w e c a n r e d u c e t h e p r o b l e m t o
A s i n S e c t i o n 4.2
where
&i [ a l j ( x , t )
-
(2.84)
Ak =
(2.87)
v < p .
,
k = 1,2
,
By c h a n g i n g t h e scale o f t i m e , w e see t h a t w e a r e r e d u c e d t o proving t h e theorem f o r
(2.88)
a t + A 1u + u = 0 on C
and w i t h (2.841,.
A u = f
2
r
=
x
,
u(0) = 0
]O,T[
,
f s a t i s f y i n g (2.81)
,
. .,( 2 . 8 7 ) .
W e set
(2.89)
a
P = - - A .
at
I t i s known t h a t , g i v e n F
E
Yp,
( w h e r e p i s a r b i t r a r y f o r the
t i m e being), t h e r e e x i s t s a unique u such t h a t P u = F i n (2.90)
U E X
P '
q , u(0) = 0
.
,
274
EVOLUTION OPERATORS Equation (2.88) is equivalent to
(2.91)
Pu
f
(A1 + A ) u - + A2u = f
,
and if the solution u of (2.90) is denoted by (2.92)
u = P-1F
then (2.91) is equivalent to (2.93)
u
+
(P-l(A 1
+
A ) + P-lA2)u = P-’f
.
We introduce
and the Theorem will be proven if we verify that (2.95)
one can find p > 2 such that k ( p ) < 1
.
We have
But exactly as in Chapter 1, Section 4,3 (t plays the role of a parameter) we have
so that
(2.97)
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS It is straightforward to verify that that, since 1 (2.98)
-
p
+
1 IP-ll 1
v < 1, we shall have ( 2 . 9 5 )
2 ( Y 2 ;x2)
275
= 1 so
if we can prove that
there exists a function p ( p ) which is continuous
and p ( 2 ) = 1
.
Indeed we have then k(p) < p ( p ) and we can find p0 > 2 such that k(po) < 1.
Proof of ( 2 . 9 8 ) . We take F = div g, g
E
(Lp(
in ( 2 . 9 0 ) and we consider the
mapping (2.99)
TT
g
+
gradx u
,
Using Riesz-Thorin's theorem it follows that
Therefore, according to ( 2 . 7 2 ) and to ( 2 . 7 4 ) , we have
276
EVOLUTION OPERATORS
so that we can take p(p) = w:-'(~)
and the result follows.
Remark 2.4. We have, for arbitrary boundary conditions, a local result (as there exists p > 2 (and not too large:
in Theorem 4.3, Chapter 1): p
5
%)
such that if f and uo satisfy locally (2.81) and (2.82),
then (2.103)
$u
E
L p ( O , T ; W ~ ' p () ~)
.
Remark 2.5. .It follows of course from these estimates that, if f and uo satisfy conditions of the type (2.81), (2.82), then (2.104)
au
E -f
1
2.8
au ax i
in Lp(
s)
weakly or in Lp(O,T;Lp( 8
I ) )
weakly
,
The adjoint expansion. Let us remark that convergence can be proven for Dirichlet's
boundary conditions and when coefficients a
are smooth enough ij (cf. Section 2.3) using the method of the adjoint expansion as in
Chapter 1, Section 3.3.
We present this briefly, since the idea is
exactly analogous to the one in the elliptic case.
We can always
assume that uo = 0, and we write the equation in the equivalent form
where here (f,v) =
/
J
s
fvdxdt.
CONVERGENCE OF THE HOMOGENIZATION 6 F PARABOLIC EQUATIONS
Given v E
J
(S
),
asymptotic e x p a n s i o n s - a
we construct-using
sequence of f u n c t i o n s v
such t h a t vE
+
277
v i n L L ( )~ w e a k l y ( s a y ) , a n d
such t h a t
[- ata +
(2.106) where
i.e.
ff
*
AE*]vE
[- 2 + at
+
a * ] v i n L2 (
= a d j o i n t of t h e operator
ff
S)
weakly
,
c o n s t r u c t e d i n S e c t i o n 1,
o r f f 3 a c c o r d i n g t o t h e case k < 1, k = 2 , k > 3.
0 = Ol, ff
Then
and t h e r e s u l t f o l l o w s
U s e o f t h e maximum p r i n c i p l e .
2.9
Assuming a l l d a t a t o
be v e r y s m o o t h a n d f o r t h e D i r i c h l e t ' s
boundary c o n d i t i o n s , we have
Indeed let u s w r i t e (2.108) valid
PE =
a a t + AE
method a n d a d d a n
E
€-'Pa
+
,
E 0 P3
for k = 1 and 2.
(with d i f f e r e n t Pi's) 3
+
= €-'Pl
(For k = 3 use t h e same
u3 t e r m i n t h e proof b e l o w . )
We introduce (with
n o t a t i o n s of S e c t i o n 1 , (2.109)
ZE
-
= u
(u
+
EU1
+
E
2
u,)
and we o b s e r v e t h a t PEZE = -E(P2U2
(2.110)
I f all data m
the L (
S
)
norm.
+
P u )
3 1
-
E
2
P3U2
.
are Cm, t h e r i g h t hand s i d e of But
(2.110) i s O ( E )
in
EVOLUTION OPERATORS
and
so that, by using the maximum principle
hence (2.107) follows.
2.10
Higher order equations and systems. Except in Section 2.9 above, we never used the maximum
principle.
All results obtained sofar extend to higher order
equations and to systems, i.e. to the "parabolic analogue" of the problems considered in Chapter 1, Sections 9 and 10. Section 6, Chapter 1 also extend easily.
The results of
Let us consider an equation
of the type (2.111)
auE a at - axi
[a..(x,-,t,-)
Ek
11
aUE1
-
=
axj
f
,
where we suppose that
m
where Lrn(Rnx RT) = space of L P Y periodic. We also assume that (2.113)
a . .(x,y,t,r)C.C.> aEiCi 1 7 -
13
Then we denote by (2.114)
functions of y and
k
,
7
a > O ,
a k (x,t) the homogenized operator
(x,t) =
a k a - axi sij(x,t) ax
j
which are Y
SiER.
x T~
CONVERGENCE OF THE HOMOGENIZATION W PARABOLIC EQUATIONS k where the coefficients qij(x,t) are computed as the qij's in Section 1, with x and t playing the role of parameters.
If u
k
denotes the solution of uk
L~(o,T;v) ,
E
k
k a u + a (x,t)uk = at
(2.115)
k
u (0)
=
u
0
f
,
,
then (2.116)
u
+
2
uk in L (0,T;V) weakly as
E
+
0
.
One can also extend, always with the same type of method, the results of Chapter 1, Section 8. Let us consider now briefly the "parabolic analogue" of the situation of Section 11, Chapter 1. We assume that the space dimension equals 3 , and we denote by aE(x,t) a 3
x
We
3 matrix.
assume that
a . .( y , ~ )having the same properties than in (1.6) 13
.
I
We consider, with the notations of Chapter 1, Section 11:
and we observe that there exists a unique function uE such that
(2.119)
u
E
L~(o,T;v) ,
2 where f E L ( O , T ; V ' ) ,
uo
E
uE(0) = u 0 3
H = (L2(0))
.
,
279
EVOLUTION OPERATORS
280
Asymptotic expansions. We give only the results of the asymptotic expansion; one has to consider three cases.
Case k = 1. We define jiP(y,r) as a Y-r0 periodic solution of
-
roty a(y,r) (rot jiP Y
(2.120)
div jip = 0 Y
e
P
)
=
o ,
.
Then
U 1 (u,v) =
(2.121)
(
-
R(a
a rot ?)rot u,rot v) Y
and the homogenized equation is (%v) at
(2.122)
Case k
+ Q 1 (u,v) = (f,v)
,
tJ v E
v
-
= 2.
We define BP(y,r) as a Y - r 0
aa.tirp + (2.123)
periodic solution of
rot a(y,r) (rot GP Y Y
div BP(y,r) = 0 Y
-
e ) P
=
o ,
.
Then (2.124)
U
2 (u,v) = ( 7;1 (a
-
-
a rot 8)rot u,rot v) Y
and the homogenized equation is the analogous of ( 2 . 1 2 2 ) instead of
with 0 2
CONVERGENCE OF THE HOMOGENIZATION QF PARABOLIC EQUATIONS Case k = 3 . We define
and we define ip(y) as a Y-periodic solution of rot Z(rot Y Y (2.126) div '$I
Y
= 0
5' -
e
P
=
)
o ,
.
Then (2.127)
u
3 (u,v) =
( n (a - a 17
rot $)rot u,rot v) Y
and the homogenized equation is the analogous of (2.122) with U 3 instead of
a'.
The justification of these formulas is simple if we assume: (2.128)
,
divx f E L 2 ( S )
div uo
E
L2( 8 )
.
Indeed, we define z o as the solution of (2.129)
Azo = div u 0
,
and assuming the boundary (2.130)
zo E ~
0
z
r
E
1
HO(8)
smooth enough, we have
.
~ ( n 8H ~) ( D )
We then define z as the solution of (2.131)
a Az
=
divx f
,
(This is an elliptic problem:
Z ( O ) = zo
Az =
1 0
div
X
,
z =
f(x,u)do
o
on
c
.
+ Az 0 ) ; we have
281
EVOLUTION OPERATORS
282
ii
= UE
-
vz
we see that
aiiE
+ (a rot iiE,rot v)
[K,v] where ? = f =
div uo
-
a - at P,z0
= (f
-
=
( ~ , v ),
Since div fiE ( 0 )
Vz, so that divx f = 0.
=
a - at vz,v)
0, we see that div GE = 0.
Therefore we can assume (if (2.118) holds true) that divx f = 0 ,
divx uo = 0
,
divx u
= 0
.
Then one can prove the above formulas by arguments similar to those of Sections 2.2, 2.4, 2.5.
Remark 2.6. proof along the lines of Chapter 1, Sections 11.4, 11.5, does
A
not seem to work in the present situation.
The homogenization
problem seems to be open if we do not assume (2.128).
2.11
Correctors.
Orientation. We now return to the situation of (1.17), (1.18), (1.19). want to introduce first order correctors '8 (2.132)
u
-
(u +
OE)
2
-+
1
0 in L (0,T;H ( 8)
We
such that )
strongly
.
Remark 2.7. By using cut-off functions m E ( x ) , as in Chapter 1, Section 5.1, we can have correctors giving an approximation in LL(O,T;V) strongly.
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS Remark 2.8. We did not make any attempt to find the analogous results to those of Chapter 1, Section 18.
Notation. We shall denote by u1 the function: 1
(2.133)
&L
=
ax
u1 = - e l
if k = 1
,
if k = 2
,
j
ax j
We shall also denote by the ~ a m enotation u2,u3,..., the next terms of the asymptotic expansion, for k = 1,2,3. We shall use (2.134)
GE
=
u +
=
u
+
if k = 1,2
EU 1
+ 2u2
E U ~ E
,
if k = 3
Theorem 2.3. We assume the hypothesis of Theorem 2.1 to hold true and we assume all data (2.135)
uE
-
smooth enough.' (u
+ EU~) -f
0
&
Then L2 ( 0 , T ; V )
strongly
Proof. We introduce
'This
is made more precise in the proof below.
.
283
284
EVOLUTION OPERATORS
(2.136)
- 6E
uE
<,=
and we consider (2.137)
XE =
(T
-
t)[(
.
0
We observe that m
Since 1<,(0)l2
= 6 2 lu1(0)
we prove that X E replaced by T
-
0 as
+
or c 2 lul(0)
E -+
+ eu2(0] ', we see that, if
0 , then we shall have (2.135) with T
but since T does not play any role, this
q > 0,
q,
m
gives the general result. We can write XE =
1
(T
-
+ aE(uE)ldt
t) [ (u:,uE)
0
+
(T
-
+ aE(GE)]dt
-
+ aE(uE,GE)ldt
,
t) [(G;,GE)
YE
-
ZE
,
0
T
)
The first term in X
1
(T
-
.
equals
t) f,uE)dt
+
T
(T
0
0
+ aE(GE,uE)ldt
=
-
1
t ) (f,u)dt
(T
-
t) [ (u' ui
+ a ( ~ , ~ ) i d .tt
0
tWe set
=
a k , k = 1,2,3 and c7Xu,v
= (
a u,v).
CONVERGENCE OF THE HOMOGENIZATION QF PARABOLIC EQUATIONS
The second term in X
equals
One verifies, exactly as in Chapter 1, Section 5, T
T (T
-
t)aE(GE)dt
I
-f
n
0
tends to
-
5
1u0l2
+
285
(T
T 0
-
(T
-
(5.15), that
t) Q(u,u)dt, so that the second term
t) Q(u,u)dt
=
T 0
(T - t)[(u',u) +Q(u,u)ldt.
Therefore, if we show that (2.138)
YE
(2.139)
ZE
-t
-f
T T 0
0
(T - t) [(u',U) + Q(utu)ldt
(T
-
it will follow that X E
t) [(u',u)
+
+
Q ( ~ , ~ ) l d ,t
0, hence the Theorem follows.
Let $ be a given function in & g ( S ) , (near
r)
r
$ =
1 except on a set E
of measure /El. We define (compare to Chapter 1, (5.21),
(5.22)) T
T
We see exactly as in Chapter 1, Section 5,3, that (2.138), (2.139) will be proven if we verify that, for fixed $, Y
E$
(2.140)
+
1
(T
-
t) [(u',u$) + 0 $(u,u)ldt
(T
-
t) [(u',u$) + Q $ (u,u)ldt
0
zE$ 0
,
EVOLUTION OPERATORS
286
But
s
If we denote by q .
Ij
the coefficients of ftk, k = 1,2,3, we have
hence (2.140) follows for Y
EtJ.
We observe next that
But from the definition of GE it follows that +
au
2
+ C7u in L ( Q ) weakly,
hence the result
aii
$+
A Gc
so that (2.141) gives
follows.
Remark 2 . 9 . We can also write the equation as a first order system and develop arguments along the lines of Chapter 1, Sections 5.5 and 5.6.
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
287
For Dirichlet's boundary conditions we obtain in this manner error estimates which do not rely on the maximum principle.
2.12
Non-linear problems.
Orientation. We confine ourselves to
one aspect of
the homogenization of
non-linear parabolic equations (or inequalities), namely to Variational Inequalities (V.1.) of evolution.
Many other cases can
be treated by using the methods introduced below.
Setting of the problem. We consider operators AE as in Section 1.1; A E = A
X/E,t/E k 1 .
Formally we want to consider V.I. of the following type 3at+ A E u E (2.143) E
u
+
-
AeuE
,
f 0 '
-
UE(0
fluE = o
I
Q ,
in
being subject to usual boundary and initial conditions.
To
simplify a little, we shall take: (2.144)
uE(x,O) = 0
.
The variational form of (2.143) is as follows: 2 (O,T;H), then we are looking for uE such that: in L u (2.145)
and
E L 2 (0,T;V)
auE E at
let f be given
,
2 L (0,T;V')
,
uE 5 0
,
a.e. in Q
,
EVOLUTION OPERATORS
200 m
2 V V E L (0,T;V)
,
in S ,
v 2 0
and such that ( 2 . 1 4 4 ) holds true. We shall show below that this problem admits a unique solution. The main result we want to prove is Theorem 2 . 4 . We assume that the hypothesis of Theorem 2 . 1 hold true.
We
denote by U the homogenized operator found in Theorem 2 . 1 (i.e.
U
=
Uk, k
on V. -
and by
= 1,2,3)
U(u,v) the corresponding bilinear form
Let u be the solution of the V.I. of evolution
u
u(x,O) = 0
.
0
(2.149)
Then when E
I
-
[I%,.
(2.148)
-f
u
+
0, +
(2.150) -auE at + -
u in L2 (0,T;V) weakly
,
au . L2 (0,T;V') weakly at in
.
The proof is divided in several steps. We first introduce the penalized equation associated to the V.I. (2.146).
equation
For 11 > 0, we denote by uEn the solution of the nonlinear
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
(2.151)
u
En
(X,O)
= 0
,
u
En
E
289
L2 (0,T;V
(and as usual, the boundary conditions which correspond to the variational formulation).
The equation (2.1 1) is called the It admits a unique solu-
penalized equation associated to ( 2 . 1 4 6 ) .
tion, for which we now obtain a priori estimates. If we multiply ( 2 . 1 5 1 ) =
E
+
+
a (v ,v
=
+ aE(v ) ) :
+
E
+
by uEn, we obtain (since a (v,v
)
hence
We deduce from ( 2 . 1 5 2 ) (2.153)
1 u+ rl En
that
is bounded in L 2 ( Q
It follows from ( 2 . 1 5 3 )
)
,
V
8
and V n
I
that
(and uE satisfies initial and boundary conditions) so that the usual a priori estimates are valid:
If we let n such that
-+
0,
we can extract a subsequence, still denoted by uEn,
290
EVOLUTION OPERATORS
(2.156)
u
Erl
+
u
E
2 in L (0,T;V) weakly
au
,
au
2 3 + 2 in L (0,T;V') weakly , at at (2.157)
-+
0 in L
2
(S
)
.
2 It follows from (2.156) that uEll+ uE in L ( 0 ) strongly, so that u+
El1
-f
+ in . L2(S
u
E
)
and by comparison with (2.157) we obtain
u+ = 0, i.e. uE -< 0. Therefore uE satisfies (2.146) and (2.144). 2 Let v be in L (O,T;V), v 5 0. We multiply (2.151) by v - u we
-
El1'
obtain :
hence
or
Since
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS it follows from (2.158) that uE satisfies (2.145).
291
Therefore we have
shown the existence of uE, satisfying (2.145), (2.1461, (2.144) and such that
The uniqueness of uE is straightforward. We now establish an error estimate of independent interest:
Proof.
We have
UE
-
UE, = UE
+ u-E, - u+E n and by virtue of (2.154) we
have only to show that (2.161)
I
We take v = -uiq in (2.146) and we multiply (2.151) by uE
-
+ uE,.
Adding up, we obtain PUE
+
uEq
J
+ +
+ u,,]
-
aE(uE + ui,,)
i.e.
1 + and integrating over (O,t), we obtain (since (uE n ,-uE) 1. 0 ) r7
292
EVOLUTION OPERATORS
Using (2.154) and (2.159), the right hand side is bounded by C h i , hence (2.161) follows. We have also proved that
It follows from (2.160) that it suffices to study the limit of u
€17
as
E
+
0
for Q
fixed.
But this is immediate.
By virtue of
(2.155), we can suppose, by extracting a subsequence, still denoted by uEn, that U
(2.164)
-f
En
u
in L2 (0,T;V) weakly ,
rl
au nE:' 9 in L2 (0,T;V') weakly . +
Then uErl+ u
n
in L2 ( F
)
strongly, so that f
-
II
u+
En
f
-f
- n u+n
in
L2( Q ) and then
au ].,$I where
+
a (u17 ,v)
=
(f
-
1 + - u / rl
, v v
E
v ,
Q(u,v) is defined as in the statement of the Theorem.
proof is completed.
The
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
293
Remark 2.10. Theorem 2.4 gives, in particular, the homogenization of the Stefan's free boundary problem for a "granular" medium-i.e. coefficients are functions of
when
t aij(x/E).
X/E:
Remark 2.11. The result of Theorem 2.4 extends to the case of coefficients of x t the form aij (x,-,t,x). E
E
Remark 2.12. Theorem 2.4 can be extended to obstacles of the type b1(x,t) when $,
5
v
5 b,(x,t)
and $2 are smooth enough.
, The case of the "most general"
obstacles yi under which the analogous of Theorem 2.4 still holds does not seem to be known.
Remark 2.13. The homogenization of V.1 of evolution for "arbitrary constraints"
v
E
2.13
K(t) is largely open.
Remarks on Averaging.
Setting of the problem. We consider now the case T =
'This
+w:
is because the Stefan's free boundary problem can be
transformed into a V.I. of evolution.
294
EVOLUTION OPERATORS
s
(2.165)
=
8
x
lo,+-[
and we consider the equation
(2.166)
av
For u,v (2.167)
E
H1(S), we set now
aE(u,v) = 8
8
The precise problem we consider is then: u
(2.168)
E
L~(o,T;v),
v
T finite ,
[auc w , v ] + aE(uE,v) = (f,v) UE(0) =
u0
,
V
v E V
,
t > 0
,
.
We assume the following: (2.169)
aij
and of course
Example 2.1. We take :
,
aa! axk
,
b;
remain in a bounded set of I,-( Q
)
295
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS a . .( x , ~ ), (2.172)
a aij(x,T)
b .( x , ~ ),
axk
3
13
m
belong to L ( 8 x R T ) and are almost periodic in
T
.
We extract a subsequence such that
(2.173) in L m ( 6 ) weak star
.
We want to study the behavior of uE
as
E
-t
0.
In the case of Example 2.1 we have (2.1741
iij =
nT(aij)
,
j =,R',(bj) ,
where we have set here (2.175)
m T ( $ )=
x
\
lim
X++m
$(x,a)do
.
0
Theorem 2.5.
t We assume that (2.169), (2.170) hold true and that (2.176)
1 V = Ho(8)
1 V = H ( 8 )
.
We also assume that f , g
(2.177)
i
E
L2( 8
x
lO,T[)
,
V
T finite,
UOEV.
hypothesis is by no means indispensable. I f V consists of 1 the functions of H ( 8 )which are zero on a subset ro of r , the 'This
result is still valid with a slightly more complicated proof.
EVOLUTION OPERATORS
296 Then a s (2.178)
E
-+
u
0 , one h a s +
,
u i n L2(0,-T;V) w e a k l y f o r e v e r y T f i n i t e
where u i s t h e s o l u t i o n o f
(2.179)
I&[
+ i(u,v)
u(0) = u where (2.180)
i(u,v) =
0
= (f,v)
, v
v E
v ,
,
I
aij
ax a u ax av
8
j
i
dx
+
I Gj 0
j
vdx
.
Remark 2 . 1 4 . I n f a c t w e s h a l l p r o v e more: n a m e l y , (2.181)
uE
+
2 u i n L 2 ( 0 , T ; H ( 0 ) )w e a k l y
, v
T <
m
,
Remark 2 . 1 5 . I n case o f Example 2 . 1
(formulas (2.171)) t h e theorem h o l d s
a
( 2 . 1 8 2 ) ) w i t h o u t c o n d i t i o n s on axk a i j ' u s e s t h e s e m e t h o d s a l o n g t h e l i n e s o f t h o s e of S e c t i o n 2.2. t r u e (without (2.181),
P r o o f o f Theorem 2 . 5 . Thewhole problem i s reduced t o p r o v i n g t h a t , VT f i n i t e ,
One
CONVERGENCE OF THE HOMOGENIZATION OF PARABOLIC EQUATIONS
297
Indeed, if we admit (2.183), then one can extract a subsequence, still denoted by uE, such that one has (2.178), (2.181) and (2.182). It follows that, V T finite auc
+
1
au ax
in L2 ( 8 x ]O,T[) strongly
j
and therefore
in L2 ( 0
x
]O,T[) weakly
and the result follows. Estimates (2.183) are obtained by using first local mappings, to reduce the problem to the same question when (2.184)
0
=
Ixlx > 0 1 n
.
Then one uses the method of translations parallel to the boundary. t We suppress the index h
=
"E"
to simplify the writing.
Given
{hl,...,hn-l,Oj, we define
We have
, g + ~ u = f i nS ,
tThis is a classical method, introduced by L. Nirenberg 111 for the study of the regularity of elliptic equations.
When V does not
satisfy (2.176) but consists of functions which are zero on
ro
C
r,
then there is a difficulty because V is not "translation invariant". One can then use a compensation method, due to Aronszajn and Smith, and reported in'lions 121.
298
EVOLUTION OPERATORS
and with obvious notations
3 at + AhUh
fh in
=
q
.
Therefore (2.185)
a(uh at
U)
+ Ah(uh - U) + (Ah - A)u
=
f h
-
f
.
It follows from (2. 8 5 ) and from the hypothesis that
1 IUh so that by letting (2.186)
hl
+
0, we see that when
2
a remains axjax
E
-+
0
in a bounded set of L2 ( 8
x
]O,T[)
,
j
where x‘ i = x i ,
i L n - 1 .
We now return to the equation, which we write (2.187)
au
-
ann
2 2
=
a xn
g *
where q remains in a bounded set of L2 ( 8 x lO,T[) (we use ( 2 . 1 8 6 ) ) ; au we obtain (since a > y > 0) multiplying (2.187) by 1 at nn -
au = 0 on (since u = 0 or axn (2.188)
r)
hence it follows that
au remains in a bounded set of L2 ( 8 x lO,T[)
But then ( 2 . 1 8 7 ) ,
which can be written:
.
OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE
299
imp1ies 2 9
E
bounded set of L2( 8
lO,T[)
x
a xn
and (2.183) is proven.
Remark 2.16. In the case of almost periodic coefficients, Theorem 2.5 is related to the classical theory of averaging for ordinary differential equations with almost periodic coefficients.
Cf. Bogoliubov
and Mitropolsky 111 and the Bibliography therein.
Remark 2.17. We confine ourselves to remarks made in the comments of the last section of this chapter for the case of coefficients aij(x/E,t/E k ) , where the aij's are almost periodic in
3.
T
(and periodic in y).
Evolution operators of hyperbolic, Petrowsky or Schrodinger type. I
3.1
Orientation. We present now some very simple remarks connected with the
homogenization problem for hyperbolic operators.
Further results,
using some different ideas and techniques, are given in Chapter 4 of this book.
We also give in this section some results for operators of
Petrowsky or of Schrodinger type, the hyperbolicity not being used in an essential manner.
We give next some examples of "non-local"
operators which appearin the homogenization process.
300
EVOLUTION OPERATORS
3.2
Linear operators with coefficients which are regular in t. Let us consider the operator
where the a . . ' s satisfy: 13
,
a. , (y) E Lm(R;) 17
(3.3)
a. . is Y-periodic 13
aij(y)SiEj 2 clSiSi
a > 0
I
1 We consider V as usual (Ho(Q) _c V
,
,
a.e. in y
5 H 1 ( 6 ) )and
.
with the usual
notations we consider the following problem:
0
where f, u
, u1 are given such that
(3.7)
f
E
L~(o,T:H)
,
uo
E
v ,
u1 E H
.
It is known (cf. for instance Lions [l], Lions-Magenes [11) that problem ( 3 . 4 ) , (3.5), ( 3 . 6 ) admits a unique solution.
A
priori estimates. We shall write $',$I"
for -, at
a2e at2'
Taking v = u~ in (3.5) gives
OPERATORS OF HYPERBOLIC, PETROWSKY'OR SCHRODINGER TYPE
30 1
We emphasize that in order to obtain (3.8) we used in an essential manner the fact that aE(u,v) is symmetric and does not depend on t.
It follows immediately from (3.8) that
The behavior of uE
E
-t
0 is very simple to study.
It follows
from (3.9) that we can extract a subsequence, still denoted by uE, such that uE
+
u
in L~(o,T;v)weak star
,
u'
+.
u'
in Lm(O,T;H) weak star
.
(3.10)
We use next the method of Remark 1.6 which is entirely general. With the notation (1.46) one has, V
@ E
C,"(]O,T[):
in H strongly, one has: uE ( @ ) + u ( @ ) in
V weakly
, where
where Q(u,v) = homogenized form associated to aE(u,v). u is the solution of 2
IF,.]
(3.13)
u
(3.14)
u(0) = u0
E
v v
6
L-(o,T;H)
,
+ Q(U,V) = (f,v)
(3.12)
L~(o,T;v) ,
,
u'
E
,
u' ( 0 ) = u1
Let us consider now the case when
.
v ,
Therefore
EVOLUTION OPERATORS
302 (3.15) We assume that (3.16)
aij(x,y,t) is Y-periodic (3.17)
a1.7.E Co( 5
We set, t/ u,v (3.18)
E
X
[O,T];Lm(Rn)) Y
x,t
,
,
H1 ( 8 ) :
aE(t;u,v) =
J
8 We can
, v
au av aij(x,x,t) E ax axi dx j
.
efine
(3.19)
8 We consider again (3.4), (3.5),
(3.6) (with aE(u,v) replaced by
aE (t;u,v)), and we have existence and uniqueness of the solution. The a priori estimates are now the following.
Equation (3.8)
is replaced by
and by virtue of the last condition in (3.17) we obtain the same a priori estimates (3.9). By the same technique
as
in Chapter 1, Section 6, one proves
that uE satisfies (3.10) where u is the solution of (3.13), (3.14) and
OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE (3.22)
( a (t)u,v) if
u(t;u,v) =
u,v E
303
cr(8) ,
where G(t) is the homogenized operator of AE(t) for fixed t.
Remark 3.1.
a2 + AE is a second order hyperbolic operator. 7
The operator
at
But the results indicated so far really do not depend at all on the hyperbolicity.
All what we sa d remains unchanged (except obvious
modifications) i f one considers (3.23)
aE(t;u,v) =
I
aaB(x,E X
t)DauDBvdx ,
la1 =
161 = m
,
8 a
aB
a
aB
(3.24)
aaB
= a
'
(x,y,t) being Y-periodic in y Co( 5
E
aa at
Ba
E
Lm( 8
[O,T];Lm(Ry))
x
x
(0,T)
x
Rn) Y
,
,
.
The homogenized operator is computed for every fixed t according to the results of Chapter 1, Section 9. We remark that the corresponding operator is not oft hyperbolic type when m > 1; it is an operator of Petrowskytype.
Remark 3.2. One can also add to the operator AE lower order terms. return to that in Section 3.3 which follows.
We
304 3.3
EVOLUTION OPERATORS Linear operators with coefficients which are irregular in t. A natural problem is-the following:
Instead of the situation
( 3 . 1 ) we consider functions a..(y,T) which satisfy ( 3 . 2 ) and 11
(3.25)
aij(y,r)
E
Lm(Rn Y
x
RT)
,
and the usual ellipticity condition.
Y - r 0 periodic
We then consider the operator
AE defined by
If we assume that (3.27)
a
- aij
E
L-(R;
x
R ~ ),
then problem ( 3 . 4 ) , ( 3 . 5 ) , ( 3 . 6 ) admits a unique solution. are nof a priori estimates independent of
But there
Indeed if we define
E.
(3.28)
then ( 3 . 2 0 ) is replaced by
which does not give estimates independent of
as E
-t
E.
0 does not seem to be known in general.
The behavior of uE We shall give in
Section 3 . 4 below some formal computations (which can be justified in
cases). It is possible to consider lower order terms which have
coefficients irregular in t.
Let us consider
+At least there do not seem to be known a priori estimates strong enough to pass to the limit in
E.
OPERATORS OF HYPERBOLIC, PETROWSKY.OR SCHRODINGER TYPE A'
(3.30)
given by (3.1)
305
,
a x t BE = b. (-,-) 7 E E k ax. 3
where (3.31)
b.(y,r) is Y--r0 periodic 3
,
b. E L m ( R y 3
x
RT)
.
there exists a function uE and only one which
We have then:
satisfies (3.4), (3.6) (3.32)
(u2.v) + aE(uE,v) + (B'U',~)
=
(f,v)
The a priori estimates are as follows.
, v v
E
v
Taking v = u
I
. in (3.32),
we obtain
But
so that (3.33) gives:
t
and we obtain (3.9). Therefore one can extract a subsequence, still denoted by uE, such that one has (3.10) and (3.36)
BEuE
+
rl
Then (3.32) gives, V
m
in L (0,T;H) weak star C$
EC;(]O,T[)
.
306
EVOLUTION OPERATORS
and ( B E u E )( @ )
+
i n H-weakly,
Q(@)
follows f r o m (3.37) t h a t ,
t h e r e f o r e i n V' s t r o n g l y a n d it
a ( u , v ) b e i n g t h e homogenized f o r m
a s s o c i a t e d t o aE
Therefore
and t h e p r o b l e m i s now t o compute
We s t u d y t h i s q u e s t i o n i n t h e
Q.
following s e c t i o n s .
3.4
.
A s y m p t o t i c e x p a n s i o n s (I)
a purely formal fashion-the
L e t us consider f i r s t - i n
(3.4),
(3.5),
( 3 . 6 ) f o r A E g i v e n by ( 3 . 2 6 ) .
W e have w i t h t h e u s u a l
notations: (3.39)
a2 + A E
+
= E-2Q1
+
E-lQ2
E 0Q 3
,
at2
Q1 =
a2 2 + aT
Q3 =
at2+ A 3
a
a a 11 . . (y,T) A 1 = - -ayi aYj
A 1 '
I
a2
'
1
Then, i f w e look f o r uE = uo (3.40)
QIUO
=
(3.41)
QlUl
+ Q2UO = 0
0
r
r
+
E
U
+ ~
problem
..., w e
f i n d as u s u a l :
OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE (3.42)
Q1u2 +
+ Q3u0 = f
Q2Ul
307
.
We are thus lead to the following question:
given a function F
such that (3.43)
F
E
L2(Y
( 0 , ~ ~ ) )
x
is it possible to find (3.44)
9 a-r
+ A 1@
@
which is Y--ro periodic and satisfies
,
= F
i.e., @ takes equal values on opposite faces of Y and @(Y,O) = @(YtTo) , A necessary condition for
(3.45)
7?( (F) = 0
@
a@
(Y,O) =
a@
(y,-ro)?
to exist is
.
Let usassume that this condition is sufficient (so that (3.44) defines @ up to an additive constant). "generically" the case.
It can be proven that this is
Then (3.40) is equivalent to uo = u(x,tl and
(3.41) reduces to Q u + A 2 u = 0 . 1 1
If we introduce XJ as some Y--r0 periodic solution of
then
and (3.42) can be solved for u2 iff
+ ,Q ~ u =~ f) ~ ( Q ~ u
,
i.e.
,
308
EVOLUTION OPERATORS
Of course we add the appropriate initial and boundary conditions.
In
which sense u is an “approximation“ of uE is an open question.
3.5
Asymptotic expansions (11). We now consider the situation (3.30) and we take
(3.49)
k = l .
Then
We find
,
(3.51)
RluO = 0
(3.52)
Rlul + R2u0 = 0
(3.53)
R1u2 + R2u1 + R3u0 = f
,
-
We are lead again to (3.44) but this time with
of T.
A1
independent
It follows from de Simon [l] that for almost every T ~ condi,
tion (3.45) is necessary and sufficient for (3.44) to admit a Y-ro periodic solution.
We therefore assume
OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE (3.54)
309
condition (3.45) is necessary and sufficient for the existence of $ which is a Y - T ~ periodic solution of (3.44)
.
Then (3.52) gives u1 by formula (3.47), (3.46); but since A1 does not depend on
T,
being defined by
we have
and
Then (3.53) admits a solution for u2 iff
7/1 (R2u1 + R3UO)
= f
,
i.e. 2
*+
(3.56)
nC
a u +
at2
This can actually be proven: Theorem 3.1. I
We assume that AE is given by (3.1), (3.2), (3.3) and that BE is given by (3.30)
that
bj
E
C1(?
x
with
k = 1.
[O,T~].
solution of (3.38)
with
0
Then
We assume that (3.54) holds true and uE satisfies (3.10) where u is the
given by
Sketch of Proof. We introduce BE* = adjoint of B i.e., BE*v = such that
a (b.(x,t)v). € - fax J j
in the sense of distributions,
We construct a family of functions wE
EVOLUTION OPERATORS
310
+
a2
+ BE*]wE
AE
=
€4,
,
(3.58) w
+ 1
o
-+
in L ' ( Q )
as
E
,
gE bounded in L 2 ( Q )
-+
o
.
For the construction of w, we need all coefficients aij and b smooth-we
j
to be
can always reduce the problem to this case by
approximation.
We look for wE in the form:
(3.59)
-1
W
=
+ E 2@ ( y , T ) ,
ECL( Y,T
4-
a and B being Y--to
periodic
.
*
We find, using (3.50) and since R1 = R1:
*
(3.60)
R a 1
+ R2(-l)
(3.61)
RIB
+
= 0
,
*
*
R2a + R3(-l) = 0
and g,
=
*
.
R2B
We define a as a Y--r0 periodic solution of (3.60), i.e. (3.62)
R1a = -
4 ab
(YtT)
*
Then (3.61) gives (using the fact that R3(-l) = 0):
*
R1B = - R2a which admits a Y - T ~ periodic solution by virtue of (3.54).
Then we
have (3.58). We now use $
E&,;(
Q )
and we take v = Ow, in (3.32).
multiply (3.58) by $uE. We obtain, by subtracting
We
OPERATORS OF HYPERBOLIC, PETROWSKY,OR SCHRODINGER TYPE
The first term in ( 3 . 6 3 ) equals
In ( 3 . 6 4 ) , the first term converges to 0, since u
aw = aa strongly and 2 at aT
+
E
aB + aT
Therefore ( 3 . 6 4 ) converges to If we set E:
=
aij
aa
-
I
(=)
?
2 = 0 in L (
-+
u in L
(u,@")dt.
0
au
$, the
second term in ( 3 . 6 3 ) equals
The third term in ( 3 . 6 3 ) equals b;uEwE
s
$ dxdt 1
(s)
s ) weakly.
j
-
2
+
(bj)
1
s
u'$
j
dxdt
.
311
EVOLUTION OPERATORS
312
The right hand side in (3.63) converges to
(3.38)) =
-
(u",@)dt 0
(citq) lo!u,@), =
f 0
Q(u,@)dt =
T 0
-
(n,@)dt.
T 0
(f,$)dt = (using
Since
it follows that
It remains to show the identity of (3.65) with (3.57), i.e. that
We multiply (3.62) by
xi.
It follows that
hence the result follows.
3.6
Remarks on correctors. We present here only a very preliminary result on the question
of correctors. (3.67)
1
=
We consider the case (3.1).
2~ ax ,
We have
xJ(y) being solution of A1(xJ
-
and being Y-periodic
,
j
y . ) = 0 3
and we use cut-off functions mE(x) as in Chapter 1, Section 5.2. set (3.68)
-
u
= U + E ~ U E
l
'
More generally we suppose that ue is the solution of
We
OPERATORS OF HYPERBOLIC, PETROWSKY OR SCHRODINGER TYPE (3.69)
313
(uE,v) + aE(uE,v) = (fE,v)
where (3.70)
fE,f E L2 (0,T;H) ,
fE
f in L1 (0,T;H) strongly
+
We have again (same proof), (3.10) and (3.12).
.
If we assume
that (3.71)
u E L ~ ( o , T ; H ~ ( o ),)
at
E
L~(o,T;v),
then -
-
= uE - uE
z
(3.72)
I
zE
+
-t
2
0 in L (0,T;V strongly
,
0 in L 2 0,T;H) strongly
The proof of this result relies on the following identity. 1 (T - t)2 $(t) = 2 We have
.
We set
uE,$)]dt
To prove this identity, we can assume that f
.
and f are smooth and witt
values in H (by approximation of these functions).
It is then
obtained by integration by parts. Since T does not play any role here, the result will be proved if we verify that X E
+
0.
314
One verifies that
T
EVOLUTION OPERATORS
0
@'aE(iiE)dt +
T 0
@ ' a(u)dt and one can pass
to the limit directly in all other terms.
3.7
The result follows.
Remarks on nonlinear problems. Let us give a simple example.
We consider the situation of
Section 3 . 6 and we set (3.73)
2
@(v) =
lgrad vI dx
.
8
Let us consider the nonlinear equation (ui,v) + aE(uE,v) + @(uE)(udfv)= (f,v) (3.74)
uE E L-(o,T;v) UE(0) = 0
,
,
u:
E
UE(0) = 0
,
L ~ ( o ~ T ; H,)
.
We assume that (3.75)
af
f,E
E
2
L (0,T;H)
.
Then there exists a unique solution of ( 3 . 7 4 ) which satisfies (3.76)
u :
E
L~(o,T;v),
u" E L-(o,T;H) E
.
The proof of this fact is standard. One obtains the a priori estimates (3.77)
I bElI L- ( 0, T;v)
+
1l~:Il
L~(o,T;v)
OPERATORS OF HYPERBOLIC,
PETROWSKP OR SCHRODINGER TYPE
315
Therefore we can extract a subsequence, still denoted by uE, and such that u
+
u and u'
+
u
(3.78) 1
uE
I,
+
u' in L m ( O , T ; V )
m
in L ( 0 , T ; H )
weak star
,
weak star
.
Since grad uE is bounded in Lm(O,T;(L2 ( 8 ) )"), $(uE) is bounded in L m ( O , T ) and we can assume that (3.79)
since u
$(uE) + 5 in Lm(O,T) weak star I
* u' in L2 ( S
)
;
strongly, we have $(uE
weakly and if we write (3.80)
+ aE(uE,v) = (f
(u:,v)
-
$(uE)u;,v)
gu' in L 2 ( S ) weakly.
we have fE * f
But one can improve this
result; indeed d
u
;iT@
= 2
)
i
I
grad u .grad u dx
8 m
remains in a bounded set of L (0,T) so that we can assume that $(uE) * g in Lm(O,T) strongly and therefore (3.81)
fE
+
f
-
CU' in L 2 ( S
)
strongly
.
Therefore, using (3.72), we see that $(UE)
-
1 $(fiE )+ 0 in L (0,T) for instance
.
But
a 8
(u
+
Em u E
1
)
a (u + Ern u )dx axi E l
,
EVOLUTION OPERATORS
316
so that
.
in L 1 ( 0 , ~ ) Therefore u is the solution of
where $(u) is given in ( 3 . 8 2 ) .
Remark 3 . 3 .
Homogenization of variational inequalities.
The general question of the homogenization in variational inequalities of "hyperbolic type" is largely open. simple result along the following lines. (3.84)
j(v) =
lvldx
g
I
We give only a
We set
g > 0
8
and, always with the conditions of Section 3 . 6 , we consider the solution uE of the V.I.
OPERATORS OF HYPERBOLIC, PETROWSKF OR SCHRODINGER TYPE We assume that f satisfies (3.75).
317
It is known (cf. for instance
Duvaut-Lions [l]) that (3.85) admits a unique solution.
Then one
can prove that (3.78) holds true, with u being the solution of the homogenized V.I. ( u " , -~ u') + U(U,V (3.86)
-
u')
+ j(v) - j(u') L (f,v - u')
u(0) = 0 ,
V V E V ,
u'(0)
=
0
.
Remark 3.4. If in the V.I. of the preceding Remark, we replace j(v) given by (3.84) by (3.87)
j,(v)
=
g
I
lgrad vldx
9
it is likely-but
it is not proved-that
one obtains for the limit a
V.I. of type (3.86) with j replaced by
3.8
Remarks on Schrodinger type equations. We can consider, now with complex valued functions, the equation
(of the Schrodinger type)t
'Of course one has now: (f,v) =
I
8
f Gdx
,
aE(u,v) = 8
318
EVOLUTION OPERATORS
where aE is given as in (3.1)
(3.2), (3.3).
[We can also consider
other cases considered in the preceding sections for aE . I
If we
assume (3.75), we can obtain the following a priori estimate: v = u
I
taking
in (3.89) and taking the real part
1 2 dt
aE(uE) = Re(f,ui)
Hence it follows that (one obtains a preliminary estimate by taking v = u (3.90)
in (3.89))
I bElI L~ ( o ,T ;v) -<
c .
Then we can extract a subsequence, still denoted by uEf such that (3.91)
uE
co
-t
u in L (0,T;V) weak star
and (3.92)
3.9
Nonlocal operators.
Setting of the problem. We consider: (3.93)
A E given with (3.1), (3.3)t
tThe symmetry is now necessary.
OPERATORS OF HYPERBOLICf PETROWSKY OR SCHRODINGER TYPE
bij = bji
(3.94)
m
,
b1.3. E L (R") Y
,
bij Y-periodic
319
,
B > 0
bij(y)S.S. 1 3 2 @Sici and we taket V = H 1o ( b )
(3.95)
.
We set bE (u,v) =
(3.96)
b : j
(x)
au av ax.
dx
3
8
and we consider the equation
(3.97)
UE(0) = 0
.
Remark 3.5. The ellipticity of A E is not necessary in what follows. The problem (3.97) admits a unique solution uE (3.98)
f E L'(O~T;V~)
E L
2 (O,T;V), if
,
and (3.99)
'We
I lUEl I
L~ ( o T;V)
1 can take HO(Q
)
< c .
5V 5
1
H ( 6 )if 1 9 V.
1 If V = H ( a ) , what
we are going to say is valid if one replaces BE by BE + bo (x/E) bo(y) 2 B 0 > O.,
EVOLUTION OPERATORS
320
Indeed taking v = uE in (3.97) we have
hence (3.99) follows.
Therefore one can extract a subsequence, still
denoted by uE, such that uE
(3.100)
+
u in Lm(O,T;V) weak star
.
The problem is now to see what is the "homogenized" equation which characterizes u.
We are going to show it is not a partial
differential equation but an equation with pseudo-differential operator. We take the Laplace transform in t of (3.97).
I
Let us set
W
(3.101)
uE(p) =
e-PtuE(t)dt
,
R e p > O .
0
We do not restrict the generality by assuming that (3.97) is taken for every t > 0 and with f E LL(O,m;H) with compact support. also consider (3.97) as an equation over Rt, with u t < 0.
We can
and f zero for
We then set
and we write (3.97)
This leads to "homogenization with parameter" as in Chapter 1, Sections 8.1 and 15.
We define xJ(p) as a Y-periodic solution of
T
HYPERBOLIC, PETROWSKYiOR SCHRODINGER TYPE
normalized by, say,
I’
x’(p)dy = 0 ; in (3.103) we use the standard
%].
I .
1
notation for A1 and B1 = - -[bij a (3.104)
321
Then we define
qij(p) =
and
Q (p) =
(3.105)
We can take the inverse Laplacetransform of q..(p) and set 17
(3.106)
Q
=
2 - l Q (PI
e - l = inverse Laplace transform
I
Then the homogenized equation is u
E
L~(R;v),
(3.107) Q(t)u = f
u =
o
for t <
o ,
.
The proof proceeds by methods similar to those of Chapter 1, Section 15.
Remark 3.6. The operator Q(;)
is not, in general, a partial differential
operator.
Remark 3.7. Let us consider materials which have a composite periodical structure and with a long memory (cf. for this last point, for instance, Duvaut-Lions [l], Chapter 3 , Section 7). simple model.
We consider a
These general cases will be treated with similar
.
322
EVOLUTION OPERATORS
methods.
We are given functions aij(y) satisfying (3.1), (3.2),
(3.3). We are also given functions bij(y,t) such that
bij - bji
, v
(No ellipticity is assumed). (3.109)
BE(t)(;)v =
.
i,j
Given v
- axi a j
E
Lioc (Rt;V) , we set
-
x bij(;,t
S)
av ax. (x,s)ds 3
0
and we consider the problem:
u
= O f o r t < O ,
where (3.111)
2 f E Lloc(Rt;H)
,
f = 0 for t < 0
This problem admits a unique solution.
.
One has the following
a priori estimates: (3.112)
I luElI
L-(o,T;V)
L- (0,T;H)
< c .
Indeed, if we set b E t;u,v) =
1 8
';1
t;u,v) =
x
bij(F,t)
au av ax 5 dx
(E,t) ax 8
I
j
j
$ dx =
,
OPERATORS OF HYPERBOLICf PETROWSKY OR SCHRODINGER TYPE
323
we obtain
and
so that (3.113) gives:
and therefore, for t 5 T finite (arbitrary)
I t
1 lu,'(t)
l2 +
1
aE (uE(t)) +
bE(t-s;uE(s)fuE(t))ds
0
hence
*
hence (3.112) follows. Therefore we can extract a subsequence, still denoted by uE, which satisfies (3.10).
We show now that u is characterized by the
solution of a pseudo differential operator equation.
EVOLUTION OPERATORS
324 We define, for Re p
J
0
m
2. ,(y;p) = 13 (3.1
-
,
e Ptb. .(y,t)dt
0
13
Then (3.110) can be written (3.115)
(AE
+
gE(p)
+ p 2 I)GE(p) = g(p)
.
(We can always assume that f admits a Laplace transform in t.) We use now "homogenization with parameter".
I
the Y-periodic solution (normalized by
We define
Xl(p)dy = 0) of
Y
we have set
We observe that (3.118)
A1
+ g,(p)
Therefore (3.116
gij (y;p)I
5
& so that the operator
is elliptic for Re p large enough
.
defines XJ (y;p)
(3.119)
and
4 (p) and
Q as in (3.105), (3.106).
We define
.
x3 (p) as
COMMENTS AND PROBLEMS
325
Then the homogenized equation is
(3.120) 2
u E Lloc(Rt;V)
,
u = 0 for t < 0
.
Comments and problems.
4.
Most of the results presented in this chapter are given for the first time.
Preliminary announcements were made in notes by the
authors (cf. Bibliography of Chapter 1). The Lp estimates of Section 2.7 were obtained, in the symmetric case, by Pulvirenti [l], [2].
The method given here extends to
parabolic equations of higher order or to systems. Given a family of elliptic operators, say A E , one can say that A E "G converges" to
c 7 if
V f, (AE)-lf -+
a-lf in a weak Sobolev
space, where the inverse are taken for a given set of homogeneous boundary conditions.
If one considers next a family of elliptic
operators depending on t, say AE (t), one can say that A' converges" to U -1 + A E ) f -t
(where "P" stands for "parabolic") if V f, -1 + a f in a weak Sobolev space (where again the
[A
[&
"P-G
]
inverse are taken for a given set of homogeneous boundary conditions). Then one can compare the "P-G convergence" with the "G-convergence V t " (in general these are different notions!) S.
Spagnolo [l].
Cf. F. Colombini and
Cf. also for this general approach, Spagnolo 111,
121, Sbordone [l]. In case the coefficients a..(y,-r)are Y-periodic in y and almost 13
periodic in
T,
one obtains formulas similar to those of the text,
with a difficulty for k = 2.
In the case k = 1, one considers
xJ(y,-r) as the solution of (1.67) which satisfies, say
EVOLUTION OPERATORS
326
I'
Y
XI (y,T)dy = 0. Then XI is almost periodic in
T
(T
plays the role
of a parameter) and one obtains
where
In case k = 3 , one introduces (compare to (1.86))
and
U
1
is given again as in (1.94) where @ =
lim
A+=
-
lYlA y 0
@(y,o)dody
.
In case k = 2 , one has to consider the T-almost periodic solution of
which is periodic in y, and many problems remain open in this direction.
327
COMMENTS AND PROBLEMS One can of course in the .r-almost periodic case consider correctors, as in the case of periodic coefficients.
Let us give now some indications on possible extensions of the resu ts given in the text. b is Y--r0
(4.5
Let b(y,r) be given satisfying
periodic
b E Lm(Rn Y
,
RT)
x
b(y,r) '> B > 0
,
,
et us consider the operator
and (4.6
a
bE&--
axi(arj
a
k b" = b(x/Elt/E 1
x) , j
aFj = aij (x/E,t/Ek
,
.
(If one considers the case bE(x,t) = b(x/E) when b does not depend on
T,
the homogenization is immediate and is given by
711
Y
(b)
a + ak ,
k = 1,2,3
)
.
The general case (4.6) leads to a number of interesting questions. In case k = 1, the asymptotic expansion, with usual notations, leads to
,
(4.7)
AluO = 0
(4.8)
A U + b -auO +Au = O , 1 1 aT 2 0
(4.9)
A1u2
+
(b
a + A2)u1 + aT
(b
a +
A3)u0 = f
.
It follows from (4.7) that uo does not depend on y . Taking I Y au (b) 2 = 0, so that uo = u(x,t) and with the of (4.8) gives:
Y
aT
notation (1.67), one will have:
328
EVOLUTION OPERATORS We use (4.10) into (4.9).
m
Taking
Y
of the result, we obtain:
1 (b) and we take the average in 2 y
We then multiply by
~
(4.11)
a
1
my(aij
-
aik
T:
el]% 7
One sees the appearance of first order derivatives in x in the homogenization process.
One also sees that one needs neither
regularity on the a. ' s of the type: lj
&
a
to make sense nor regularity of b:=ab
can integrate by parts in
'I
a. . E Lm(Rn 13 Y E
Lm(Ry
x
x
RT)
, in order
R T ) (since one
au in the coefficients of F ). j
The
justification can then be made along the lines of Section 2 .
If
k = 2, one runs into the difficulty of finding Y - T ~periodic solutions of
a question whose general study does not seem to have been made (the same is a fortiori the case for the almost periodic situation). In the case k = 3 , one defines:
and the functions
d
by (1.88) (with this new
El).
Then the
329
COMMENTS AND PROBLEMS homogenized equation is
Singular perturbations and homogenization. Let us consider first the problem
u
being subject, say, to Dirichlet's boundary conditions (with usual
boundary conditions).
Then the homogenized equation is obtained by
a
formulas analogous to those for homogenizing ; i i + AE but where A1 replaced by A1 + A 2 . For instance if k = 2, one considers the Y solution x 3 ( y , ~ ) , which is Y - T ~periodic, of
and
a2= a
2 p A 2 is now given by
(4.17) 2td2 =
qij
[aij
-
d].
aik ayk
One can also consider the problem: (4.18)
au
E
k 2 at + AE(t/E ) u E = f ,
with usual boundary conditions and with uE(0) = 0. If k = 2 , one has (4.19)
uE
+
u in L2 (0,T;V) weakly
&
EVOLUTION OPERATORS
330
where u is the solution of (4.20)
C7 u = f = f (x,t)
,
(t plays the role of a parameter)
(More precisely, (4.21)
u E
If k = 3
v , a 1 (u,v)=
or
(f,v)
,v
v
E
v
)
.
k = 4, one has again (4.19), with u given by
(4.22)
C12(u,v) = (f,v)
,V
E
V
,
if k = 3
,
(4.23)
0 3 (u,v) = (f,v)
, V'v E
V
,
if k = 4
.
v
Actually the proof of (4.22) has been obtained only for the Dirichlet's boundary condition by the method of the adjoint expansion, and assuming all data
to be smooth.
Proofs of (4.21), (4.23)
proceed along the lines of Section 2. There is one more a priori estimate one can obtain when
[One will obtain the same estimates for the equation
Indeed if we multiply (4.18), for k = 1, by AE(t/E)UE, we obtain (we use the symmetry): (4.26) Hence (4.27)
caE [2%~ , u € +] IAE(t/E)u,12 = (f,AE(t/E)uE)
.
.
331
COMMENTS AND PROBLEMS
=
. E (a) (uE,uE). It follows from (4.27) that
2
AE(t/c)uE remains in a bounded set of L ( 0 , T ; H ) .
(4.28)
Using (4.18) and (4.28) we see that (4.29)
E
au 2 in a bounded set of L ( 0 , T ; H ) 2 at remains
.
The behavior of the solution of (4.18) for k = 1 is unsettled.
Reiteration. Let us consider for instance the equation
with the usual boundary conditions and initial conditions, where the aij's satisfy the following conditions: aij(ylz,Tl,T2,T3)is Y-Z periodic in y,z and admits (4.31)
the period T O in the variable j
'lj,
j = 1,2,3 ;
and the usual hypothesis
-
'One
could weaken this hypo~esis but the weakest hypothesis is
not known.
332
EVOLUTION OPERATORS In (4.30) we assume that
(4.33)
0 < kl < k2 < k3
. 2
we have then the "usual" result that uE + u in L (0,T;V) weakly, where u is the solution of
au at + au
=
f subject to appropriate
boundary conditions and where 2 is computed as follows. One considers the operator with coefficients (4.34)
X t a. .(l,I,,v,v,x)
13
E
where k = k3/2.
One homogenizes the corresponding operator according
to the rules of Section 1; therefore we have three cases to consider: k < 2, k = 2 or k > 2,
(4.35)
k3 < 4
,
i.e.
k3 = 4
or
k3 > 4
.
Let us denote by b..(A,iJ,v) the coefficients of the homogenized 17
operator obtained in this way.
We then consider the operator with
coefficients
and we homogenize the corresponding operator: therefore there are again three cases to consider: (4.37)
k2 < 2
,
k2=2
Let us denote by c .
11
k 2 > 2 .
the coefficients of the homogenized operator
(11)
obtained in this way.
or
The coefficients of the operator we are looking
for are then given by 0
I
71 (4.38) 71
o
~
~
~
( = ~dij~
.)
d
7
~
333
COMMENTS AND PROBLEMS The proof is technically long but the ideas are those given in Chapters 1 and 2 . In case of coefficients of the form
one will apply the same rule to aij(x0,y,z,t0,~l,~2,T3) and one will obtain coefficients d . . (xo,to). Then the homogenized operator is 11
d . .(x,t). 11
Homogenization with rapidly oscillating potentials. One can extend the considerations of Chapter 1, Section 12, to the evolution problems, and this for parabolic and hyperbolic equations, and also forschrodinger equations.
For instance let uE
be the solution of
-at(4.40)
AuE + -1 W Eu
uE = 0 on
C
=
f
O
in
E
E
,
uE(xIO) = 0
Q
= Ox
lO,T[
,
.
2 We assume that WE = W(x/E,t/E ) (one can also consider 2) ) is almost W(X/E,~/Ek ) , k # 2 , and a function W (x,~,x/E,~/E periodic. y and in
Then one considers the almost periodic solution T)
and we set
of
=
average in y and A
Then
T
of Wx A
x
(a.p. in
334
EVOLUTION OPERATORS
(4.42
uE
.+
u in L2(0,T;Hi(8)) weakly
,
where u is the solution of
cf. Bensoussan, Lions, Papanicolaou [21 (where one will find analogous formulas also for the hyperbolic case).
Homogenization an8 penalty. Let us consider
(4.44)
where the aijk(y,r) satisfy the usual hypotheses.
We consider the
system : aUEl
at
+
1 (u A h 1 + 2 E
l
-
UE2) =
1'
(4.45)
with boundary conditions, say the Dirichlet's boundary conditions but this is without importance, (4.46)
uEl = uE2 = 0 on E
and (4.47)
UEl(X,0) = Uc2(X,O) = 0
.
This problem admits a unique solution.
Multiplying the first
(respectively, second) equation (4.45) by uEl (respectively, uc2) orLe obtains:
335
COMMENTS AND PROBLEMS
Hence it follows that
and
-
1 (uEl
(4.49)
uE2) is bounded in L2 (0,T;H)
Estimate (4.49) comes from the penalty terms 5 (4.35).
.
1 7
(uEl
-
uE2) in
It follows from (4.38), (4.39) that one can extract a
subsequence still denoted by uEi such that uEi
(4.50)
+
L2 (0,T;V) weakly (where we have the same
u
limit for uEl and for uE2)
.
The limit u is the solution of the following homogenized We give the formulas only for the case k = 2.
problem.
Alj, A2j by
and
{;x
x
;I
A:
= E-2~11
A :
=
E-2
+
E-1~12
+
,
E0
A21 + E - ~ +A E~ 0A23 ~
is the solution which is Y - T ~periodic of
Then the equation for u is (4.52)
au
2 - at -
qij
a 2u ZQT
j
= fl + f2
#
One defines
336
EVOLUTION OPERATORS
where (4.53)
qij =
nm
1
ax;
a i k l - K + aij2
-
aik2
$1
.
The proof proceeds along the lines of Section 2. In (4.52) the boundary condition is (4.54)
u = O o n C .
If in (4.45) one takes boundary conditions which are the same for u
El and for uc2, that is if we take a variational formulation with a space (4.55)
V = W
x
W
,
Hi(0)
5 W 5 H1(S) ,
(with the same W), the result extends:
u will be characterized by
u(t) E W and
In case we take V = V1
x
Vz with different spaces V
and V 12
one obtains (4.57)
u(t)
E
v1
n
v2 ,
and
Problems where one has rapi ly oscillating coefficients in t an a penalty term are studied by Simonenko [l], [21. One can also study the evolution problems with degenerate elliptic part and also with domains with "periodic holes".
Homogenization and regularization. It is known that the solution of a parabolic equation can be obtained as the limit of the solution of elliptic equations, via the
COMMENTS AND PROBLEMS
337
elliptic regularization. Applying this idea to operators of the type
a + A' at
leads to the behavior as
(4.58)
-Em
aLuc
au
at2
at
-- k E + A E u
Ue(O) = 0
E +
0 of equations of the type
= f ,
au EaT( T ) = 0
,
.
The behavior of uE will depend on m and on k, if ' A k cients a. . ( x / ~t/E , )
.
13
is with coeffi-
For instance if we consider
(4.59)
-€
2 3% at2
au au +at > - -axia aij ( E L2 ) E ax ' f , E'
j
then the limit value u of uE is solution of a new parabolic problem obtained in the following manner.
One defines $J(y,.r)as the Y - T ~
periodic solution of
Then (4.61)
& + B u = f at
where B u =
-
,
77i
The results presented for the homogenization of V.1 are sketchy. Other results are announced in the first note of the authors in the Bibliography of this chapter. The problem studied in Section 3.9 corresponds to operators which have been studied (without homogenization) by Showalter and Ting [l]; cf. also other equations of this type, or which present somewhat analogous properties, in Carroll-Showalter 111. results for the homogenization of V . I .
Some
connected with these operators
338
EVOLUTION OPERATORS
are given in the first note of the authors in the Bibliography.
Many
questions arise for these equations. It would be interesting to understand "intrinsically" when the homogenization of partial differential operators leads to non-local operators in the limit (this question could also have a physical interest).
For instance let us
consider the equation
[bij (y) "aYj -1
BE given as in Section 3.9 and the aijvs not necessarily elliptic. We set:
B1 =
a - ayi
and let us assume that there exists
a function ~ j ( ~ , rwhich ) is Y-r0 periodic and which satisfies
"
(i.e., 0
aa.. ay,
(y,-c)dT= 0). Then the homogenized equation (found by
asymptotic expansion) is formally
For the homogenization of first order hyperbolic systems, we refer to Chapter 4.
One can also study (cf. Bensoussan, Lions and Papanicolaou
[ 3 1 ) a case when one has simultaneously "homogenization" and
"penalty":
COMMENTS AND PROBLEMS
339
a u ~ 2 aE a u ~ 2 bE E ax 2 (UEl - UE2) = f2 at (4.64)
(0,l)
x E
,
UEl(O,t) = 0
u (x,O) = 0 Ei
,
,
aE,bE 1. a > 0
1 m - -+ p in L (0,l) weak star aE
,
v in L (0,l) weak star
.
(4.65) -+
afi at E L2((0,1) x Then, assuming that fi, __ m
UEi
-+
UE2(l,t) = 0
,
,
aE,bE E Lm(O,l)
aC
I
,
(O,T)), one has
u in L (0,T;L2(0,1)) weak star
,
(same limit u for i = 1,2) and where u is the solution of the parabolic equation
(4.66)
In ( 4 . 6 4 ) we have only ( 4 . 6 5 ) ,
without periodic, or almost
periodic, structure. For problems of this type for parabolic equations, we refer to Markov and Oleinik [l]. Very many questions arise in connection with the homogenization of second order hyperbolic operators. the text.
Some of them are indicated in
We shall return to this question in Chapter 4 .
For a
340
EVOLUTION OPERATORS
recent work on the regularity of the solution of hyperbolic equations with irregular coefficients we refer to Colombini 1 1 1 . One can also study the homogenization of coupled hyperbolic parabolic systems: cf. Bensoussan, Lions, Papanicolaou [l]. Another problem is the following. Consider the equation
where k E i (4.68)
E
Lm( a ) , k :
-+
Gl
in Lm( 8 ) weak star
.
We assume that (4.69)
kT,B>O
and that (4.70)
kz 1. 0 (but kh can be zero)
.
In (4.67) A is a fixed second order elliptic operator.
Then one
can add to (4.67) standard boundary conditions on uE and for the initial conditions
(4.72)
k9
at
auE where uo and v1 are given: (4.72) gives a condition on ar (X,O) only on the set where ki # 0. This is an equation of "hyperbolic-
parabolic" type.
Cf. V. N. Vragov [l] for the case of fixed
coefficients kl, k2.
One can show the following (cf. Bensoussan,
Lions, Papanicolaou [4]): we do notrestrict the generality in assuming that
341
COMMENTS AND PROBLEMS
@* x
( 4 . 73)
in Lm(B ) weak star
.
Then us * u in Lm(O,T;V) weak star, ui * u' in L2 (0,T;H) weak star, where u is the solution of
+
(4.74)
i2u"
(4.75)
u(0) = u0 ,
(4.76)
u'(0) =
1u'
+
;1L Vl
Au = f
,
.
k2 We remark that in ( 4 . 7 6 ) , domain 0
.
u' (x,O) = u' ( 0 ) is given over the whole
Comparing with ( 4 . 7 2 )
we see that there is, here, some
sort of "increase" in the initial data. Other evolution equations are studied from the new point of homogenization in separated papers by the authors.
For the transport
equations, cf. already the authors' [51 and the Bibliography therein. For other physical aspects of these questions, cf. Sanchez-
.
Palencia [l]
We also refer to the survey of O.A. Oleinik [I], to appear in the Ouspechi Mat. Nauk. We also mention applications of the asymptotic expansion method to problems in turbulence?
.
-t P. Perrier and 0. Pironneau, C.R. Acad. Sc. Paris, 1978.
BIBLIOGRAPHY OF CHAPTER 2 A. Bensoussan, J.L; Lions and G. Papanicolaou [l] Sur quelques problbmes asymptotiques d'dvolution. C.R.A.S. Paris, t. 281 (1975), p. 317-322. [2] Sur la convergence d'opgrateurs diffgrentiels avec potentiel oscillant, C.R.A.S., Paris, February 7, 1977. 131 Remarques sur le comportement asymptotique de systsmes d'6volution. French-Japan Symposium, Tokyo, September 1976. [ E l Perturbations et "augmentation" des conditions initiales. Lyon, December 1976. 151 Boundary layers and homogenization of transport processes. R.I.M.S. Kyoto, 1978. N.N. Bogoliubov and Y.A. Mitropolsky [l] Asymptotic methods in the theory of non linear oscillations. (Translated from the Russian) Hindustan Pub. Corp. Delhi 1961. R.W. Carroll and R.E. Showalter [l] Singular and degenerate Cauchy problems. Math. in Sc. and Eng., Vol. 127, Acad. Press 1976. F. Colombini [l] On the regularity of solutions of hyperbolic equations with discontinuous coefficients variable in time. Scuola Normale Superiore Report, 1976. F. Colombini and S. Spagnolo 111 Sur la convergence de solutions d'gquations paraboliques. To appear. G. Duvaut and J.L. Lions [l] Les Idquations en Mdcanique et en Physique, Paris, Dunod 1972. English translation, Springer, 1975 J.L. Lions [l] Equations differentielles op6rationnelles et problsmes aux limites. Springer, 1961. [2] Lectures on elliptic differential equations. Tata Institute, Bombay, 1957. J.L. Lions and E. Magenes 111 ProblSmes aux limites non homogenes et applications. Vol. 1, 2, Paris, Dunod, 1968. English translation, Springer 1970. P. Marcellini and C. Sbordone [l] An approach to the asymptotic behaviour of elliptic parabolic operators. To appear A. Marino and S. Spagnolo [l] Un tipo di approssimazione dell' operatore Ann. S.N. Sup. Pisa, XXIII (1969), p. 657-673. V . G . Markov and O.A. Qleinik [l] On the heat diffusion in a one dimensional dispersive medium. P.M.M. 39 (19751, p. 1073-1081.
...
342
BIBLIOGRAPHY
343
L. Nirenberq 111 Remarks on strongly elliptic partial differential equations. C.P.A.M. 8 ( 1 9 5 5 ) , p. 6 4 8 - 6 7 6 . O.A. Oleinik [l] Survey on homogenization. Ouspechi Mat. Nauk. to appear. G. Pulvirenti [ l l Sulla sommabilits Lp.. Le Matematiche. 2 2 ( 1 9 7 1 ) , p. 2 5 0 - 2 6 5 . [ 2 1 Ancora sulla sommabilits Lp.. Le Matematiche, 2 3 ( 1 9 6 8 ) , p. 1 6 0 - 1 6 5 . E. Sanchez-Palencia 111 Comportement local et macroscopique d'un type de milieux physiques h6t6rogSne.s. Int. J. Enqrg. Sciences, 1 2 ( 1 9 7 h ) , p. 3 3 1 - 3 5 1 . C. Sbordone [ l l Sulla G-convergenza di equazioni ellittiche e paraboliche. Ric. di Mat. 2 4 ( 1 9 7 5 ) , p. 7 6 - 1 3 6 . R.E.Showalter and T.W. Ting [l] Pseudo parabolic partial dif-
.
.
ferential equations. SIAM J. Math. Anal. 1 ( 1 9 7 0 ) , p. 1 - 2 6 . L. de Simon [l] Sull'equazione delle onde con termine noto periodico. Ren. 1st. di Mat. Univ. Trieste, Vol. 1, fasc. 11, ( 1 9 6 9 ) , p. 1 5 0 - 1 6 2 . I.B. Simonenko [ l ] A justification of the averaging method for abstract parabolic equations. Mat. Sbornik. ( 1 2 3 ) ( 1 9 7 0 ) , p. 5 3 - 6 1 . [ 2 ] A justi€ication of the averaging method ..., Mat. Sbornik, ( 1 2 9 ) ( 1 9 7 2 ) , p. 2 4 5 - 2 6 3 . S. Spaqnolo [l] Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore. Ann. Sc. Norm. Sup. Pisa, 2 1 ( 1 9 6 7 1 , 657-699. [ 2 ] Sulla convergenza di soluzioni di equazioni pakaboliche ed ellittiche. Ann. Sc. Norm. Sup. Pisa, 2 2 ( 1 9 6 8 ) , 5 7 1 - 5 9 7 . V.N. Vragov [ l ] On a mixed problem for a class of hyperbolicparabolic equations. Doklady, 2 2 2 ( 1 9 7 4 ) , Soviet Math. Doklady, 16 (1975),
1179-1183.