Two-Scale Convergence of Unsteady Stokes Type Equations

0.22 MB·English

by  Lazarus Signing

TWO-SCALE CONVERGENCE OF UNSTEADY STOKES
TYPE EQUATIONS
1
LAZARUSSIGNING
1
0
2 University of Ngaoundere
Department of Mathematics and
n
Computer Science, P.O.Box 454
a
Ngaoundere, Cameroon
J
email: [email protected]
4
1
Abstract. In this paper we study the homogenization of unsteady
]
Stokes type equations in the periodic setting. The usual Laplace op-
P
erator involved in the classical Stokes equations is here replaced by a
A
linear elliptic differential operator of divergence form with periodically
.
h oscillating coefficients. Our mean tool is the well known two-scale con-
t vergence method.
a
m
[

1 1. Introduction
v
1 Let Ω be a smooth bounded open set in RN (the N-numerical space RN
x
8 of variables x = (x ,...,x )), where N is a given positive integer, and let T
1 N
7
and ε be real numbers with T > 0 and 0 < ε < 1. We consider the partial
2
. differential operator
1
0 N
∂ ∂
1 Pε = − aε
1 ∂x ij∂x
i j
: iX,j=1 (cid:18) (cid:19)
v
Xi in Ω×]0,T[, where aεij(x) = aij xε (x ∈ Ω), aij ∈ L∞ RNy ;R (1 ≤ i,j ≤
N) with
r (cid:0) (cid:1) (cid:0) (cid:1)
a (1.1) a = a ,
ij ji
and the assumption that there is a constant α > 0 such that
N
(1.2) a (y)ζ ζ ≥ α|ζ|2 for all ζ = ζ ∈ RN and
ij j i j
i,j=1
X (cid:0) (cid:1)
for almost all y ∈ RN, where RN is the N-numerical space RN of variables
y
y = (y ,...,y ),andwhere|·|denotestheEuclideannorminRN. Theopera-
1 N
tor Pε acts on scalar functions, say ϕ ∈ L2 0,T;H1(Ω) . However, we may
(cid:0) (cid:1)
2000 Mathematics Subject Classification. 35B27, 35B40, 73B27, 76D30.
Keywordsandphrases. Homogenization,Two-scaleconvergence,UnsteadyStokestype
equations.
1
2 LAZARUSSIGNING
as well view Pε as acting on vector functions u = ui ∈ L2 0,T;H1(Ω)N
in a diagonal way, i.e., (cid:16) (cid:17)
(cid:0) (cid:1)
(Pεu)i = Pεui (i = 1,...,N).
For any Roman character such as i, j (with 1 ≤ i,j ≤ N), ui (resp.
uj) denotes the i-th (resp. j-th) component of a vector function u in
L1 (Ω×]0,T[)N or in L1 RN ×R N where R is the numerical space
loc loc y τ τ
R of variables τ. Further, for any real 0 < ε< 1, we define uε as
(cid:0) (cid:1)
x t
uε(x,t) = u , ((x,t) ∈ Ω×]0,T[)
ε ε
(cid:18) (cid:19)
for u ∈ L1 RN ×R , as is customary in homogenization theory. More
loc y τ
generally, for u ∈ L1 Q×RN ×R with Q = Ω×]0,T[, it is customary
(cid:0) loc(cid:1) y τ
to put
(cid:0) (cid:1)
x t
uε(x,t) = u x,t, , ((x,t) ∈Ω×]0,T[)
ε ε
(cid:18) (cid:19)
whenever the right-hand side makes sense (see, e.g., [8, 9]).
After these preliminaries, let f = fi ∈ L2 0,T;H−1(Ω;R)N . For any
fixed 0 < ε < 1, we consider the initial bounda(cid:16)ry value problem (cid:17)
(cid:0) (cid:1)
∂u
(1.3) ε +Pεu +gradp = f in Ω×]0,T[,
ε ε
∂t
(1.4) divu = 0 in Ω×]0,T[,
ε
(1.5) u =0 on ∂Ω×]0,T[,
ε
(1.6) u (0) = 0 in Ω.
ε
We will later see that as in [18], (1.3)-(1.6) uniquely define (u ,p ) with
ε ε
u ∈ W(0,T) and p ∈ L2 0,T;L2(Ω;R)/R , where
ε ε
W(0,T) = u(cid:0)∈ L2(0,T;V) :u(cid:1)′ ∈ L2 0,T;V′
V being the space of fun(cid:8)ctions u in H1(Ω;R)N wi(cid:0)th divu(cid:1)=(cid:9) 0 (V′ is the
0
topological dual of V) and where
L2(Ω;R)/R = v ∈ L2(Ω;R): vdx = 0 .
(cid:26) ZΩ (cid:27)
Let us recall that W(0,T) is provided with the norm
1
kuk = kuk2 + u′ 2 2 (u ∈ W(0,T)),
W(0,T) L2(0,T;V) L2(0,T;V′)
which makes it a(cid:16)Hilbert space (cid:13)wit(cid:13)h the foll(cid:17)owing properties (see [18]):
(cid:13) (cid:13)
W(0,T) is continuously embedded in C [0,T];L2(Ω)N and is compactly
embedded in L2 0,T;L2(Ω)N . (cid:16) (cid:17)
(cid:16) (cid:17)
UNSTEADY STOKES TYPE EQUATIONS 3
Our aim here is to investigate the asymptotic behavior, as ε → 0, of
(u ,p ) under the assumption that the functions a (1≤ i,j ≤ N) are peri-
ε ε ij
odic in the space variable y. The steady state version of this problem (i.e.,
the homogenization of stationary Stokes type equations) was first investi-
gated by Bensoussan, Lions and Papanicalaou [2]. These authors use the
well-known approach of asymptotic expansions combined with Tartar’s en-
ergy method. We also mention the paper of Nguetseng and the author [16],
on the sigma-convergence of stationary Navier-Stokes type equations.
Thepresent studydeals with the periodichomogenization of an evolution
problem for Stokes type equations.
Thisstudyis motivated by thefactthat thehomogenization of (1.3)-(1.6)
is connected with the modelling of heterogeneous fluid flows (see, e.g., [19]
for more details about such models).
Our approach is the well-known two-scale convergence method.
Unless otherwise specified, vector spaces throughout are considered over
the complex field, C, and scalar functions are assumed to take complex
values. Let us recall some basic notation. If X and F denote a locally
compact space and a Banach space, respectively, then we write C(X;F)
for continuous mappings of X into F, and B(X;F) for those mappings
in C(X;F) that are bounded. We shall assume B(X;F) to be equipped
with the supremum norm kuk = sup ku(x)k (k·k denotes the norm
∞ x∈X
in F). For shortness we will write C(X) = C(X;C) and B(X) = B(X;C).
LikewiseinthecasewhenF = C,theusualspacesLp(X;F)andLp (X;F)
loc
(X provided with a positive Radon measure) will be denoted by Lp(X) and
Lp (X), respectively. Finally, the numerical space RN and its open sets are
loc
each provided with Lebesgue measure denoted by dx = dx ...dx .
1 N
The rest of the paper is organized as follows. Section 2 is devoted to the
preliminary results on existence and uniqueness, and some estimates on the
velocity u , the pressure p and the accelaration ∂uε of the fluid, whereas in
ε ε ∂t
Section 3 one convergence theorem is established.
2. Preliminaries
Let Ω be a smooth bounded open set in RN, let T > 0 be a real number
and let f = fj ∈ L2 0,T;H−1(Ω)N . For 0 < ε < 1, it is not apparent
that the init(cid:0)ial(cid:1)bounda(cid:16)ry value problem(cid:17) (1.3)-(1.6) has a solution (uε,pε),
and that the latter is unique. With a view to elucidating this, we introduce,
for fixed 0 < ε< 1 the bilinear form aε on H1(Ω;R)N ×H1(Ω;R)N defined
0 0
by
N N ∂uk ∂vk
aε(u,v) = aε dx
ij∂x ∂x
k=1i,j=1ZΩ j i
X X
4 LAZARUSSIGNING
for u = uk and v = vk in H1(Ω;R)N. By virtue of (1.1), the form aε is
0
symmetric. Further, in view of (1.2),
(cid:0) (cid:1) (cid:0) (cid:1)
(2.1) aε(v,v) ≥αkvk2
H1(Ω)N
0
for every v = vk ∈ H1(Ω;R)N and 0 < ε < 1, where
0
(cid:0) (cid:1) 1
N 2
kvk = ∇vk dx
H1(Ω)N
0 Xk=1ZΩ(cid:12) (cid:12) !
(cid:12) (cid:12)
with ∇vk = ∂vk,..., ∂vk . On the other (cid:12)hand(cid:12), it is clear that a constant
∂x1 ∂xN
c0 > 0 exists(cid:16)such that (cid:17)
(2.2) |aε(u,v)|≤ c kuk kvk
0 H1(Ω)N H1(Ω)N
0 0
for every u, v ∈ H1(Ω;R)N and 0< ε < 1.
0
We are now in a position to verify the following result.
Proposition 2.1. Suppose f lies in L2 0,T;L2(Ω;R)N . Under the hy-
potheses (1.1)-(1.2), the initial boundary(cid:16) value problem (cid:17)(1.3)-(1.6) deter-
mines a unique pair (u ,p ) with
ε ε
u ∈ L2 0,T;H1(Ω;R)N ∩C [0,T];L2(Ω;R)N and
ε 0
p ∈ L2 (cid:16)0,T;L2(Ω;R)/R(cid:17). (cid:16) (cid:17)
ε
Proof. For(cid:0)fixed 0 < ε < 1, w(cid:1)e consider the Cauchy problem
u′ (t)+A u (t) = ℓ(t) in ]0,T[
(2.3) ε ε ε
u (0) = 0,
ε
(cid:26)
where A is the linear operator of V into V′ defined by
ε
(A u,v) = aε(u,v) for all u,v ∈ V
ε
and ℓ is the function in L2(0,T;V′) defined by
(ℓ(t),v) = (f (t),v) for all v ∈ V
and for almost all t ∈]0,T[, and where (,) denotes the duality pairing be-
tween V′ and V as well as between H−1(Ω;R)N and H1(Ω;R)N. Thanks
0
to (2.1)-(2.2) the Cauchy problem (2.3) admits a unique solution u in
ε
W(0,T), as is easily seen by following [5, Chap.3, Th´eor`eme 1.2, p.116],
see also [18, pp.254-260]. Now, let us check that the abstract parabolic
problem (2.3) is equivalent to (1.3)-(1.6). Let U (t) = tPεu (s)ds and
ε 0 ε
t
F(t) = f (s)ds for 0 ≤ t ≤ T, where u satisfies (2.3). It is evident that
0 ε R
U and F ∈ C [0,T];H−1(Ω;R)N . By the first equality of (2.3) we have
ε R
(cid:16) (cid:17)
d
(2.4) (u (t),ϕ) = (ℓ(t)−A u (t),ϕ) for all ϕ ∈ V,
ε ε ε
dt
UNSTEADY STOKES TYPE EQUATIONS 5
where
V = ϕ ∈ D(Ω;R)N : divϕ = 0 .
n o
Integrating (2.4), we have
hu (t)+U (t)−F(t),ϕi =0, 0 ≤ t ≤ T, ϕ ∈ V.
ε ε
Thus, using a classical argument (see, e.g., [18, p.14]), we get a function
P ∈ C [0,T];L2(Ω;R)/R such that
ε
(cid:0) u(cid:1) +U +gradP = F.
ε ε ε
Hence p = ∂Pε ∈ D′(Q) and the pair (u ,p ) verifies (1.3) (in the distribu-
ε ∂t ε ε
tion sense on Q), with in addition (1.4)-(1.6), of course. Futhermore, by us-
ingthefactthatf ∈ L2 0,T;L2(Ω;R)N wehaveu′ ∈ L2 0,T;L2(Ω;R)N ,
ε
asiseasilyseenbyfollow(cid:16)ing[18,p.268]. T(cid:17)hereforep liesin(cid:16)L2 0,T;L2(Ω;R(cid:17))/R
ε
and is unique. Conversely, it is an easy exercise to verify that if (u ,p ) is a
ε ε
solution of (1.3)-(1.6) with u ∈ W(0,T) and p ∈ L2 0,T;L2(cid:0)(Ω;R) , then (cid:1)
ε ε
u satisfies (2.3). The proof is complete. (cid:3)
ε
(cid:0) (cid:1)
The following regularity result is fundamental for the estimates of the
solution (u ,p ) of (1.3)-(1.6).
ε ε
Lemma 2.1. Suppose f,f′ ∈ L2(0,T;V′) and f (0) ∈L2(Ω;R)N. Then the
solution u of (2.3) verifies:
ε
u′ ∈ L2(0,T;V)∩L∞(0,T;H),
ε
where H is the closure of V in L2(Ω;R)N.
The proof of the above lemma follows by the same line of argument as in
the proof of [18, p.299, Theorem 3.5]. So we omit it. We are now able to
prove the result on the estimates.
Proposition 2.2. Under the hypotheses of Lemma 2.1, there exists a con-
stant c > 0 (independent of ε) such that the pair (u ,p ) solution of (1.3)-
ε ε
(1.6) in W(0,T)×L2 0,T;L2(Ω;R)/R satisfies:
(2.5) (cid:0) ku k (cid:1)≤ c
ε W(0,T)
∂u
ε
(2.6) ≤ c
∂t
(cid:13) (cid:13)L2(0,T;H−1(Ω)N)
(cid:13) (cid:13)
(cid:13) (cid:13)
and
(cid:13) (cid:13)
(2.7) kp k ≤ c.
ε L2(0,T;L2(Ω))
Proof. Let (u ,p ) be the solution of (1.3)-(1.6). We have
ε ε
(2.8) u′ (t),v +aε(u (t),v) = (f (t),v) (v ∈ V)
ε ε
(cid:0) (cid:1)
6 LAZARUSSIGNING
for almost all t ∈ [0,T]. By taking in particular v = u (t) in (2.8), we have
ε
for almost all t ∈[0,T]
d 1
|u (t)|2+2αku (t)k2 ≤ kf (t)k2 +αku (t)k2
dt ε ε α V′ ε
where|·|andk·karerespectivelythenormsinL2(Ω)N andH1(Ω)N. Hence,
0
for every s ∈ [0,T]
s 1 T
|u (s)|2+α ku (t)k2dt ≤ kf (t)k2 dt
ε ε α V′
Z0 Z0
since u (0) = 0. By the preceding inequality, we see that
ε
T 1 T
(2.9) α ku (t)k2dt ≤ kf (t)k2 dt.
ε α V′
Z0 Z0
On the other hand, the abstract parabolic problem (2.3) gives
u′ = f −A u .
ε ε ε
Hence, in view of (2.2)
(2.10) u′ ≤ kfk +c ku k .
ε L2(0,T;V′) L2(0,T;V′) 0 ε L2(0,T;V)
Thus, by (2.9)(cid:13)an(cid:13)d (2.10) one quickly arrives at (2.5). Let us show (2.6).
(cid:13) (cid:13)
We are allowed to differentiate (2.8) in distribution sense on ]0,T[, and by
virtue of the hypotheses of Lemma 2.1, we get u′′ ∈ L2(0,T;V′) and
ε
(2.11) u′′,v +aε u′,v = f′,v (v ∈ V).
ε ε
In view of Lemma(cid:0)2.1, w(cid:1)e take(cid:0)in pa(cid:1)rtic(cid:0)ular v(cid:1) = u′ (t) in (2.8). This yields
ε
u′ (t) 2+aε u (t),u′ (t) = f (t),u′ (t) (t ∈ [0,T]).
ε ε ε ε
Further,(cid:12)since(cid:12)u′ ∈ (cid:0)L2(0,T;V) a(cid:1)nd (cid:0)u′′ ∈ L2(0(cid:1),T;V′), we have u′ ∈
(cid:12) (cid:12) ε ε ε
C([0,T];H). Hence, by taking in particular t = 0 in the preceding equality
and using (1.6) one quickly arrives at
u′ (0) 2 ≤ |f (0)| u′ (0) ,
ε ε
i.e., (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(2.12) u′ (0) ≤ |f (0)|.
ε
The inequality (2.12) shows t(cid:12)hat u′(cid:12)(0) lies in a bounded subset of H. On
ε
the other hand, by taking in p(cid:12)articu(cid:12)lar v =u′ (t) in (2.11), we get
ε
d 1
u′ (t) 2+2α u′ (t) 2 ≤ f′(t) 2 +α u′ (t) 2
dt ε ε α V′ ε
for almost all(cid:12)t ∈ [0(cid:12),T]. In(cid:13)tegrati(cid:13)ng the p(cid:13)reced(cid:13)ing ineq(cid:13)uality(cid:13)on [0,t] (t ∈
(cid:12) (cid:12) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
[0,T]) leads to
t 1
u′ (t) 2+α u′ (s) 2ds ≤ f′ 2 + u′ (0) 2.
ε ε α L2(0,T;V′) ε
Z0
(cid:12) (cid:12) (cid:13) (cid:13) (cid:13) (cid:13) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:13) (cid:13) (cid:13) (cid:13) (cid:12) (cid:12)
UNSTEADY STOKES TYPE EQUATIONS 7
It follows from (2.12) and the preceding inequality that u′ belongs to a
ε
bounded subset of L2 0,T;L2(Ω;R)N . Hence (2.6) is immediate. Let us
prove (2.7). For almos(cid:16)t all t ∈ [0,T], p(cid:17)(t) ∈ L2(Ω;R)/R. Thus, by [17, p.
ε
30] there exists v (t) ∈ H1(Ω;R)N such that
ε 0
(2.13) divv (t)= p (t)
ε ε
(2.14) kv (t)k ≤ c |p (t)| ,
ε 1 ε L2(Ω)
wheretheconstantc dependssolely on Ω. Multiplying(1.3)byv (t)yields
1 ε
u′ (t),v (t) +aε(u (t),v (t))− p (t)divv (t)dx = (f (t),v (t))
ε ε ε ε ε ε ε
ZΩ
(cid:0) (cid:1)
for almost all t ∈ [0,T]. Integrating the preceding equality on [0,T] and
using (2.13)-(2.14) lead to
kp k2 ≤ c c u′ kp k +c kfk kp k
ε L2(Q) 1 ε L2(0,T;L2(Ω)N) ε L2(Q) 1 L2(0,T;H−1(Ω)) ε L2(Q)
(cid:13) (cid:13)
(cid:13) (cid:13) +c c ku k kp k ,
1 0 ε L2(0,T;V) ε L2(Q)
wherecistheconstantinthePoincar´einequality, c andc aretheconstants
0 1
in (2.2) and (2.14) respectively. Thus,
(2.15)
kp k ≤c c u′ +c kfk +c c ku k .
ε L2(Q) 1 ε L2(0,T;L2(Ω)N) 1 L2(0,T;H−1(Ω)) 1 0 ε L2(0,T;V)
Combining (2.15(cid:13)(cid:13)), ((cid:13)(cid:13)2.5) and (2.6) leads to (2.7). (cid:3)
3. A convergence result for (1.3)-(1.6)
We set Y = −1, 1 N, Y considered as a subset of RN (the space RN
2 2 y
of variables y = (y ,...,y )). We set also Z = −1, 1 , Z considered as
(cid:0) 1 (cid:1) N 2 2
a subset of R (the space R of variables τ). Our purpose is to study the
τ
(cid:0) (cid:1)
homogenization of (1.3)-(1.6) under the periodicity hypothesis on a .
ij
3.1. Preliminaries. Let us first recall that a function u ∈ L1 RN ×R
loc y τ
is said to be Y × Z-periodic if for each (k,l) ∈ ZN × Z (Z denotes the
(cid:0) (cid:1)
integers), we have u(y+k,τ +l) = u(y,τ) almost everywhere (a.e.) in
(y,τ) ∈ RN ×R. If in addition u is continuous, then the preceding equality
holds for every (y,τ) ∈ RN ×R, of course. The space of all Y ×Z-periodic
continuous complex functions on RN ×R is denoted by C (Y ×Z); that
y τ per
of all Y ×Z-periodicfunctions in Lp RN ×R (1≤ p < ∞) is denoted by
loc y τ
p
L (Y ×Z). C (Y ×Z) is a Banach space under the supremum norm on
per per
(cid:0) (cid:1)
RN ×R, whereas Lp (Y ×Z) is a Banach space under the norm
per
1
p
kuk = |u(y,τ)|pdydτ u∈ Lp (Y ×Z) .
Lp(Y×Z) per
(cid:18)ZZZY (cid:19)
(cid:0) (cid:1)
8 LAZARUSSIGNING
We will need the space H1 (Y) of Y-periodic functions u ∈ H1 RN =
# loc y
W1,2 RN such that u(y)dy = 0. Provided with the gradient norm,
loc y Y (cid:0) (cid:1)
(cid:0) (cid:1) R 1
2
kuk = |∇ u|2dy u∈ H1 (Y) ,
H1(Y) y #
# (cid:18)ZY (cid:19)
(cid:0) (cid:1)
where ∇ u = ∂u,..., ∂u , H1 (Y) is a Hilbert space. We will also need
y ∂y1 ∂yN #
the space L2 (cid:16)Z;H1 (Y)(cid:17)with the norm
per #
(cid:16) (cid:17)
1
2
kuk = |∇ u(y,τ)|2dydτ u∈ L2 Z;H1 (Y)
L2 (Z;H1(Y)) y per #
per # (cid:18)ZZZY (cid:19)
(cid:0) (cid:0) (cid:1)(cid:1)
which is a Hilbert space.
Beforewecanrecalltheconceptoftwo-scale convergence, letusintroduce
one further notation. The letter E throughout will denote a family of real
numbers 0 < ε < 1 admitting 0 as an accumulation point. For example,
E may be the whole interval (0,1); E may also be an ordinary sequence
(ε ) with 0 < ε < 1 and ε → 0 as n → ∞. In the latter case E will be
n n∈N n n
referred to as a fundamental sequence.
Let Ω be a bounded open set in RN and Q = Ω×]0,T[ with T ∈ R∗, and
x +
let 1 ≤ p < ∞.
Definition 3.1. A sequence (u ) ⊂ Lp(Q) is said to:
ε ε∈E
(i) weakly two-scale converge in Lp(Q) to some u ∈ Lp(Q;Lp (Y ×Z))
0 per
if as
E ∋ ε → 0,
(3.1)
u (x,t)ψε(x,t)dxdt → u (x,t,y,τ)ψ(x,t,y,τ)dxdtdydτ
ε 0
ZQ Z Z ZQ×Y×Z
for all ψ ∈ Lp′(Q;C (Y ×Z)) 1 = 1− 1 , where ψε(x,t) =
per p′ p
ψ x,t, x, t ((cid:16)(x,t) ∈ Q);(cid:17)
ε ε
(ii)stronglytwo-scalecon(cid:0)vergein(cid:1)Lp(Q)tosomeu0 ∈Lp(Q;Lpper(Y ×Z))
if the following property is verified:
Given η > 0 and v ∈ Lp(Q;C (Y ×Z)) with
per
ku −vk ≤ η, there is some α> 0 such
 0 Lp(Q×Y×Z) 2
that ku −vεk ≤ η provided E ∋ ε ≤ α.
 ε Lp(Q)
We willbriefly express weak and strong two-scale convergence by writing
u → u in Lp(Q)-weak 2-s and u → u in Lp(Q)-strong 2-s, respectively.
ε 0 ε 0
Remark 3.1. It is of interest to know that if u → u in Lp(Q)-weak 2-s,
ε 0
then (3.1) holds for ψ ∈ C Q;L∞ (Y ×Z) . See [9, Proposition 10] for the
per
proof.
(cid:0) (cid:1)
UNSTEADY STOKES TYPE EQUATIONS 9
Instead of repeating here the main results underlying two-scale conver-
gence, we find it more convenient to draw the reader’s attention to a few
references, see, e.g., [1], [7], [9] and [20].
However, we recall below two fundamental results. First of all, let
Y(0,T) = v ∈ L2 0,T;H1(Ω;R) :v′ ∈ L2 0,T;H−1(Ω;R) .
0
Y(0,T) is provi(cid:8)ded with(cid:0)the norm (cid:1) (cid:0) (cid:1)(cid:9)
1
kvk = kvk2 + v′ 2 2 (v ∈ Y(0,T))
Y(0,T) L2(0,T;H1(Ω)) L2(0,T;H−1(Ω))
0
which makes it(cid:16)a Hilbert space. (cid:13) (cid:13) (cid:17)
(cid:13) (cid:13)
Theorem 3.1. Assume that 1 < p < ∞ and further E is a fundamental
sequence. Let a sequence (u ) be bounded in Lp(Q). Then, a subsequence
ε ε∈E
E′ can be extracted from E such that (u ) weakly two-scale converges in
ε ε∈E′
Lp(Q).
Theorem 3.2. Let E be a fundamental sequence. Suppose a sequence
(u ) is bounded in Y(0,T). Then, a subsequence E′ can be extracted
ε ε∈E
from E such that, as E′ ∋ ε→ 0,
u → u in Y(0,T)-weak,
ε 0
u → u in L2(Q)-weak 2-s,
ε 0
∂u ∂u ∂u
ε → 0 + 1 in L2(Q)-weak 2-s (1 ≤ j ≤ N),
∂x ∂x ∂y
j j j
where u ∈ Y(0,T), u ∈ L2 Q;L2 Z;H1 (Y) .
0 1 per #
(cid:16) (cid:16) (cid:17)(cid:17)
The proof of Theorem 3.1 can be found in, e.g., [1], [7], whereas Theorem
3.2 has its proof in, e.g., [9] and [15].
3.2. A global homogenization theorem. Before we can establish a so-
called global homogenization theorem for (1.3)-(1.6), we require a few basic
notation and results. To begin, let
V = ψ ∈ C∞ (Y;R)N : ψ(y)dy = 0, div ψ =0 ,
Y per y
(cid:26) ZY (cid:27)
V = w ∈ H1 (Y;R)N :div w =0 ,
Y # y
where: C∞ (Y;R) = C∞n RN;R ∩ C (Y), div deonotes the divergence
per per y
operator in RN. We provide V with the H1 (Y)N-norm, which makes it
y (cid:0) Y (cid:1) #
a Hilbert space. There is no difficulty in verifying that V is dense in V
Y Y
(proceed as in [14, Proposition 3.2]). With this in mind, set
F1 = L2(0,T;V)×L2 Q;L2 (Z;V ) .
0 per Y
This is a Hilbert space with norm (cid:0) (cid:1)
1
kvk = kv k2 +kv k2 2 , v =(v ,v )∈ F1.
F10 0 L2(0,T;V) 1 L2(Q;L2per(Z;VY)) 0 1 0
(cid:16) (cid:17)
10 LAZARUSSIGNING
On the other hand, put
F∞ = D(0,T;V)× D(Q;R)⊗ C∞ (Z;R)⊗V ,
0 per Y
where Cp∞er(Z;R) = C∞(R;R)(cid:2)∩ Cper(Z), C(cid:2)p∞er(Z;R) ⊗ VY (cid:3)s(cid:3)tands for the
space of vector functions w on RN ×R of the form
y τ
w(y,τ) = χ (τ)v (y) τ ∈R, y ∈ RN
i i
finite
X (cid:0) (cid:1)
with χ ∈ C∞ (Z;R), v ∈ V , and where D(Q;R)⊗ C∞ (Z;R)⊗V is
i per i Y per Y
the space of vector functions on Q×RN ×R of the form
y (cid:0) (cid:1)
ψ(x,t,y,τ) = ϕ (x,t)w (y,τ) (x,t)∈ Q, (y,τ)∈ RN ×R
i i
finite
X (cid:0) (cid:1)
with ϕ ∈ D(Q;R), w ∈ C∞ (Z;R)⊗V . Since V is dense in V (see [18,
i i per Y
p.18]), it is clear that F∞ is dense in F1.
0 0
Now, let
U = V ×L2 Ω;L2 (Z;V ) .
per Y
Provided with the norm
(cid:0) (cid:1)
1
kvk = kv k2+kv k2 2 (v = (v ,v ) ∈ U),
U 0 1 L2(Ω;L2per(Z;VY)) 0 1
U is a Hilbert(cid:16)space. Let us set (cid:17)
N ∂uk ∂uk ∂vk ∂vk
a (u,v) = a 0 + 1 0 + 1 dxdydτ
Ω ij
∂x ∂y ∂x ∂y
i,j,k=1Z Z ZΩ×Y×Z (cid:18) j j(cid:19)(cid:18) i i(cid:19)
X
fobr u =(u ,u ) and v =(v ,v ) in U. This defines a symmetric continuous
0 1 0 1
bilinear form a on U×U. Furthermore, a is U-elliptic. Specifically,
Ω Ω
(3.2) a (u,u) ≥ αkuk2 (u ∈ U)
Ω U
b b
∂uk
as is easily checked by using (1.2) and the fact that 1 (x,y,τ)dy = 0.
b Y ∂yj
Here is one fundamental lemma.
R
Lemma 3.1. Under the hypotheses (1.1)-(1.2). The variational problem
(3.3)
u ∈ W(0,T) with u (0) = 0;
0 0
u = (u ,u ) ∈ F1 :
0 1 0

T (u′ (t),v (t))dt+ T a (u(t),v(t))dt = T (f (t),v (t))dt
 0 0 0 0 Ω 0 0
 for all v =(v ,v ) ∈ F1
R R 0 1 0 R
hasat most one solution. b

Proof. Let v = (v ,v ) ∈ U and ϕ ∈ D(]0,T[). By taking v =ϕ⊗v in
∗ 0 1 ∗
(3.3), we arrive at
(3.4) u′ (t),v +a (u(t),v ) = (f (t),v ) (v ∈U)
0 0 Ω ∗ 0 ∗
(cid:0) (cid:1)
b