Homogenized models for a short-time filtration and for acoustic waves propagation in a porous media

0.3 MB·English

by  Anvarbek Meirmanov

HOMOGENIZED MODELS FOR A SHORT-TIME FILTRATION AND
FOR ACOUSTIC WAVES PROPAGATION IN A POROUS MEDIA
ANVARBEK MEIRMANOV∗
Abstract. Weconsideralinearsystemofdifferentialequationsdescribingajointmotionofelas-
7 ticporousbodyandfluidoccupyingporousspace. Therigorousjustification,undervariousconditions
0 imposedonphysicalparameters,isfulfilledforhomogenizationproceduresasthedimensionlesssize
0 oftheporestendstozero,whiletheporousbodyisgeometricallyperiodicandacharacteristictime
2 of processes is small enough. Such kind of models may describe, for example, hydraulic fracturing
or acoustic or seismic waves propagation. As the results, we derive different types of homogenized
n
equationsinvolvingnon-isotropicStokessystemforfluidvelocitycoupledwithacousticequationsfor
a
the solid component or different types of acoustic equations, depending on ratios between physical
J
parameters. TheproofsarebasedonNguetseng’s two-scaleconvergence methodof homogenization

2 inperiodicstructures.
2
Keywords. Stokesequations,Lam´e’sequations,waveequation,hydraulicfracturing,two-scale

] convergence, homogenizationofperiodicstructures
P
A AMS subject classifications. 35M20,74F10, 76S05
.
h
1. Introduction. The paper addresses the problem of a joint motion of a de-
t
a formable solid (elastic skeleton), perforated by system of channels or pores (porous
m
space) and a fluid, occupying porous space. In dimensionless variables (without
[ primes)

1
v
3 x′ =Lx, t′ =τt, w′ =Lw, ρ′s =ρ0ρs, ρ′f =ρ0ρf, F′ =gF,
1
6 differential equations of the problem in a domain Ω R3 for the dimensionless dis-
∈
1 placement vector w of the continuum medium have a form:
0
7 ∂2w
0 (1.1) ατρ¯∂t2 =divxP+ρ¯F,
/
h ∂w
(1.2) P=χ¯α D x, +(1 χ¯)α D(x,w) (q+π)I,
t µ λ
a ∂t − −
m (cid:16) (cid:17) αν ∂p
(1.3) q =p+ ,
: αp ∂t
v
(1.4) p+χ¯α div w =0,
i p x
X
(1.5) π+(1 χ¯)α div w =0.
η x
r −
a
The problem is endowed with homogeneous initial and boundary conditions
∂w
(1.6) w =0, =0, x Ω
t=0 t=0
| ∂t | ∈
(1.7) w =0, x S =∂Ω, t 0.
∈ ≥
Here and further we use notations
D(x,u)=(1/2) u+( u)T , ρ¯=χ¯ρ +(1 χ¯)ρ .
x x f s
∇ ∇ −
∗Math. Dept., Belgorod S(cid:0)tate University, (cid:1)ul.Pobedi 85, 308015 Belgorod, Russia
([email protected]).
1
2 A.Meirmanov
Inthismodelthecharacteristicfunctionoftheporousspaceχ¯(x)andadimensionless
vector F(x,t) of distributed mass forces are known functions.
Dimensionless constants α (i=τ,ν,...) are defined by the formulas
i
L ν 2µ
α = , α = , α = ,
τ gτ2 ν τLgρ µ τLgρ
0 0
c2ρ η 2λ
f
α = , α = , α = ,
p η λ
Lg Lgρ Lgρ
0 0
where µ is the viscosity of fluid or gas, ν is the bulk viscosity of fluid or gas, λ and η
are elastic Lam´e’s constants, c is a speed of sound in fluid, L is a characteristic size
of the domain in consideration and τ is a characteristic time of the process.
For more details about Eqs.(1.1)– (1.5) see [7], [2], [10].
Ouraimistoderiveallpossiblelimitingregimes(homogenizedequations)forthe
problem (1.1)– (1.7) as ε 0.
ց
To do that we accept the following constraints
Assumption 1.1. domain Ω = (0,1)3 is a periodic repetition of an elementary
cellYε =εY, whereY =(0,1)3 andquantity1/εis integer, sothatΩ always contains
an integer number of elementary cells Yε. Let Y be a ”solid part” of Y, and the
i s
”liquid part” Y – is its open complement. We denote as γ = ∂Y ∂Y and γ is
f f s
∩
C1-surface. A porous space Ωε is the periodic repetition of the elementary cell εY ,
f f
and solid skeleton Ωε is the periodic repetition of the elementary cell εY . A boundary
s s
Γε = ∂Ωε ∂Ωε is the periodic repetition in Ω of the boundary εγ. The ”solid
s ∩ f
skeleton” Ωε and the ”porous space” Ωε are connected domains. In these
s f
assumptions
χ¯(x)=χε(x)=χ(x/ε),
ρ¯=ρε(x)=χε(x)ρ +(1 χε(x))ρ .
f s
−
Let ε be a characteristicsize of pores l divided by the characteristicsize L of the
entire porous body:
l
ε= .
L
Suppose that all dimensionless parameters depend on the small parameter ε and
there exist limits (finite or infinite)
limα (ε)=µ , limα (ε)=λ , limα (ε)=τ ,
µ 0 λ 0 τ 0
εց0 εց0 εց0
limα (ε)=η , limα (ε)=p , limα (ε)=ν ,
η 0 p ∗ ν 0
εց0 εց0 εց0
α α
µ λ
lim =µ , lim =λ .
εց0 ε2 1 εց0 ε2 1
The first research with the aim of finding limiting regimes in the case when the
skeleton was assumed to be an absolutely rigid body was carried out by E. Sanchez-
Palencia and L. Tartar. E. Sanchez-Palencia[10, Sec. 7.2] formally obtained Darcy’s
HOMOGENIZEDMODELSFORFILTRATIONANDACOUSTIC 3
law of filtration using the method of two-scale asymptotic expansions, and L. Tar-
tar[10,Appendix]mathematicallyrigorouslyjustifiedthehomogenizationprocedure.
Using the same method of two-scale expansions J. Keller and R. Burridge [2] derived
formallythesystemofBiot’sequationsfromtheproblem(1.1)–(1.7)inthecasewhen
the parameter α was of order ε2, and the rest of the coefficients were fixed indepen-
µ
dent of ε. Under the same assumptions as in the article [2], the rigorous justification
of Biot’s model was given by G. Nguetseng [9] and later by Th. Clopeaut et al. [3].
The most general case of the problem (1.1)– (1.7) when
τ , µ , λ−1, ν , p−1, η−1 <
0 0 0 0 ∗ 0 ∞
has been studied in [7].
All these authors have used Nguetseng’s two-scale convergence method [8, 6].
In the present work by means of the same method we investigate the rest of all
possible limiting regimes in the problem (1.1)– (1.7). Namely, if τ = , which is a
0
∞
case of short-time processes, then re-normalizing the displacement vector by setting
w α w,
τ
→
wereduce the problemto the caseτ =1 andµ < . Here the only onecaseλ =0
0 0 0
∞
needs an additional consideration.
Therefore we restrict ourself by the case, when
ν , µ < ; λ =0, τ =1, 0<p , η .
0 0 0 0 ∗ 0
∞
We show that in the case µ > 0 the homogenized equations are non-isotropic
0
Stokes equations for fluid velocity coupled with acoustic equations for the solid com-
ponent, or non-isotropic Stokes system for the one-velocity continuum (theorem2.2).
Inthecaseµ =0thehomogenizedequationsaredifferenttypesofacousticequations
0
for two-velocity or one-velocity continuum (theorem 2.3).
2. Main results. As usual, equation (1.1) is understood in the sense of distri-
butions. It involvesthe properequation(1.1) ina usualsense in the domainsΩε and
f
Ωε and the boundary conditions
s
(2.1) [w]=0, x Γε, t 0,
0
∈ ≥
(2.2) [P]=0, x Γε, t 0
0
∈ ≥
on the boundary Γε, where
[ϕ](x )=ϕ (x ) ϕ (x ),
0 (s) 0 (f) 0
−
ϕ (x )= lim ϕ(x), ϕ (x )= lim ϕ(x).
(s) 0 (f) 0
x→x0 x→x0
x∈Ωεs x∈Ωεf
There are various equivalent in the sense of distributions forms of representation
ofequation(1.1)andboundaryconditions(2.1)–(2.2). Inwhatfollows,itisconvenient
to write them in the form of the integral equalities.
Wesaythatfunctions(wε,pε,qε,πε)arecalledageneralizedsolutionoftheprob-
lem (1.1)– (1.7), if they satisfy the regularity conditions
∂pε
(2.3) wε, D(x,wε), div wε, qε, pε, , πε L2(Ω )
x T
∂t ∈
4 A.Meirmanov
inthedomainΩ =Ω (0,T),boundaryconditions(1.7)inthetracesense,equations
T
×
(1.3)– (1.5) a.e. in Ω and integral identity
T
∂2ϕ ∂ϕ
α ρεwε χεα D(x,wε):D(x, )+
(2.4) ZΩT(cid:16) τ · ∂t2 − µ ∂t 
(1 χε)αλD(x,wε) (qε+πε)I :D(x,ϕ) dxdt=0 
{ − − }
(cid:17)
forallsmoothvector-functionsϕ=ϕ(x,t)suchthatϕ∂Ω =ϕt=T =∂ϕ/∂tt=T =0.
| | |
In (2.4) by A : B we denote the convolution (or, equivalently, the inner tensor
product)oftwosecond-ranktensorsalongthebothindexes,i.e.,A:B =tr(B∗ A)=
◦
3 A B .
i,j=1 ij ji
In what follows all parameters may take all permitted values. If, for example,
Pη−1 =0 , then all terms in final equations containing this parameter disappear.
0
The following theorems 2.1–2.3 are the main results of the paper.
Theorem 2.1. Let F, ∂F/∂t and ∂2F/∂t2 are bounded in L2(Ω). Then for all
ε>0 on the arbitrary time interval [0,T] there exists a unique generalized solution of
the problem (1.1)– (1.7) and
∂2wε
(2.5) max (t) C ,
0≤t≤Tk ∂t2 k2,Ω ≤ 0
∂wε ∂wε
(2.6) max √αµχε x (t) +(1 χε)√αλ x (t) 2,Ω C0,
0≤t≤Tk |∇ ∂t | − |∇ ∂t |k ≤
α ∂pε
(2.7) qε + pε + ν C
k k2,ΩT k k2,ΩT α k ∂t k2,ΩT ≤ 0
p
where C does not depend on the small parameter ε.
0
Theorem 2.2. Assume that the hypotheses in theorem 2.1 hold, and µ > 0.
0
Then functions ∂wε/∂t admit an extension vε from Ωε =Ωε (0,T) into Ω such
f,T f× T
that the sequence vε converges strongly in L2(Ω ) and weakly in L2((0,T);W1(Ω))
{ } T 2
to the function v. At the same time, sequences wε , (1 χ)wε , pε , qε , and
{ } { − } { } { }
πε converge weakly in L2(Ω ) to w, ws, p, q, and π, respectively.
T
{ }
I) If λ = , then ∂ws/∂t = (1 m)v = (1 m)∂w/∂t and weak and strong
1
∞ − −
limits q, p, π and v satisfy in Ω the initial-boundary value problem
T
∂v
ρˆ = div µ Af :D(x,v)+Bfπ+Bfdiv v+
(2.8) ∂t x{ 0 0 0 1 x
tBf(t τ)div v(x,τ)dτ (q+π)+ρˆF, 
0 2 − x }−∇ 
R

p−1∂p/∂t+Cf :D(x,v)+afπ+(af +m)div v
∗ 0 0 1 x
(2.9)
+ 0taf2(t−τ)divxv(x,τ)dτ =0, )
R
ν ∂p 1 ∂p 1 ∂π
(2.10) q =p+ 0 , + +div v =0,
x
p ∂t p ∂t η ∂t
∗ ∗ 0
where ρˆ=mρ +(1 m)ρ , m= χdy and the symmetric strictly positively defined
f − s Y
constant fourth-rank tensor Af, matrices Cf,Bf, Bf and Bf(t) and scalars af, af
0 R 0 0 1 2 0 1
and af(t) are defined below by formulas (5.29), (5.31) - (5.32).
2
HOMOGENIZEDMODELSFORFILTRATIONANDACOUSTIC 5
Differential equations (2.8) are endowed with homogeneous initial and boundary
conditions
(2.11) v(x,0)=0, x Ω, v(x,t)=0, x S, t>0.
∈ ∈
II) If λ < , then weak and strong limits ws, q, p, π and v satisfy in Ω the
1 T
∞
initial-boundary value problem, which consists of Stokes like system
ρ m∂v/∂t+ρ ∂2ws/∂t2+ (q+π) ρˆF =div Bfπ+
(2.12) f s ∇ − x{ 0
µ0Af0 :D(x,v)+B1fdivxv+ 0tB2f(t−τ)divxv(x,τ)dτ}, )
R
p−1∂p/∂t+Cf :D(x,v)+afπ+(af +m)div v
∗ 0 0 1 x
(2.13)
+ 0taf2(t−τ)divxv(x,τ)dτ =0, )
R
ν ∂p
0
(2.14) q =p+ ,
p ∂t
∗
for the liquid component coupled with a continuity equation
1 ∂p 1 ∂π ∂ws
(2.15) + +div +mdiv v =0,
x x
p ∂t η ∂t ∂t
∗ 0
the relation
∂ws t
(2.16) =(1 m)v(x,t)+ Bs(t τ) z(x,τ)dτ,
∂t − 1 − ·
Z0
1 ∂v
z(x,t)= π(x,t)+ρ F(x,t) ρ (x,t)
x s s
−1 m∇ − ∂t
−
in the case of λ >0, or the balance of momentum equation in the form
1
∂2ws ∂v 1
(2.17) ρ =ρ Bs +((1 m)I Bs) ( π+ρ F)
s ∂t2 s 2· ∂t − − 2 · −1 m∇x s
−
in the case of λ = 0 for the solid component. The problem is supplemented by
1
boundary and initial conditions (2.11) for the velocity v of the liquid component and
by the homogeneous initial conditions and the boundary condition
(2.18) ws(x,t) n(x)=0, (x,t) S, t>0,
· ∈
for the displacement ws of the solid component. In Eqs. (2.16)–(2.18) n(x) is the
unit normal vector to S at a point x S, and matrices Bs(t) and Bs are given
∈ 1 2
below by Eqs. (5.39) and (5.41), where the matrix ((1 m)I Bs) is symmetric and
− − 2
positively definite.
Theorem 2.3. Assume that the hypotheses in Theorem 2.1 hold, and
µ =0; p , η < .
0 ∗ 0
∞
Then there exist functions wε, wε L∞(0,T;W1(Ω)) such that
f s ∈ 2
wε =wε in Ωε (0,T), wε =wε in Ωε (0,T)
f f × s s×
6 A.Meirmanov
and sequences pε , qε , πε , wε , χεwε , wε and wε converge weakly in
{ } { } { } { } { } { f} { s}
L2(Ω ) to functions p, q, π, w, wf, w and w respectively as ε 0.
T f s
ց
I) If µ = λ = , then w = w = w and functions w, p, q and π satisfy in
1 1 f s
∞
Ω the system of acoustic equations
T
∂2w 1
(2.19) ρˆ = π+ρˆF,
∂t2 −(1 m)∇x
−
1 1
(2.20) p+ π+div w =0,
x
p η
∗ 0
ν ∂p 1 1
0
(2.21) q =p+ , q = π,
p ∂t m 1 m
∗
−
homogeneous initial conditions
∂w
(2.22) w(x,0)= (x,0)=0, x Ω
∂t ∈
and homogeneous boundary condition
(2.23) w(x,t) n(x)=0, x S, t>0.
· ∈
II) If µ = and λ < , then functions w =w, ws, p, q and π satisfy in Ω the
1 1 f T
∞ ∞
system of acoustic equations, which consist of the state equations (2.21) and balance
of momentum equation
∂2w ∂2ws 1
f
(2.24) ρ m +ρ = π+ρˆF,
f ∂t2 s ∂t2 −(1 m)∇x
−
for the liquid component, continuity equation
1 1
(2.25) p+ π+mdiv w +div ws =0,
x f x
p η
∗ 0
and the relation
∂ws ∂w t
(2.26) =(1 m) f + Bs(t τ) zs(x,τ)dτ,
∂t − ∂t 1 − ·
Z0
1 ∂2w
zs(x,t)= π(x,t)+ρ F(x,t) ρ f(x,t)
−1 m∇x s − s ∂t2
−
in the case of λ >0, or the balance of momentum equation in the form
1
∂2ws ∂2w 1
(2.27) ρ =ρ Bs f +((1 m)I Bs) ( π+ρ F)
s ∂t2 s 2· ∂t2 − − 2 · −1 m∇x s
−
in the case of λ = 0 for the solid component. The problem (2.21), (2.24)–(2.27) is
1
supplemented by homogeneous initial conditions (2.22) for the displacements in the
liquid and the solid components and homogeneous boundary condition (2.23) for the
displacements w =mw +ws.
f
HOMOGENIZEDMODELSFORFILTRATIONANDACOUSTIC 7
In Eqs.(2.26)–(2.27) matrices Bs(t) and Bs are the same as in theorem 2.2.
1 2
III) If µ < and λ = , then functions wf, w = w, p, q and π satisfy in
1 1 s
∞ ∞
Ω the system of acoustic equations, which consist of the state equations (2.21) and
T
the balance of momentum equation
∂2wf ∂2w 1
(2.28) ρ +ρ (1 m) s = π+ρˆF,
f ∂t2 s − ∂t2 −(1 m)∇x
−
for the solid component, the continuity equation
1 1
(2.29) p+ π+div wf +(1 m)div w =0,
x x s
p η −
∗ 0
and the relation
∂wf ∂w t
(2.30) =m s + Bf(t τ) zf(x,τ)dτ,
∂t ∂t 1 − ·
Z0
1 ∂2w
zf(x,t)= q(x,t)+ρ F(x,t) ρ s(x,t)
−m∇x f − f ∂t2
in the case of λ >0, or the balance of momentum equation in the form
1
∂2wf ∂2w 1
(2.31) ρ =ρ Bf s +(mI Bf) ( q+ρ F)
f ∂t2 f 2 · ∂t2 − 2 · −m∇x f
in the case of λ = 0 for the liquid component. The problem (2.21), (2.28)–(2.31)
1
is supplemented by homogeneous initial conditions (2.22) for the displacements in the
liquid and the solid components and homogeneous boundary condition (2.23) for the
displacements w =wf +(1 m)w .
s
−
In Eqs.(2.30)–(2.31) matrices Bf(t) and Bf are given below by formulas (6.32)–
1 2
(6.33), where the matrix (mI Bf) is symmetric and positively definite.
− 2
IV) If µ < and λ < , then functions w, p, q and π satisfy in Ω the
1 1 T
∞ ∞
system of acoustic equations, which consist of the continuity and the state equations
(2.20) and (2.21) and the relation
∂w t
(2.32) = Bπ(t τ) π(x,τ)dτ +f(x,t),
∂t − ·∇
Z0
where Bπ(t) and f(x,t) are given below by Eqs.( 6.40) and (6.41).
The problem (2.20), (2.21), (2.32) is supplemented by homogeneous initial and
boundary conditions (2.22) and (2.23).
3. Preliminaries.
3.1. Two-scale convergence. Justification of theorems 2.1–2.3 relies on sys-
tematic use of the method of two-scale convergence, which had been proposed by
G. Nguetseng [8] and has been applied recently to a wide range of homogenization
problems (see, for example, the survey [6]).
Definition 3.1. A sequence ϕε L2(Ω ) is said tobe two-scale convergent to
T
{ }⊂
a limit ϕ L2(Ω Y) if and only if for any 1-periodic in y function σ =σ(x,t,y)
T
∈ ×
the limiting relation
(3.1) lim ϕε(x,t)σ(x,t,x/ε)dxdt= ϕ(x,t,y)σ(x,t,y)dydxdt
εց0ZΩT ZΩT ZY
8 A.Meirmanov
holds.
Existence andmain propertiesof weaklyconvergentsequences areestablishedby
the following fundamental theorem [8, 6]:
Theorem 3.2. (Nguetseng’s theorem)
1. Any bounded in L2(Q) sequence contains a subsequence, two-scale convergent
to some limit ϕ L2(Ω Y).
T
∈ ×
2. Letsequences ϕε and ε ϕε beuniformlyboundedinL2(Ω ). Thenthereexist
x T
{ } { ∇ }
a 1-periodic in y function ϕ=ϕ(x,t,y) and a subsequence ϕε such that ϕ, ϕ
y
{ } ∇ ∈
L2(Ω Y), and ϕε and ε ϕε two-scale converge to ϕ and ϕ, respectively.
T x y
× ∇ ∇
3. Let sequences ϕε and ϕε be bounded in L2(Q). Then there exist functions
x
{ } {∇ }
ϕ L2(Ω ) and ψ L2(Ω Y) and a subsequence from ϕε such that ψ is 1-
T T
∈ ∈ × { }
periodic in y, ψ L2(Ω Y), and ϕε and ϕε two-scale converge to ϕ and
y T x
∇ ∈ × ∇
ϕ(x,t)+ ψ(x,t,y), respectively.
x y
∇ ∇
Corollary 3.3. Let σ L2(Y) and σε(x) := σ(x/ε). Assume that a sequence
∈
ϕε L2(Ω ) two-scale converges to ϕ L2(Ω Y). Then the sequence σεϕε
T T
{ } ⊂ ∈ ×
two-scale converges to σϕ.
3.2. An extension lemma. The typical difficulty in homogenizationproblems
while passing to a limit in Model Bε as ε 0 arises because of the fact that the
ց
bounds on the gradient of displacement wε may be distinct in liquid and rigid
x
∇
phases. The classical approach in overcoming this difficulty consists of constructing
of extension to the whole Ω of the displacement field defined merely on Ω . The
s
following lemma is valid due to the well-knownresults from[1, 4]. We formulate it in
appropriate for us form:
Lemma3.4. Supposethatassumption1.1ongeometryofperiodicstructureholds,
ψε W1(Ωε) and ψε = 0 on Sε = ∂Ωε ∂Ω in the trace sense. Then there exists
∈ 2 s s s ∩
a function σε W1(Ω) such that its restriction on the sub-domain Ωε coincide with
∈ 2 s
ψε, i.e.,
(3.2) (1 χε(x))(σε(x) ψε(x))=0, x Ω,
− − ∈
and, moreover, the estimate
(3.3) σε C ψε , σε C ψε
2,Ω 2,Ωε x 2,Ω x 2,Ωε
k k ≤ k k s k∇ k ≤ k∇ k s
hold true, where the constant C depends only on geometry Y and does not depend on
ε.
3.3. Friedrichs–Poincar´e’s inequality in periodic structure. The follow-
inglemmawasprovedbyL.Tartarin[10,Appendix]. ItspecifiesFriedrichs–Poincar´e’s
inequality for ε-periodic structure.
Lemma 3.5. Suppose that assumptions on the geometry of Ωε hold true. Then
f
◦
for any function ϕ W1 (Ωε) the inequality
∈ 2 f
(3.4) ϕ2dx Cε2 ϕ2dx
x
| | ≤ |∇ |
ZΩεf ZΩεf
holds true with some constant C, independent of ε.
HOMOGENIZEDMODELSFORFILTRATIONANDACOUSTIC 9
3.4. Some notation. Further we denote
1)
Φ = Φdy, Φ = χΦdy, Φ = (1 χ)Φdy,
h iY h iYf h iYs −
ZY ZY ZY
ϕ = ϕdx, ϕ = ϕdxdt.
h iΩ h iΩT
ZΩ ZΩT
2) If a and b are two vectors then the matrix a b is defined by the formula
⊗
(a b) c=a(b c)
⊗ · ·
for any vector c.
3) If B and C are two matrices, then B C is a forth-rank tensor such that its
⊗
convolution with any matrix A is defined by the formula
(B C):A=B(C :A)
⊗
.
4) By Iij we denote the 3 3-matrix with just one non-vanishing entry, which is
×
equal to one and stands in the i-th row and the j-th column.
5) We also introduce
1 1
Jij = (Iij +Iji)= (e e +e e ),
i j j i
2 2 ⊗ ⊗
where (e ,e ,e ) are the standard Cartesian basis vectors.
1 2 3
4. Proof of theorem 2.1. Estimates (2.5)-(2.6) follow from
max (√αη divx∂wε/∂t(t) 2,Ωε +√αλ x∂wε/∂t(t) 2,Ωε
0<t<T k k s k∇ k s
(4.1) +√ατ ∂2wε/∂t2(t) 2,Ω+√αp divx∂wε/∂t(t) 2,Ωε) 
k k k k f
+√αµkχε∇x∂2wε/∂t2k2,ΩT +√ανkχεdivx∂2wε/∂t2k2,ΩT ≤C0/√ατ, 
whereC isindependentofε. Lastestimatesweobtainifwedifferentiateequationfor
0
wεwithrespecttotime,multiplyby∂2wε/∂t2andintegratebypartsusingcontinuity
and state equations (1.3)– (1.5). The same estimates guarantee the existence and
uniqueness of the generalized solution for the problem (1.1)– (1.7).
If p +η < , then estimate (2.7) for pressures follows from estimate (4.1) and
∗ 0
∞
continuity and state equations (1.3)– (1.5).
For the case p +η = estimate (2.7) follows from integral identity (2.4) and
∗ 0
∞
estimates (4.1) as an estimate of the corresponding functional, if we re-normalized
pressures, such that
(qε(x,t)+πε(x,t))dx=0.
ZΩ
.
Indeed, integral identity (2.4) and estimates (4.1) imply
(qε+πε)div ψdx C ψ .
x 2,Ω
| |≤ k∇ k
ZΩ
10 A.Meirmanov
Choosing now ψ such that (qε+πε)=div ψ we get the desired estimate for the
x
sum of pressures (qε+πε). Such a choice is always possible (see [5]), if we put
ψ = ϕ+ψ0, divxψ0 =0, ϕ=qε+πε, ϕ∂Ω =0, ( ϕ+ψ0)∂Ω =0.
∇ △ | ∇ |
Note that the re-normalizationof the pressures(qε+πε) transforms continuity equa-
tions (1.4)-(1.5) for pressures into
1 1
(4.2) pε+χεdiv wε = βεχε,
x
α m
p
1 1
(4.3) πε+(1 χε)div wε = βε(1 χε),
x
α − −(1 m) −
η
−
where
βε = χεdiv wε .
x Ω
h i
In what follows we will use equations (4.2) and (4.3) only if p +η = .
∗ 0
∞
Note that for the lastcasethe basicintegralidentity (2.4) permits to boundonly
thesum(qε+πε). Butthankstothepropertythattheproductofthesetwofunctions
is equal to zero, it is enough to get bounds for each of these functions. The pressure
pε is bounded from the state equation(1.3), if we substitute the term (α /α )∂pε/∂t
ν p
from the continuity equation 4.2 and use estimate (4.1).
5. Proof of theorem 2.2.
5.1. Weak and two-scale limits of sequences of displacement and pres-
sures. On the strength of theorem 2.1, the sequences pε , qε , πε and wε
{ } { } { } { }
are uniformly in ε bounded in L2(Ω ). Hence there exist a subsequence of small
T
parameters ε>0 and functions p, q, π and w such that
{ }
(5.1) pε p, qε q, πε π, wε w
→ → → →
weakly in L2(Ω ) as ε 0.
T
ց
Moreover, due to lemma 3.4 there is a function vε L∞(0,T;W1(Ω)) such that
∈ 2
vε = ∂wε/∂t in Ω (0,T), and the family vε is uniformly in ε bounded in
f
× { }
L∞(0,T;W1(Ω)). Therefore it is possible to extract a subsequence of ε > 0 such
2 { }
that
(5.2) vε v weakly in L2(0,T;W1(Ω))
→ 2
as ε 0.
ց
Note also, that
(5.3) (1 χε)α D(x,wε) 0.
λ
− →
strongly in L2(Ω ) as ε 0.
T
ց
Relabeling if necessary, we assume that the sequences converge themselves.
On the strength of Nguetseng’s theorem, there exist 1-periodic in y functions
P(x,t,y), Π(x,t,y), Q(x,t,y), W(x,t,y) and V(x,t,y) such that the sequences
pε , πε , qε , wε and vε two-scaleconvergetoP(x,t,y),Π(x,t,y),Q(x,t,y),
x
{ } { } { } { } {∇ }
W(x,t,y) and v+ V(x,t,y), respectively.
x y
∇ ∇ ◦
Notethatthesequence div wε weaklyconvergestodiv wandv L2(0,T;W1
{ x } x ∈ 2
(Ω)). Last assertion follows from the Friedrichs–Poincar´e’sinequality for vε in the ε-
layer of the boundary S and from convergence of sequence vε to v strongly in
{ }
L2(Ω ) and weakly in L2((0,T);W1(Ω)).
T 2