on di2usion of a phase, slightly compressible 8uid

8/8/2019 On Di2usion of a Phase, Slightly Compressible 8uid

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MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2001; 24:805825 (DOI: 10.1002/mma.243)MOS subject classication: primary 76 R 50; secondary 76 N 99

On diusion of a single-phase, slightly compressible uidthrough a randomly ssured medium

Steve Wright;

Department of Mathematics and Statistics; Oakland University; Rochester; Michigan 48309-4485; U.S.A.

Communicated by W. Sproig

SUMMARY

In this paper, the DouglasPeszynskaShowalter model of diusion through a partially ssured mediumis given a stochastic formulation using the framework for problems in random media as set forth byJikov, Kozlov and Oleinik. The concept of stochastic two-scale convergence in the mean is then usedto homogenize the randomized micromodels which result. As a consequence of this homogenization procedure, exact stochastic generalizations of results obtained by Clark and Showalter on diusionthrough periodically ssured media are derived. Copyright ? 2001 John Wiley & Sons, Ltd.

1. INTRODUCTION

In several recent papers, diusion of a single-phase, slightly compressible uid through a s-sured porous medium is studied [18]. Generally speaking, a ssured porous medium consists

of a permeable material or matrix intricately interlaced by a system of highly permeable s-sures. The bulk of the ow occurs through the ssure system, and the complement of thessure system is a matrix of permeable cells. Depending on the degree of connectiveness ofthe matrix structure, there may be partial ow directly within the cell matrix, and when theconnections between the cells is suciently extensive, this inter-matrix ow can have a non-trivial eect on the dynamics of the global ow. When these inter-matrix eects are present,the medium is said to be partially ssured.

Exact mathematical models of the uid-ow system in a ssured porous medium typicallytreat the regions occupied by the ssures and porous matrix as two Darcy media determined

by dierent physical parameters (for a nice discussion of these models, consult the article[9] of Showalter). Severe discontinuities across the interface of the two media consequentlyresult, and the discontinuities are exacerbated to such an extent by the typically very small

size and complicated structure of the ssures that analytical or even numerical treatment of the

Correspondence to: Steve Wright, Department of Mathematics and Statistics, Oakland University, Rochester,Michigan 48309-4485, U.S.A.

E-mail: [email protected]

Copyright ? 2001 John Wiley & Sons, Ltd. Revised 11 December 2000


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806 S. WRIGHT

models is intractable. Homogenization procedures are hence required. In [3], Clark and Showal-ter use two-scale convergence to rigorously homogenize an exact micro-model that was in-troduced in [4] for ow through a partially ssured porous medium. Their work incorporatesquasi-linear features and establishes both variational and strong forms of the macroscopic

equations.Two-scale convergence, introduced by Nguetseng [10] and rened and further developed by

Allaire [11], is a homogenization technique that has proven to be very useful in a wide varietyof problems in porous media (the article of Allaire [12] contains a brief discussion of themethod and some typical applications). Although two-scale convergence is simpler to applyand analyse than some of the more general homogenization procedures such as -, G-, or H-convergence, it is restricted to situations in which the cell and=or ssure structure is periodic.In an attempt to develop a homogenization technique which enjoys the ease and eciencyof two-scale convergence but which applies to problems which incorporate more general andmore realistic random eects, the method of stochastic two-scale convergence in the mean wasintroduced and developed in [13]. In References [1416] it was further developed and appliedto the study of incompressible Stokes ow through a randomly perforated porous medium.

In the paper before the reader, stochastic two-scale convergence in the mean will be used toderive a stochastic analog of the results of Clark and Showalter.

In Section 2, the DouglasPeszynskaShowalter model [4] for diusion through a partiallyssured porous medium is given a stochastic formulation based on the framework for such

problems as set out in [17]. We emphasize that the underlying stochastic process is notnecessarily assumed to be ergodic. In Section 3, the results that are required from the theoryof stochastic two-scale convergence in the mean are described. Section 4 is concerned with thestochastic homogenization of the randomized model described in Section 2. The main resultof the paper (Theorem 4.4) gives an exact stochastic analogue of the two-scale convergenceresults of Clark and Showalter, deriving weak and strong forms of the homogenized equationswhich incorporate both the deterministic and stochastic eects present in the macroscopicow.

2. RANDOMIZATION OF THE DOUGLASPESZY NSKASHOWALTER MODEL

Let (; ; ) denote a separable probability space, with probability measure and-measurable sets . An n-dimensional dynamical system is a family {T(x) : x Rn} ofinvertible maps T(x) : with T(x) and T(x)1 both measurable with respect to (; ; ),for each x Rn, and which satises the following conditions:

(a) T(0) is the identity map on and for (x;y) Rn Rn, T(x + y) = T(x)T(y);

(b) for each x Rn and measurable set F , (T(x)1F) = (F), i.e., is an invariantmeasure for T;

(c) for each F , the set {(x;!) Rn : T(x)! F} is a dx -measurable subset ofRn , where dx denotes Lebesgue measure on Rn.

We now consider -measurable subsets F and G of , each of positive -measure, suchthat =F G and (F G) = 0. We set F1 =F\G; F2 = G\F. Then (Fi)0 for i = 1; 2 and

Copyright ? 2001 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2001; 24:805825


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DIFFUSION OF A SINGLE-PHASE FLUID 807

F1 + F2 = 1, -a.e. on , where i = Fi denotes the characteristic function ofFi, i = 1; 2. Foreach ! , we dene the homogeneous random set determined by Fi at ! by

Fi(!) = {x Rn : T(x)! Fi}

and for each 0, we let

Fi (!) = {x Rn : 1x Fi(!)}; i = 1; 2

We suppose that Fi(!) is a Lipschitz domain for each ! , and for a bounded Lipschitzdomain Q in Rn(n3), which will remain xed for the rest of the paper, we consider the opensets Q i(!) = Q F

i (!); i = 1; 2. Suppose that Q

i(!); i = 1; 2 divide Q into complementary

subdomains, with Q1 (!) thought of as representing a system of randomly distributed ssuresinterlacing the porous matrix Q2 (!), all located within the ambient medium Q. Let

1; 2 (!) = Q 9Q1 (!) 9Q

2 (!)

denote the interface lying inside Q of the domains Q1 (!) and Q2 (!), and let ,

i(!) denote

the outward unit vector eld normal to 9Q and 9Q i(!), respectively, so that 1 (!) = 2 (!)on 1; 2 (!).

We can now randomize the exact micromodel for diusion through a ssured porousmedium introduced in [4]. Denote the ow potential of the uid through the ssures Q1 (!)

by u1(t ;x;!). The ow potential in the matrix Q2 (!) is a convex combination of two com-

ponents u2(t ;x;!) and u3(t ;x;!), which, respectively, account for global diusion through the

pore system of the matrix and the very high-frequency spatial variations in the ow whichlead to local storage in the matrix. The total ow potential in Q2 (!) is thus u

2 + u

3, where

0; 0, and + = 1.We need to preface our specication of the ux of each ow potential by a discussion of

some issues concerning measurability. Let A denote the -algebra generated in Rn Rn by all sets of the form L B S, where L is a Lebesgue-measurable subset ofRn, B is a Borelsubset of Rn, and S is a -measurable subset of (we will call a set of the form L B San A-measurable rectangle). A function :Rn Rn R is said to be A-measurable iffor each rR, 1[(; r)] A. IfL(Rn); B(Rn), and M() denote the set of real-valuedfunctions that are, respectively, Lebesgue measurable on Rn, Borel measurable on Rn, and -measurable on , it is easy to see that any function that is the pointwise limit on Rn Rn of a sequence of functions from L(Rn) B(Rn) M() is A-measurable. It follows that theclass ofA-measurable functions is large enough to include most of the functions that willarise when the results that we obtain are used in practical applications.

We will say that a function :Rn Rn Rn is A-measurable if each of its co-ordinatefunctions is A-measurable. The following lemma shows that A-measurability is the globalmeasurability criterion that will suce for the work in this paper.

Lemma 2.1. Let be a xed positive number, and let :Rn Rn Rn be anA-measurable function. Then for each 0 and dt dx -measurable function f : (0; ) Q Rn, the function

(t ;x;!) (x;f(t ;x;!); T(1x)!)

is dt dx -measurable on (0; ) Q .



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808 S. WRIGHT

Proof. From the denition of Rn-valued A-measurability, we may suppose that is realvalued. We will rst verify the lemma when is the characteristic function of a set in A.To that end, consider the collection E of all subsets E of Rn Rn for which the charac-teristic function E of E satises the conclusion of the lemma. E is clearly a -algebra. If

F=L B S is an A-measurable rectangle, is a positive number, and f: (0; ) Q Rnis a dt dx -measurable function, then

F(x;f(t ;x;!); T(1x)!) = L(x)B(f(t ;x;!))S(T(

1x)!)

The rst factor on the right-hand side of this equation is clearly dt dx -measurable on(0; ) Q ; since B is a Borel set, the second factor there is likewise measurable, and thethird factor is likewise measurable by property (c) of the dynamical system T. Hence FE,and therefore AE.

If is now an arbitrary A-measurable function, then is the pointwise limit everywhereon Rn Rn of a sequence (k) of A-simple functions. By what we have just proved,each k satises the conclusion of the lemma, and so therefore does .

In order to specify the ux of the ow potentials, we suppose givenA-measurable functions

i :Rn Rn Rn; i = 1; 2; 3

which satisfy the following conditions: for a xed non-negative function kL2(), positiveconstants C1; C2 and for dx -a.e.(x;!) R

n ,

|i(x;;!)|6C1|| + k(!); Rn (1)

(i(x;;!) i(x;;!)) ( )0; (; ) Rn Rn (2)

i(x;;!) C2||2

k(!); Rn

(3)

where the phrase dx -a.e. (x;!) Rn means here that (1) (3) hold for all (x;!)outside of a set of dx -measure zero which lies in the -algebra L generated by allsets of the form L S, where L is a Lebesgue-measurable subset of Rn and S . In lightof the A-measurability assumption on the is, this is the most natural meaning of almosteverywhere for the situation occurring here.

For each 0, we dene the scaled coecients

i (x;;!) = i(x;;T(1x)!); i = 1; 2; 3

It follows from properties (b), (c) of the dynamical system T and Fubinis theorem that theinverse image under the mapping

(x;!) (x;T(1x)!); (x;!) Rn

of each element E of L i s dx -measurable with the same dx -measure as E.We conclude from this that for each 0, i (x; ; !) satises (1)(3) with k(!) replaced

by k(T(1x)!), for dx -a.e. (x;!) Rn . We now assume that the ux of the owcomponent u1(t ;x;!) in Q

1 (!) is given by

1(x; xu

1(t ;x;!); !) and the ux of the ow



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components u2(t ;x;!) and u3(t ;x;!) in Q

2 (!) are given, respectively, by

2(x; xu

2(t ;x;!);

!) and 3(x;xu3(t ;x;!); !).

The dynamics of the diusion through the ssures and matrix is governed by the conserva-tion-of-mass equations given as follows: for i = 1; 2; 3, we x functions ci L

(Q ) such

that for a positive constant C0; ciC0; dx -a.e. on Q ; i = 1; 2; 3. If we set

ci (x;!) = ci(x;T(1x)!)

then by [14, Lemma 2.2], upon the choice of an appropriate dx -representative of ci ifnecessary, we have ci L

(Q ) with ci L(Q) = ciL(Q) for each 0. It alsofollows from the T-invariance of that ciC0; dx -a.e. on Q for each 0. We thustake

c1( ; !)u1; t( ; ; !) div

1( ; u

1( ; ; !); !) = 0 in Q

1 (!) (4)

c2( ; !)u2; t( ; ; !) div

2( ; u

2( ; ; !); !) = 0 in Q

2 (!) (5)

c3( ; !)u3; t( ; ; !) div 3( ; u3( ; ; !); !) = 0 in Q2 (!) (6)

The conservation-of-mass equations are augmented by the interface conditions

u1 = u2 + u

3 on

1; 2 (!) (7)

1( ; u1( ; ; !); !)

1 (!) =

2( ; u

2( ; ; !); !)

1 (!) on

1; 2 (!) (8)

1( ; u1( ; ; !); !)

1 (!) =

3( ; u

3( ; ; !); !)

1 (!) on

1; 2 (!) (9)

which specify continuity of the ow potential and an appropriate partition of the ux acrossthe interface of Q1 (!) and Q

2 (!). As in the periodic case, the boundary conditions will

play no essential role in our subsequent work, and so for simplicity we assume homogeneous

Neumann conditions on the boundary:

i ( ; ui ( ; ; !); !)

i(!) = 0 on 9Q 9Q

i(!); i = 1; 2 (10)

3( ; u3( ; ; !); !)

2 (!) = 0 on 9Q 9Q

2 (!) (11)

Finally, we choose u0i L2(Q); i = 1; 2; 3, and impose the initial conditions

ui (0; ; !) = u0i ; i = 1; 2; 3 (12)

The next step is to describe the required variational formulation of problem (4)(12). LetH(!) denote the Hilbert space

L2(Q1 (!)) L2(Q2 (!)) L

2(Q2 (!))

equipped with the weighted inner product

((u1; u2; u3); (v1; v2; v3))H(!)

=

Q1 (!)

c1( ; !)u1v1 dx +3

i=2

Q2 (!)

ci ( ; !)uivi dx



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810 S. WRIGHT

Let i (!) : H1(Q i(!)) L

2(9Q i(!)); i = 1; 2 denote the standard trace maps, and let V(!)denote the energy space dened by

V(!) = {(v1; v2; v3) H1(Q1 (!)) H

1(Q2 (!)) H1(Q2 (!)) :

1(!)v1

= 2(!)v2 + 2(!)v3 on

1; 2 (!)}

equipped with the usual H1-norm. If we now suppose that (u1( ; ; !); u2( ; ; !); u

3( ; ; !))

L2((0; ); V(!)), take (v1; v2; v3) V(!), multiply (4), (5), and (6) by v1; v2; and v3, re-spectively, integrate over the corresponding domains, apply Greens theorem, and use (7)(11) in the equations that result, one nds that u = (u1( ; ; !); u

2( ; ; !); u

3( ; ; !)) satises

(ut(t); (v1; v2; v3))H(!) + A(!) (u) (v1; v2; v3) = 0

where the operator A(!) mapping V(!) into the dual V(!) of continuous linear functionals

on V(!) is given by

A(!) (u) (v1; v2; v3) =

Q1 (!)

1( ; u1; !) v1 dx

+

Q2 (!)

2( ; u2; !) v2 dx

+

Q2 (!)

3( ; u3; !) v3 dx

The mapping property ofA(!) is easily veried after observing that the constant k(T(1x)!)

in the estimate for i (x; ; !) that comes from (1) is in L2(Q ), and hence for -a.e.

!; x k(T(1

x)!) is in L

2

(Q). It follows that the variational formulation of problem (4)(12) can be expressed as a non-linear abstract Cauchy problem: for each 0; ! , andu0 = (u01; u

02; u

03) L

2(Q)3, nd u() in L2((0; ); V(!)) such that

du

dt+ A(!)u

= 0 in L2((0; ); V(!)) (13)

u(0)= u0 in H(!) (14)

Conversely, a suciently smooth solution of problem (13) (14) will be a solution of(4)(12).

In order to solve problem (13)(14), we apply [18, Proposition III.4.1], obtaining

Theorem 2.2. Problem (13)(14) has a unique solution u

( ; !) = (u1( ; !); u

2( ; !); u

3( ;

!)) in L2(0; ); V(!)). Moreover, u( ; !) C([0; ]; H(!)); and consequently (14) holds

pointwise.

For each ! , u1( ; !) is dened uniquely on (0; ) Q1 (!) and u

i ( ; !) is dened

uniquely on (0; ) Q2 (!); i = 2; 3. For each t (0; ), we extend u1( ; !) (resp. u

i ( ; !);

i = 2; 3) to Q by zero outside of Q1 (!) (resp. Q2 (!)), and we continue to denote this



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extension by ui ; i = 1; 2; 3. It now follows from property (c) of the dynamical system Tand the construction of u( ; ; !) by means of Galerkins method [18, Section III.4] that(t ;x;!) u(t ;x;!) is dt dx -measurable on (0; ) Q . In addition, if we set

i (x;!) = Fi (T(

1

x)!) = Fi (!)(x); i = 1; 2

then on (0; ) Q ,

1u1 = u

1;

2u

i = u

i ; i = 2; 3

We will maintain this notation throughout the sequel by writing i f for the extension by zeroto all of Q of any function f dened on Q i(!); i = 1; 2.

3. STOCHASTIC TWO-SCALE CONVERGENCE IN THE MEAN

Let T denote an n-dimensional dynamical system dened on (; ; ) as in Section 2.

A sequence (u) in L2((0; ) Q ) is said to stochastically two-scale converge in themean to u L2((0; ) Q ) if for all v L2((0; ) Q ),

lim0

(0; )Q

u(t ;x;!)v(t ;x;T(1x)!) dtdx d

=

(0; )Q

uv dtdx d

This denition diers slightly from the one introduced in [14], in that the time variable entersas a free parameter. This does not eect the verication of any of the results stated below inany essential way. In particular, it follows from the proof of [14, Lemma 2:2] that for eachv L2((0; ) Q ), there exists a dt dx -representative of v such that the functiondened by the second factor in the integrand on the left-hand side of the above equationusing this representative determines a unique element of L2((0; )) Q ) for each 0.

In order to describe precisely the technology that we require from the theory of stochastictwo-scale mean convergence, we need to discuss a certain type of stochastic partial derivativethat comes from the action of T on .

It follows from property (c) of T that if f is a measurable function dened on , the func-tion (x;!) f(T(x)!) is dx d-measurable on Rn and so using (a) and (b) we can de-ne a group {U(x) : x Rn} of isometries on L2() by (U(x)f) (!) = f(T(x)!); x Rn; ! ; f L2(). It follows from Reference [17, Section 7.1] that the function x U(x) is con-tinuous in the strong operator topology, i.e. for each f L2(); U(x)f f strongly in L2()

as x 0.We now use T to dene a stochastic dierential calculus in L2() which comes from theindividual co-ordinate actions arising from the isometry group {U(x) : x Rn}. Towards thatend, we note that when each co-ordinate ofx = (x1; : : : ; xn) varies over R with the other co-ordinates held equal to 0 in U(x), we obtain n one-parameter, strongly continuous, groupsof isometries Ui on L

2() which pairwise commute. Let D1; : : : ; Dn denote the innitesimalgenerators in L2() of these one-parameter groups and let D1; : : : ;Dn denote their respective



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on , for all x Rn. The set of all functions in L2() invariant for T is a closed subspace ofL2(), which we denote by I2(). We set M2()=I2().

Lemma 3.1. (a) If f L2(), then f I2() if and only if !f = 0.

(b) If D

(), then for -a.e. ! , the function x (T(x)!) is in C

(Rn

), and forany multi-index = (1; : : : ; n),

@(T(x)!) = (D) (T(x)!)

where @ = @||=@x11 : : : @ xnn .

(c) C() is strongly dense in L2() and H1().

The next theorem collects together all of the compactness properties enjoyed by sto-chastic two-scale convergence in the mean that will be required for the derivation of thehomogenization results in the next section. Its proof is obtained from a straightforward mod-ication of the relevant arguments from [13; 14].

Theorem 3.2. (a) Every bounded sequence in L2((0; ) Q ) has a subsequence that sto-chastically two-scale converges in the mean to an element ofL2((0; ) Q ).

(b) Suppose that (u) and (xu) are bounded sequences in their respective L2-spaces

on (0; ) Q . Then there exists u L2((0; ) Q; H1()) and a subsequence, which wecontinue to denote by (u), such that (u) (resp. (xu) stochastically two-scale converges inthe mean to u (resp. !u).

(c) Suppose that (u) is a bounded sequence in L2((0; ) ; H1(Q)). Then there exists

u L2((0; ), H1(Q; I2()); L2((0; ) Q; M2()n) and a subsequence (u), which we con-tinue to denote by (u), such that(i) curl! (t;x; )=0 on , for dt dx-a.e (t; x) (0; ) Q;(ii) is contained in the L2((0; ) Q )n-norm closure of L2((0; ) Q) (range of !);(iii) (u) (resp. (xu)) stochastically two-scale converges in the mean to u (resp. + xu).

If F is a -measurable subset of , we denote by C(F) the set of all functions in C()which vanish on \F. If f L1(), we will say that Dif vanishes on F if (Dif) =0, forall C(F). It follows from Lemma 3.1(a) that if f L2() agrees on F with an elementof I2(), then !f vanishes on F. We are interested in formulating a converse to this fact.In order to do that, we say that F is T-open in if C(F) is strongly dense in the set

L2(F) of all elements of L2() which vanish on \F and that F is T-connected in ifwhenever f C() has a stochastic gradient which vanishes on F, then f agrees on Fwith an element of I2(). The following lemma gives us the desired converse ([15, Lemma2:2]).

Lemma 3.3. If F is a -measurable subset of that is T-open and T-connected, then

f L2() has a stochastic gradient which vanishes on F if and only if f agrees on F withan element of I2().

Finally, we need to make some observations about certain spaces of invariant functions. Iffor each y Rn, we dene the mapping T(y) : ( 0; ) Q (0; ) Q by

T(y) (t ;x;!) = (t ;x;T(y)!)



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814 S. WRIGHT

then {T(y) : y Rn} is an n-dimensional dynamical system on (0; ) Q with an in-variant measure given by dt dx . If I2((0; ) Q ) denotes the space of functions in

L2((0; ) Q ) invariant with respect to T, then

I2((0; ) Q ) =L2((0; ) Q; I2())

Furthermore, if E (resp. E) denotes the orthogonal projection of L2() (resp. L2((0; ) Q )) onto I2() (resp. I2((0; ) Q )), then

( Ef) (t ;x;!) =E[f(t;x; )] (!)

dt dx -a.e on (0; ) Q , and the following module property is satised:

E(fg) =f E(g); f L2((0; ) Q); g L2((0; ) Q )

4. STOCHASTIC HOMOGENIZATION OF THE RANDOMIZEDDOUGLASPESZY NSKASHOWALTER MODEL

We make the following assumption throughout this section:

each set Fi is T-open and T-connected in ; i = 1; 2

Let u = (u1; u2; u

3) L

2((0; ) Q ))3 denote the solution of the variational problem(13)(14) that was obtained in Section 2. The rst step in the homogenization procedure for

problem (4)(12) is contained in the following lemma:

Lemma 4.1. There exists ui L2((0; ); H1(Q; L2())); i L2((0; ) Q; M2()n); i=1; 2;

u3 L2((0; ) Q; H1()); gi L

2((0; ) Q ), ui L2(Q ); i = 1; 2; 3, and a subse-

quence (u) such that

(i) ui agrees on (0; ) Q Fi with an element of I2((0; ) Q ); i = 1; 2;

(ii) curl! i(t;x; ) = 0 on ; for dt dx-a.e. (t; x) (0; ) Q; i = 1; 2;(iii) i is in the L

2((0; ) Q )n-norm closure of L2((0; ) Q)(range of !), i = 1; 2;(iv) ui ;

i xu

i ; i = 1; 2; u

3;

2xu

3,

i

i ( ; xu

i ; ), i = 1; 2,

2

3( ; xu

3; ), u

1(; ; ), and

ui (; ; ); i = 2; 3 stochastically two-scale converge in the mean to, respectively, ui= iui, i(i + xui), i = 1; 2, u3 = 2u3, 2!u3, igi, i = 1; 2, 2g3, u

1 = 1u

1 , and

ui = 2ui ; i = 2; 3.

Proof. It follows from (13)(14) that for 0, ! , t (0; ),

1

2u(t)2H(!)

1

2u(0)2H(!) +

t0

A(!) (u)u d = 0 (15)

From (3), we deduce the estimate

t0

A(!) (u)u dC2

t0

(1xu1

2L2(Q) +

2xu

2

2L2(Q) +

2xu

3

2L2(Q)) d

3t

Q

k(T(1x)!) dx; t (0; ) (16)



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It follows from (15) and (16) that for a positive constant C that does not depend on ,

(0; )(u(t)2H(!) +

1xu

1

2L2(Q) +

2xu

2

2L2(Q) +

2xu

3

2L2(Q)) dtd

6 C((u01; u02; u

03)

2L2(Q)3 + |Q| kL1()) (17)

where |Q| = Lebesgue measure of Q. By (17),

(ui ); (i xu

i ); i = 1; 2; (u

3); (

2xu

3) are bounded sequences in their

respective L2-spaces on (0; ) Q (18)

In order to obtain the appropriate estimates on i i ( ; xu

i ; ); i = 1; 2, and

2

3( ; xu

3; ),

we observe that (1) implies that if : (0; ) Q Rn is a xed measurable function, thenfor 0 and ! ,

i ( ; !)

j( ; ( ; ; !); !)L2((0; )Q) 6 Ci ( ; !)( ; ; !)L2((0; )Q)

+C

Q

|k(T(1x)!)|2 dx

1=2; i = 1; 2; j = 1; 2; 3

and so

i

j( ; ; )L2((0; )Q)

6 C(i L2((0;)Q) +

|Q| kL2()); i = 1; 2; j = 1; 2; 3 (19)

It thus follows from (18) and (19) that(i

i ( ; xu

i ; )); i = 1; 2; (

2

3( ; xu

3; )) are bounded sequences in L

2((0; ) Q )n

(20)

Finally, it follows from (15) and (16) that for t [0; ),

u(t)2H(!) 6 C((u01; u

02; u

03)

2L2(Q)3 + t

Q

k(T(1x)!) dx)

Since t u(t)2H(!) is a continuous function on [0; ], we can take t= in this inequality;hence,

u()L2(Q)61

C0

u()2H(!) d

6C

C0((u01; u

02; u

03)

2L2(Q)3 + |Q| kL1())

and so

(u(; ; )) is a bounded sequence in L2(Q ()) (21)



12/21


13/21


The next step is to establish a variational equation for the limits u1; u2; u3; 1; 2 which willbe the basic relation from which both the weak and strong form of the homogenized equationswill be derived. We begin by determining an appropriate set of test functions.

Let (v1; v2; w1; w2; w3) be an element of L2((0; ); H1(Q; I2()))2 L2((0; ) Q; H1())3

satisfying the following conditions:vi; t L

2((0; ); H1(Q; L2())); i = 1; 2

w3; t L2((0; ) Q; H1())

and dt dx-a.e. on (0; ) Q,

!(1v1 + 2(v2 + w3)) = 2!w3 on (25)

For each 0; set

w1 = v1 + wT; 1 ; w

2 = v2 + w

T; 2

w

3 = w

T;

3 +

w

T;

1

w

T;

2

Lemma 4.3. For 0 and ! ;

(w1 ( ; ; !); w2( ; ; !); w

3(; ; !)) V(!)

Proof. We will show that dt -a.e. on (0; ) ;

2(!) (wT; 3 (t; ; !)) +

2(!) (v

T; 2 (t; ; !)) =

1(v

T; 1 (t; ; !)) on

1; 2 (!) (26)

An easy calculation will then establish that

1(!) (w

1 (t; ; !)) = 2(!) (w

2(t; ; !)) +

2(!) (w

3(t; ; !)) on

1; 2 (!)

and the conclusion of Lemma 4.3 will follow.We begin the verication of (26) by concluding from Lemma 3.1(c) and (25) that for all

L2((0; ); H10 (Q; H1())n),

(0; )Q

2!w3 dtdx d

=

(0; )Q

(1v1 + 2(v2 + w3))div! dtdx d (27)

As varies over (C0 (0; ) C0 (Q) H

1())n, T; varies over L2((0; ); H10 (Q; H1())n),

and so (27) holds with replaced by T;

as varies over (C0 (0; ) C

0 (Q) H

1

())n

.From Lemma 3.1(b), it follows that

(!w3)T; = 1x(w

T; 3 )

1(xw3)T; (28)

div!(T;) = 1 divx(

T;) + 1(divx)T; (29)



14/21

818 S. WRIGHT

It now follows from (27) (with replaced by T;), (28), (29), and the T-invariance of that

(0; )

Q2 (!)

x

(wT;

3) (

xw

3)T; ) dxd dt

=

(0; )

Q1 (!)

vT; 1 divx dx

+

Q2 (!)

(vT; 2 + wT; 3 )divx dx

d dt

+

(0; )Q

(1v1 + 2(v2 + w3))divx(T;) dtdx d (30)

From Greens theorem, we have that

Q2 (!)

x(wT; 3 ) dx

=

Q2 (!)

wT; 3 divx dx +

9Q2 (!)

wT; 3 2 (!) ds (31)

Q

(1v1 + 2(v2 + w3)) divx(T;) dx d

= Q(1xv1 + 2(xv2 + xw3)) T; dx d

=

Q1 (!)

(xv1)T; dx

+

Q2 (!)

((xv2)T; + (xw3)

T; ) dx

d (32)

where in (32), we have used the fact that T; vanishes on 9Q.Since vi L2((0; ); H1(Q; I2)); i = 1; 2; we have that

vT; i = vi; (xvi)T; = xvi; i = 1; 2

and so it follows from (32) and another application of Greens theorem that

Q

(1v1 + 2(v2 + w3))divx(T;) dx d

=

Q1 (!)

vT; 1 divx dx

9Q1 (!)

vT; 1 1 (!) ds

d



15/21


+

Q2 (!)

vT; 2 divx dx

9Q2 (!)

vT; 2 2 (!) ds

d

Q2 (!) (xw3)

T;

dx

d (33)

We now substitute (31) and (33) into (30) and cancel like terms to obtain

(0; )

9Q2 (!)

wT; 3 2 (!) ds

d dt

=

(0; )

9Q1 (!)

vT; 1 1 (!) ds +

9Q2 (!)

vT; 2 2 (!) ds

d dt (34)

Equation (26) is now a consequence of (34), the arbitrariness of, and the fact that 1 (!) =

2 (!) on 1; 2 (!).

By virtue of Lemma 4.3, we may apply (14) and (15) to (w1 ; w2; w

3), integrate by parts

in t, integrate over , and pass to the limit using the stochastic two-scale mean convergencein the equation that results in order to obtain the fundamental variational equation

2

i=1

(0; )Q

iciuivi; t dtdx d

(0; )Q

2c3u3w3; t dtdx d

+2

i=1

Q

iciui vi(; ; ) dx d +

Q

2c3u3 w3(; ; ) dx d

2i=1

Q

iciu0i vi(0; ; ) dx d

Q

2c3u03w3(0; ; ) dx d

+2

i=1

(0; )Q

igi (xvi+!wi) dtdx d

+

(0; )Q

2g !w3 dtdx d = 0 (35)

Following the terminology and reasoning of [3], we next proceed to decouple identity (35) by specifying appropriate choices of the test functions (v1; v2; w1; w2; w3).

First, take v1 = v2 = w1 = w2 = 0 and choose w3 to vanish at t= 0; , and such that !(2w3)

= 2!w3. Then from (35), we have(0; )Q

2c3u3w3; t dtdx d

+

(0; )Q

2g3 !w3 dtdx d = 0 (36)



16/21

820 S. WRIGHT

In particular, if we choose w3 C0 ([0; ] C

( Q) C(F2), then these conditions aresatised, and so it follows that (u1; u2; u3) is a solution of the cell equations dx-a.e. on Qgiven by

2

c3u

3; t div

!(

2g

3) = 0 on (0; ) F

2

!(u1 + u2 + u3) = 2!u3 on

Next, let v1 vanish at t= 0; , take v2 = w1 = w2 0 and choose w3 so that w3 = v1on F2. Since v1 L

2((0; ) Q; I2()), it follows from Lemma 3.1(a) that 2!w3 0 on(0; ) Q ; hence, from (35) we get

(0; )Q

1c1u1v1; t dtdx d 1

(0; )Q

2c3u3v1; t dtdx d

+(0; )Q

1g1 xv1 dtdx d = 0 (37)

Letting v1 vary over C0 ([0; ]) C

( Q) I2() and applying the reasoning in the proofof [14, Theorem 3:1], it follows that (u1; u3; g1) is a solution to the macro-ssure equation-a.e. on given by

9

9tE(1c1u1) +

1

9

9tE(2c3u3)=divx E(1g1) on (0; ) Q

where E is the orthogonal projection of L2((0; ) Q ) onto I2((0; ) Q ).Now let v2 vanish at t= 0; , let v1 = w1 = w2 0, and choose w3 so that w3 = v2 on

F2. Then 2!w3 0 on (0; ) Q , and so

(0; )Q

2c2u2v2; t dtdx d +

(0; )Q

2c3u3v2; t dtdx d

+

(0; )Q

2g2 xv2 dtdx d = 0 (38)

This says that (u2; u3; g2) satises the macro-matrix equation -a.e. on given by

9

9tE(2c2u2)

9

9tE(2c3u3)=divx E(2g2) on (0; ) Q

Finally taking v1 = v2 = w2 = w3 0 with w1 free and then v1 = v2 = w1 = w3 0 with w2

free, we deduce that (0; )Q

igi !wi dtdx d = 0; i = 1; 2 (39)

i.e. dt dx-a.e. on (0; ) Q,

div! igi = 0 on ; i = 1; 2



17/21


If we now substitute (36)(39) back into (35), it is straightforward to deduce that (g1; g2)satises dt -a.e. on (0; ) the boundary conditions

E(igi) = 0 on 9Q; i = 1; 2

and (u1; u2; u3) satises the initial and nal conditions

ui(0; ; ) = iu0i ; ui(; ; ) = u

i on Q ; i = 1; 2 (40)

u3(0; ; ) = 2u03; u3(; ; ) = u

3 on Q (41)

Following the relevant argument in [3], we can now determine the ux terms g1; g2; g3.The main step for us is an approximation of (u1; u2; u3) in the right manner by test func-tions for (35). In order to do that, we note rst from Lemma 4 :1(i) that there is a func-tion i I2((0; ) Q ) such that ui = i on (0; ) Q Fi; i = 1; 2. If (k)0 is a familyof compactly supported, C regularizing kernels for Rn+1 and if denotes convolution in

L1(Rn+1),

k i L2((0; ); H1(Q; I2()); i = 1; 2; 0

Moreover, it follows from Lemma 4.2 that

!(1k 1 + 2(k 2 + k u3)) = 2!(k u3)

on for each 0. After noting that !(k u3) = k !u3, we can thus employ a C

partition of unity on (0; ) Q in the usual way [19, Proof of Theorem 2.3.2] to construct asequence (vm1 ; v

m2 ; w

m3 ) of test functions for (35) such that as m +,

(ivmi ; ixv

mi ; 2w

m3 ; 2!w

m3 ) (iui; ixui; 2u3; 2!u3); i = 1; 2 (42)

strongly in the appropriate L2-spaces on (0; ) Q; -a.e. on . But recalling the norm estimate

k fL2(Rn+1)6fL2(Rn+1); 0

we may arrange things so that the sequences (ivmi ); (ixv

mi ), (2w

m3 ), and (2!

wm3 ), where the norm here is taken in L2((0; ) Q) are, respectively, dominated pointwise

-a.e. on by a xed constant multiple of iui, ixui, 2u3, and 2!u3. The dom-inated convergence theorem thus implies that the convergence in (42) can be taken stronglyin the appropriate L2-spaces on (0; ) Q .

Using the analogous regularization of distributions, a similar argument proves also that asm +,

(iv

m

i; t; 2w

m

3; t) (iui; t; 2u3; t); i = 1; 2

strongly in the appropriate L2-spaces of distributions on (0; ) Q .By making use of Lemma 4.1(iii), we can nd a sequence (wmi ) L

2((0; ) Q; H1()) forwhich !w

mi i strongly in L

2((0; Q ); i = 1; 2.We now substitute (40), (41) and the test functions (vm1 ; v

m2 ; w

m1 ; w

m2 ; w

m3 ) into (35), inte-

grate by parts in t (using [18, Proposition III.1.2, Corollary III.1.1] applied to u1; u2; u3),



18/21

822 S. WRIGHT

pass to the limit as m + in the equation that results, and obtain

1

2

2

i=1

Q

ici|ui(; ; )|2 dx d +

1

2 Q

2c3|u3(; ; )|2 dx d

1

2

2i=1

Q

ici|u0i |

2 dx d 1

2

Q

2c3|u03|

2 dx d

+2

i=1

(0;)Q

igi (xui + i) dtdx d

+

(0;)Q

2g3 !u3 dtdx d = 0 (43)

We can hence use (43), the monotonicity of 1; 2; 3 given by (2), property (iii) of iin Lemma 4.1, i = 1; 2, and [13, Proposition 3.5(c)] in a straightforward modication of the

argument on pp. 1315 of [3] to deduce that

gi = i( ; i + xui; ) on (0; ) Q Fi; i = 1; 2

g3 = 3( ; !u3; ) on (0; ) Q F2 (44)

We can now state and prove the principle result of this paper:

Theorem 4.4. Assume that Fi is T-open and T-connected in ; i = 1; 2, and let (u1; u

2; u

3)

be the solution of problem (4)(12) obtained in Section 2. Then there exists (u1; u2; u3; 1; 2)L2((0; ); H1(Q; L2()))2 L2((0; ) Q; H1()) L2((0; ) Q; M2()n)2 such that, as 0, (ui ); (

i xu

i ); i = 1; 2; (u

3), and (

2xu

3) stochastically two-scale converge in the mean

to, respectively, ui; i(i +xui); i = 1; 2; u3, and 2!u3, and which also satises the followingconditions:

(a) ui = iui i = 1; 2; u3 = 2u3 on (0; ) Q ;(b) ui agrees on (0; ) Q Fi with an element of I

2((0; ) Q ); i = 1; 2;(c) curl! i = 0 on ; dt dx-a.e. on (0; ) ; i = 1; 2;(d) i is in the L

2((0; ) Q )n-norm closure of L2((0; ) Q) (range of !); i = 1; 2;(e) !(u1 + u2 + u3) = 2!u3 on ; dt dx-a.e. on (0; ) Q;(f) (u1; u2; u3; 1; 2) is the unique solution of the following weak homogenized system: for

each element (v1; v2; w1; w2; w3) of L2((0; ); H1(Q; I2()))2 L2((0; ) Q; H1())3

which satises

vi; t L2((0; ); H1(Q; L2())); i = 1; 2

w3; t L2((0; ) Q; H1())

!(1v1 + 2(v2 + w3)) = 2!w3 on

and

vi(; ; ) = w3(; ; ) = 0 on Q ; i = 1; 2



19/21


the equation

(0;)Q

2

i=1iciuivi; t + 2c3u3w3; t

dtdx d

Q

2

i=1

iciu0i vi(0; ; ) + 2c3u

03w3(0; ; )

dx d

+

(0;)Q

2

i=1

ii( ; i + xui; ) (xvi + !wi)

+ 23( ; !u3; ) !w3) dtdx d = 0

holds;(g) (u1; u2; u3) is the unique solution of the strong homogenized system given by

9

9tE(1c1u1) +

1

9

9tE(2c3u3)

= divx E(11( ; 1 + xu1; ) on (0; ) Q; -a:e: on

9

9tE(2c2u2)

9

9tE(2c3u3)

= divx E(22( ; 2 + xu2; )) on (0; ) Q; -a:e: on

2c3u3; t div!(23( ; !u3; )) =0 on F2; dt dx-a:e: on (0; ) Q

with boundary conditions

E(ii( ; i + xu1; )) =0 on 9Q; i = 1; 2

and initial conditions

ui(0; ; ) = iu0i ; u3(0; ; ) = 2u

03 on Q ; i = 1; 2

where (1; 2) is the unique solution satisfying (c), (d) above of the local problem

div!(ii( ; i + xui; ))=0 on ; dt dx-a:e: on (0; ) Q; i = 1; 2

Proof. All has been established save for the uniqueness. We proceed to verify that by lettingK denote the Hilbert space L2(Q )3 equipped with the inner product ((k1; k2; k3); (l1; l2; l3))

=3

i=1

Q

cikili dx d. Consider the energy space V dened by

V = {(v1; v2; v3) H1

(Q; L2

())2

L2

(Q; H1

())vi = ivi and vi agrees on Q Fi with an element of L

2(Q; I2()); i = 1; 2

v3 = 2v3; !(v1 + v2 + v3) = 2!v3 on }

with state space H given by the norm closure in K of V. Since L2(Q; I2()) is the space offunctions in L2(Q ) that are invariant under the n-dimensional dynamical system dened



20/21

824 S. WRIGHT

on Q by (x;!) (x;T(y)!); y Rn, it follows from [15, Lemma 2:3] that V is a closedsubspace of H1(Q; L2))2 L2(Q; H1()).

Given a xed element (v1; v2; v3) of V, we consider the problem:

nd i L2(Q; M2()n) such that

div!(ii( ; i + xvi; ) ) = 0

curl! i = 0; on ; i = 1; 2 (45)

A straightforward modication of the arguments in [13, Section 4:2] shows that this problemhas a unique solution, and so we can dene an operator A : V V by

A(v1; v2; v3) (w1; w2; w3)

=

Q2( ; !v3; ) !w3 dx d

+2

i=1

Q

ii( ; i + xvi; ) xwi dx d

where (1; 2) is the solution of problem (45) determined by (v1; v2; v3). A is monotone and bounded, and so the abstract Cauchy problem

nd u L2((0; ); V) with u L2((0; ); V) such that

u + A(u(t)) = 0 in L2((0; ); V)

u(0)= u0 in H (46)

has a unique solution [18, Proposition III.4.1]. Any solution of the weak homogenized systemin (f) of Theorem 4.4 satises the variational form of problem (46) with u0 = (1u

01; 2u

02; 2u

03)

H, and since the strong form of problem (46) is equivalent to is variational form [18, remarkat the beginning of Section III.4.1], this proves that the weak homogenized system determines(u1; u2; u3), and hence also (1; 2), uniquely. Since the strong form of problem (46) is alsoequivalent to the strong homogenized system in part (g) of Theorem 4.4, this latter systemalso has a unique solution.

In closing, we remark that the restriction in this paper to the Hilbert-space setting wasdone only for simplicity and ease of exposition. The interested reader will have no dicultyin formulating and proving the analogous results in the Lp-setting for 1p+, using [3]and the results on stochastic two-scale convergence in the mean from [14].

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on di2usion of a phase, slightly compressible 8uid

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