行政院國家科學委員會專題研究計畫 期中進度報告
二相流在破裂介質中的宏觀模式(1/2)
計畫類別： 個別型計畫
計畫編號： NSC93-2115-M-009-009-
執行期間： 93 年 08 月 01 日至 94 年 07 月 31 日
執行單位： 國立交通大學應用數學系(所)
計畫主持人： 葉立明
報告類型： 精簡報告
報告附件： 出席國際會議研究心得報告及發表論文
處理方式： 本計畫可公開查詢
中 華 民 國 94 年 6 月 1 日
f
c NCTU
Tall Block Models for Two-Phase flows in Fractured Media
Li-Ming Yeh
Department of Applied Mathematics, National Chiao Tung University Hsinchu, 30050, Taiwan, R.O.C.
Email: [email protected]
Two-phase, incompressible, immiscible flow in fractured media with tall matrix blocks is concerned. Suppose denotes horizontal size ratio of matrix blocks to whole medium, and suppose the horizontal widths of the fracture planes and matrix blocks are in same order. As goes to 0, microscopic model for the two-phase flow problem converges to 1) a dual-porosity model if permeability ratio of matrix blocks to fracture planes is of order 2; 2) a single-porosity model for fracture flow if the ratio is smaller than order 2;
3) another type of single-porosity model if the ratio is greater than order 2.
Keywords: dual-porosity model, fractured media
1. Introduction
Homogenization for two-phase, incompressible, immiscible flow in fractured media with tall matrix blocks is concerned. Within a fractured medium there is an in-terconnected system of fracture planes dividing the porous rock into a collection of matrix blocks. The fracture planes, while very thin, form paths of high perme-ability. Most of the fluids reside in matrix blocks, where they move very slow. Let be the horizontal size ratio of tall matrix blocks to the whole medium, and let the horizontal widths of the fracture planes and matrix blocks be in same order. In case permeability ratio of matrix blocks to fracture planes is of order 2,
micro-scopic models for the two-phase flow problem converge to a dual-porosity model as tends to 0. For the macroscopic model, a fractured medium is regarded as a porous medium consisting of two superimposed continua, a continuous fracture sys-tem and a discontinuous syssys-tem of matrix blocks. Matrix blocks play the role of a global source distributed over the entire medium. The immiscible two-phase flow is formulated by conservation of mass principles for each continum plus sources from tall matrix blocks. This problem was also considered by formal asymptotic expan-sion in [8]. If the ratio is smaller than order 2, the microscopic models approach
a single-porosity model for fracture flow. If the ratio is greater than order 2, then
microscopic models tend to another type of single-porosity model. Our intention is to prove the convergence of the microscopic models.
Rest of the paper is organized as follows: In next section§2, we state microscopic model for two-phase flow in fractured media. Notation and assumption will be given in§3. Then in §4, we present our main results. Some known results needed for our main results will be recalled in §5. Proof of main result is in §6. In §6, we need to use the convergence of oil saturation in matrix blocks. The proof is lengthy and tedious, so we present it in last section§7.
2. Microscopic Model for Tall Matrix Blocks
Let Y ≡ [0, 1]2 be a cell consisting of a matrix block domain Y
m completely
surrounded by a connected fracture domain Yf. Xm(y) is the characteristic function
of Ym, extended Y -periodically to all of<2. ˜Ω⊂ <2 contains two subdomains, ˜Ωf
and ˜Ω
m. ˜Ωm⊂ {˜x ∈ ˜Ω|Xm(˜x/) = 1}, ˜Ωf = ˜Ω\ ˜Ωm. Let ˜Γ ≡ ∂ ˜Ωf∩ ∂ ˜Ωm∩ ˜Ω.
Boundary of ˜Ω includes two parts ˜Γ1and ˜Γ2satisfying ˜Γ1∪˜Γ2= ∂ ˜Ω and ˜Γ◦1∩˜Γ◦2=∅.
Porous medium considered is a cylindrical aquifer Ω ≡ ˜Ω× [0, H] ⊂ <3 and is
assumed to be a two-connected domain with a periodic structure. It contains two subdomains, Ω
f ≡ ˜Ωf× [0, H] and Ωm ≡ ˜Ωm× [0, H], representing the system of
fracture planes and matrix blocks respectively. Let Γ ≡ ˜Γ× [0, H] be that part
of the interface of Ω
m with Ωf that is interior to Ω. Both Γ1 ≡ ˜Γ1× [0, H] and
Γ2≡ ˜Γ2× [0, H] are part of lateral boundary of Ω.
In fracture subdomain Ω
f, porosity is denoted by Φ, absolute permeability by
K, saturation of oil phase by S, capillary pressure by Υ(S), relative permeability
by Λα(S), phase pressure by Pα, and a density-gravity term by Gα for α = w, o.
φ, k, s, υ(s), λ
α(s), pα, gαfor α = w, o, in subdomain Ωm represent same
quan-tities as those denoted by upper case symbol in fracture subdomain. Conservation of mass in each phase are written as, in Ω
f, t > 0, −Φ∂tS− ∇ · (KΛw(S)∇(Pw − Gw)) = 0, (2.1) Φ∂tS− ∇ · (KΛo(S)∇(Po− Go)) = 0, (2.2) Υ(S) = Po− Pw, (2.3) in Ω m, t > 0, −φ∂ ts− ∇ · kI2$λw(s)∇(pw− Gw) = 0, (2.4) φ∂ts− ∇ · kI2$λo(s)∇(po− Go) = 0, (2.5) υ(s) = po− pw, (2.6) whereId
is a diagonal matrix defined byId≡
d 0 0 0 d 0 0 0 1
. Phase fluxes and pressures are required to be continuous on interface Γ, t > 0, α = w, o,
KΛ
α(S)∇(Pα− Gα)· ~ν= kI2$λα(s)∇(pα− Gα)· ~ν, (2.7)
where ~ν is the unit vector normal to Γ
. Boundary conditions are, for α = w, o,
KΛα(S)∇(Pα− Gα)· ~n = 0 on Γ1, (2.9) KΛ α(S)∂x3(P α− Gα)|x3=0,H = k λ α(s)∂x3(p α− Gα)|x3=0,H = 0, (2.10) P α= Pb,α on Γ2, (2.11)
where ~n is the unit vector outer normal to Γ1. Initial conditions are
S(0, x) = S0(x) in Ωf, (2.12)
s(0, x) = s0(x) in Ωm. (2.13)
3. Notation and Assumption For any x ∈ <3, x = (˜x, x
3) where ˜x∈ <2. ˜Ω(2)≡ {˜x ∈ ˜Ω : dist(˜x, ∂ ˜Ω) > 2},
˜ Ω
m ≡ {˜x : ˜x ∈ (Ym+ j) ⊂ ˜Ω(2) for j∈ Z2}, ˜Ωf ≡ ˜Ω\ ˜Ωm, and ˜Ω ≡ {z :
z ∈ (Y + j), (Ym+ j) ⊂ ˜Ω(2) for j∈ Z2}. Ω ≡ ˜Ω× [0, H], Ωi ≡ ˜Ωi× [0, H],
YH
m ≡ Ym×[0, H], Q ≡ Ω×Y , Qm≡ Ω×Ym,Qi≡ Ω×Yi, i = f, m. Bt≡ (0, t)×B
forB = YH
m, Γ,Q, Qm, Ω, Ωi,Qi i = f, m.
<+0 ≡ <+∪ {0}. Time difference is defined to be ∂hψ(t) ≡
ψ(t+h)−ψ(t) h . For a
setB, XBis a characteristic function ofB. ψ(t, x, y) ∈ Lr(ΩT; Lr
per(Y )), 1 < r <∞,
coincides with a function in Lr(
QT) extended by Y -periodicity in y to the whole
of <2. For
B = Yf, Ym, we define Lr(ΩT; Lperr (B)) ≡ {ψ ∈ Lr(ΩT; Lrper(Y )) :
ψ(t, x, y) = 0 if y ∈ Y \ B}. W0i,r(Ω) ≡ {ψ ∈ Wi,r(Ω) : ψ|Γ2 = 0} if i ∈ N and
r > 1,U ≡ W01,2(Ω),U2 ≡ U × U, dual X ≡ dual space of X, sl (resp. 1− sr) is
residual matrix oil (resp. water) saturation. Lq,r(ΩT)
≡ Lr(0, T ; Lq(Ω)).
If Υ : [0, 1) → <+0 (resp. υ : [sl, sr) → <+0) is onto and strictly increasing,
Υ−1 (resp. υ−1) denotes the inverse function of Υ (resp. υ). Then we define
J : [sl, sr)→ [0, 1) by J (z) ≡ Υ−1(υ(z)), and denote byJ−1 the inverse function
ofJ . Pb,c≡ Pb,o− Pb,w, Sb≡ Υ−1(Pb,c), Λ≡ Λw+ Λo, λ≡ λw+ λo, R(z) ≡R0z ΛwΛo Λ dΥ dS(ξ)dξ for z∈ [0, 1), A(z) ≡R0z q ΛwΛo Λ (Υ−1(ξ))dξ for z∈ [0, ∞), M(z) ≡Rszl λwλo λ dυ ds(ξ)dξ for z∈ [sl, sr). (3.1)
ϑ∈ (0, 1/8) is a number such that R0is increasing (resp. decreasing) in (0, ϑ) (resp.
(1− ϑ, 1)).
Next let us assume the following conditions: For α = w, o,
A1. Γ26= ∅, Ym⊂ <2is a bounded smooth domain, and Ω⊂ <3is open, bounded,
and connected with Lipschitz boundary, A2. K, G
α(x3)∈ W1,∞(Ω), ∂tPb,α∈ L2(0, T ; H1(Ω)), Pb,α ∈ C(0, T ; C1,d1(Ω)),
S
A3. K, k, Λ, λ∈ [d
3, d4], Sb, S0,J (s0)∈ (d5, 1− d5) and d5∈ (0, 1),
A4. φ= φ(x
), k= k( x
), where φ, k are smooth Y -periodic functions,
A5. Λw, λw (resp. Λo, λo) : [0, 1] → [0, 1] are continuous and decreasing (resp.
increasing), Λw(1− z) ∝ zd6, Λo(z)∝ zd7 for z∈ (0, ϑ), ΛΛα(J (z)) =λλα(z),
A6. Υ : [0, 1)→ <+0 (υ : [sl, sr)→ <+0) is onto, increasing, and a locally Lipschitz
continuous function, and inf
z∈[0,1) dΥ dS(z) > 0, dΥ dS(J (z)) dυ ds(z) , Φ, φ ∈ [d8, d9] for z ∈ [sl, sr], dd98 ∼ 1,
A7. Λ3/2o (z)≤Rz2z(A(Υ(2z)) − A(Υ(ξ)))dξ for z ∈ (0, ϑ) and
Λ3/2w (1− z) ≤R11−2z−z(A(Υ(ξ)) − A(Υ(1 − 2z)))dξ for z ∈ (0, ϑ),
A8. |Λα(z1)− Λα(z2)| ≤ d10 p (R(z1)− R(z2))(z1− z2) for any z1, z2∈ [0, 1], A9. max z∈[0,1]|Λ(z) − 1| + maxz∈[sl,sr]|λ(z) − 1| ≤ d 11 (d11 only depends on Ω, K, k), A10. ΛoΛw(z)≤ d12z|1 − z|pR0(z),R0(z)∝ zm|1 − z|m1 for z∈ (0, ϑ) ∪ (1 − ϑ, 1) and m, m1> 1,
where m, m1, di, i = 1,· · · , 12 are positive constants.
Remark 3.1 From A1, Ω
f is an open, bounded, and connected domain with
Lip-schitz boundary. In A2, the density-gravity terms G
w, Go are functions depending
on x3 variable. Initial and boundary saturations are away from two end points 0
and 1 (see A3). A5 implies that relative permeability Λw (resp. λw) in the
neigh-bor of end point 1 has similar properties as Λo (resp. λo) in the neighbor of end
point 0. Relative phase mobilities in fractures and matrix blocks behave similar. A6 requires that fracture capillary pressure increases as fast as capillary pressure of matrix blocks. Usually, derivative of capillary pressure Υ0(z) (resp. υ0(z)) tends to
infinity as z→ 0 or 1 (resp. sl or sr). A10 allows parabolic equations considered
are degenerate at end points 0 and 1, a characteristic of a porous medium equation. Indeed, it also impliesR0 ∈ L∞(0, 1). A7-8,10 are the restrictions on relative
per-meability and capillary pressure in fractures. Indeed, if d6, d7 (see A5) are large
enough (depending on capillary pressure), A7-8,10 hold. One may also note that because of A5-10, Λo andR0 at the end point 0 have similar properties as Λw and
R0 at the end point 1.
4. Main Result
In this section, we present the limit models of (2.1–2.13) as → 0. Roughly speaking, the limit models are fracture flow equations plus interior sources from matrix blocks. The source terms depend on how fast the matrix permeability tends to 0 as → 0. For 0 < $ < 1 case, matrix permeability tends to 0 very slow and saturation variation in fracture system and in matrix blocks is almost simultaneous.
So the limit model is a single-porosity model with sources from matrix blocks. For $ = 1 case, saturation variation in fracture system and in matrix blocks is not simultaneous and the limit model is a dual-porosity model. In this case, domain acts as a porous medium consisting of two superimposed continua, a continuous fracture system Ω and a discontinuous system of matrix blocksQm. Primary flow
occurs in fracture system Ω, and each point x ∈ Ω is associated with a matrix block Ym. Flow in matrix blocks plays the role of a global source in the whole
fracture system. The model includes two systems of equations, one for flow in fracture system and the other for flow in matrix block system. The two systems are coupled through nonlinear sources. For 1 < $ case, matrix permeability tends to 0 so fast that matrix blocks play no roles in the limit model. The limit model is a single-porosity model containing only fracture flow equations without matrix sources.
4.1. For $ = 1 case
Let Ω ⊂ <3 be a fractured medium. Equations for fracture flow are, for x
∈ Ω, t > 0,
−Φ∂tS− ∇ · (KΛw(S)∇(Pw− Gw)) = qw, (4.1)
Φ∂tS− ∇ · (KΛo(S)∇(Po− Go)) = qo, (4.2)
Υ(S) = Po− Pw. (4.3)
Φ is porosity, K is permeability field, S is oil saturation, Υ(S) is capillary pressure curve, Λα (α = w, o) is relative permeability curve of α-phase, Pα denotes phase
pressure, Gαis a function depending on density, gravity, and position, and qαis the
matrix-fracture source.
Above each point x ∈ Ω is suspended topologically a matrix block Ym ⊂ <2.
Equations for flow in a matrix block are, for x∈ Ω, y ∈ Ym, t > 0,
−φ∂ts− ∂y,x3· (kλw(s)∂y,x3(pw− Gw)) = 0, (4.4)
φ∂ts− ∂y,x3· (kλo(s)∂y,x3(po− Go)) = 0, (4.5)
υ(s) = po− pw. (4.6)
Here functions s, pw, p0are defined in space domainQmand ∂y,x3= (∂y1, ∂y2, ∂x3).
Each lower case symbol denotes the quantity on Ymcorresponding to that denoted
by an upper case symbol in the fracture system equations. The matrix-fracture sources are given by, for x∈ Ω, t > 0,
qα= −1
|Ym|
Z
Ym
(σαφ∂ts− ∂x3(kλα(s)∂x3(pα− Gα))) dy, (4.7)
where σw =−1, σo = 1, and|Ym| is the volume of Ym. Boundary conditions are,
for t > 0, α = w, o,
KΛα(S)∂x3(Pα− Gα)|x3=0,H = kλα(s)∂x3(pα− Gα)|x3=0,H = 0, (4.9)
Pα= Pb,α for x∈ Γ2, (4.10)
where ~n is the unit vector outward normal to Γ1. On interface, pressures are
continuous, that is, for t > 0, x∈ Ω, y ∈ ∂Ym, α = w, o,
pα(t, x, y) = Pα(t, x). (4.11)
Initial conditions are
S(0, x) = S0(x) for x∈ Ω, (4.12)
s(0, x, y) = s0(x) for x∈ Ω, y ∈ Ym. (4.13)
Theorem 4.1 Under A1−10, a subsequence of solutions of the microscopic models (2.1–2.13) converges in two-scale sense to a solution of (4.1–4.13) (see next section for the definition of convergence in two-scale sense).
4.2. For 0 < $ < 1 case Equations are, for x∈ Ω, t > 0,
−Φ∂tS− ∇ · (KΛw(S)∇(Pw− Gw)) = qw, (4.14)
Φ∂tS− ∇ · (KΛo(S)∇(Po− Go)) = qo, (4.15)
Υ(S) = Po− Pw= υ(s). (4.16)
Φ, K, S, Υ(S), υ(s), Λα, Pα, Gα, and qα (α = w, o) are the same quantities as
those in $ = 1. The matrix-fracture sources are given by, for x∈ Ω, t > 0, qα= −1
|Ym|
Z
Ym
(σαφ∂ts− ∂x3(kλα(s)∂x3(Pα− Gα))) dy, (4.17)
where σw =−1, σo = 1, and|Ym| is the volume of Ym. Boundary conditions are,
for t > 0, α = w, o,
KΛα(S)∇(Pα− Gα)· ~n = 0 for x∈ Γ1, (4.18)
KΛα(S)∂x3(Pα− Gα)|x3=0,H = 0, (4.19)
Pα= Pb,α for x∈ Γ2, (4.20)
where ~n is the unit vector outward normal to Γ1. Initial condition is
S(0, x) = S0(x) for x∈ Ω. (4.21)
Theorem 4.2 Under A1−10, a subsequence of solutions of the microscopic models (2.1–2.13) converges in two-scale sense to a solution of (4.14–4.21) (see next section for the definition of convergence in two-scale sense).
4.3. For $ > 1 case
Equations are, for x∈ Ω, t > 0,
−Φ∂tS− ∇ · (KΛw(S)∇(Pw− Gw)) = 0, (4.22)
Φ∂tS− ∇ · (KΛo(S)∇(Po− Go)) = 0, (4.23)
Υ(S) = Po− Pw. (4.24)
Φ, K, S, Υ(S), υ(s), Λα, Pα, Gα, and qα (α = w, o) are the same quantities as
those in $ = 1. Boundary conditions are, for t > 0, α = w, o,
KΛα(S)∇(Pα− Gα)· ~n = 0 for x∈ Γ1, (4.25)
KΛα(S)∂x3(Pα− Gα)|x3=0,H = 0, (4.26)
Pα= Pb,α for x∈ Γ2, (4.27)
where ~n is the unit vector outward normal to Γ1. Initial condition is
S(0, x) = S0(x) for x∈ Ω. (4.28)
Theorem 4.3 Under A1−10, a subsequence of solutions of the microscopic models (2.1–2.13) converges in two-scale sense to a solution of (4.22–4.28) (see next section for the definition of convergence in two-scale sense).
5. Some Known Results
Lemma 5.1 [1] Let 1≤ r < ∞ and A1 hold. There is a constant d13(Yf, r) and a
linear continuous extension operator Π: W1,r(Ωf)∩ L∞(Ωf)→ W1,r(Ω)∩ L∞(Ω)
such that if ϕ∈ W1,r(Ω f)∩ L∞(Ωf) and d14≤ ϕ ≤ d15, then Πϕ = ϕ in Ωf almost everywhere, kΠϕkW1,r(Ω)≤ d13kϕkW1,r(Ω f), d14≤ Πϕ≤ d15.
Definition 5.1 For a given > 0 and 1≤ r < ∞, we define a dilation operator “ ” mapping a measurable function ϕ∈ Lr(Ω,T
m ) to a measurable function ϕ∈ Lr(QTm) by, for (t, ˜x, x3, y)∈ QTm, ϕ(t, ˜x, x3, y)≡ ϕ(t, `(˜x) + y, x 3) if (`(˜x) + y, x3)∈ Ωm, 0 elsewhere, where `(˜x)
≡ j if ˜x ∈ (Y + j), j ∈ Z2, denoting the lattice translation point of
-cell domain containing ˜x.
Definition 5.2 A sequence of functions ϕ in Lr(ΩT), 1 < r <∞, is said to two-scale converge to ϕ in Lr(ΩT; Lr
per(Y )) if, for any function ψ∈ C0∞(ΩT; Cper∞ (Y )),
we have lim →0 Z ΩT ϕ(t, x)ψ(t, x, ˜x/)dxdt = Z QT ϕ(t, x, y)ψ(t, x, y)dydxdt,
denoted by ϕ * ϕ2
∈ Lr(ΩT; Lr
per(Y )). Besides lim→0kϕkLr(ΩT) =kϕkLr(QT),
ϕ is said to two-scale converge to ϕ in Lr(ΩT; Lr
per(Y )) strongly, and denoted by
ϕ 2→ ϕ ∈ Lr(ΩT; Lr
per(Y )) strongly.
6. Proof of Main Result
A1-10 are assumed from now on. Let us derive a weak formulation of (2.1–2.6). Multiplying (2.1) and (2.4) by η as well as (2.2) and (2.5) by ζ, integrating over ΩT, and employing boundary conditions (2.7) and (2.9), we obtain
− Z Ω,Tf Φ∂tSη + Z Ω,Tf KΛw(S)∇(Pw − Gw)∇η − Z Ω,T m φ∂ tsη + Z Ω,T m k I2$ λw(s)∇(pw− Gw)∇η = 0, (6.1) Z Ω,T f Φ∂ tSζ + Z Ω,T f KΛ o(S)∇(Po− Go)∇ζ + Z Ω,Tm φ∂tsζ + Z Ω,Tm kI2$ λo(s)∇(po− Go)∇ζ = 0, (6.2)
for smooth functions η, ζ∈ L2(0, T ;U). Next we define global pressure [11] as
P ≡1 2 P o+ Pw + RΥ(S) 0 Λo Λ(Υ−1(ξ))− Λw Λ (Υ−1(ξ)) dξ, p≡ 1 2 po+ pw+ Rυ(s) 0 λo λ(υ−1(ξ))− λw λ (υ−1(ξ)) dξ, (6.3) Pbis defined as Pin (6.3)1except replacing Po, Pw, Υ(S) by Pb,o , Pb,w , Pb,c
respec-tively. Then∇P=Λw
Λ (S)∇Pw+ΛΛo(S)∇Po and∇p=λλw(s)∇pw+λλo(s)∇po
by (2.3) and (2.6). (6.2) can be rewritten as Z Ω,Tf Φ∂tSζ + Z Ω,Tf K Λo(S)∇(P− Go) +∇R(S) ∇ζ + Z Ω,Tm φ∂tsζ + Z Ω,Tm kI2$ λo(s)∇(p− Go) +∇M(s) ∇ζ = 0. (6.4) See§3 for R, M. Summing (6.1) and (6.2), we obtain, for η ∈ L2(0, T ;
U), Z Ω,Tf K Λ(S)∇(P − G o)− Λw(S)∇(Gw− Go) ∇η + Z Ω,Tm kI2$ λ(s)∇(p− Go)− λw(s)∇(Gw− Go) ∇η = 0. (6.5) For ζ∈ L2(0, T ;U) ∩ H1(ΩT), ζ(T ) = 0, Z Ω,Tf Φ∂tSζ + Φ(S− S0)∂tζ =− Z Ω,Tm φ∂tsζ + φ(s− s0)∂tζ. (6.6)
(6.1–6.6), (2.3), (2.6), (2.8), (2.11) form a weak formulation of (2.1–2.13).
Next we consider a regularized problem. Let v be a small number satisfying 0 < v < d5
4. Extend Λα (α = w, o) constantly and continuously to < and define
Λα,v, Λv, λα,v, λv as Λα,v(z)≡ Λα 0.5(0.5z−v−v) , Λv ≡ Λw,v+ Λo,v, λv(z)≡ Λv(J (z)), λα,v(z) = Λα,v(J (z)). (6.7)
By A2-3, there exist smooth functions S
0,v, Sb,v, s0,v such that S0,v , Sb,v,J (s0,v)∈ (d5/2, 1− d5/2), S0,v |Γ2 = Sb,v|Γ2(t = 0), (6.8) S 0,v, Sb,v, s0,v→ S0, Sb, s0 in L2(0, T ; H1(Ω)), ∂tΥ(Sb,v)→ ∂t(Pb,o− Pb,w) in L1(ΩT), as v→ 0. (6.9) The regularized problem is: Find{S
vXΩ f + s vXΩ m, P vXΩ f+ p vXΩ m} satisfying Φ∂ tSvXΩ f + φ ∂ tsvXΩ m ∈ dual L 2(0, T ; U), (6.10) v≤ S vXΩ f+J (s v)XΩ m ≤ 1 − v, (6.11) R(SvXΩ f +J (s v)XΩ m)− R(S b), PvXΩ f+ p vXΩ m− Pb∈ L 2(0, T ; U), (6.12) Z Ω,Tf Φ∂tSvζ + Z Ω,Tf K Λo,v(Sv)∇(Pv− Go) +∇R(Sv) ∇ζ + Z Ω,Tm φ∂tsvζ + Z Ω,Tm kI2$ λo,v(sv)∇(pv− Go) +∇M(sv) ∇ζ = 0, (6.13) Z Ω,Tf K Λv(Sv)∇(Pv− Go)− Λw,v(Sv)∇(Gw− Go) ∇η + Z Ω,Tm kI2$ λv(sv)∇(pv− Go)− λw,v(sv)∇(Gw− Go) ∇η = 0, (6.14) S vXΩ f(0, x) + s vXΩ m(0, x) = S 0,vXΩ f + s 0,vXΩ m, (6.15) for any ζ, η∈ L2(0, T ;
U). It is easy to see that (6.13) is a nondegenerate (depending on v) parabolic equation, and (6.13–6.14) imply, if S
w,v≡ 1 − Sv, 0 = Z Ω,Tf Φ∂tSw,v ζ + K Λw,v(1− Sw,v)∇(Pv− Gw)− ∇R(1 − Sw,v ) ∇ζ + Z Ω,T m φ∂ tsw,vζ + kI2$ λw,v(1− sw,v)∇(pv− Gw)− ∇M(1 − sw,v) ∇ζ.(6.16) By [4, 5, 6, 9, 12, 20, 22, 29], it is known
Lemma 6.1 Under (6.8–6.9), there exist functions S
v, Pv in Ωf and sv, pv in Ωm
satisfying (6.10–6.15) for each v, as well as there exist functions S, P, P α in
Ω
f and s, p, pα in Ωm for α = w, o satisfying (6.1–6.6), (2.3), and (2.6–2.11).
ˇ S v ≡ SvXΩ f +J (s v)XΩ m is in L
and, as v→ 0, ( ˇ S v→ ˇS≡ SXΩ f +J (s ) XΩ m pointwise, R( ˇS v), PvXΩ f+ p vXΩ m → R( ˇS ), P XΩ f+ p XΩ m in L 2(0, T ; H1(Ω)). Moreover, 0 < S< 1, s l< s< sr, and X α=w,o kpΛα(S)∇PαkL2(Ω,T f )+kI $ p λα(s)∇pαkL2(Ω,T m ) +k |∇P| + |∇R(S)| + |∇A| kL2(Ω,T f ) +k |I$ ∇p| + |I$∇M(s)| + |I$∇A| kL2(Ω,T m )≤ c, whereA ≡ A(Υ(S)) if x ∈ Ω f, A(υ(s)) if x ∈ Ω m.
and c is a constant independent of .
Lemma 6.2 For any β, τ satisfying 2≤ β0≤ β − 2 ∈ N, dβ50 ≤ ϑ, and τ ≤ T , the
following inequality holds:
sup t≤τ {x ∈ Ω : ˇS(t)≤ µ or 1 − µ ≤ ˇS(t)} ≤ c0|c0τ| β−β0 (β− β0)(β−β0)fβ , (6.17) where µ≡ d5 2β, lim
β→∞fβ= 1, and c0 is a constant independent of τ, β, , µ.
Proof: Let us defineLµ,Kµ, cKµ as
Lµ(z)≡ 1 if µ≤ z ≤ 2µ, 0 elsewhere,
Kµ(z)≡RA(Υ(2µ))z Lµ(Υ−1(A−1(ξ)))dξ for z∈ [0, A(∞)),
c Kµ(z)≡R
z
A(Υ(2µ))(LµΛΛo)◦ (Υ−1(A−1(ξ)))dξ for z∈ [0, A(∞)).
By 2µ ≤ d5
2 and A2-3,5, we take ζ = Kµ(A) ∈ L2(0, T ;U) in (6.4) and η =
c Kµ(A)∈ L2(0, T ;U) in (6.5) to obtain Z Ω,τ f Φ Kµ(A)∂tS+ Z Ω,τ f KΛ o(S)Lµ(S)∇Υ(S)∇A + Z Ω,τm φKµ(A)∂ts+ Z Ω,τm kI2$ Λo(u)Lµ(u)∇υ(s)∇A ≤ c1 Z Ω,τf KΛo(S)Lµ(S)|∂x3A | + Z Ω,τm kΛo(u)Lµ(u)|∂x3A | , (6.18)
where u≡ J (s) and constant c
1 is independent of , µ. Suppose
Z
Kµ(A) Φ∂tSXΩ,τf + φ∂tsXΩ,τm
(6.18–6.19) imply Z Ω,τf KΛo(S)Lµ(S)|∂x3A | + Z Ω,τm kΛo(u)Lµ(u)|∂x3A | ≤ c2 Z Ω,τf KΛ3/2o Lµ(S) 1 2 Z Ω,τf KΛo(S)Lµ(S)∂x3Υ(S )∂ x3A 1 2 +c2 Z Ω,τm kΛ3/2o Lµ(u) 1 2 Z Ω,τm kΛo(u)Lµ(u)∂x3υ(s )∂ x3A 1 2 , (6.20) where constant c2is independent of , µ. A3 and (6.18–6.20) imply
Z Kµ(A) Φ∂tSXΩ,τ f + φ ∂ tsXΩ,τ m ≤ c3 Z Ωτ Λ3/2o Lµ( ˇS). (6.21) Let us define Z(S, s, µ)≡ ( ΦRS 2µ Kµ(A(Υ(ξ)))dξ in Ω f, φRs J−1(2µ)Kµ(A(υ(ξ)))dξ in Ωm. (6.21) implies Z Ωτ ∂tZ(S, s, µ)≤ c4 Z Ωτ Λ3/2o Lµ( ˇS). (6.22)
(6.22) and A6-7 yield that, if 0≤ t1≤ t2≤ T ,
Z t2 t1 Z Ω ∂tZ(S, s, µ)≤ c4 Z t2 t1 Z ΩZ(S , s, 2µ), (6.23)
where c4 is independent of t1, t2, µ, . Define
F(τ, µ) ≡ Λ 1 o(µ)3/2 sup t≤τ Z ΩZ(S , s, µ).
A5 and (6.23) imply that, for 0≤ t1≤ t2≤ T ,
F(t2, µ)− F(t1, µ)≤ c5(t2− t1)F(t2, 2µ),
where c5 is independent of t1, t2, µ, . By induction and A3, one obtains, for j ∈
N, jh≤ T , F(jh,d5 2β)≤ (β − β0+ 1) j−1|c 5h|β−β0F(jh, d5 2β0). (6.24) If j = β−β0
log(β−β0) and τ = jh in (6.24), then
F(τ,d5 2β)≤ |c5τ|β−β0 (β− β0)(β−β0)fβF (τ, d5 2β0), (6.25)
where fβ→ 1 as β → ∞. Define B(t) ≡ {x ∈ Ω : ˇS(t, x)≤d2β5}. (6.25) implies sup t≤τ Z XB(t)≤ c6F(τ, d5 2β)≤ c6|c5τ|β−β0 (β− β0)(β−β0)fβF (τ, d5 2β0),
where constant c6 is independent of τ, β, , µ. So proof of first part of (6.17) is
completed. The other part can be proved in a similar way, so we skip it. Lemma 6.3 If r∈ (1, 2), kP αkLr(0,T ;W1,r(Ω f))+kI $ ∇pαkLr(Ω,T m )≤ c, where α =
w, o and c is a constant independent of . Moreover, if $≤ 1, then kp
αkLr(Ω,T m )≤ c.
Proof: We define, for 2≤ β0∈ N,
( B1+β0 ≡ {(t, x) ∈ Ω ,T f : d5 22+β0 ≤ S}, Bβ≡ {(t, x) ∈ Ω,Tf : 2dβ+15 ≤ S< d2β5} if 2 + β0≤ β ∈ N.
A5, Lemmas 6.1-6.2, and H¨older inequality imply k∇P okrLr(Ω,T f )≤ k p Λo(S)∇PokrL2(Ω,T f )kΛ −1 o (S)k r/2 Lr/(2−r)(Ω,T f ) ≤ c1 Z Ω,Tf |Λ o(S)| −r 2−r ∞ X β=1+β0 XBβ 2−r 2 ≤ c2 (indep. of ). (6.26)
Similar argument will give k∇P
wkLr(Ω,T f )+
P
α=w,okI$∇pαkLr(Ω,T
m ) ≤ c. By
boundary condition A2, kP αkLr(Ω,T f ) ≤ c, α = w, o. By Lemma 5.1, (2.8), and $ ≤ 1, kp α− ΠPαkLr(Ω,T m ) ≤ k∂x1(p α− ΠPα)kLr(Ω,T m ) ≤ c. So kp αkLr(Ω,T m ) is bounded.
Lemma 6.4 For r∈ [1, ∞) and sufficiently small δ, kδ2∂−δS ∂−δAkLr((δ,T )×Ω f)+kδ 2∂−δs ∂−δA kLr((δ,T )×Ω m)≤ cδ 1/r, (6.27)
where c is independent of , δ. See§4 for notation ∂−δ.
Proof: Note ζ(t, x) ≡ Rmax(t,δ)min(t+δ,T )δ ∂−δ A
− A(Pb,c)(τ, x)dτ ∈ L2(0, T ;U) by
A2-3 and Lemma 6.1. Take ζ above in (6.2) to get, by Fubini’s theorem, A2, and Lemma 6.1, Z T δ Z Ω f Φδ2∂−δS∂−δA(τ, x) + Z T δ Z Ω m φδ2∂−δs∂−δA(τ, x) = Z Ω,Tf Φ∂tS(t, x)ζ + Z Ω,Tm φ∂ts(t, x)ζ + Z T δ Z Ω f Φδ2∂−δS∂−δA(Pb,c) + Z T δ Z Ω m φδ2∂−δs∂−δA(Pb,c)≤ cδ,
where c is independent of , δ. So we prove (6.27) for r = 1 case. (6.27) for r > 1 case follows directly becauseA, ˇSare bounded and (6.27) for r = 1 holds.
Lemma 6.5 A subsequence of Π(A|Ω
f) converges toA
∗in L2(ΩT) and pointwise.
Proof: This is due to A6,10, Lemmas 5.1, 6.1-6.4, and compactness principle. Lemma 6.6 s, p, p
α (α = w, o) satisfy, for almost all x∈ ˜Ω,
φ∂ts− ∂y,x3· kI 2$−2 ∂y,x3M(s) + λo(s)∂y,x3(p− Go) = 0, (6.28) ∂y,x3· kI2$−2 λ(s)∂y,x3p− X λα(s)∂y,x3Gα = 0, (6.29) −φ∂ts− ∂y,x3· kI 2$−2 λw(s)∂y,x3(p w− Gw) = 0, (6.30) φ∂ts− ∂y,x3· kI 2$−2 λo(s)∂y,x3(p o− Go) = 0, (6.31) in L2(0, T ; H−1(YH m)). Proof: Let ˆζ∈ L2(0, T ; C∞ 0 (YmH)). For x∈ Ω, y ∈ <2, we define ˇ ζ(t, x, y)≡ ˆ ζ(t,y−`(˜x), x3) for y∈ Ym+ `(˜x), 0 elsewhere. Then we plug ζ(t, x)≡ X(Ym+j)(˜x)ˇζ(t, x, ˜x) for j ∈ Z
2 into (6.4). Since supp ζ
⊂ (0, T )× (Ym+ j)× [0, H], Z T 0 Z H 0 Z (Ym+j) φ∂tsζ + kI2$ λo(s)∇(p− Go) +∇M(s) ∇ζ = 0.
Since ˜x∈ (Ym+ j), `(˜x) = j. Changing variable y = x−`˜ (˜x) gives Z T 0 Z YH m φ∂tsζ + kˆ I2$−2 ∂y,x3M(s) + λo(s)∂y,x3(p− Go) ∂y,x3ζ = 0,ˆ (6.32)
for almost all ˜x∈ (Ym+ j), j ∈ Z2. Actually, by Definition 5.1, (6.32) holds for
˜
x∈ ˜Ω, i.e., (6.28). (6.29–6.31) can be proved in a similar way.
Remark 6.2 By Lemmas 5.1, 6.5, if we define S
≡ Υ−1(A−1(Π (A|Ω f))) and S≡ Υ−1(A−1(A∗)) if A∗<A(∞), 1 if A∗=A(∞), then 0≤ S, S≤ 1.
Lemma 6.7 There is a r∈ (1, 2) and a subsequence of {S, s, S
0, s0, φ, k, Pα, p α, α = w, o} such that, as → 0, XΩ fP α 2 *XYf(y)Pα(t, x) where Pα∈ L r(0, T ; W1,r(Ω)), P α= Pb,α in Γ2, XΩ f∇P α 2
*XYf(y)(∇Pα+ ∂yPα,1(t, x, y)) where Pα,1∈ L
r(ΩT; Lr per(Yf)), XΩ fS 0 2 * S0∈ L2(Ω; L2per(Yf)), S
→ S strongly in L2(ΩT) and pointwise,
XΩ fS 2→ X Yf(y)S(t, x) strongly, XΩ ms 0 2 → s0∈ L2(Ω; L2per(Ym)) strongly, p α* pα weakly in Lr(ΩT; W1,r(Ym)).
Proof: By Lemma 5.1 and Lemma 6.3, ΠPα is bounded in Lr(0, T ; W1,r(Ω)). So
a subsequence of ΠPα converges weakly to limit Pα ∈ Lr(0, T ; W1,r(Ω)). Since
ΠPα = Pb,αin Γ2, Pα= Pb,α in Γ2. Rest of proof are due to A2-4,6,10, Lemmas
6.1, 6.3, 6.5, and [3].
Lemma 6.8 s converges to s in L2(
QT
m) if 0 < $≤ 1.
Proof of this lemma is lengthy, and will be postponed untill the last five sections. Lemma 6.9 If $ = 1, then po− pw = υ(s), Po− Pw = Υ(S), and pα(t, x, y) =
Pα(t, x) for x∈ Ω, y ∈ ∂Ym, α = w, o. If $ < 1, then υ(s) = Po− Pw= Υ(S) and
pα(t, x, y) = Pα(t, x) for x∈ Ω, y ∈ Ym, α = w, o.
Proof: First we consider $ = 1 case. Note 0≤ S < 1, sl ≤ s < sr by Egoroff’s
theorem [25] and Lemmas 6.1-6.2, 6.7-6.8. Since p
o− pw= υ(s), we get po− pw=
υ(s) by Lemmas 6.7-6.8. Similarly, one can derive Po−Pw= Υ(S). By Lemmas 5.1,
6.3 and (2.8), (ΠPα)|Ω m−p
α∈ Lr(ΩT; W 1,r
0 (Ym)) for 1 < r < 2. So, a subsequence
of (ΠPα)|Ω m− p
α converges weakly to XYm(y)Pα(t, x)− pα ∈ L
r(ΩT; W1,r 0 (Ym))
by Lemma 6.7. So, pα(t, x, y) = Pα(t, x) for y ∈ ∂Ym. Results for $ < 1 case can
be obtained by similar argument as above, so we skip it.
Now we consider the limit model of (2.1–2.13) as → 0. Plug into (6.1) and (6.6) a test function η = ˆζ(t, x) + ˆη(t, x,x˜) where ˆζ ∈ C∞
0 (ΩT), ˆη ∈ C0∞(ΩT; Cper∞(Y )) to obtain 0 = Z Ω,Tf ΦS(∂tζ + ∂ˆ tη) + Kˆ Λw(S)∇(Pw − Gw)(∇ˆζ + ∂xη + ∂ˆ yη)ˆ + Z Ω,Tm φs(∂tζ + ∂ˆ tη) + kˆ I2$λw(s)∇(pw− Gw)(∇ˆζ + ∂xη + ∂ˆ yη)ˆ + Z Ω f ΦS0(ˆζ + ˆη)(0) + Z Ω m φs0(ˆζ + ˆη)(0). By A2 and Lemma 6.7, KΛ w(S) converges to K∗Λw(S) in Lr(ΩT), r <∞ strongly.
Passing to two-scale limit, we get, by A2-4, Lemmas 6.3-6.9, Theorem 2.28 of [2], Theorem 1.8 of [3], and [8, 10], Z QT f Φ∗S∂tζ + Kˆ ∗Λw(S)(∇Pw+ ∂yPw,1− ∇Gw)(∇ˆζ + ∂yη)ˆ =− Z QT m φs∂tζ +ˆ Fw∗∂x3ζˆ− Z Qf Φ∗S0ζ(0)ˆ − Z Qm φs0ζ(0),ˆ where Fw∗ ≡ (kλ w(s)∂x3(Pw− Gw) if 0 < $ < 1, kλw(s)∂x3(pw− Gw) if $ = 1, An L2 function if $ > 1. (6.33) Apply Green’s theorem in t variable to get
− Z
QT f
= Z QT m φ∂ts ˆζ− Fw∗∂x3ζ +ˆ Z Qf Φ∗(S(0)− S0)ˆζ(0) + Z Qm φ(s(0)− s0)ˆζ(0). So we have, in ΩT, (S(0)− S0) Z Yf Φ∗dy + Z Ym φ(s(0)− s0)dy = 0, (6.34)
and the choice of ˆη = 0 gives, in ΩT,
Z Yf Φ∗dy∂tS +∇ Z Yf K∗Λw(S)(∇Pw+ ∂yPw,1− ∇Gw) =− Z Ym (φ∂ts + ∂x3F ∗ w) dy. (6.35)
The choice of ˆζ = 0 gives, by A2-3 and Lemma 6.7,
∂2
yPw,1= 0 inQf,
(∂x˜Pw+ ∂yPw,1)· ~νy = 0 on ∂Ym, (6.36)
where ~νy is the unit vector outward normal to ∂Ym. Let ~ej be the unit vector in
jth direction. We denote by Ξ the tensor whose (i, j) component is ∂ϕj/∂yi, where
ϕj is a periodic solution in Y of the auxiliary problem
∆yϕj= 0 in Yf,
∂yϕj· ~νy=−~ej· ~νy on ∂Ym.
Pw,1 of (6.36) is given by the product Pw,1=Pjϕj(y)∂xjPw. So (6.35) becomes
Φ∂tS +∇ · (KΛw(S)∇(Pw− Gw)) = −1 |Ym| Z Ym (φ∂ts + ∂x3F ∗ w)dy, (6.37) where Φ≡ 1 |Ym| R YfΦ
∗dy and K is a diagonal matrix satisfying
K11= K22= K∗ |Ym| Z Yf (I + Ξ(y))dy, K33= |Yf|K ∗ |Ym| .
Proceeding as the proof of (6.37), we obtain, by (6.2), −Φ∂tS +∇ · (KΛo(S)∇(Po− Go)) = 1 |Ym| Z Ym (φ∂ts− ∂x3F ∗ o)dy, (6.38) where Fo∗≡ (kλ o(s)∂x3(Po− Go) if 0 < $ < 1, kλo(s)∂x3(po− Go) if $ = 1, An L2 function if $ > 1. (6.39) Matrix sources for 0 < $ < 1 case is clear from (6.33), (6.39), and Lemma 6.9. Next we consider the matrix source terms for $≥ 1 cases.
6.1. For $ = 1 case
By (6.30) of Lemma 6.6, we have, for any η∈ L2(ΩT; H1 0(Ym)), Z Q,T m φ∂tsη + Z Q,T m kλo(s)∂y,x3(po− Go)∂y,x3η = 0.
As → 0, by Lemmas 6.7-6.8, one obtains Z QT m φ∂ts η + Z QT m kλo(s)∂y,x3(po− Go)∂y,x3η = 0. (6.40)
In a similar way, we obtain, by (6.31), Z QT m φ∂ts η− Z QT m kλw(s)∂y,x3(pw− Gw)∂y,x3η = 0. (6.41)
By (6.37–6.41) and Lemmas 6.7-6.9, it is easy to show Theorem 4.1. 6.2. For $ > 1 case
By (6.30) of Lemma 6.6, we have, for any η∈ L2(ΩT; H1 0(Ym)), Z Q,T m φ∂tsη + Z Q,T m kI2$−2λo(s)∂y,x3(p o− Go)∂y,x3η = 0.
As → 0, by Lemmas 6.7-6.8, one obtains Z QT m φ∂ts η + Z QT m Fo∗∂x3η = 0.
So we get φ∂ts− ∂x3Fo∗ = 0. In a similar way, we obtain φ∂ts + ∂x3Fw∗ = 0.
Therefore we prove Theorem 4.3.
Rest of this work is to prove Lemma 6.8. 7. Convergence of s Remark 7.3 Define G ≡ υ−1( A−1(Π (A|Ω f))) if ΠA < A(∞), sr if ΠA=A(∞), G ≡ υ−1(A−1(A∗)) if A∗<A(∞), sr if A∗=A(∞).
See Lemma 6.5 forA∗. By Lemma 6.7, A1,3, Theorem 2.28 of [2], and [3, 8, 10],
it is easy to see that
kM(G)
kL2(0,T ;H1(Ω))are bounded independently of ,
M(G| Ω m)→ M(G) strongly in L 2( QT m), M(G| Ω m)− M(s )∈ L2(ΩT; H1 0(Ym)). (7.1)
Assume that si, pi, i = 1, 2 are two solutions of (6.28–6.29), and ζ, η are smooth
functions satisfying
ζ(T ) = 0, ζ|∂Ym×[0,H]= η|∂Ym×[0,H]= ∂x3ζ|x3∈{0,H}= ∂x3η|x3∈{0,H}= 0. (7.2)
Let x∈ Ω1∩ Ω2. By subtracting one solution from the other and integration by
parts, we obtain Z YH,T m (s1− s2) φ∂ tζ +F1∂y,x3(kI 2$−2
∂y,x3ζ)− F2∂y,x3ζ− F3∂y,x3η
+ Z YmH,T (p1− p2) ∂ y,x3kI 2$−2 (λ(s1)∂y,x3η + λo(s 1)∂ y,x3ζ) =F4+F5, (7.3) where F1≡ µ + M(s1)−M(s2) s1−s2 if s1 6= s2, 0 otherwise, (7.4) F2≡ ( k(λo(s1)−λo(s2))I2$−2∂y,x3(p2−G2o ) s1−s2 if s1 6= s2, 0 otherwise, (7.5) F3≡ ( P α k(λα(s1)−λα(s2))I2$−2∂y,x3(p2−G2α) s1−s2 if s1 6= s2, 0 otherwise, (7.6) F4≡ µ Z YmH,T (s1− s2)∂ y,x3(kI 2$−2 ∂y,x3ζ), (7.7) F5≡ Z YmH,T 2$−2∂y M(G1| Ω1m)− M(G 2| Ω2m) k∂yζ + Z YmH,T 2$−2∂y (Π1P 1| Ω1m − Π2P 2| Ω2m) kλ(s 1)∂ yη + kλo(s1)∂yζ − X α∈{w,o} Z YmH,T kλα(s1)∂x3(G 1 α − Gα2)∂x3η − Z YmH,T kλo(s1)∂x3(G 1 o − Go2)∂x3ζ− Z YH m (s1 0 − s02)φζ(0). (7.8) Define eU1 ≡ {ζ : ζ ∈ H1(YmH,T)∩ L∞(0, T ; H1(YmH)), ζ|∂Ym×[0,H] = ∂x3ζ|x3=0,H =
ζ(0) = 0}. We consider the following auxiliary problem for fixed µ:
Lemma 7.1 LetF2,F3∈ L∞(YmH,T) and 0 < d18<F1< d19<∞. For (f1, f2)∈
L2(YH,T
m )× L2(YmH,T), there is a unique (ζ, η)∈ eU1× L2(0, T ; H1(YmH)) such that
−φ∂tζ +F1∂y,x3(kI 2$−2
∂y,x3ζ)− F2∂y,x3ζ− F3∂y,x3η = f1, (7.9)
∂y,x3 kI 2$−2 (λ∂y,x3η + λo∂y,x3ζ) = f2. (7.10) Moreover, sup τ≤TkI $−1 ∂y,x3ζ(τ )kL2(YmH)+k|I $−1 ∂y,x3η| + d 1/2 18 |∂y,x3(kI 2$−2 ∂y,x3ζ)|kL2(YH,T m ) ≤ cd19,k(|F2| + |F3|)/F11/2kL∞(YmH,T) k |f1|/F11/2+|f2| kL2(YH,T m ). (7.11)
Proof: This is proved by following the argument of Lemma 5.1 [29]. Finally we give the proof of Lemma 6.8.
Proof: For x∈ Ω1∩Ω2, we take f
1=M(s1)−M(s2) in (7.9) and f2= p1−p2
in (7.10) to obtain solution (ζµ, ηµ) for each µ by (7.4–7.6), Remark 7.3, and Lemma
7.1. After substitution t→ T − t for the solution (ζµ, ηµ), we plug it into (7.3) to
obtain Z YmH,T (s1− s2)(M(s1)− M(s2)) + Z YmH,T |p1− p2|2=F 4+F5. (7.12)
By Lemmas 6.1, 7.1 and [13, 15, 26, 27], we see 1)F4 is bounded by cõ, where
c is a constant independent of µ, 1, 2; and 2) For fixed µ, F5 converges to 0 as
1, 2 tend to 0. So it is not difficult to show that M(s2) is a Cauchy sequence in
L2(QT
m), which implies s2 is a Cauchy sequence in L2(QTm) as well.
Acknowledgement
This research is supported by the grant number NSC 93-2115-M-009-09 from the research program of National Science Council of Taiwan.
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