March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
Applicable Analysis
Vol. 00, No. 00, March 2008, 1{22
RESEARCH ARTICLE
Laplace transform approach to the rigorous upscaling of the in¯nite adsorption rate reactive °ow under dominant Peclet number through a pore z Catherine Choquet a and Andro Mikeli¶c b ¤ aUniversit¶e P. C¶ezanne, LATP UMR 6632,
Facult¶e des Sciences et Techniques de Saint-J¶er^ome,
13397 Marseille Cedex 20, FRANCE bUniversit¶e de Lyon, Lyon, F-69003, FRANCE;
Universit¶e Lyon 1, Institut Camille Jordan, UFR Math¶ematiques,
Site de Gerland, B^at. A, 50, avenue Tony Garnier
69367 Lyon Cedex 07, FRANCE
(submitted on March 31, 2008)
In this paper we undertake a rigorous derivation of the upscaled model for reactive °ow through a narrow and long two-dimensional pore. The transported and di®used solute par- ticles undergo the in¯nite adsorption rate reactions at the lateral tube boundary. At the inlet boundary we suppose Danckwerts' boundary conditions. The transport and reaction pa- rameters are such that we have dominant Peclet number. Our analysis uses the anisotropic singular perturbation technique, the small characteristic parameter " being the ratio between the thickness and the longitudinal observation length. Our goal is to obtain error estimates for the approximation of the physical solution by the upscaled one. They are presented in the energy norm. They give the approximation error as a power of " and guarantee the validity of the upscaled model. We use the Laplace transform in time to get better estimates than in our previous article [20] and to undertake the study of important Danckwerts' boundary conditions. Keywords: Taylor's dispersion; large Peclet number; singular perturbation; Laplace's transform; adsorption chemical reaction; Danckwerts' boundary conditions.
AMS Subject Classi¯cation: 35B25; 92E20; 76F25; 44A10
1. Introduction
We consider the transport of a reactive solute by molecular di®usion and convec- tion in a semi-in¯nite two-dimensional channel. We suppose that the characteristic numbers are large (Peclet's number, Damkohler's number, : : :) and study the dis- persion e®ects.
Dispersion expresses the °uctuation of a quantity with respect to its mean behav- ior. It is induced by motion of a transported solute in a °uid (molecular di®usion, z Research of the authors was partially supported by the GDR MOMAS (Mod¶elisation Math¶ematique et
Simulations num¶eriques li¶ees aux problµemes de gestion des d¶echets nucl¶eaires) (PACEN/CNRS, ANDRA,
BRGM, CEA, EDF, IRSN) as a part of the project "Modµeles de dispersion e±cace pour des problµemes de
Chimie-Transport: Changement d'¶echelle dans la mod¶elisation du transport r¶eactif en milieux poreux, en pr¶esence des nombres caract¶eristiques dominants. " .
¤Corresponding author. Email: Andro.Mikelic@univ-lyon1.fr
ISSN: 0003-6811 print/ISSN 1563-504X online
°c 2008 Taylor & Francis
DOI: 10.1080/0003681YYxxxxxxxx http://www.informaworld.comMarch 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
2 Catherine Choquet and Andro Mikeli¶c convection and their interaction) or by the chemical reactions which that solute undergoes. At the pore level we have a) the molecular di®usion, expressed by Fick's law, b) the convective dispersion, which corresponds to the spreading of particles by the velocity ¯eld and c) the creation (or destruction) of the solute particles induced by chemical reactions.
Next, for the solute particles subject to convection and molecular di®usion, a complicated interaction of di®usion and convection is observed. The overall behav- ior heavily depends on the ratios of characteristic times.
In the literature usually we ¯nd three distinct regimes: a) di®usion-dominated mixing, b) Taylor dispersion-mediated mixing and c) chaotic advection.
Our goal is the study of reactive °ows through slit channels in the regime of Tay- lor dispersion-mediated mixing, using anisotropic singular perturbations. Contrary to the regime a), where we could quote papers [9], [10],[11], [14], [15] and [18] and references therein, and despite a vaste literature on the subject, with over 2000 citations here to date, mathematical results on the subject are rare. The derivation of the model without chemical reactions is in the original paper [24] by Taylor. It was formally justi¯ed, using the method of moments in [3]. We also mention the mathematically rigorous paper [6] and the papers with formal asymptotic expan- sion [17], [22] and [23]. Nevertheless, they address the mechanical dispersion in the absence of chemical reactions. In presence of chemical reactions we mention the following papers:
(i) Flow with chemistry, as described by equation (2), is considered by Paine,
Carbonell and Whitaker in [21]. They noted that the equation for the di®erence between the physical and averaged concentrations is not closed, since it contains a dispersive source term
@
@x
< q x c >. Then they multiplied the equation for c by q x and got an equation for < q x c >. Nevertheless, a dispersive transport term @
@x
< q
2
x c > is present and clearly the procedure enters the same di±culty as the method of moments: there is an in¯nite system of equations. Paine et al used the "single-point" closure schemes of turbulence modeling by Launder to obtain a closed model for the averaged concentration. We note that their e®ective equations contain non-local terms depending of the solution and in fact the e®ective coe±cients are not explicitly given.
(ii) The center manifold approach of Mercer and Roberts (see the article [17] and the subsequent article [22] by Rosencrans) allowed to calculate approxima- tions at any order for the original Taylor's model with no chemistry. Even if the error estimate was not obtained, this approach gives a very plausible argument for the validity of the e®ective model. This approach was applied to reactive
°ows in the article [4] by Balakotaiah and Chang. A number of e®ective mod- els for di®erent Damkohler numbers were obtained. Some generalizations to reactive °ows through porous media are in [16] and the preliminary results on their mathematical justi¯cation are in [1] .
(iii) Another approach consisting of the Liapounov-Schmidt reduction coupled with a perturbation argument is developed in the articles [5] and [7]. It allows developing multi-mode hyperbolic upscaled models.
(iv) For the case of reactive °ows with an irreversible, ¯rst order, heterogeneous chemical reaction with equilibrium between the liquid and the concentrations of adsorbed solutes, we refer to [19], where the problem is rigorously solved. It covers also the classical Taylor's dispersion, which corresponds to absence of the chemistry. The case of general chemical reactions was considered from theMarch 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
Laplace transform approach to the rigorous upscaling of the reactive °ow 3 point of view of formal expansions in [12] and the results were justi¯ed through numerical simulations.
In this article we continue our research from the article [20], where a slit °ow under dominant Peclet and Damkohler numbers was considered in the case when the adsorption rate constant is in¯nitely large.
Let us write the precise setting of the problem: We consider the transport of a reactive solute by di®usion and convection by Poiseuille's velocity in a semi-in¯nite
2D channel. The solute particles do not react among themselves. Instead they undergo an adsorption process at the lateral boundary. We consider the following model for the solute concentration c
¤
:
a) transport through channel ­
¤ = f(x
¤
; y
¤
) : 0 < x
¤ < +1; jy
¤
j < Hg
@c
¤
@t
¤
+ q(y
¤
)
@c
¤
@x
¤
¡ D¤ @
2
c
¤
@(x
¤
)
2
¡ D¤ @
2
c
¤
@(y
¤
)
2
= 0 in ­
¤
; (1) where q(z) = Q¤
(1¡(z=H)
2
) and where Q¤
(velocity) and D¤
(molecular di®usion) are positive constants.
b) reaction at channel wall ¡
¤ = f(x
¤
; y
¤
) : 0 < x
¤ < +1; jy
¤
j = Hg
¡D¤
@y
¤ c
¤
= Ke
@c
¤
@t
¤ on ¡
¤
; (2) where Ke is the linear adsorption equilibrium constant.
c) in¯ltration with a pulse of water containing a solute of concentration c
¤
f
,
followed by solute-free water is stated using the Danckwerts boundary condition from [13]
¡D¤
@x¤ c
¤
+ q(y
¤
)c
¤
=
½
q(y
¤
)c
¤
f
; for 0 < t
¤ < t
¤
0
0; for t > t
¤
0
:
(3)
The natural way of analyzing this problem is to introduce appropriate scales. This requires characteristic or reference values for the parameters in variables involved.
The obvious transversal length scale is H. For all other quantities we use reference values denoted by the subscript R. Setting c = c ¤ c^ ; x = x ¤
LR
; y = y ¤
H
; t = t ¤
TR
; Q =
Q¤
QR
; D =
D¤
DR
; (4) where LR is the " observation distance ", we obtain the dimensionless equations
@c
@t
+
QRTR
LR
Q(1 ¡ y
2
)
@c
@x
¡
DRTR
L
2
R
D
@
2 c @x
2
¡
DRTR
H2 D
@
2 c @y
2
= 0 in ­ (5) and ¡
DDRTR
HKe
@c
@y
=
@c
@t
on ¡; (6) where ­ = (0; +1) £ (¡1; 1) and ¡ = (0; +1) £ f¡1; 1g: (7)March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
4 Catherine Choquet and Andro Mikeli¶c
The equations involve the time scales:
TL = characteristic longitudinal time scale =
LR
QR
,
TT = characteristic transversal time scale =
H2
DR
,
TC = super¯cial chemical reaction time scale =
LRKe
HQR
,
and the non-dimensional number Pe =
LRQR
DR
(Peclet number). In this paper we
¯x the reference time by setting TR = Tc = TL and K = Ke=H = TC=TL = O(1).
We are going to investigate the behavior of the two-dimensional system (5)-(6) with respect to the small parameter " =
H
LR
. Speci¯cally, as in [19], we will derive expressions for the e®ective values of the dispersion coe±cient and velocity, and an e®ective 1-D convection-di®usion equation for small values of ". To carry out the analysis we need to compare the dimensionless numbers with respect to ". For this purpose we set Pe = "
¡®
. Introducing the dimensionless numbers in equations
(5)-(6) yields the problem:
@c
"
@t
+ Q(1 ¡ y
2
)
@c
"
@x
= D"
® @
2
c
"
@x
2
+ D"
®¡2 @
2
c
"
@y
2
in ­
+
£ (0; T); (8)
¡D"
®¡2 @c
"
@y
= ¡D
1
"
2Pe
@c
"
@y
= K
@c
"
@t
on ¡
+
£ (0; T); (9) c "
(x; y; 0) = 0 for (x; y) 2 ­
+
; (10)
¡D"
®
@xc
"
+ Q(1 ¡ y
2
)c
"
=
½
Q(1 ¡ y
2
)cf ; for 0 < t < t0
0; for t > t0: at fx = 0g; (11)
@c
"
@y
(x; 0; t) = 0; for (x; t) 2 (0; +1) £ (0; T): (12)
The latter condition results from the y¡symmetry of the solution. Further
­
+
= (0; +1) £ (0; 1); ¡
+
= (0; +1) £ f1g; and T is an arbitrary chosen positive number.
We study the behavior of this problem as " & 0, while keeping the coe±cients
Q; D and K all O(1).
Continuing the work from [20], in this paper we prove that the correct upscaling of the problem (8)-(12) gives the following 1D parabolic problem :
(EF F)
8
<
:
@tc +
2Q
3(1 + K)
@xc = D"
~ ® @xxc
1 + K in (0; +1) £ (0; T);
¡D"
®
@xcjx=0 +
2Q
3
(cjx=0 ¡ cfÂt ® ¸ 1, since the case ® 2 (0; 1) does not pose di±culties. Contrary to the particular Dirichlet boundary conditions, which were chosen in [20] and which allow an explicit solution having the form of moving Gaussian, here we consider the general Danckwerts boundary condition. After [13], it is very important in application and more realistic than Dirichlet's boundary condition. In di®erence to [20], we are not able to give an explicit solution and investigate its properties when di®usion coe±cient becomes small. Consequently, we use the vector-valued
Laplace's transform in time. It permits to calculate the Laplace transform of the solution and to get precisely its limit behavior when " tends to zero. The estimates depends on the compatibility between the boundary and initial data and on the direction of the °ow.
Then in Section x4 we give a justi¯cation of a lower order approximation, using the energy argument. In fact such approximation does not use Taylor's dispersion formula and, for ® ¸ 2=3 gives an error of the same order as the solution to the linear transport equation.
In Section x5, we use a formal derivation of the upscaled problem (EFF), obtained in [20] and [12], to set up the correction. We prove that the e®ective concentration satisfying the corresponding 1D parabolic problem, with Taylor's di®usion coe±- cient and the reactive correction, is an approximation in C(L
2
) for the physical concentration. We give the corresponding error estimate. We note that we were able to obtain a better approximation than in [20], without using the boundary layer correction for the Danckwerts boundary condition. Furthermore, using the elementary parabolic theory one concludes that the problem (8)-(12) has a unique bounded variational solution c
"
, with square integrable gradient in x and y. Func- tion c
"
belongs to C1 for x > 0 and it stabilizes to 0 exponentially fast when x ! 1.
Let us announce our main result.
Theorem 1.1 : Let ® ¸ 1 and let cf 2 C
1
0
(0; T). Let c be given by (EFF). Then we have kc "
¡ ckC([0;T];L2
(­+)) · C"
2¡®
; (14) k@yc " kC([0;T];L2 (­+)) · C"
3¡3®=2
; (15) k@x ¡ c "
¡ c
¢
kC([0;T];L2
(­+)) · C"
2¡3®=2
: (16)
For ill-prepared data see Corollary 5.4. Note that in estimate (15) c has disap- peared since it is only x and t dependent. This estimate is superior to estimate (16)March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
6 Catherine Choquet and Andro Mikeli¶c because of the large O("
®¡2
) transversal di®usivity. Our result could be restated in dimensional form:
Theorem 1.2 : Let us suppose that LR > maxfDR=QR; QRH2
=DR; Hg. Then the upscaled dimensional approximation for (1) reads
(1+K)
@c
¤;ef f
@t
¤
+
2
3
Q
¤ @c
¤;ef f
@x
¤
= D¤
³
1+(
8
945
+
4
135
K(7K + 2)
(1 + K)
2
)Pe
2
T
´
@
2
c
¤;ef f
@(x
¤
)
2
; (17) where PeT =
Q¤H
D¤ is the transversal Peclet number and K = Ke=H is the transversal Damkohler number.
We note that the powers of " obtained in Theorem 1.1, are superior to the corresponding results in [20]. Furthermore, we obtain better functional spaces in time. 2. Vector valued Laplace transform and applications to PDEs
We start this section by recalling the basic facts about applications of Laplace's transform to linear parabolic equations. The Laplace's transform method is widely used in solving engineering problems. In applications it is usually called the oper- ational calculus or Heaviside's method.
For locally integrable function f 2 L
1
loc
(R) such that f(t) = 0 for t < 0 and jf(t)j · Ae at as t ! +1, the Laplace transform of f, denoted ^ f, is de¯ned as
^
f(¿ ) =
Z +1
0
f(t)e
¡¿ t dt; ¿ = » + i ´ 2 C: (18)
It is closely linked with Fourier's transform in R. We note that
^
f(¿ ) = F
¡
f(t)e
¡»t
¢
(¡´); » > a; (19) where the Fourier's transform of a function g 2 L
1
(R) is given by
F
¡ g(t) ¢
(!) =
Z
R g(t)e i!t dt; ! 2 R:
It is well-known (see e.g. [25] or [8]) that
^
f de¯ned by (18) is analytic in the half-plane fRe(¿ ) = » > ag and it tends to zero as Re(¿ ) ! +1.
For real applications, Laplace's transform of functions is not well-adapted and it is natural to use Laplace's transform of distributions. It is de¯ned for distributions with support on [a; +1) i.e. for f 2 D0
+(a), where D0
+(a) = ff 2 D0
(R); supp f ½
[a; +1)g. If S
0
(R) denotes the space of distributions of slow growth, then we in- troduce S
0
+(R) by
S
0
+(R) = D
0
+(0) \ S
0
(R) (20) and we use the formula (19) to de¯ne Laplace's transform for f 2 D0
+(a) such that fe
¡»t
2 S
0
+(R) for all » > a. This approach permits the rigorous operational calculus. For details we refer to classical textbooks as [25] by Vladimirov.
Laplace's transform is applier to linear ODEs and PDEs, the transform problem is solved and its solution ^ f is calculated. Then the important question is how toMarch 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
Laplace transform approach to the rigorous upscaling of the reactive °ow 7 inverse the Laplace's transform. First we need a suitable space for image functions.
It is the algebra H(a) de¯ned by
H(a) = f g 2 Hol
¡
f¿ 2 C; Re(¿ ) > ag
¢
satisfying the growth condition : for any ¾o > a there are real numbers C(¾o) > 0 and m = m(¾o) ¸ 0 such that jg(¿ )j · C(¾o) (1 + j¿ j m ); Re(¿ ) > ¾og:
(21)
For elements of H(a) we have the following classical result.
Theorem 2.1 : ([25] pp. 162-165) Let
^
f 2 H(a) be absolutely integrable with respect to ´ on R for certain » > a. Then the following formula holds true. f(t) =
1
2¼i
Z
»+i1
»¡i1
^ f(z)e zt dz: (22)
These classical results are not su±cient for our purposes. We need results for re°exive Sobolev space X valued Laplace's transform. Furthermore we need an inversion theorem in L p ((0; +1); X). The corresponding theory could be found in
Arendt [2] and we give only results directly linked to our needs. For a re°exive
Banach space X we set
C
1 w (R+; X) = fr 2 C
1
((0; +1); X); krkw = sup n2N sup
¸>0
¸ n+1 n! d n d¸n r(¸)
X
< +1g:
(23)
Then we have the following result.
Theorem 2.2 : ([2], Chapter 2) Let X be a re°exive Banach space. Then the
(real) Laplace's transform f 7! ^ f is an isometric isomorphism between L1(R+; X) and C1 w (R+; X).
In many instances the growth condition in (23) is too di±cult to check. It is easier to use the complex Laplace's transform. We have the su±cient condition by
PrÄuss:
Theorem 2.3 : ([2], Chapter 2) Let q : fRe(¿ ) > 0g ! X be analytic. If there exists a real number M > 0 such that k¸q(¸)kX · M and k¸
2
q
0
(¸)kX · M for
Re(¸) > 0, then there exists a bounded function f 2 C(0; +1; X) such that q(¸) =
Z +1
0
f(t)e
¡¸t
dt:
In particular q 2 C1 w (R+; X).
Even Theorem 2.3 represents a complicated criterium and, following ideas from
[8], we will use a direct approach based on the link to Fourier's transform. We apply this approach in the study of the upscaled equations and then in the error estimates.
We derive estimates for the solutions of the Laplace transformed problem. We use that if image is in L
1
, the original is in C.
3. Study of the upscaled di®usion-convection equation on the half-line
In Sections x4 and x5, we will prove that the original problem can be approxi- mated by some upscaled 1-D di®usion-convection equation. The present section is thus devoted to the study of this type of equation in the half-line. The results ofMarch 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
8 Catherine Choquet and Andro Mikeli¶c
Subsection x3.1 (respectively Subsection x3.2) are used in Section x4 (respectively
Section x5).
3.1. Danckwerts boundary condition
For Q;
¹ D¹ and ° > 0, we consider the problem
8
<
:
@tu + Q@¹ xu = °D@¹ xxu in (0; +1) £ (0; T);
@xu 2 L
2
((0; +1) £ (0; T)); u(x; 0) = 0 in (0; +1); ¡°D@¹ xu + Qu¹ = Qg
¹
at x = 0:
(24)
Let ­l = R+ £ fRe(¿ ) > 0g. After applying the Laplace transform with respect to the time variable we get the following equation for the Laplace transform ^u(x; ¿ ) of u:
8
<
:
¿u^ + Q@¹ xu^ = °D@¹ xxu^ in ­l
;
@xu^ 2 L
2
(R+); Re(¿ ) > 0;
¡°D@¹
xu^ + Q¹u^ = Q¹ g^ at x = 0;
(25)
where ¿ = » + i ´ 2 C, » > 0. Problem (25) allows the following explicit solution: u^(x; ¿ ) =
2Q¹
Q¹ + p Q¹2 + 4°¿D¹ e ¡2¿x
Q¹ + p Q¹2 + 4°¿D¹ g^(¿ ): (26)
This explicit formula allows us to ¯nd the exact behavior of u with respect to °.
For the sake of simplicity, we write u^(x; ¿ ) =
2Q¹
l(¿ ) e ¡
2¿x
l(¿ ) g^(¿ ) where 8
<
: l(¿ ) = Q¹ + q Q¹2 + 4°¿D;
¹ Re(¿ ) = » > 0;
R(¿ ) =
¡
Q¹4
+ (4°D¹
)
2 j¿ j
2
+ 8°D»
¹ Q¹2
¢
1=4
;
Á 2
£
¡
¼
2
;
¼
2
¤
; cos Á =
Q¹2 + 4°D»
¹
R2
(¿ )
; sin Á =
4°D´
¹
R2
(¿ )
:
(27)
Then cos(Á=2) > 0 and
(
l(¿ ) = Q¹ + R(¿ ) cos(Á=2) + iR(¿ ) sin(Á=2); jl(¿ )j
2
= Q¹2
+ R
2
(¿ ) + p 2Q¹ q Q¹2 + R2
(¿ ) + 4°D»:
¹
(28)
Consequently,
R
2
(¿ ) + Q¹2
· jl(¿ )j
2
· 3(R
2
(¿ ) + Q¹2
): (29)March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
Laplace transform approach to the rigorous upscaling of the reactive °ow 9
Furthermore, we note that for » > 0
Re
³
¿
l(¿ )
´
= Re
³
¿ l(¿ ) jl(¿ )j
2
´
=
»Q¹ + (»= p 2) p R2
(¿ ) + Q¹2 + 4°D»
¹ + p 2°D´
¹ 2
=
p
R2
(¿ ) + Q¹2 + 4°D»
¹
jl(¿ )j
2
¸
C1»
(R(¿ )
2 + Q¹2
)
1=2
+
C2´
2
(R(¿ )
2 + Q¹2
)
3=2
¸
C1»
(R(¿ )
2 + Q¹2
)
1=2
> 0: (30)
Now we compute some explicit estimates for ^u. First, by the maximum principle and for 0 · g(t) · Cg, we have
0 · u(x; t) · Cg: (31)
If, in addition, @tg ¸ 0, then
0 · u(x; t) · g(t): (32)
Next we estimate the di®erence between ^g expf¡
¿x
Q¹ g and ^u. We have the following approximation result.
Lemma 3.1 : Function u^ satis¯es the estimate g^ expf¡
¿x
Q¹ g ¡ u^
Lp
((0;+1))
· °
Cjg^(¿ )j
»
1=p j¿ j
Q¹ + °D¹ j¿ j
; 81 · p < +1: (33)
Proof : It is enough to make the calculations with ^g = 1. Let q(x; ¿ ) = ^u(x; ¿ ) ¡ e
¡¿x=Q¹
=
2Q¹
l(¿ ) e ¡2¿x=l(¿ )
¡ e
¡¿x=Q¹
:
Then we have with (25)
8
<
:
¿ q(x; ¿ ) + Q@¹ xq(x; ¿ ) = °D¹
8¿
2Q¹ l(¿ )
3
e
¡2¿x=l(¿ )
; x 2 R+; q(0; ¿ ) = ¡
4¿D°¹
l(¿ )
2
:
(34)
We look for the solution of (34) in the form q(x; ¿ ) = qH(x; ¿ ) + qP (x; ¿ ); (35) with qH(x; ¿ ) = ¡
4¿D°¹
l(¿ )
¡
l(¿ ) ¡ 2Q¹
¢e
¡¿x=Q¹
; (36) qP (x; ¿ ) =
8¿D¹Q°¹
l(¿ )
2
¡ l(¿ ) ¡ 2Q¹
¢e
¡2¿x=l(¿)
: (37)March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
10 Catherine Choquet and Andro Mikeli¶c
Then we compute the L p -norms, 1 · p < +1, kqH(¢; ¿ )kLp
(R+) = °
4D¹
j¿ j jl(¿ )jjl(¿ ) ¡ 2Q¹ j ³
Q¹
p»
´1=p
; (38) kqP (¢; ¿ )kLp
(R+) · C° j¿ j l(¿ )
2
¡ l(¿ ) ¡ 2Q¹
¢
1
¡
Re(¿ =l(¿ ))
¢
1=p
: (39)
Since jl(¿ )j ¸ p 2=3(R(¿ )+Q¹
), jl(¿ )¡2Q¹ j ¸ (1¡ p 2=2) p 2=3(R(¿ )+Q¹
), R(¿ )+Q¹ ¸
CQ¹ + p °D¹ j¿ j and Re(¿ =l(¿ )) ¸ C»=(Q¹ + p °D¹ j¿ j), we infer from (35)-(39): kq(¢; ¿ )kLp
(R+) ·
C°
»
1=p
³ j¿ j
Q¹2 + °D¹ j¿ j
+
j¿ j
(Q¹ + p °D¹ j¿ j)
3¡1=p
´
: (40)
The lemma is proved. ¤
The following result follows as the consequence of Lemma 3.1.
Corollary 3.2 : Let g 2 C1
0
(R+), then u ¡ g(t ¡ x Q¹
)
C(R+;Lp
(R+))
· C°; 1 < p < +1:
If g 2 W1;1(R+) is with compact support in [0; +1), but g(0) = 0 6 , then u ¡ g(t ¡ x Q¹
)
Lr
(R+;Lp
(R+))
· C°
1¡±
; 1 < p; r < +1; 0 < ± < 1; r(1 ¡ ±) < 1:
Remark 1 : Presence of the contact discontinuity due to g(0) = 0 diminishes 6 precision. Furthermore, the case of g = 1 on (0; T) is covered by Corollary 3.2, since it could be extended to a Lipschitz function on R+, with compact support in [0; +1). Hence, it is easy to compare the result of Corollary 3.2 with the cor- responding results from [19] and [20] and see that we have now a more precise estimate. We now aim to give explicit estimates with respect to ¿ for ^u in Hp
((0; +1)).
Since jl(¿ )j ¸ p 2=3(R(¿ ) + Q¹
), Re(¿ =l(¿ )) ¸ C»=(Q¹ + p °D¹ j¿ j) and R(¿ ) + Q¹ ¸
CQ¹ + p °D¹ j¿ j, we begin by noting that ^u(x; ¿ ) = (2Q=l
¹
(¿ ))exp(¡2¿x=l(¿ ))^g(¿ ) satis¯es ku^(¢; ¿ )kLp
(R+) ·
Cjg^(¿ )j
»
1=p
1
Q¹ + p °D¹ j¿ j
(Q¹ + q °D¹ j¿ j)
1=p
: (41)
We also compute
@xu^(x; ¿ ) = ¡
4Q¿
¹ l(¿ )
2
e
¡2¿x=l(¿ ) g^(¿ );
@
k xu^(x; ¿ ) = 2Q¹
³
¡2¿ l(¿ )
´k
u^(x; ¿ ); k 2 N:
The following are then straightforward.March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
Laplace transform approach to the rigorous upscaling of the reactive °ow 11
Lemma 3.3 : Function u^ satis¯es the estimates ku^(¢; ¿ )kL2
((0;+1)) ·
Cjg^(¿ )j p »(1 + °
1=4
j¿ j
1=4
)
; (42) j@xu^(¢; ¿ ) jx=0 j ·
Cj¿g^(¿ )j
1 + °j¿ j
; (43) k@ k xu^(¢; ¿ )kL2
((0;+1)) ·
Cjg^(¿ )j p »(1 + °
1=4
j¿ j
1=4
)
³
j¿ j
2
1 + °j¿ j
´k=2
k ¸ 1; (44) with ¿ = » + i ´, » > 0.
3.2. Perturbed Danckwerts boundary condition
For Q;
¹ D¹
, ° > 0 and ± 2 R such that D¹ ¡ j±j ¸ C0 > 0 , we consider the problem
8
<
:
@tu + Q@¹ xu = °D@¹ xxu in (0; +1);
@xu 2 L
2
((0; +1)); u(x; 0) = 0 in (0; +1); ¡°(D¹ + ±)@xu + Qu¹ = Qg
¹
at x = 0:
(45)
We have the corresponding equation for the Laplace transform ^u(x; ¿ ) of u:
8
<
:
¿u^ + Q@¹ xu^ = °D@¹ xxu^ in ­l
;
@xu^ 2 L
2
(R+); Re(¿ ) > 0;
¡°(D¹ + ±)@xu^ + Q¹u^ = Q¹ g^ at x = 0;
(46)
where ¿ = » + i ´ 2 C. Problem (46) allows the following explicit solution: u^(x; ¿ ) =
2D¹Q¹
( p Q¹2 + 4°¿D¹ + Q¹
)(D¹ + ±) ¡ 2Q±
¹
e
Q¹ ¡ p Q¹2 + 4°¿D¹
2°D¹
x g^(¿ )
=
2D¹Q¹ l(¿ )(D¹ + ±) ¡ 2Q±
¹
e
¡2¿x
l(¿ ) g^(¿ ): (47)
The explicit formula allows us to ¯nd the exact behavior of u with respect to °.
We emphasize that we can prove similar results as in the previous subsection in spite of the ± perturbation. We follow the lines of Subsection 3.1. We bear in mind the auxiliary computations (27)-(30). We also note that l(¿ )(D¹ + ±) ¡ 2Q±
¹
2
= Q¹2
(D¹ ¡ ±)
2
+ R(¿ )
2
(D¹ + ±)
2
+ p 2Q¹
(D¹ 2
¡ ±
2
) q R(¿ )
2 + Q¹2 + 4°D»
¹
¸ (D¹ ¡ j±j)
2
(Q¹2
+ R(¿ )
2
): (48)
Next we estimate the di®erence between ^g expf¡
¿x
Q¹ g and ^u.
Lemma 3.4 : Let ± 2 R be such that D¹ ¡ j±j ¸ C0 > 0. Then u^ satis¯es theMarch 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
12 Catherine Choquet and Andro Mikeli¶c estimate kg^ expf¡
¿x
Q¹ g ¡ u^kLp
((0;+1))) · °
Cjg^(¿ )j
»
1=p j¿ j
Q¹ + °D¹ j¿ j
; 81 · p < +1: (49)
Proof : It is enough to make the calculations with ^g = 1. Let q(x; ¿ ) = ^u(x; ¿ ) ¡ e
¡¿x=Q¹
=
2D¹Q¹
l(¿ )(D¹ + ±) ¡ 2Q±
¹
e
¡2¿x=l(¿ )
¡ e
¡¿x=Q¹
:
In view of (46), function q is solution of
8
<
:
¿ q(x; ¿ ) + Q@¹ xq(x; ¿ ) = °D¹
8¿
2Q¹D¹ l(¿ )
2
(l(¿ )(D¹ + ±) ¡ 2Q±
¹
) e ¡2¿x=l(¿ )
; x 2 R+; q(0; ¿ ) = ¡
4¿D¹
(D¹ + ±)° l(¿ )(l(¿ )(D¹ + ±) ¡ 2Q±
¹
)
:
(50)
We look for the solution of (50) in the form q(x; ¿ ) = qH(x; ¿ ) + qP (x; ¿ ); (51) with qH(x; ¿ ) = ¡
4¿D°¹
l(¿ )
¡
l(¿ ) ¡ 2Q¹
¢e
¡¿x=Q¹
; (52) qP (x; ¿ ) =
8¿D¹ 2Q°¹ l(¿ )
¡
l(¿ ) ¡ 2Q¹
¢¡
l(¿ )(D¹ + ±) ¡ 2Q±
¹
¢e
¡2¿x=l(¿)
: (53)
Then we compute the L p -norms, 1 · p < +1, kqH(¢; ¿ )kLp
(R+) =
4D°¹
j¿ j jl(¿ )jjl(¿ ) ¡ 2Q¹ j ³
Q¹
p»
´1=p
; (54) kqP (¢; ¿ )kLp
(R+) ·
C°j¿ j l(¿ )
¡
l(¿ ) ¡ 2Q¹
¢¡
l(¿ )(D¹ + ±) ¡ 2Q±
¹
¢
1
¡
Re(¿ =l(¿ ))
¢
1=p
:(55)
Using jl(¿ )j ¸ p 2=3(R(¿ ) + Q¹
), jl(¿ ) ¡ 2Q¹ j ¸ (1 ¡ p 2=2) p 2=3(R(¿ ) + Q¹
), R(¿ ) +
Q¹ ¸ CQ¹ + p °D¹ j¿ j, Re(¿ =l(¿ )) ¸ C»=(Q¹ + p °D¹ j¿ j) and (48), we infer from
(51)-(55):
kq(¢; ¿ )kLp
(R+) ·
C°
»
1=p
³ j¿ j
Q¹ + °D¹ j¿ j
+
j¿ j
(D¹ ¡ j±j)(Q¹ + p °D¹ j¿ j)
3¡1=p
´
: (56)
Estimate (49) follows. ¤
Corollary 3.5 : Let ± 2 R be such that D¹ ¡ j±j ¸ C0 > 0: Let g 2 C1
0
(R+). Then we have u ¡ g(t ¡ x Q¹
)
C(R+;Lp
(R+))
· C°; 1 < p < +1:March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
Laplace transform approach to the rigorous upscaling of the reactive °ow 13
If g 2 W1;1(R+) is with compact support in [0; +1), but g(0) = 0 6 , then u ¡ g(t ¡ x Q¹
)
Lr
(R+;Lp
(R+))
· C°
1¡±
; 1 < p; r < +1; 0 < ± < 1; r(1 ¡ ±) < 1:
We now aim to give explicit estimates with respect to ¿ for ^u in Hp
((0; +1)).
Following the lines of Subsection 3.1 and using (48) we get directly the following set of estimates.
Lemma 3.6 : Function u^ satis¯es the estimates ku^(¢; ¿ )kL2
((0;+1)) ·
Cjg^(¿ )j p »(1 + °
1=4
j¿ j
1=4
)
; (57) j@xu^(¢; ¿ ) jx=0 j ·
Cj¿g^(¿ )j
1 + °j¿ j
; (58) k@ k xu^(¢; ¿ )kL2
((0;+1)) ·
Cjg^(¿ )j p »(1 + °
1=4
j¿ j
1=4
)
³
j¿ j
2
1 + °j¿ j
´k=2
k ¸ 1; (59) with ¿ = » + i ´, » > 0.
4. A simple L2 error estimate
The simplest way to average the problem (8)-(12) is to take the mean value with respect to y. Supposing that the mean of the product is the product of the means, which is in general wrong, we get the following problem for the " averaged " concentration c
L;ef f
0
(x; ¿ ):
8
<
:
(1 + K)¿ c
L;ef f
0 +
2Q
3
@c
L;ef f
0
@x
= "
®D
@
2
c
L;ef f
0
@x
2
in (0; +1);
@xc
L;ef f
0 2 L
2
((0; +1));
¡D"
®
@xc
L;ef f
0 + 2Qc
L;ef f
0
=3 = 2Qc^f =3; for x = 0:
(60)
This is the problem (24) with ^g = ^cf , Q¹ =
2
3
Q
1 + K and D¹ =
D
1 + K
. The small parameter ° is equal to "
®
. We will call this problem the "simple closure ap- proximation". Let the operator L
"
be given by
L
"
³ = ¿ ³ + Q(1 ¡ y
2
)
@³
@x
¡ D"
®
µ
@
2
³
@x
2
+ "
¡2 @
2
³
@y
2
¶
: (61)
The non-dimensional physical concentration c
"
satis¯es (8)-(12). Its Laplace trans- form ^c
"
is thus solution of
L
" c^ "
= 0 in (0; +1) £ (0; 1) (62)
¡D"
®
@xc^
"
+ Q(1 ¡ y
2
)^c
"
= Q(1 ¡ y
2
)^cf ; for (x; y) 2 f0g £ (0; 1); (63)
¡D"
®¡2
@yc^
"
(x; 1; ¿ ) = K¿ c^
"
(x; 1; ¿ ) in (0; +1): (64)March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
14 Catherine Choquet and Andro Mikeli¶c
We want to approximate ^c
"
by c
L;ef f
0
. Then, if
L
"
(c
L;ef f
0
) = ¡K¿ c
L;ef f
0 + Q@xc
L;ef f
0
(1=3 ¡ y
2
) = R
"
; we have to consider
L
"
(^c
"
¡ c
L;ef f
0
) = ¡R
"
in (0; +1) £ (0; 1); (65)
¡D"
®¡2
@y(^c
"
(x; 1; ¿ ) ¡ c
L;ef f
0
) = K¿ c^
"
(x; 1; ¿ ) on (0; +1); (66)
¡D"
®
@x(^c
"
¡ c
L;ef f
0
) + Q(1 ¡ y
2
)(^c
"
¡ c
L;ef f
0
)
= Q(1=3 ¡ y
2
)(^cf ¡ c
L;ef f
0
); on f0g £ (0; 1): (67)
The weak formulation for the system (65)-(67) reads: for any ¿ = » + i´, ¯nd c^ " ¡ c
L;ef f
0 = w 2 H1
(­
+
) such that
Z
­+
¿w' dxdy ¡
Z
­+
Q(1 ¡ y
2
)@x'w dxdy +
Z
­+
D"
®
(@xw@x' + "
¡2
@yw@y') dxdy
+K
Z +1
0
¿wjy=1'jy=1 dx = ¡
Z
­+
Q@xc
L;ef f
0
(1=3 ¡ y
2
)' dxdy
¡
Z
1
0
Q(1=3 ¡ y
2
)(^cf ¡ c
L;ef f
0
jx=0)'jx=0 dy
+K
Z +1
0
¿ c
L;ef f
0
Z
1
0
(' ¡ 'jy=1) dydx; 8 ' 2 H1
(­
+
): (68)
Next we test (68) by ' = w = c^
" ¡ c
L;ef f
0
. The real part of the corresponding relation is
Z
­+
»jwj
2 dxdy +
Z
­+
D"
®
(j@xwj
2
+ "
¡2
j@ywj
2
) dxdy + K
Z +1
0
»jwjy=1j
2
dx
¡Re
Z
­+
Q(1 ¡ y
2
)@xww dxdy = ¡Re
Z
­+
Q@xc
L;ef f
0
(1=3 ¡ y
2
)w dxdy
¡Re
Z
1
0
Q(1=3 ¡ y
2
)(^cf ¡ c
L;ef f
0
jx=0)w jx=0 dy
+KRe
Z +1
0
¿ c
L;ef f
0
Z
1
0
(w ¡ wjy=1) dydx: (69)
We ¯nd out immediately that
¡Re
Z
­+
Q(1 ¡ y
2
)@xww dxdy =
1
2
Z
1
0
Q(1 ¡ y
2
)jw jx=0 j
2
dy ¸ 0:March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
Laplace transform approach to the rigorous upscaling of the reactive °ow 15
The terms in the right hand side of (69) are estimated as follows. Using
Z +1
0
Z
1
0
Q@xc
L;ef f
0
(1=3 ¡ y
2
)w dxdy
= ¡
Z +1
0
Z
1
0
Q@xc
L;ef f
0
(y=3 ¡ y
3
=3)@yw dxdy; (70) we obtain
Re
Z +1
0
Z
1
0
Q@xc
L;ef f
0
(1=3 ¡ y
2
)w ·
Z +1
0
Z
1
0
Q@xc
L;ef f
0
(1=3 ¡ y
2
)w
· Ck"
®=2¡1
@ywkL2
(­+)"
1¡®=2 k@xc L;ef f
0
kL2
((0;+1))
: (71)
Next, let ! = ^cf ¡ c
L;ef f
0
jx=0. We have
Z
1
0
Q(1=3 ¡ y
2
)(^cf ¡ c
L;ef f
0
jx=0)w jx=0 dy
=
Z
­+
Q(1=3 ¡ y
2
)!@x(we
¡x="
) dxdy
·
Z
­+
Q(1=3 ¡ y
2
)!@xwe
¡x="
dxdy + "
¡1
Z
­+
Q(y=3 ¡ y
3
=3)!@ywe
¡x="
dxdy
· Cj!j"
¡®=2
ke
¡x="
kL2
(­+)
¡ k" ®=2
@xwkL2
(­+) + k"
®=2¡1
@ywkL2
(­+)
¢
:
We bear in mind that ke
¡x="
kL2
(­+) · C"
1=2
. We also compute ! using (26) and obtain j!j = °
¡4¿D¹
l(¿ )
2
c^f · C"
®
j@xc
L;ef f
0
jx=0 j:
We get
Z
1
0
Q(1=3 ¡ y
2
)(^cf ¡ c
L;ef f
0
jx=0)w jx=0 dy
· C"
(1+®)=2
j@xc
L;ef f
0
jx=0 j
¡
k"
®=2
@xwkL2
(­+) + k"
®=2¡1
@ywkL2
(­+)
¢
: (72)
The last term in the right hand side of (69) is treated as follows.
K
Z +1
0
¿ c
L;ef f
0
Z
1
0
(w ¡ wjy=1) dydx = K
Z
­+
¿ c
L;ef f
0
¡
Z
y
1
@yw dz
¢
dxdy
· Ck"
®=2¡1
@ywkL2
(­+)"
1¡®=2 j¿ j kc
L;ef f
0
kL2
((0;+1))
: (73)March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
16 Catherine Choquet and Andro Mikeli¶c
Inserting estimates (71)-(73) in (69), we obtain
Z
­+
»jwj
2 dxdy +
Z
­+
D"
®
(j@xwj
2
+ "
¡2
j@ywj
2
) dxdy + K
Z +1
0
»jwjy=1j
2
dx
+
1
2
Z
1
0
Q(1 ¡ y
2
)jw jx=0 j
2
dy · C"
(1+®)=2
j@xc
L;ef f
0
jx=0 j k"
®=2
@xwkL2
(­+)
+"
1¡®=2
¡
C"
®¡1=2 j@xc L;ef f
0
jx=0 j + Cj¿ jkc
L;ef f
0
kL2
((0;+1))
+Ck@xc
L;ef f
0
kL2
((0;+1))
¢ k" ®=2¡1
@ywkL2
(­+)
:
The terms j@xc
L;ef f
0
jx=0 j, kc
L;ef f
0
kL2
((0;+1)) and k@xc
L;ef f
0
kL2
((0;+1)) are esti- mated through (42)-(44). We thus infer from the latter relation the following error estimates. Proposition 4.1 : kc^ "
¡ c
L;ef f
0
kL2
(IR+£(0;1)) · "
¯ Cj¿ jjc^f j
1 + ("
®
j¿ j)
1=4
; (74) k@x(^c "
¡ c
L;ef f
0
)kL2
(IR+£(0;1)) · "
¯¡®=2 Cj¿ jjc^f j
1 + ("
®
j¿ j)
1=4
; (75) k@y(^c "
¡ c
L;ef f
0
)kL2
(IR+£(0;1)) · "
¯+1¡®=2 Cj¿ jjc^f j
1 + ("
®
j¿ j)
1=4
; (76) where ¯ = 1 ¡ ®=2 if ® ¸ 1=2 and ¯ = (1 + ®)=2 if ® < 1=2.
Note that the presence of the given source term ^cf is su±cient to control the behavior in ¿ of the right hand side terms.
Corollary 4.2 : Let cf 2 D(0; T) and let c ef f
0
be such that c^ ef f
0 = c
L;ef f
0
. Let ¯ be de¯ned as in Proposition 4.1. Then we have kc "
¡ c ef f
0
kC(R+;L2
(­+)) · C"
¯
:
Next let cf 2 W1;1(R+) with compact support in [0; +1), such that cf (0) = 0 6 .
Then for 1 < r < +1, we have kc "
¡c
ef f
0
kLr
(R+;L2
(­+)) ·
½
C"
1¡®=2¡®±
; 2 > ® ¸ 1=2; 0 < ± < 1=4; r(1 ¡ ±) < 1:
C"
(1+®)=2
; 1=2 > ® ¸ 0:
Remark 1 : Presence of the contact discontinuity due to cf (0) = 0 diminishes 6 precision. Furthermore, the case of cf = 1 on (0; T) is covered by Corollary 4.2, since it could be extended to a Lipschitz function on R+, with compact support in [0; +1). Hence, it is easy to compare the result of Corollary 4.2 with the cor- responding results from [19] and [20] and see that we have now a more precise estimate, which gives convergence for any ® 2 [0; 2). Other possible comparaison is with the case ® = 0 from [11], but they have more complex chemical reactions.
In the next section we provide another "average" problem which leads to better error estimates.March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
Laplace transform approach to the rigorous upscaling of the reactive °ow 17
5. Next order corrector
We now limit ourself to the more realistic and critical case ® ¸ 1. A formal asymp- totic expansion of a solution c
"
of system (8)-(12) (see [19] or [20] for the details of the corresponding computation) leads to consider the following function for ap- proximating ^c
"
: c L;ef f
1
(x; y; ") = c0(x; ") + "
2¡® Q
D
³ y 2
6
¡ y 4
12
¡
7
180
¡
2
45
K(7K + 2)
(1 + K)
2
´
@xc0(x; ")
+"
2¡®K
D
³
1
6
+
K
3(1 + K)
¡
y
2
2
´
¿ c0(x; ") (77) where c0 2 H1
(­
+
) is the solution of the following e®ective problem
8

:>
(1 + K)¿ c0 +
2Q
3
@xc0 ¡ "
®D@~ 2 xx c0 = 0 in (0; +1);
¡D"
®
@xc0 +
2Q
3 c0 =
2Q
3 c^f x = 0;
(78)
with
D~ = D +
8
945
Q2
D
"
2(1¡®)
+
4Q2
135D
K(7K + 2)
(1 + K)
2
"
2(1¡®)
:
After some computations, we assert that ^c
²¡c
L;ef f
1
satis¯es the following problem.
L
"
(^c
²
¡ c
L;ef f
1
) = ¡©
"
in ­
+
; (79)
¡D"
®¡2
@y(^c
²
¡ c
L;ef f
1
) jy=1= K¿ (^c
²
¡ c
L;ef f
1
) jy=1 +g
"
jy=1 in (0; +1); (80)
@y(^c
²
¡ c
L;ef f
1
) jy=0= 0 in (0; +1); (81)
¡D"
®
@x(^c
²
¡ c
L;ef f
1
) jx=0 +Q(1 ¡ y
2
)(^c
²
¡ c
L;ef f
1
) jx=0
= Q(
1
3
¡ y
2
)(^cf ¡ c0 jx=0) + ´
"
0 jx=0; (82) where functions ©
"
, g
"
and ´
"
0 are de¯ned by
©
"
=
X5 i=1 F
"
i ¡ g
"
; (83)
F
"
1 = "
2¡®
@
2
xx c0 Q2
D
³
8
945
+ (1 ¡ y
2
)
¡y
2
6
¡ y 4
12
¡
7
180
¢
´
; (84)
F
"
2 = "
2¡®
¿ @xc0
QK
D
³
¡
2
45
+ (1 ¡ y
2
)
¡1
6
¡
y
2
2
¢
´
; (85)
F
"
3 = "
2¡®
¿ @xc0
Q
D
³
y
2
6
¡
y
4
12
¡
7
180
´
; (86)
F
"
4 = "
2¡®
¿
2
c0
K
D
³
1
6
¡ y 2
2
´
; (87)March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
18 Catherine Choquet and Andro Mikeli¶c
F
"
5 = ¡"
2¡® Q
D
¡1
3
¡ y
2
¢
³
2Q
45
@
2
xx c0 K(7K + 2)
(1 + K)
2
¡
K2
3(1 + K)
¿ @xc0
´
; (88) g "
= "
2¡®K¿
D
³
2Q
45
@xc0
¡
1 ¡
K(7K + 2)
(1 + K)
2
¢
¡
K
3(1 + K)
¿ c0
´
; (89)
´
"
0 = "
2¡®Q2
D
(1 ¡ y
2
)@xc0
¡y
2
6
¡ y 4
12
¡
7
180
¡
2
45
K(7K + 2)
(1 + K)
2
¢
+"
2¡®QK
D
(1 ¡ y
2
)¿ c0
¡1
6
¡
y
2
2
+
K
3(1 + K)
¢
¡"
2
³
K¿ @xc0
¡1
6
¡
y
2
2
+
K
3(1 + K)
¢
+Q@
2
xx c0 ¡y
2
6
¡
y
4
12
¡
7
180
¡
2
45
K(7K + 2)
(1 + K)
2
¢
´
: (90)
The variational formulation corresponding to problem (79)-(82) is
Z
­+
¿wÁ dxdy +
Z
­+
D"
®
(@xw@xÁ + "
¡2
@yw@yÁ) dxdy
+K
Z +1
0
¿wjy=1Á jy=1 dx +
Z
­+
Q(1 ¡ y
2
)Á@xw dxdy
+
Z
1
0
Q(1 ¡ y
2
)w jx=0 Á jx=0 dy
= ¡
Z
1
0
Q(1=3 ¡ y
2
)(^cf ¡ c0 jx=0)Á jx=0 dy +
Z
1
0
´
"
0 jx=0 Á jx=0 dy
¡
Z
­+
X5 i=1 F
"
i Á dxdy +
Z +1
0
g
"
Z
1
0
(Á ¡ Ájy=1) dydx: (91)
We note that the source terms ©
"
and g
"
satisfy the following properties.
Lemma 5.1 : Let Á 2 H1
(­
+
). The following estimates hold true.
Z
­+
X5
i=1
F
" i Á dxdy · C"
3(1¡®=2)
C(c0)k"
®=2¡1
@yÁkL2
(­+)
; (92)
Z +1
0
g
"
¡
Z
1
0
Á ¡ Á jy=1 dy
¢
dx · C"
3(1¡®=2)
C(c0)k"
®=2¡1
@yÁkL2
(­+)
; (93) where the quantity C(c0) is de¯ned by
C(c0) = k@
2
xx c0kL2 ((0;+1)) + (1 + j¿ j)k@xc0kL2
((0;+1)) + j¿ j
2
kc0kL2
((0;+1))
:
Proof : On the one hand, we note that
P5
i=1 F
"
i can be written as
X5
i=1
F
" i = "
2¡®
¡
@y(P0(y))¿
2 c0 + @y(P1(y))¿ @xc0 + @y(P2(y))@
2
xx c0 ¢March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
Laplace transform approach to the rigorous upscaling of the reactive °ow 19 where the polynomials Pj , 0 · j · 2, have zero traces in y = 0; 1. We thus have
Z
­+
X5
i=1
F
" i Á dxdy
=
Z
­+
"
2¡®
¡
@y(P0(y))¿
2 c0 + @y(P1(y))¿ @xc0 + @y(P2(y))@
2
xx c0 ¢
Á dxdy
= ¡
Z
­+
"
2¡®
¡
P0(y)¿
2
c0 + P1(y)¿ @xc0 + P2(y)@
2
xx c0 ¢
@yÁ dxdy
· C"
2¡®
k"
®=2¡1
@yÁkL2
(­+)"
1¡®=2
¡
j¿ j
2
kc0kL2
((0;+1)) + j¿ jk@xc0kL2
((0;+1))
+k@
2
xx c0kL2 ((0;+1))
¢
:
Estimate (92) is proved. On the other hand, we write
Z +1
0
g
"
¡
Z
1
0
Á ¡ Á jy=1 dy
¢
dx =
Z
­+ g "
¡
Z y 1
@yÁ dz
¢
dxdy
· C"
2¡®
(k@xc0kL2
((0;+1)) + j¿ jkc0kL2
((0;+1))
)"
1¡®=2
k"
®=2¡1
@yÁkL2
(­+)
:
This ends the proof of the lemma. ¤
Let us now study the terms in (91) coming from the boundary condition at x = 0.
Lemma 5.2 : The following estimates hold true.
Z
1
0
Q(1=3 ¡ y
2
)(^cf ¡ c0 jx=0)Á jx=0 dy
· C"
(1+®)=2
j@xc0 jx=0 j(k"
®=2
@xÁkL2
(­+) + k"
®=2¡1
@yÁkL2
(­+)
); (94)
Z
1
0
´
"
0 jx=0 Á jx=0 dy · C"
2¡®
k p 1 ¡ y
2Á jx=0 kL2
(0;1)
¡ j@xc0 jx=0 j + j¿ jjc0 jx=0 j
¢
+C"
2¡®=4
¡ j@ 2 xx c0 jx=0 j + j¿ jj@xc0 jx=0 j
¢¡
k"
®=2
@xÁkL2
(­+) + kÁkL2
(­+)
¢
: (95)
Proof : Let ! = jc^f ¡ c0 jx=0 j. We have
Z
1
0
Q(1=3 ¡ y
2
)(^cf ¡ c0 jx=0)Á jx=0 dy =
Z
­+
Q(1=3 ¡ y
2
) ! @x(Áe
¡x="
) dxdy
· Cj!j
Z
­+
1
3
¡ y
2
j@xÁje
¡x="
dxdy + Cj!j
Z
­+
"
¡1
¡1
3
¡ y
2
¢
Áe
¡x=" dxdy · Cj!j
³
k"
®=2
@xÁkL2
(­+)"
¡®=2 ke ¡x=" kL2 (­+) +
Z
­+
"
¡1
¡y
3
¡
y
3
3
¢
@yÁe
¡x="
dxdy
´
· Cj!j"
(1¡®)=2
¡ k" ®=2
@xÁkL2
(­+) + k"
®=2¡1
@yÁkL2
(­+)
¢
:
Then, using the explicit solution of problem (78) given in (3.2) with Q¹ =
2
3
Q
1 + K
,
D¹ =
D
1 + K
, ^g = ^cf , ° = "
®
and ± = ¡"
2(1¡®)
(8Q2
=(945D) + (4Q2
=135D)K(7K +March 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
20 Catherine Choquet and Andro Mikeli¶c
2)=(1 + K)
2
), we compute j!j = jc^f ¡ c0 jx=0 j =
2D¹Q¹ ¡ l(¿ )(D¹ + ±) + 2Q±
¹
l(¿ )(D¹ + ±) ¡ 2Q±
¹
jcf j · C"
®
j@xc0 jx=0 j:
We thus get
Z
1
0
Q(1=3 ¡ y
2
)(^cf ¡ c0 jx=0)Á jx=0 dy
· C"
(1¡®)=2
"
®
j@xc0 jx=0 j(k"
®=2
@xÁkL2
(­+) + k"
®=2¡1
@yÁkL2
(­+)
):
Estimate (94) is proved.
We now prove (95). We write
Z
1
0
"
2¡®
(1 ¡ y
2
)
Q
D
³
Q@xc0
¡y
2
6
¡ y 4
12
¡
7
180
¡
2
45
K(7K + 2)
(1 + K)
2
¢
+K¿ c0
¡1
6
¡
y
2
2
+
K
3(1 + K)
¢
´ jx=0 ¢Á jx=0 dy
· C"
2¡®
k p 1 ¡ y
2Á jx=0 kL2
(0;1)
¡ j@xc0 jx=0 j + j¿ jjc0 jx=0 j
¢
: (96)
The remaining term to estimate is
Z
1
0
"
2
³
K¿ @xc0
¡1
6
¡
y
2
2
+
K
3(1 + K)
¢
+ Q@
2
xx c0 ¡y
2
6
¡
y
4
12
¡
7
180
¡
2
45
K(7K + 2)
(1 + K)
2
¢
´
jx=0 Á jx=0 dy
=
Z
1
0
"
2
(p1(y)¿ @xc0 + p2(y)@
2
xx c0) jx=0 Á jx=0 dy; with Z
1
0
"
2
(p1(y)¿ @xc0 + p2(y)@
2
xx c0) jx=0 Á jx=0 dy
=
Z
­+
"
2
(p1(y)¿ @xc0 + p2(y)@
2
xx c0) jx=0 @x(Áe
¡x="
®=2
) dxdy
· "
2
Z
­+
(p1(y)¿ @xc0 + p2(y)@
2
xx c0) jx=0 @xÁe
¡x="
®=2 dxdy +"
2¡®=2
Z
­+
(p1(y)¿ @xc0 + p2(y)@
2
xx c0) jx=0 Áe
¡x="
®=2 dxdy · C"
2¡®=4
¡ j@ 2 xx c0 jx=0 j + j¿ jj@xc0 jx=0 j
¢¡
k"
®=2
@xÁkL2
(­+) + kÁkL2
(­+)
¢
: (97)
Estimate (95) follows from (96)-(97).
¤
Now let w = ^c
" ¡ c
L;ef f
1
. We write the variational formulation (91) for the testMarch 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
Laplace transform approach to the rigorous upscaling of the reactive °ow 21 function Á = w. We obtain
Z
­+
»jwj
2 dxdy +
Z
­+
D"
®
(j@xwj
2
+ "
¡2
j@ywj
2
) dxdy + K
Z +1
0
»jwjy=1j
2
dx
+
1
2
Z
1
0
Q(1 ¡ y
2
)jw jx=0 j
2
dy = ¡Re
Z
1
0
Q(1=3 ¡ y
2
)(^cf ¡ c0 jx=0)w jx=0 dy
+Re
Z
1
0
´
"
0
jx=0 w jx=0 dy ¡ Re
³Z
­+
X5
i=1
F
" i w dxdy ¡
Z +1
0
g
"
Z
1
0
(w ¡ wjy=1) dydx
´
:
The terms in the right hand side of the latter relation are estimated in Lemmas
5.1 and 5.2. The L
2
error estimate is thus kwkL2 (­+) = kc^
²
¡ c
L;ef f
1
kL2
(­+) · C"
2¡®
³
"
1¡®=2
¡
k@
2
xx c0kL2 ((0;+1))
+(1 + j¿ j)k@xc0kL2
((0;+1)) + j¿ j
2
kc0kL2
((0;+1))
¢
+ j@xc0 jx=0 j + j¿ jjc0 jx=0 j
+"
3®=4
¡
j@
2
xx c0 jx=0 j + j¿ jj@xc0 jx=0 j
¢
´
:
The terms containing c0 are estimated in Subsection 3.2. Problem (78) is in- deed Problem (46) where Q¹
2Q
3(1 + K)
, D¹ =
D~
1 + K
, ^g = ^cf , ° = "
®
and
± = ¡"
2(1¡®)
¡ 8
945
Q2
D
+
4Q2
135D
K(7K + 2)
(1 + K)
2
¢
. We thus claim the following result.
Proposition 5.3 : k(^c "
¡ c
L;ef f
1
)(¿ )kL2
(IR+£(0;1)) · C"
2¡®
j¿
2
cf j
1 + ("
®
j¿ j)
1=4
; (98) k@x(^c "
¡ c
L;ef f
1
)kL2
(IR+£(0;1)) · C"
2¡3®=2
j¿
2
cf j
1 + ("
®
j¿ j)
1=4
; (99) k@y(^c "
¡ c
L;ef f
1
)kL2
(IR+£(0;1)) · C"
3¡3®=2
j¿
2
cf j
1 + ("
®
j¿ j)
1=4
: (100)
Corollary 5.4 : Let cf 2 D(0; T) and let c ef f
0
be such that c^ ef f
0 = c
L;ef f
0
. Let ¯ be de¯ned as in Proposition 5.3. Then we have kc "
¡ c ef f
0
kC(R+;L2
(­+)) · C"
2¡®
:
Next let cf 2 W1;1(R+) with compact support in [0; +1), such that cf (0) = 0 6 .
Then for 1 < r < +1, we have k Z t 0
(c
"
¡ c ef f
0
) dtkLr
(R+;L2
(­+)) · C"
2¡®¡®±
; 0 < ± < 1=4; r(1 ¡ ±) < 1:
Remark 1 : As before, presence of the contact discontinuity due to cf (0) = 0 6 diminishes precision. Nevertheless, main deterioration of the approximation comes from the boundary condition. Without inlet boundary, we would have an approxi- mation of order "
3¡3®=2
. The case of cf = 1 on (0; T) is covered by the CorollaryMarch 28, 2008 22:40 Applicable Analysis ArticleApplAnalHomogVolChoquetMikeliclast
22 REFERENCES
5.4, since it could be extended to a Lipschitz function on R+, with compact sup- port in [0; +1). Hence, it is easy to compare the result of the Corollary 5.4 with the corresponding results from [19] and [20] and see that we have now a better estimate, even without constructing boundary layers.
Theorem 1.1 follows from Corollary 5.4.
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