
As an important part of machine learning, classification learning has been applied in many practical fields. It is valuable that to discuss class imbalance learning in several fields. In this research, we provide a review of class imbalanced learning methods from the data driven methods and algorithm driven methods based on numerous published papers which studied class imbalance learning. The preliminary analysis shows that class imbalanced learning methods mainly are applied both management and engineering fields. Firstly, we analyze and then summarize resampling methods that are used in different stages. Secondly, we provide a detailed instruction on different algorithms, and then we compare the results of decision tree classifiers based on resampling and empirical cost sensitivity. Finally, some suggestions from the reviewed papers are incorporated with our experiences and judgments to offer further research directions for the class imbalanced learning fields.
Citation: Cui Yin Huang, Hong Liang Dai. Learning from class-imbalanced data: review of data driven methods and algorithm driven methods[J]. Data Science in Finance and Economics, 2021, 1(1): 21-36. doi: 10.3934/DSFE.2021002
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As an important part of machine learning, classification learning has been applied in many practical fields. It is valuable that to discuss class imbalance learning in several fields. In this research, we provide a review of class imbalanced learning methods from the data driven methods and algorithm driven methods based on numerous published papers which studied class imbalance learning. The preliminary analysis shows that class imbalanced learning methods mainly are applied both management and engineering fields. Firstly, we analyze and then summarize resampling methods that are used in different stages. Secondly, we provide a detailed instruction on different algorithms, and then we compare the results of decision tree classifiers based on resampling and empirical cost sensitivity. Finally, some suggestions from the reviewed papers are incorporated with our experiences and judgments to offer further research directions for the class imbalanced learning fields.
We consider a viscous fluid obeying the Stokes system in a thin porous medium
The domain: the periodic porous medium is defined by a domain
The microscale of a porous medium is a small positive number
We denote by
The fluid part of the bottom
Ωε={(x1,x2,x3)∈ωε×R:0<x3<ε}. | (1) |
We make the following assumption:
The obstacles τ(¯T′k′,ε) do not intersect the boundary ∂ω. |
We define
Sε=⋃k′∈Kε¯Tεk′,ε. |
We define
˜Ωε=ωε×(0,1),Ω=ω×(0,1),Λε=ω×(0,ε). | (2) |
We observe that
The problem: let us consider the following Stokes system in
{−μΔuε+∇pε=fε in Ωε,divuε=0 in Ωε,uε=0 on ∂Λε,−pε⋅n+μ∂uε∂n+αεγuε=gε on ∂Sε, | (3) |
where we denote by
This choice of
Problem (3) models in particular the flow of an incompressible viscous fluid through a porous medium under the action of an exterior electric field. This system is derived from a physical model well detailed in the literature. As pointed out in Cioranescu et al. [12] and Sanchez-Palencia [32], it was observed experimentally in Reuss [31] the following phenomenon: when a electrical field is applied on the boundary of a porous medium in equilibrium, a motion of the fluid appears. This motion is a consequence of the electrical field only. To describe such a motion, it is usual to consider a modified Darcy's law considering of including an additional term, the gradient of the electrical field, or consider that the presence of this term is possible only if the electrical charges have a volume distribution. However, this law contains implicitly a mistake, because if the solid and fluid parts are both dielectric, such a distribution does not occur, the electrical charges act only on the boundary between the solid and the fluid parts and so they have necessarily a surface distribution. If such hypothesis is done, we can describe the boundary conditions in terms of the stress tensor
σε⋅n+αεγuε=gε, |
which is precisely the non-homogeneous slip boundary condition (3)
On the other hand, the behavior of the flow of Newtonian fluids through periodic arrays of cylinders has been studied extensively, mainly because of its importance in many applications in heat and mass transfer equipment. However, the literature on Newtonian thin film fluid flows through periodic arrays of cylinders is far less complete, although these problems have now become of great practical relevance because take place in a number of natural and industrial processes. This includes flow during manufacturing of fibre reinforced polymer composites with liquid moulding processes (see Frishfelds et al. [22], Nordlund and Lundstrom [29], Tan and Pillai [34]), passive mixing in microfluidic systems (see Jeon [23]), paper making (see Lundström et al. [25], Singh et al. [33]), and block copolymers self-assemble on nanometer length scales (see Park et al. [30], Albert and Epps [1], Farrell et al. [21]).
The Stokes problem in a periodically perforated domain with holes of the same size as the periodic has been treated in the literature. More precisely, the case with Dirichlet conditions on the boundary of the holes was studied by Ene and Sanchez-Palencia [20], where the model that describes the homogenized medium is a Darcy's law. The case with non-homogeneous slip boundary conditions, depending on a parameter
The earlier results relate to a fixed height domain. For a thin domain, in [4] Anguiano and Suárez-Grau consider an incompressible non-Newtonian Stokes system, in a thin porous medium of thickness
Using a combination of the unfolding method (see Cioranescu et al. [13] and Cioranescu et al. [16] for perforated domains) applied to the horizontal variables, with a rescaling on the height variable, and using monotonicity arguments to pass to the limit, three different Darcy's laws are obtained rigorously depending on the relation between
The behavior observed when
One of the main difficulties in the present paper is to treat the surface integrals. The papers mentioned above about problems with non-homogeneous boundary conditions use a generalization (see Cioranescu and Donato [9]) of the technique introduced by Vanninathan [36] for the Steklov problem, which transforms the surface integrals into volume integrals. In our opinion, an excellent alternative to this technique was made possible with the development of the unfolding method (see Cioranescu et al. [13]), which allows to treat easily the surface integrals. In the present paper, we extend some abstract results for thin domains, using an adaptation of the unfolding method, in order to treat all the surface integrals and we obtain directly the corresponding homogenized surface terms. A similar approach is made by Cioranescu et al. [14] and Zaki [37] with non-homogeneous slip boundary conditions, and Capatina and Ene [7] with non-homogeneous pure slip boundary conditions for a fixed height domain.
In summary, we show that the asymptotic behavior of the system (3) depends on the values of
We observe that we have obtained the same three regimes as in Cioranescu et al. [14] (see Theorems 2.1 and 2.2), and Zaki [37] (see Theorems 14 and 16). Thus, we conclude that the fact of considering the thin domain does not change the critical size of the parameter
We also remark the differences with the result obtained in [4] where Dirichlet boundary conditions are prescribed on the cylinders in the case
The paper is organized as follows. We introduce some notations in Section 2. In Section 3, we formulate the problem and state our main result, which is proved in Section 4. The article closes with a few remarks in Section 5.
Along this paper, the points
In order to apply the unfolding method, we need the following notation: for
κ(x′)=k′⟺x′∈Y′k′,1. | (4) |
Remark that
κ(x′ε)=k′⟺x′∈Y′k′,ε. |
For a vectorial function
(Dεv)i,j=∂xjvifori=1,2,3,j=1,2,(Dεv)i,3=1ε∂y3vifori=1,2,3,∇εw=(∇x′w,1ε∂y3w)t,divεv=divx′v′+1ε∂y3v3. |
We denote by
For every bounded set
MO[φ]=1|O|∫Oφdx. | (5) |
Similarly, for every compact set
M∂K[φ]=1|∂K|∫∂Kφdσ, |
is the average of
We denote by
L2♯(Y)={v∈L2loc(Y):∫Y|v|2dy<+∞,v(y′+k′,y3)=v(y)∀k′∈Z2,a.e. y∈Y}, |
and
H1♯(Y)={v∈H1loc(Y)∩L2♯(Y):∫Y|∇yv|2dy<+∞}. |
We denote by
Finally, we denote by
In this section we describe the asymptotic behavior of a viscous fluid obeying (3) in the geometry
The variational formulation: let us introduce the spaces
Hε={φ∈H1(Ωε) : φ=0 on ∂Λε},H3ε={φ∈H1(Ωε)3 : φ=0 on ∂Λε}, |
and
˜Hε={˜φ∈H1(˜Ωε) : ˜φ=0 on ∂Ω},˜H3ε={˜φ∈H1(˜Ωε)3 : ˜φ=0 on ∂Ω}. |
Then, the variational formulation of system (3) is the following one:
{μ∫ΩεDuε:Dφdx−∫Ωεpεdivφdx+αεγ∫∂Sεuε⋅φdσ(x)=∫Ωεf′ε⋅φ′dx+∫∂Sεg′ε⋅φ′dσ(x), ∀φ∈H3ε,∫Ωεuε⋅∇ψdx=∫∂Sε(uε⋅n)ψdσ(x), ∀ψ∈Hε. | (6) |
Assume that
Our aim is to study the asymptotic behavior of
y3=x3ε, | (7) |
in order to have the functions defined in the open set with fixed height
Namely, we define
˜uε(x′,y3)=uε(x′,εy3), ˜pε(x′,y3)=pε(x′,εy3), a.e. (x′,y3)∈˜Ωε. |
Using the transformation (7), the system (3) can be rewritten as
{−μΔx′˜uε−ε−2μ∂2y3˜uε+∇x′˜pε+ε−1∂y3˜pεe3=fε in ˜Ωε,divx′˜u′ε+ε−1∂y3˜uε,3=0 in ˜Ωε,˜uε=0 on ∂Ω, | (8) |
with the non-homogeneous slip boundary condition,
−˜pε⋅n+μ∂˜uε∂n+αεγ˜uε=gε on ∂Tε, | (9) |
where
Taking in (6) as test function
{μ∫˜ΩεDε˜uε:Dε˜φdx′dy3−∫˜Ωε˜pεdivε˜φdx′dy3+αεγ∫∂Tε˜uε⋅˜φdσ(x′)dy3=∫˜Ωεf′ε⋅˜φ′dx′dy3+∫∂Tεg′ε⋅˜φ′dσ(x′)dy3,∀˜φ∈˜H3ε,∫˜Ωε˜uε⋅∇ε˜ψdx′dy3=∫∂Tε(˜uε⋅n)˜ψdσ(x′)dy3,∀˜ψ∈˜Hε. | (10) |
In the sequel, we assume that the data
εf′ε⇀f′ weakly in L2(ω)2. | (11) |
Observe that, due to the periodicity of the obstacles, if
χΩεf′ε=εf′ε⇀θf′ in L2(ω)2, |
assuming
θ:=|Y′f||Y′|. |
We also define the constant
Main result: our goal then is to describe the asymptotic behavior of this new sequence
Our main result referred to the asymptotic behavior of the solution of (8)-(9) is given by the following theorem.
Theorem 3.1. Let
˜Πε˜uε⇀0 in H10(Ω)3. |
Moreover,
{˜v′(x′)=−θαμ1∇x′˜p(x′)˜v3(x′)=0, in ω, | (12) |
where
εγ+12˜Πε˜uε⇀0 in H10(Ω)3. |
Moreover,
{˜v′(x′)=θαμ1(f′−∇x′˜p(x′)+μ1M∂T′[g′])˜v3(x′)=0, in ω, | (13) |
where
ε˜Πε˜uε⇀0 in H10(Ω)3. |
Moreover,
{˜v′(x′)=−θμA∇x′˜p(x′)˜v3(x′)=0, in ω, | (14) |
where
Aij=1|Y′f|∫Y′fDwi(y′):Dwj(y′)dy,i,j=1,2. |
For
{−Δy′wi+∇y′qi=ei in Y′fdivy′ˆwi=0 in Y′f∂wi∂n=0 on ∂T′,wi,qiY′−periodic,MYf[wi]=0. | (15) |
Remark 1. We observe that in the homogenized problems related to system (8)-(9), the limit functions do not satisfy any incompressibility condition, so (12), (13) and (14) do not identify in a unique way
∫Ω˜Uε⋅∇ε˜ψdx′dy3=∫∂Tε(˜uε⋅n)˜ψdσ(x′)dy3,∀ψ∈˜Hε, |
and the term on the right-hand side is not necessarily zero. Therefore, by weak continuity, it is not possible to obtain an incompressibility condition of the form
In the context of homogenization of flow through porous media Arbogast et al. [5] use a
The a priori estimates independent of
Some abstract results for thin domains: let us introduce the adaption of the unfolding method in which we divide the domain
ˆφε(x′,y)=˜φ(εκ(x′ε)+εy′,y3), a.e. (x′,y)∈R2×Yf, | (16) |
where the function
Remark 2. The restriction of
y′=x′−εk′ε, | (17) |
which transforms
Proposition 1. We have the following estimates:
‖ˆφε‖Lp(R2×Yf)3=|Y′|1p‖˜φ‖Lp(˜Ωε)3, | (18) |
where
‖Dyˆφε‖Lp(R2×Yf)3×3=ε|Y′|1p‖Dε˜φ‖Lp(˜Ωε)3×3. | (19) |
Proof. Let us prove
∫R2×Yf|ˆφε(x′,y)|pdx′dy=∑k′∈Z2∫Y′k′,ε∫Yf|ˆφε(x′,y)|pdx′dy=∑k′∈Z2∫Y′k′,ε∫Yf|˜φ(εk′+εy′,y3)|pdx′dy. |
We observe that
∫R2×Yf|ˆφε(x′,y)|pdx′dy=ε2|Y′|∑k′∈Z2∫Yf|˜φ(εk′+εy′,y3)|pdy. |
For every
k′+y′=x′ε, dy′=dx′ε2 ∂y′=ε∂x′, | (20) |
and we obtain
∫R2×Yf|ˆφε(x′,y)|pdx′dy=|Y′|∫ωε×(0,1)|˜φ(x′,y3)|pdx′dy3 |
which gives (18).
Let us prove
∫R2×Yf|Dy′ˆφε(x′,y)|pdx′dy=ε2|Y′|∑k′∈Z2∫Yf|Dy′˜φ(εk′+εy′,y3)|pdy. |
By (20), we obtain
∫R2×Yf|Dy′ˆφε(x′,y)|pdx′dy=εp|Y′|∑k′∈Z2∫Y′fk′,ε∫10|Dx′˜φ(x′,y3)|pdx′dy3=εp|Y′|∫ωε×(0,1)|Dx′˜φ(x′,y3)|pdx′dy3. | (21) |
For the partial of the vertical variable, proceeding similarly to (18), we obtain
∫R2×Yf|∂y3ˆφε(x′,y)|pdx′dy=|Y′|∫ωε×(0,1)|∂y3˜φ(x′,y3)|pdx′dy3=εp|Y′|∫ωε×(0,1)|1ε∂y3˜φ(x′,y3)|pdx′dy3, |
which together with (21) gives (19).
In a similar way, let us introduce the adaption of the unfolding method on the boundary of the obstacles
ˆφbε(x′,y)=˜φ(εκ(x′ε)+εy′,y3), a.e. (x′,y)∈R2×∂T, | (22) |
where the function
Remark 3. Observe that from this definition, if we consider
Observe that for
We have the following property.
Proposition 2. If
‖ˆφbε‖Lp(R2×∂T)3=ε1p|Y′|1p‖˜φ‖Lp(∂Tε)3, | (23) |
where
Proof. We take
∫R2×∂T|ˆφbε(x′,y)|pdx′dσ(y)=ε2|Y′|∑k′∈Z2∫∂T|˜φ(εk′+εy′,y3)|pdσ(y). |
For every
∫R2×∂T|ˆφbε(x′,y)|pdx′dσ(y)=ε|Y′|∫∂Tε|˜φ(x′,y3)|pdσ(x′)dy3, |
which gives (23).
Now, let us give two results which will be useful for obtaining a priori estimates of the solution
Proposition 3. Let
|∫R2×∂Tg(y′)⋅ˆφε(x′,y)dx′dσ(y)|≤C|M∂T′[g]|(‖˜φ‖L2(˜Ωε)3+ε‖Dε˜φ‖L2(˜Ωε)3×3). | (24) |
In particular, if
|∫R2×∂Tˆφε(x′,y)dx′dσ(y)|≤C(‖˜φ‖L1(˜Ωε)3+ε‖Dε˜φ‖L1(˜Ωε)3×3). | (25) |
Proof. Due to density properties, it is enough to prove this estimate for functions in
|∫R2×∂Tg(y′)⋅ˆφε(x′,y)dx′dσ(y)|=|∫R2×∂Tg(y′)⋅˜φ(εκ(x′ε)+εy′,y3)dx′dσ(y)|≤|∫R2×∂Tg(y′)⋅˜φ(εκ(x′ε),y3)dx′dσ(y)|+|∫R2×∂Tg(y′)⋅(˜φ(εκ(x′ε)+εy′,y3)−˜φ(εκ(x′ε),y3))dx′dσ(y)|≤C|M∂T′[g]|(‖˜φ‖L2(˜Ωε)3+ε‖Dx′˜φ‖L2(˜Ωε)3×3)≤C|M∂T′[g]|(‖˜φ‖L2(˜Ωε)3+ε‖Dε˜φ‖L2(˜Ωε)3×3), |
which implies (24). In particular, if
|∫R2×∂Tˆφε(x′,y)dx′dσ(y)|=|∫R2×∂T˜φ(εκ(x′ε)+εy′,y3)dx′dσ(y)|≤|∫R2×∂T˜φ(εκ(x′ε),y3)dx′dσ(y)| |
+|∫R2×∂T(˜φ(εκ(x′ε)+εy′,y3)−˜φ(εκ(x′ε),y3))dx′dσ(y)|≤C(‖˜φ‖L1(˜Ωε)3+ε‖Dx′˜φ‖L1(˜Ωε)3×3)≤C(‖˜φ‖L1(˜Ωε)3+ε‖Dε˜φ‖L1(˜Ωε)3×3), |
which implies (25).
Corollary 1. Let
|∫∂Tεg(x′/ε)⋅˜φ(x′,y3)dσ(x′)dy3|≤Cε(‖˜φ‖L2(˜Ωε)3+ε‖Dε˜φ‖L2(˜Ωε)3×3). | (26) |
In particular, if
|∫∂Tε˜φ(x′,y3)dσ(x′)dy3|≤Cε(‖˜φ‖L1(˜Ωε)3+ε‖Dε˜φ‖L1(˜Ωε)3×3). | (27) |
Proof. Since
|∫∂Tεg(x′/ε)⋅˜φ(x′,y3)dσ(x′)dy3|=1ε|Y′||∫R2×∂Tg(y′)⋅ˆφε(x′,y)dx′dσ(y)|, |
and by Proposition 3, we can deduce estimates (26) and (27).
Moreover, for the proof of the a priori estimates for the velocity, we need the following lemma due to Conca [17] generalized to a thin domain
Lemma 4.1. There exists a constant
‖φ‖L2(Ωε)3≤C(ε‖Dφ‖L2(Ωε)3×3+ε12‖φ‖L2(∂Sε)3). | (28) |
Proof. We observe that the microscale of the porous medium
∫Yf|φ|2dz≤C(∫Yf|Dzφ|2dz+∫∂T|φ|2dσ(z)), | (29) |
where the constant
Then, for every
k′+z′=x′ε, z3=x3ε, dz=dxε3, ∂z=ε∂x, dσ(x)=ε2dσ(z), | (30) |
we rescale (29) from
∫Qfk′,ε|φ|2dx≤C(ε2∫Qfk′,ε|Dxφ|2dx+ε∫T′k′,ε×(0,ε)|φ|2dσ(x)), | (31) |
with the same constant
∫Ωε|φ|2dx≤C(ε2∫Ωε|Dxφ|2dx+ε∫∂Sε|φ|2dσ(x)). |
In fact, we must consider separately the periods containing a portion of
Considering the change of variables given in (7) and taking into account that
Corollary 2. There exists a constant
‖˜φ‖L2(˜Ωε)3≤C(ε‖Dε˜φ‖L2(˜Ωε)3×3+ε12‖˜φ‖L2(∂Tε)3). | (32) |
The presence in (3) of the stress tensor in the boundary condition implies that the extension of the velocity is no longer obvious. If we consider the Stokes system with Dirichlet boundary condition on the obstacles, the velocity would be extended by zero in the obstacles. However, in this case, we need another kind of extension for the case in which the velocity is non-zero on the obstacles. Since in the extension required, the vertical variable is not concerned, therefore the proof of the required statement is basically the extension of the result given in Cioranescu and Saint-Jean Paulin [8,11] to the time-depending case given in Cioranescu and Donato [10], so we omit the proof. We remark that the extension is not divergence free, so we cannot expect the homogenized solution to be divergence free. Hence we cannot use test functions that are divergence free in the variational formulation, which implies that the pressure has to be included.
Lemma 4.2. There exists an extension operator
Πεφ(x)=φ(x),if x∈Ωε,‖DΠεφ‖L2(Λε)3×3≤C‖Dφ‖L2(Ωε)3×3,∀φ∈H3ε. |
Considering the change of variables given in (7), we obtain the following result for the domain
Corollary 3. There exists an extension operator
˜Πε˜φ(x′,y3)=˜φ(x′,y3),if (x′,y3)∈˜Ωε,‖Dε˜Πε˜φ‖L2(Ω)3×3≤C‖Dε˜φ‖L2(˜Ωε)3×3,∀˜φ∈˜H3ε. |
Using Corollary 3, we obtain a Poincaré inequality in
Corollary 4. There exists a constant
‖˜φ‖L2(˜Ωε)3≤C‖Dε˜φ‖L2(˜Ωε)3×3. | (33) |
Proof. We observe that
‖˜φ‖L2(˜Ωε)3≤‖˜Πε˜φ‖L2(Ω)3,∀˜φ∈˜H3ε. | (34) |
Since
‖˜Πε˜φ‖L2(Ω)3≤C‖D˜Πε˜φ‖L2(Ω)3×3≤C‖Dε˜Πε˜φ‖L2(Ω)3×3≤C‖Dε˜φ‖L2(˜Ωε)3×3. |
This together with (34) gives (33).
Now, for the proof of the a priori estimates for the pressure, we also need the following lemma due to Conca [17] generalized to a thin domain
Lemma 4.3. There exists a constant
divφ=q in Ωε, | (35) |
‖φ‖L2(Ωε)3≤C‖q‖L2(Ωε),‖Dφ‖L2(Ωε)3×3≤Cε‖q‖L2(Ωε). | (36) |
Proof. Let
Q(x)={q(x)ifx∈Ωε−1|Λε−Ωε|∫Ωεq(x)dxifx∈Λε−Ωε. |
It is follows that
‖Q‖2L2(Λε)=‖q‖2L2(Ωε)+1|Λε−Ωε|(∫Ωεq(x)dx)2. | (37) |
Since
‖Q‖L2(Λε)≤C‖q‖L2(Ωε). | (38) |
Besides that, since
divφ=Q in Λε, | (39) |
‖φ‖L2(Λε)3≤C‖Q‖L2(Λε),‖Dφ‖L2(Λε)3×3≤Cε‖Q‖L2(Λε). | (40) |
Let us consider
Considering the change of variables given in (7), we obtain the following result for the domain
Corollary 5. There exists a constant
divε˜φ=˜q in ˜Ωε,‖˜φ‖L2(˜Ωε)3≤C‖˜q‖L2(˜Ωε),‖Dε˜φ‖L2(˜Ωε)3×3≤Cε‖˜q‖L2(˜Ωε). |
A priori estimates for (
Lemma 4.4. We distinguish three cases:
‖˜uε‖L2(˜Ωε)3≤Cε,‖Dε˜uε‖L2(˜Ωε)3×3≤C. | (41) |
‖˜uε‖L2(˜Ωε)3≤Cε−γ,‖Dε˜uε‖L2(˜Ωε)3×3≤Cε−1+γ2. | (42) |
‖˜uε‖L2(˜Ωε)3≤Cε−1,‖Dε˜uε‖L2(˜Ωε)3×3≤Cε−1. | (43) |
Proof. Taking
μ‖Dε˜uε‖2L2(˜Ωε)3×3+αεγ‖˜uε‖2L2(∂Tε)3=∫˜Ωεf′ε⋅˜u′εdx′dy3+∫∂Tεg′ε⋅˜u′εdσ(x′)dy3. | (44) |
Using Cauchy-Schwarz's inequality and
∫˜Ωεf′ε⋅˜u′εdx′dy3≤C‖˜uε‖L2(˜Ωε)3, |
and by using that
|∫∂Tεg′ε⋅˜u′εdσ(x′)dy3|≤Cε(‖˜uε‖L2(˜Ωε)3+ε‖Dε˜uε‖L2(˜Ωε)3×3). |
Putting these estimates in (44), we get
μ‖Dε˜uε‖2L2(˜Ωε)3×3+αεγ‖˜uε‖2L2(∂Tε)3≤C(‖Dε˜uε‖L2(˜Ωε)3×3+ε−1‖˜uε‖L2(˜Ωε)3). | (45) |
In particular, if we use the Poincaré inequality (33) in (45), we have
‖Dε˜uε‖L2(˜Ωε)3×3≤Cε, | (46) |
therefore (independently of
‖˜uε‖L2(˜Ωε)3≤Cε. | (47) |
These estimates can be refined following the different values of
ε−1‖˜uε‖L2(˜Ωε)3≤C(‖Dε˜uε‖L2(˜Ωε)3×3+ε−12‖˜uε‖L2(∂Tε)3). |
Using Young's inequality, we get
ε−12‖˜uε‖L2(∂Tε)3≤ε−1+γ2εγ2‖˜uε‖L2(∂Tε)3≤2αε−1−γ+α2εγ‖˜uε‖2L2(∂Tε)3. |
Consequently, from (45), we get
μ‖Dε˜uε‖2L2(˜Ωε)3×3+α2εγ‖˜uε‖2L2(∂Tε)3≤C(‖Dε˜uε‖L2(˜Ωε)3×3+ε−1−γ), |
which applying in a suitable way the Young inequality gives
μ‖Dε˜uε‖2L2(˜Ωε)3×3+αεγ‖˜uε‖2L2(∂Tε)3≤C(1+ε−1−γ). | (48) |
For the case when
‖Dε˜uε‖L2(˜Ωε)3×3≤C,‖˜uε‖L2(∂Tε)3≤Cε−γ2. |
Then, estimate (32) gives
‖˜uε‖L2(˜Ωε)3≤C(ε+ε1−γ2)≤Cε, |
since
For
‖Dε˜uε‖L2(˜Ωε)3×3≤Cε−1+γ2,‖˜uε‖L2(∂Tε)3≤Cε−12−γ. |
Applying estimate (32), we get
‖˜uε‖L2(˜Ωε)3≤C(ε1−γ2+ε−γ)≤Cε−γ |
since
We will prove now a priori estimates for the pressure
Lemma 4.5. We distinguish three cases:
‖˜pε‖L2(˜Ωε)≤Cεγ. | (49) |
‖˜pε‖L2(˜Ωε)≤Cε−1. | (50) |
‖˜pε‖L2(˜Ωε)≤Cε−2. | (51) |
Proof. Let
divε˜φ=˜Φ in ˜Ωε, ‖˜φ‖L2(˜Ωε)3≤C‖˜Φ‖L2(˜Ωε), ‖Dε˜φ‖L2(˜Ωε)3×3≤Cε‖˜Φ‖L2(˜Ωε). | (52) |
Taking
|∫˜Ωε˜pε˜Φdx′dy3|≤μ‖Dε˜uε‖L2(˜Ωε)3×3‖Dε˜φ‖L2(˜Ωε)3×3+αεγ|∫∂Tε˜uε⋅˜φdσ(x′)dy3|+C‖˜φ‖L2(˜Ωε)3+|∫∂Tεg′ε⋅˜φ′dσ(x′)dy3|. | (53) |
By using that
|∫∂Tεg′ε⋅˜φ′dσ(x′)dy3|≤C(ε−1‖˜φ‖L2(˜Ωε)3+‖Dε˜φ‖L2(˜Ωε)3×3). |
Analogously, using estimate (27) and the Cauchy- Schwarz inequality, a simple computation gives
αεγ|∫∂Tε˜uε⋅˜φdσ(x′)dy3|≤εγ−1C‖˜uε‖L2(˜Ωε)‖˜φ‖L2(˜Ωε)+εγC‖˜uε‖L2(˜Ωε)‖Dε˜φ‖L2(˜Ωε)+εγC‖Dε˜uε‖L2(˜Ωε)‖˜φ‖L2(˜Ωε). |
Then, turning back to (53) and using (52), one has
|∫˜Ωε˜pε˜Φdx′dy3|≤C(ε−1+εγ)‖Dε˜uε‖L2(˜Ωε)3×3‖˜Φ‖L2(˜Ωε)+C(εγ−1‖˜uε‖L2(˜Ωε)3+ε−1)‖˜Φ‖L2(˜Ωε). | (54) |
The a priori estimates for the pressure follow now from (54) and estimates (41)-(42) and (43), corresponding to the different values of
A priori estimates of the unfolding functions
Lemma 4.6. We distinguish three cases:
‖ˆuε‖L2(R2×Yf)3≤Cε,‖Dyˆuε‖L2(R2×Yf)3×3≤Cε, | (55) |
‖ˆpε‖L2(R2×Yf)≤Cεγ. | (56) |
‖ˆuε‖L2(R2×Yf)3≤Cε−γ,‖Dyˆuε‖L2(R2×Yf)3×3≤Cε1−γ2, | (57) |
‖ˆpε‖L2(R2×Yf)≤Cε−1. | (58) |
‖ˆuε‖L2(R2×Yf)3≤Cε−1,‖Dyˆuε‖L2(R2×Yf)3×3≤C, | (59) |
‖ˆpε‖L2(R2×Yf)≤Cε−2. | (60) |
Proof. Using properties (18) and (19) with
Let us remember that, for the velocity, we denote by
Lemma 4.7. There exists an extension operator
˜Πε˜uε⇀0 in H10(Ω)3. | (61) |
Moreover,
ε−1˜Uε⇀˜u in H1(0,1;L2(ω)3), | (62) |
ε−γ˜Pε⇀˜p in L2(Ω), | (63) |
εγ+12˜Πε˜uε⇀0 in H10(Ω)3. | (64) |
Moreover,
εγ˜Uε⇀˜u in H1(0,1;L2(ω)3), | (65) |
ε˜Pε⇀˜p in L2(Ω), | (66) |
ε˜Πε˜uε⇀0 in H10(Ω)3. | (67) |
Moreover,
˜Uε⇀˜u in H1(0,1;L2(ω)3), | (68) |
ε2˜Pε⇀˜p in L2(Ω). | (69) |
Proof. We proceed in three steps:
Step 1. For
Moreover, we have
||˜uε||L2(˜Ωε)3≤C, |
and we can apply Corollary 3 to
On the other side, we have the following indentity:
χ˜Ωε(˜Πε˜uε)=εε−1˜Uε in Ω. |
Due the periodicity of the domain
Step 2. For
Moreover, as
‖εγ+12˜uε‖L2(˜Ωε)3≤C, |
and using Corollary 3, we have
εγ+12˜Πε˜uε⇀u∗ in H1(Ω)3. |
Consequently,
εγ+12˜Πε˜uε→u∗ in L2(Ω)3, |
and passing to the limit in the identity
χ˜Ωε(εγ+12˜Πε˜uε)=ε1−γ2εγ˜Uε in Ω, |
we deduce that
Step 3. For
||˜Uε||L2(Ω)3≤||∂y3˜Uε||L2(Ω)3≤C, |
and we have immediately, after eventual extraction of subsequences, the convergence (68). From estimate (51), we have immediately, after eventual extraction of subsequences, the convergence (69).
Moreover, we can apply Corollary 3 to
On the other side, we have the following indentity:
χ˜Ωε(ε˜Πε˜uε)=ε˜Uε in Ω. |
We can pass to the limit in the term of the left hand side. Thus,
Finally, we give a convergence result for
Lemma 4.8. We distinguish three cases:
ε−1ˆuε⇀ˆu in L2(R2;H1(Yf)3), | (70) |
ε−1ˆuε⇀ˆu in L2(R2;H12(∂T)3), | (71) |
|Y′f||Y′|MY′f[ˆu]=˜ua.e. in Ω, | (72) |
εγˆuε⇀ˆu in L2(R2;H1(Yf)3), | (73) |
εγˆuε⇀ˆu in L2(R2;H12(∂T)3), | (74) |
|Y′f||Y′|ˆu=˜ua.e. in Ω, | (75) |
ˆuε−MYf[ˆuε]⇀ˆu in L2(R2;H12(∂T)3). | (76) |
Dyˆuε⇀Dyˆu in L2(R2×Yf)3×3. | (77) |
|Y′f||Y′|MY′f[ˆu]=˜ua.e. in Ω, | (78) |
divyˆu=0in ω×Yf. | (79) |
Proof. We proceed in three steps:
Step 1. For
∫R2×Yf|ˆu|2dx′dy≤C,∫R2×Yf|Dyˆu|2dx′dy≤C, |
which, once we prove the
It remains to prove the
ˆuε(x1+ε,x2,−1/2,y2,y3)=ˆuε(x′,1/2,y2,y3) |
a.e.
ˆu(x′,−1/2,y2,y3)=ˆu(x′,1/2,y2,y3) a.e. (x′,y2,y3)∈R2×(−1/2,1/2)×(0,1). |
Analogously, we can prove
ˆu(x′,y1,−1/2,y3)=ˆu(x′,y1,1/2,y3) a.e. (x′,y1,y3)∈ω×(−1/2,1/2)×(0,1). |
These equalities prove that
Finally, using Proposition 1, we can deduce
1|Y′|∫R2×Yfˆuε(x′,y)dx′dy=∫˜Ωε˜uε(x′,y3)dx′dy3, |
and multiplying by
1ε|Y′|∫R2×Yfˆuε(x′,y)dx′dy=1ε∫Ω˜Uε(x′,y3)dx′dy3. |
Using convergences (62) and (70), we have (72).
Step 2. For
On the other hand, since
Step 3. For
∫Yf|ˆuε−MYf[ˆuε]|2dy≤C∫Yf|Dyˆuε|2dy,a.e. in ω, |
we deduce that there exists
ˆUε=ˆuε−MYf[ˆuε]⇀ˆu in L2(R2;H1(Yf)3), |
and (77) holds. Convergence (76) is straightforward from the definition (22) and the Sobolev injections.
It remains to prove the
ˆuε(x1+ε,x2,−1/2,y2,y3)=ˆuε(x′,1/2,y2,y3) |
a.e.
ˆUε(x′,−1/2,y2,y3)−ˆUε(x′,1/2,y2,y3)=−MYf[ˆuε](x′+εe1)+MY[ˆuε](x′), |
which tends to zero (see the proof of Proposition 2.8 in [15]), and so
ˆu(x′,−1/2,y2,y3)=ˆu(x′,1/2,y2,y3) a.e. (x′,y2,y3)∈R2×(−1/2,1/2)×(0,1). |
Analogously, we can prove
ˆu(x′,y1,−1/2,y3)=ˆu(x′,y1,1/2,y3) a.e. (x′,y1,y3)∈ω×(−1/2,1/2)×(0,1). |
These equalities prove that
Step 4. From the second variational formulation in (10), by Proposition 2, we have that
{∫˜Ωε(˜u′ε⋅∇x′˜ψ+ε−1˜uε,3∂y3˜ψ)dx′dy3=ε−1|Y′|∫ω×∂T(ˆuε⋅n)ˆψεdx′dσ(y′)dy3,∀˜ψ∈˜Hε, | (80) |
and using the extension of the velocity, we obtain
{∫Ω(˜U′ε⋅∇x′˜ψ+ε−1˜Uε,3∂y3˜ψ)dx′dy3=ε−1|Y′|∫ω×∂T(ˆuε⋅n)ˆψεdx′dσ(y′)dy3,∀˜ψ∈˜Hε. |
We remark that the second term in the left-hand side and the one in the right-hand side have the same order, so in every cases when passing to the limit after multiplying by a suitable power of
divx′(∫10˜u′(x′,y3)dy3)=0on ω. |
On the other hand, we focus in the third case. Thus, using Proposition 1 in the left-hand side of (80), we have
ε−1|Y′|∫ω×Yfˆuε⋅∇yˆψεdx′dy=ε−1|Y′|∫ω×∂Tˆuε⋅ˆψεdx′dσ(y′)dy3, | (81) |
which, multiplying by
∫ω×Yf(ˆuε−MYf[ˆuε])⋅∇yˆψεdx′dy=∫ω×∂T[(ˆuε−MYf[ˆuε])⋅n]⋅ˆψεdx′dσ(y′)dy3 | (82) |
Thus, passing to the limit using convergences (77), we get condition (79).
In this section, we will multiply system (10) by a test function having the form of the limit
Proof of Theorem 3.1: We proceed in three steps:
Step 1. For
μ∫˜ΩεDε˜uε:Dε˜φdx′dy3−∫˜Ωε˜pεdivε˜φdx′dy3+αεγ−1|Y′|∫ω×∂Tˆuε⋅ˆφεdx′dσ(y)=∫˜Ωεf′ε⋅˜φ′dx′dy3+ε−1|Y′|∫ω×∂T˜g′⋅ˆφ′εdx′dσ(y), |
i.e.,
μ∫˜ΩεDx′˜uε:Dx′˜φdx′dy3+με2∫˜Ωε∂y3˜uε⋅∂y3˜φdx′dy3−∫˜Ωε˜pεdivx′˜φ′dx′dy3−1ε∫˜Ωε˜pε∂y3˜φ3dx′dy3+αεγ−1|Y′|∫ω×∂Tˆuε⋅ˆφεdx′dσ(y)=∫˜Ωεf′ε⋅˜φ′dx′dy3+ε−1|Y′|∫ω×∂T˜g′⋅ˆφ′εdx′dσ(y), | (83) |
Next, we prove that
με−γ+1∫˜Ωε∇x′˜uε,3⋅∇x′˜φ3dx′dy3+με−γ−1∫˜Ωε∂y3˜uε,3∂y3˜φ3dx′dy3−ε−γ∫˜Ωε˜pε∂y3˜φ3dx′dy3+α|Y′|∫ω×∂Tˆuε,3⋅ˆφε,3dx′dσ(y)=0. |
Taking into account that
∫˜Ωε˜pε∂y3˜φ3dx′dy3=∫Ω˜Pε∂y3˜φ3dx′dy3, |
and by the second a priori estimate (41), the convergences (63) and (71), passing to the limit we have
∫Ω˜p∂y3˜φ3dx′dy3=0, |
so
Let
με−γ∫˜ΩεDx′˜u′ε:Dx′˜φ′dx′dy3+με−γ−2∫˜Ωε∂y3˜u′ε⋅∂y3˜φ′dx′dy3−ε−γ∫˜Ωε˜pεdivx′˜φ′dx′dy3+αε−1|Y′|∫ω×∂Tˆu′ε⋅ˆφ′εdx′dσ(y)=ε−γ∫˜Ωεf′ε⋅˜φ′dx′dy3+ε−γ−1|Y′|∫ω×∂T˜g′⋅ˆφ′εdx′dσ(y), |
and
με−γ∫˜Ωε∇x′˜uε,3⋅∇x′˜φ3dx′dy3+αε−1|Y′|∫ω×∂Tˆuε,3ˆφε,3dx′dσ(y)=0. |
Taking into account that
∫˜Ωε˜pεdivx′˜φ′dx′dy3=∫Ω˜Pεdivx′˜φ′dx′dy3. |
Using that
−∫Ω˜p(x′)divx′˜φ′(x′,y3)dx′dy3+α|Y′|∫ω×∂T′∫10ˆu′(x′,y)⋅˜φ′(x′,y3)dx′dσ(y′)dy3=0, |
and
α|Y′|∫ω×∂Tˆu3(x′,y)˜φ3(x′)dx′dσ(y)=0, |
which implies that
Taking into account that
∫ω×∂T′∫10ˆu′(x′,y)⋅˜φ′(x′,y3)dx′dσ(y′)dy3=|∂T′|∫ΩM∂T′[ˆu′](x′,y3)⋅˜φ′(x′,y3)dx′dy3, |
implies that
∫Ω∇x′˜p(x′)⋅φ′(x′,y3)dx′dy3=−α|∂T′||Y′|∫ΩM∂T′[ˆu′](x′,y3)⋅˜φ′(x′,y3)dx′dy3. | (84) |
In order to obtain the homogenized system (12), we introduce the auxiliary problem
{−Δy′χ(y′)=−|∂T′||Y′f|MY′f[ˆu](x′,y3), in Y′f,∂χ∂n=ˆu,on ∂T′,MY′f[χ]=0,χ(y)Y′−periodic, |
for a.e.
∫ΩM∂T′[ˆu]⋅˜φdx′dy3=∫ΩMY′f[ˆu]⋅˜φdx′dy3, | (85) |
which together with (84) and
MY′f[ˆu′](x′,y3)=−|Y′|α|∂T′|∇x′˜p(x′), |
and
MY′f[ˆu3]=0, |
which together with (72) gives
˜u′(x′,y3)=−|Y′f|α|∂T′|∇x′˜p(x′),˜u3(x′,y3)=0. |
This together with the definition of
Step 2. For
Let
−∫Ω˜p(x′)divx′˜φ′(x′,y3)dx′dy3+α|Y′|∫ω×∂T′∫10ˆu′(x′)⋅˜φ′(x′,y3)dx′dσ(y′)dy3=∫Ωf′(x′)⋅˜φ′(x′,y3)dx′dy3+1|Y′|∫ω×∂T′∫10g′(y′)⋅˜φ′(x′,y3)dx′dσ(y′)dy3, |
and
α|Y′|∫ω×∂Tˆu3(x′)˜φ3(x′)dx′dσ(y)=0, |
which implies that
Taking into account that
∫ω×∂T′∫10ˆu′(x′)⋅˜φ′(x′,y3)dx′dσ(y′)dy3=|∂T′|∫Ωˆu′(x′)⋅˜φ′(x′,y3)dx′dy3, |
implies that
∫Ω∇x′˜p(x′)⋅φ′(x′,y3)dx′dy3+α|∂T′||Y′|∫Ωˆu′(x′)⋅˜φ′(x′,y3)dx′dy3=∫Ωf′(x′)⋅˜φ′(x′,y3)dx′dy3+|∂T′||Y′|∫ΩM∂T′[g′]⋅˜φ′(x′,y3)dx′dy3, |
which together with (75) gives (13) after integrating the vertical variable
Step 3. For
με2∫ω×YfDyˆuε:Dy′ˆφdx′dy−∫ω×Yfˆpεdivx′ˆφ′dx′dy−ε−1∫ω×Yfˆpεdivyˆφdx′dy+αεγ−1∫ω×∂Tˆuε⋅ˆφdx′dσ(y)=∫ω×Yff′ε⋅ˆφ′dx′dy+ε−1∫ω×∂T˜g′⋅ˆφ′dx′dσ(y)+Oε. | (86) |
First, we remark that thanks to (60), there exists
∫ω×Yfˆpdivyˆφdx′dy3=0, |
which shows that
Now, we consider
μ∫ω×YfDyˆuε:Dy′ˆφdx′dy−ε2∫ω×Yfˆpεdivx′ˆφ′dx′dy+αεγ+1∫ω×∂Tˆuε⋅ˆφdx′dσ(y)=ε2∫ω×Yff′ε⋅ˆφ′dx′dy+ε∫ω×∂T˜g′⋅ˆφ′dx′dσ(y)+Oε. | (87) |
Reasoning as Step 1, and using the convergences (11), (76), (77) and the convergence of
μ∫ω×YfDyˆu:Dyˆφdx′dy−∫ω×Yfˆp(x′)divx′ˆφ′dx′dy=0. | (88) |
By density, this equation holds for every function
{−μΔyˆu+∇yˆq=−∇x′ˆp in ω×Yf,divyˆu=0 in ω×Yf,∂ˆu∂n=0on ω×∂T,ˆu=0on y3=0,1,y′→ˆu(x′,y),ˆq(x′,y)Y′−periodic. |
We remark that
Finally, we will eliminate the microscopic variable
Aij=1|Yf|∫YfDywi(y):Dywj(y)dy=∫Yfwi(y)ejdy,i,j=1,2. |
By definition
The behavior of the flow of Newtonian fluids through periodic arrays of cylinders has been studied extensively, mainly because of its importance in many applications in heat and mass transfer equipment. However, the literature on Newtonian thin film fluid flows through periodic arrays of cylinders is far less complete, although these problems have now become of great practical relevance because take place in a number of natural and industrial processes. This paper deals with the modelization by means of homogenization techniques of a thin film fluid flow governed by the Stokes system in a thin perforated domain
The main novelty here are the combination of the mixed boundary condition considered on the obstacles and the thin thickness of the domain. Namely, a standard (no-slip) condition is imposed on the exterior boundary, whereas a non-standard boundary condition of Robin type which depends on a parameter
By means of a combination of homogenization and reduction of dimension techniques, depending on the parameter
We would like to thank the referees for their comments and suggestions.
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