Proper water resource management is essential for maintaining a sustainable supply chain and meeting water demand. The urgent need to preserve river ecosystems by sustaining environmental flow (EF) in the realm of environmental management has been highlighted by the drastic changes to river ecosystems and upstream flow dynamics brought about by careless river exploitation in the last few decades. To optimize EF in river basin management, we present an integrated modeling approach. We focused on the Pir Khezran River basin. Our objective was to estimate EF and generalize the findings to adjacent rivers using modeling techniques, thus providing valuable insights for environmental management applications. The assessment and optimization of EF under uncertain conditions was achieved by combining physical habitat simulation (PHABSIM) modeling with advanced techniques like Adaptive Neuro-Fuzzy Inference Systems (ANFIS) and Multilayer Perceptron (MLP) neural networks. This integrated modeling approach contributes to sustainable solutions for river basin management and environmental conservation by effectively optimizing EF, as demonstrated by the results. This research, therefore, makes valuable contributions to environmental management in various areas such as ecological preservation, modeling and optimizing environmental systems, and policy considerations.
Citation: Seiran Haghgoo, Jamil Amanollahi, Barzan Bahrami Kamangar, Shahryar Sorooshian. Decision models enhancing environmental flow sustainability: A strategic approach to water resource management[J]. AIMS Environmental Science, 2024, 11(6): 900-917. doi: 10.3934/environsci.2024045
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Proper water resource management is essential for maintaining a sustainable supply chain and meeting water demand. The urgent need to preserve river ecosystems by sustaining environmental flow (EF) in the realm of environmental management has been highlighted by the drastic changes to river ecosystems and upstream flow dynamics brought about by careless river exploitation in the last few decades. To optimize EF in river basin management, we present an integrated modeling approach. We focused on the Pir Khezran River basin. Our objective was to estimate EF and generalize the findings to adjacent rivers using modeling techniques, thus providing valuable insights for environmental management applications. The assessment and optimization of EF under uncertain conditions was achieved by combining physical habitat simulation (PHABSIM) modeling with advanced techniques like Adaptive Neuro-Fuzzy Inference Systems (ANFIS) and Multilayer Perceptron (MLP) neural networks. This integrated modeling approach contributes to sustainable solutions for river basin management and environmental conservation by effectively optimizing EF, as demonstrated by the results. This research, therefore, makes valuable contributions to environmental management in various areas such as ecological preservation, modeling and optimizing environmental systems, and policy considerations.
We consider the Stokes system in which the viscosity is a periodically varying function of the space variable with small period
The first goal of this paper is to study homogenization of the above systems via Bloch Wave Method which is based on the fact that the homogenized operator can be defined using differential properties of the bottom of the so-called Bloch spectrum. The second goal of the paper is to explore this regularity issue which is delicate for the systems under consideration because of the presence of the incompressibility condition. These points are elaborated below.
Through out the paper, we will follow the usual Einstein summation convention with respect to repeated indices. We introduce now our first model. Assuming that the viscosity (denoted by
−∇⋅(μϵ∇uϵ)+∇pϵ=f in Ω,∇⋅uϵ=0 in Ω,uϵ=0 on ∂Ω.} | (1) |
As usual,
V={v∈H10(Ω)d;∇⋅v=0 in Ω}, | (2) |
Here
∫Ωμϵ∇uϵ⋅∇v dx=∫Ωf⋅v dx∀v∈V. | (3) |
The classical Lax-Milgram Lemma (essentially, Riesz Representation Theorem due to the symmetry of our bilinear form) ensures existence and uniqueness of a solution
V⊥:={w∈H−1(Ω)d; ⟨w,v⟩H−1(Ω)d,H10(Ω)d=0,∀v∈V}={∇p; p∈L2(Ω)}, | (4) |
which implies that the pressure
||uϵ||(H10(Ω))d+||pϵ||L2(Ω)≤C||f||(L2(Ω))d. | (5) |
We are interested here in the homogenization limit of (1), that is the asymptotic limit of the solution
(A∗)klαβ=∫Tdμ(y)∇(χkα+yαek):∇(χlβ+yβel) dy, | (6) |
in which figure the cell test functions
−∇⋅(μ∇(χkα+yαek))+∇Πkα=0 in Td∇⋅χkα=0 in Td(χkα,Πkα)is Y−periodic. } | (7) |
We impose
(A∗)klαβ=(A∗)lkβα, | (8) |
which corresponds to the fact that the fourth-order tensor
Theorem 1.1. The homogenized limit of the problem (1) is
−∂∂xβ((A∗)klαβ∂uk∂xα)+∂p∂xl=fl in Ω, for l=1,2,...,d,∇⋅u=0in Ω,u=0 in ∂Ω.} | (9) |
More precisely, we have the convergence of solutions:
(uϵ,pϵ)⇀(u,p) in H10(Ω)d×L20(Ω) weak. |
Note that the simple symmetry (8) does not imply that
Let us next consider the second model of incompressible elasticity :
−∇⋅(μϵE(uϵs))+∇pϵs=f in Ω,∇⋅uϵs=0 in Ω,uϵs=0 on ∂Ω.} | (10) |
Here the strain rate tensor is given by
E(v)=12(∇v+∇tv) namely Ekl(v)=12(∂vk∂xl+∂vl∂xk). |
As before, there exists a unique solution
||uϵs||(H10(Ω))d+||pϵs||L2(Ω)≤C||f||(L2(Ω))d, | (11) |
where the constant
(A∗s)klαβ=∫Tdμ(y)E(˜χkα+yαek):E(˜χlβ+yβel) dy | (12) |
where the cell test functions
−∇⋅(μE(˜χkα+yαek))+∇˜Πkα=0 in Td∇⋅˜χkα=0 in Td(˜χkα,˜Πkα)is Y− periodic } | (13) |
We impose
(A∗s)klαβ=(A∗s)αlkβ=(A∗s)kβαl=(A∗s)lkβα, | (14) |
which corresponds to the fact that the fourth-order tensor
The first goal of this paper is to give an alternate proof of Theorem 1.1 using the Bloch Wave Method instead of two-scale asymptotic expansions and the method of oscillating test functions. The notion of Bloch waves is well-known in physics and mathematics [8,11,19,22]. Bloch waves are eigenfunctions of a family of "shifted" spectral problems in the unit cell
The Bloch wave method for scalar equations and systems without differential constraints (like the incompressibility condition) was studied in [12,13,14,21]. In such cases, this approach gives a spectral representation of the homogenized tensor
Let us end this discussion with two general remarks on Bloch wave method. First one is about the nature of convergence of the homogenization process. It is well-known in the homogenization theory that the convergence in Theorem 1.1 is only weak and not strong. To have strong convergence, we need the so-called correctors [8]. Within Block wave theory, correctors are discussed in [12] for the scalar equation. We do not construct explicitly correctors for Stokes system in this paper, even though all necessary ingredients are presented. Because of the lack of smoothness of the bottom level Bloch spectrum, corrector issue is worth considering separately. The second remark is about non-periodic coefficients. Bloch wave approach to homogenization is well developed only in the case of periodic coefficients. It is known that for some restricted class of locally-periodic/modulated coefficients, new phenomena (like localization) may appear [2,3,4]. We are not aware of a Bloch wave approach for more general coefficients.
The plan of this paper is as follows. In section 2, we recall from [7] the properties of Bloch waves associated with the Stokes operator. It turns out that the Bloch waves and their energies can be chosen to be directionally regular, upon modifying the spectral cell problem at zero momentum. Bloch transform using eigenfunctions lying at the bottom of the spectrum is also introduced in this section. Its asymptotic behaviour for low momenta is also described. Next, Section 3 is devoted to the computation of directional derivatives of Bloch spectral data. Even though these results are essentially borrowed from [7], some new ones are also included because of their need in the sequel. In particular we derive the so-called propagation relation linking the homogenized tensor
Note added in proof. At the end of the introduction we claimed that we were not aware of a Bloch wave approach for non-periodic coefficients. We recently learned about a new work in this direction by A. Benoit and A. Gloria, ''Long-time homogenization and asymptotic ballistic transport of classical waves", which will appear in Annales Scientifiques de l'Ecole Normale Supérieure.
In this section, we introduce Bloch waves associated to the Stokes operator following the lead of [7]. The Bloch waves are defined by considering the shifted (or translated) eigenvalue problem in the torus
−D(η)⋅(μD(η)ϕ)+D(η)Π=λ(η)ϕ in Td,D(η)⋅ϕ=0 in Td,(ϕ,Π) is Y− periodic,∫Y|ϕ|2dy=1.} | (15) |
The solutions of (15)
D(η)=∇y+iη |
the shifted gradient operator, with
The main feature of (15) is that the state space keeps varying with
As discussed in introduction, lack of regularity of the Bloch spectrum at
−D(δe)⋅(μ(y)D(δe)ϕ(y;δ))+D(δe)q(y;δ)+q0(δ)e=λ(δ)ϕ(y;δ) in Td,D(δe)⋅ϕ(y;δ)=0 in Td,e⋅∫Tdϕ(y;δ) dy=0,(ϕ,q) is Y− periodic,∫Td|ϕ(y;δ)|2 dy=1.} | (16) |
Note that if
It is natural to consider the system (16) with
−∇⋅(μ∇w)+∇q+q0ˆη=ν(ˆη)w in Td,∇⋅w=0 in Td,ˆη⋅∫Yw dy=0,(w,q) is Y− periodic,∫Y|w|2 dy=1.} | (17) |
Existence of eigenvalues and eigenvectors for either (16) or (17) is proved in [7]. Let us recall their result, by specializing to the eigenvalue
Theorem 2.1. Fix
(ⅰ)
(ⅱ)
(ⅲ) The set
(ⅳ) For each interval
The above theorem says that there are
Bϵm,ˆηg(ξ)=∫Rdg(x)⋅¯ϕm,ˆη(xϵ,δ)e−ix⋅ξ dx, | (18) |
where
For later purposes we need the Bloch transform for
Bϵm,ˆηF(ξ):=∫Rdg0(x)⋅¯ϕm,ˆη(xϵ;δ)e−ix⋅ξ dx+∫Rdid∑j=1ξjgj(x)⋅¯ϕm,ˆη(xϵ;δ)e−ix⋅ξ dx−ϵ−1∫Rdd∑j=1gj(x)⋅∂¯ϕm,ˆη∂yj(xϵ;δ)e−ix⋅ξ dx. | (19) |
Definition (19) is independent of the representation used for
Remark 1. Due to the property
Bϵm,ˆη(F+∇ψ)(ξ)=Bϵm,ˆηF(ξ), for all ψ∈L2(Rd). | (20) |
In fact, by considering
Our next result is concerned with the asymptotic behavior of these Bloch transforms as
Theorem 2.2. Let
χϵ−1Td(ξ)Bϵm,ˆηgϵ(ξ)⇀ϕ0m,ˆη⋅ˆg(ξ),weakly in L2loc(Rdξ) for 1≤m≤d−1 | (21) |
where
Proof. Let us remark that
Bϵm,ˆηgϵ(ξ)=χϵ−1Td(ξ)ϕ0m,ˆη⋅gϵ(ξ)+∫Kgϵ(x)⋅(¯ϕm,ˆη(xϵ;δ)−¯ϕm,ˆη(xϵ;0))e−ix⋅ξ dx. |
By using Cauchy-Schwarz, the second term on the above right hand side can be estimated by the quantity
CK‖ϕm,ˆη(y;δ)−ϕm,ˆη(y;0)‖(L2(Y))d |
where
Since compactly supported elements are dense in
Corollary 1. In the setting of Theorem 2.2, if
χϵ−1TdBϵm,ˆηgϵ(ξ)→ϕ0m,ˆη⋅ˆg,strongly in L2loc(Rdξ) for 1≤m≤d−1. | (22) |
We recall the classical orthogonal decomposition [15] :
L2(Rd)d={∇ψ:ψ∈H1(Rd)}⊕{ϕ∈L2(Rd)d:∇⋅ϕ=0}. | (23) |
Let us denote
X={∇ψ:ψ∈H1(Rd)}, so that, X⊥={ϕ∈L2(Rd)d:∇⋅ϕ=0}. | (24) |
By our choice,
Proposition 1. If
Proof. The proof is immediate, as
Corollary 2. In the setting of Theorem 2.2 and Proposition 1 if
Proof. The proof simply follows as
Theorem 2.2 and its corollaries give some sufficient conditions and specify the sense in which the first Bloch transform tends to Fourier transform, which is a sign of homogenization on the Fourier side. We do not believe that these conditions are sharp and exhaust all possibilities. It would be interesting to explore further in this direction.
Remark 2. Bloch waves being incompressible are transversal. Longitudinal direction is missing and it has to be added to get the full basis. Naturally, asymptotics of the Bloch transform contains information of the Fourier transform only in transversal directions. It contains no information in the longitudinal direction. Because of this feature, in the homogenization limit also, there is no information in the longitudinal direction. This is however proved to be enough to complete the homogenization process because the limiting velocity field is incompressible. See Section 5.
In this section, we give the expressions of the derivatives (at
−D(δˆη)⋅(μ(y)D(δˆη)ϕm,ˆη(y;δ))+D(δˆη)qm,ˆη(y;δ)+q0,m,ˆη(δ)ˆη=λm,ˆη(δ)ϕm,ˆη(y;δ) in Td,D(δˆη)⋅ϕm,ˆη(y;δ)=0 in Td,ˆη⋅∫Tdϕm,ˆη(y;δ) dy=0,(ϕm,ˆη,qm,ˆη) is Y− periodic.} | (25) |
Zeroth order derivatives : For
First order derivatives : Let us differentiate (25) once with respect to
−D(δˆη)⋅(μ(y)D(δˆη)ϕ′m,ˆη(y;δ))+D(δˆη)q′m,ˆη(y;δ)+q′0,m,ˆη(δ)ˆη−λm,ˆη(δ)ϕ′m,ˆη(y;δ)=f(δ) in Td,D(δˆη)⋅ϕ′m,ˆη(y;δ)=g(δ) in Td,ˆη⋅∫Tdϕ′m,ˆη(y;δ) dy=0,(ϕ′m,ˆη,q′m,ˆη) is Y− periodic} | (26) |
where,
f(δ)=λ′m(δ)ϕm,ˆη(y;δ)−iqm,ˆη(y;δ)ˆη+iˆη⋅μ(y)D(δˆη)ϕm,ˆη(y;δ)+iD(δˆη)⋅(μ(y)ϕm,ˆη(y;δ)⊗ˆη),g(δ)=−iˆη⋅ϕm,ˆη(y;δ). |
We put
q′0,m,ˆη(0)ˆη=λ′m,ˆη(0)ϕ0m,ˆη. |
Taking scalar product with
Using the above information in (26), we find that
−∇⋅(μ(y)∇ϕ′m,ˆη(y;0))+∇q′m,ˆη(y;0)=i∇⋅(μ(y)ϕ0m,ˆη⊗ˆη) in Td,∇⋅ϕ′m,ˆη(y;0)=0 in Td,ˆη⋅∫Tdϕ′m,ˆη(y;0) dy=0,∫Tdq′m,ˆη(y;0) dy=0(ϕ′m,ˆη(y;0),q′m,ˆη(y;0)) is Y− periodic.} | (27) |
Comparing this with (7), it can be seen that that
ϕ′m,ˆη(y;0)=iˆηαχrα(y)(ϕ0m,ˆη)r+ζm,ˆη | (28) |
where
In a similar manner, the derivative of the eigenpressure
q′m,ˆη(y;0)=iˆηαΠrα(y)(ϕ0m,ˆη)r, | (29) |
That is, the
Second order derivatives : Next we differentiate (26) with respect to
−D(δˆη)⋅(μ(y)D(δˆη)ϕ′′m,ˆη(y;δ))+D(δˆη)q′′m,ˆη(y;δ)+q′′0,m,ˆη(δ)ˆη−λm,ˆη(δ)ϕ′′m,ˆη(y;δ)=F(δ) in Td,D(δˆη)⋅ϕ′′m,ˆη(y;δ)=G(δ) in Tdˆη⋅∫Tdϕ′′m,ˆη(y;δ) dy=0,(ϕ′′m,ˆη,q′′m,ˆη) is Y− periodic} | (30) |
where
F(δ)=−2μ(y)ϕm,ˆη(y;δ)+2iˆη⋅μ(y)D(δˆη)ϕ′m,ˆη(y;δ)+2iD(δˆη)⋅(μ(y)ϕ′m,ˆη(y;δ)⊗ˆη)−2iˆηq′m,ˆη(y;δ)+λ′′m,ˆη(δ)ϕm,ˆη(y;δ)+2λ′m,ˆη(δ)ϕ′m,ˆη(y;δ),G(δ)=−2iˆη⋅ϕ′m,ˆη(y;δ). | (31) |
We consider (30) at
q′′0,m,ˆη(0)ˆηk=−2∫Tdμ(y)(ϕ0m,ˆη)k dy−2∫Td[ˆηβμ(y)∇yχlβ(y)(ϕ0m,ˆη)l]kαˆηα dy+λ′′m,ˆη(0)(ϕ0m,ˆη)k |
or,
−12(q′′0,m,ˆη(0)ˆηk−λ′′m,ˆη(0)(ϕ0m,ˆη)k)=∫Tdμ(y)[δlkδαβ+(∇χlβ)kα]dy ˆηαˆηβ(ϕ0m,ˆη)l dy=∫Tdμ(y)[∇(yβel):∇(yαek)+∇χlβ:∇(yαek)] dy ˆηαˆηβ(ϕ0m,ˆη)l dy=(A∗)klαβˆηαˆηβ(ϕ0m,ˆη)l.=[(ϕ0m,ˆη)tM(ˆη,A∗)]k=[M(ˆη,A∗)(ϕ0m,ˆη)]k | (33) |
where
M(ˆη,A∗)kl=(A∗)klαβˆηαˆηβ. |
This is nothing but a contraction of the homogenized tensor
−12q′′0,m,ˆη(0)=M(ˆη,A∗)ϕ0m,ˆη⋅ˆη and 12λ′′m,ˆη(0)=M(ˆη,A∗)ϕ0m,ˆη⋅ϕ0m,ˆη. |
It also follows that
By summarizing the above computations, we have
Theorem 3.1. For
(ⅰ)
(ⅱ)
(ⅲ) The derivative of the eigenfunction
ϕ′m,ˆη(y;0)=iˆηαχrα(y)(ϕ0m,ˆη)r+ζm,ˆη |
where
(ⅳ) The derivative of the eigenfunction
q′m,ˆη(y;0)=iˆηαΠrα(y)(ϕ0m,ˆη)r. |
(v) The second derivative of the eigenvalue
12λ′′m,ˆη(0)ϕ0m,ˆη=12q′′0,m,ˆη(0)ˆη+M(ˆη,A∗)ϕ0m,ˆη | (34) |
where
M(ˆη,A∗)kl=(A∗)klαβˆηαˆηβ. |
Remark 3. The above matrix
Remark 4. In the linearized elasticity system, the propagation relation is an eigenvalue relation. Here, relation (34) can again be seen as an eigenvalue problem, posed in the
Case of Symmetrized gradient : We recall the incompressible elasticity system (10) with the symmetrized gradient introduced in Section 1
−∇⋅(μϵE(uϵs))+∇pϵs=f in Ω,∇⋅uϵs=0 in Ω,uϵs=0 on ∂Ω} | (35) |
where
We introduce Bloch waves associated to the Stokes operator defined in (35).
Find
−D(η)⋅(μE(η)ϕs)+D(η)Πs=λs(η)ϕs in RdD(η)⋅ϕs=0 in Rd(ϕs,Πs) is Y− periodic∫Y|ϕs|2 dy=1.} | (36) |
As usual
2E(η)ψ=(∇+iη)ψ+(∇+iη)tψ,(2E(η)ψ)kl=(∂ψk∂xl+iηlψk)+(∂ψl∂xk+iηkψl). |
As earlier, we modify the spectral problem (36) as follows : Find
−D(δe)⋅(μ(y)E(δe)ϕs(y;δ))+D(δe)qs(y;δ)+ q0,s(δ)e=λs(δ)ϕs(y;δ) in TdD(δe)⋅ϕs(y;δ)=0 in Tde⋅∫Tdϕs(y;δ) dy=0,(ϕs,qs) is Y− periodic,∫Td|ϕs(y;δ)|2 dy=1.} | (37) |
As before, we can compute directional derivatives of the solution of (37) and prove a result completely analogous to Theorem 3.1. In particular, we will have the following propagation relation : For
12λ′′s,m,ˆη(0)ϕ0s,m,ˆη=12q′′0,s,m,ˆη(0)ˆη+M(ˆη,A∗s)ϕ0s,m,ˆη, | (38) |
where
M(ˆη,A∗s)jl=(A∗s)jlαβˆηαˆηβ. |
In the scalar self-adjoint case, it is known that the homogenized matrix is equal to one-half the Hessian of the first Bloch eigenvalue at zero momentum [13]. In the general (non-symmetric) scalar case, treated in [21], it was shown that only the symmetric part of the homogenized matrix is determined by the Bloch spectrum and it is given again by the same one-half of the Hessian of the first Bloch eigenvalue (which exists by virtue of the Krein-Rutman theorem). The fact that only the symmetric part of the homogenized matrix plays a role is not a big surprise since, the homogenized tensor
∇⋅A∗∇=d∑k,l=1A∗kl∂2∂xk∂xl |
depends only on the symmetric part of
In the case of systems, another phenomenon takes place. For example, the linearized elasticity system (in which there are no differential constraints) was treated in [14] where it was recognized that not only Bloch eigenvalues but also Bloch eigenfunctions at zero momentum are needed to determine the homogenized tensor. More precisely, this connection between Bloch eigenvalues and eigenfunctions, on the one hand, and the homogenized tensor, on the other hand, was expressed via a relation called propagation relation in [14] which uniquely determines the homogenized tensor.
In the case of Stokes system, a new phenomenon arises because of the presence of a differential constraint (the incompressibility condition). Even though there is an analogue of the propagation relation (see (34) above), it does not determine uniquely the homogenized tensor. In fact the propagation relation (34) is unaltered if we add a multiple of
Our concern now is the following question: to what extent do the Bloch spectral elements determine the homogenized tensor
Proposition 2. Let
B∗−A∗=c(I⊗I)+N | (39) |
where
Njlαβ=−Njlβα=−Nljαβwhenever, (α,β)≠(j,l) and (β,α)≠(j,l)Niiii=0.} | (40) |
Proof. Let us observe that the addition of
M(ˆη,A∗+c(I⊗I)+N)jl=(A∗)jlαβˆηαˆηβ+cδαjδβlˆηαˆηβ+Njlαβˆηαˆηβ=M(ˆη,A∗)jl+cˆηjˆηl. |
Since
M(ˆη,A∗+c(I⊗I)+N)ϕ0m,ˆη=M(ˆη,A∗)ϕ0m,ˆη. |
Conversely, let us assume that there are two fourth-order tensors
Step 1. We begin with showing the matrix
M(ˆη,A∗)jl=(A∗)jlαβˆηαˆηβ=(A∗)jlβαˆηβˆηα=(A∗)ljαβˆηαˆηβ=M(ˆη,A∗)lj | (41) |
which shows the required symmetry.
Step 2. For
˜Njlαβˆηαˆηβ=cˆηjˆηl1≤j,l≤d. | (42) |
Step 3. Under condition (42), we verify that
˜Niiii=c ∀ i. | (43) |
and˜Njlik+˜Njlki =0 if (i,k)≠(j,l) and (k,i)≠(j,l). | (44) |
For this purpose, let us take
˜Niiii = c | (45) |
and ˜Njlii= 0 if i≠j or i≠l. | (46) |
In particular, (43) is proved. Next, choosing
˜Njlii+˜Njlkk+˜Njlik+˜Njlki=c(δji+δjk)(δli+δlk). | (47) |
To check (44), there are several cases to consider.
(ⅰ)
(ⅱ) Similarly, for
(ⅲ)
˜Njljj+˜Njlij+˜Njlji=c(δli+δlj). | (48) |
Now together with
˜Njljj+˜Njlij+˜Njlji=cδlj. | (49) |
Then both
(ⅳ) Similarly, for
˜Njiii+˜Njiik+˜Njiki=c(δji+δjk). | (50) |
Together with
˜Njiii+˜Njiik+˜Njiki=cδji. | (51) |
Then both
Step 4. Now we consider the two remaining cases not covered in (44).
(ⅰ)
˜Nikii+˜Nikkk+˜Nikik+˜Nikki=c(1+δik)2. |
For
˜Nikik+˜Nikki=c. | (52) |
(ⅱ) Similarly, for
˜Nkiik+˜Nkiki=c | (53) |
Step 5. Let us set
Njlik=−Njlki=−Nljik.whenever, (i,k)≠(j,l) and (k,i)≠(j,l) | (54) |
From its very definition
Next we extend Proposition 2 to the Stokes system (10), featuring a symmetric gradient tensor. In this case the propagation relation (34) is replaced by (38) and the homogenized tensor is denoted by
Proposition 3. The propagation relation (38) characterizes uniquely the tensor
B∗s−A∗s=c(I⊗I). | (55) |
Proof. The proof continues from the Step 5 of the previous proof of Proposition 2. We defined
Njlik=−Njlki=−Nljik.whenever, (i,k)≠(j,l) and (k,i)≠(j,l) |
Now as
Njlik=Niljk=Nljki=Njkilfor all i,j,k,l. | (56) |
This symmetry combined with the anti-symmetry established in the previous step implies that
This can be seen as follows: whenever
Njlik=−Nljik=−Nijlk=Njilk=Nijkl=Nkjil=−Njkil=−Njlik | (57) |
ThusNjlik=0. | (58) |
Similarly, whenever
˜Nikik+˜Nikki = c = ˜Nkiik+˜Nkiki. |
Then using (56) and (46) we clearly have
Nikik= 0 = Nkiki. | (59) |
Therefore (58), (59) imply that
Remark 5. The conclusion of the above proposition was conjectured in [7] and it is proved here to be true whenever we are working with the system (10) with symmetrized gradient. However, it is not true with the full gradient Stokes system (1) as shown by Proposition 2. However, in both of these cases the propagation relation fixes the homogenized operator(9) uniquely, as is stated in the following proposition.
Proposition 4. If (39) is satisfied, then
Proof. We have to check that
u=(uk)1≤k≤d → (−∂∂xβ((A∗−B∗)klαβ∂uk∂xα))1≤l≤d |
is
This section is devoted to a proof of Theorem 1.1, our main homogenization result stated in the first section. It is based on the tools that we have introduced so far. A similar proof is given for the linear elasticity problem in [21]. However, the presence of a pressure and a differential constraint in the Stokes system seriously complexifies the analysis and has a non-trivial effect in the homogenization process. Besides, we also bring some simplifications to the proof given in [21].
We consider a sequence of solutions
||uϵ||(H10(Ω))d+||pϵ||L2(Ω)≤C||f||(L2(Ω))d, | (60) |
where
There are several steps in the proof. First, we localize the Stokes system (1) by applying a cut-off function technique to the velocity
Notation. in the sequel L.H.S. stands for left hand side, and R.H.S. for right hand side.
Step 1. Localization of the velocity
−∂∂xα(μϵ∂∂xα)(vuϵl)+∂pϵ∂xlv=vfl+gϵl+hϵlin Rd, | (61) |
where,
gϵl=−2μϵ∂uϵl∂xα∂v∂xα−μϵ∂2v∂xα∂xαuϵlandhϵl=−∂μϵ∂xα∂v∂xαuϵl. | (62) |
Note that,
Step 2. Limit of
ϕϵm,ˆη(x;δ)=ϕm,ˆη(xϵ;ϵ(ξ⋅ˆη)), and λϵm,ˆη(δ)=ϵ−2λm,ˆη(ϵ(ξ⋅ˆη))qϵm,ˆη(x;δ)=ϵ−1qm,ˆη(xϵ;ϵ(ξ⋅ˆη)), and qϵ0,m,ˆη(δ)=ϵ−2q0,m,ˆη(ϵ(ξ⋅ˆη)). |
They satisfy the following system because of (25) :
−D(δˆη)⋅(μϵ(x)D(δˆη)ϕϵm,ˆη(x;δ))+D(δˆη)qϵm,ˆη(x;δ)+ qϵ0,m,ˆη(δ)ˆη=λϵm,ˆη(δ)ϕϵm,ˆη(x;δ) in Rd,D(δˆη)⋅ϕϵm,ˆη(x;δ)=0 in Rd,ˆη⋅∫Rdϕϵm,ˆη(x;δ) dx=0,(ϕϵm,ˆη,qϵm,ˆη) is ϵY− periodic,∫ϵTd|ϕϵm,ˆη(x;δ)|2 dx=1.} | (63) |
Let us first consider the L.H.S. of (61). For
Bϵm,ˆη(−∂∂xα(μϵ∂∂xα)g)(ξ)=⟨eix⋅ξϕϵm,ˆη(.;δ),−∂∂xα(μϵ∂∂xα)g⟩=⟨g,−∂∂xα(μϵ∂∂xα)(eix⋅ξϕϵm,ˆη(.;δ))⟩=⟨g,λϵm,ˆη(δ)eix⋅ξϕϵm,ˆη(.;ξ)−∇(qϵm,ˆη(.;δ)eix⋅ξ)−qϵ0,m,ˆη(δ)ˆηeix⋅ξ⟩=λϵm,ˆη(δ)Bϵm,ˆηg(ξ)−⟨g,∇(qϵm,ˆη(.;δ)eix⋅ξ)⟩−⟨g,qϵ0,m,ˆη(δ)ˆηeix⋅ξ⟩. |
In the previous equation the duality bracket is between
Therefore,
λϵm,ˆη(δ)Bϵm,ˆη(vuϵ)(ξ)−⟨vuϵ,∇(qϵm,ˆη(.;δ)eix⋅ξ)⟩−⟨vuϵ,qϵ0,m,ˆη(δ)ˆηeix⋅ξ⟩+Bϵm,ˆη(v∇pϵ)(ξ). | (64) |
Below, we treat each term of (64) one by one.
1st term of (64) : By using the Taylor expansion
λϵm,ˆη(δ)=ϵ−2λm,ˆη(ϵ(ξ⋅ˆη))=12λ′′m,ˆη(0)(ξ⋅ˆη)2+O(ϵ(ξ⋅ˆη)3) | (65) |
and then using Theorem 2.2, we get
χϵ−1Td(ξ)λϵm,ˆη(δ)Bϵm,ˆη(vuϵ)(ξ)→12λ′′m,ˆη(0)(ξ⋅ˆη)2ϕ0m,ˆη⋅^(vu)(ξ)in L2loc(Rdξ) strongly, | (66) |
where we recall that
(ξ⋅ˆη)2(12q′′0,m,ˆη(0)ˆη+M(ˆη,A∗)ϕ0m,ˆη)⋅^(vu)(ξ)=(ξ⋅ˆη)212q′′0,m,ˆη(0)ˆηk^(vuk)+(ξ⋅ˆη)2(A∗)klαβˆηαˆηβ(ϕ0m,ˆη)l(^vuk)(ξ). | (67) |
2nd term of (64) :
−⟨vuϵ,∇(qϵm,ˆηeix⋅ξ)⟩=⟨∇⋅(vuϵ),eix⋅ξqϵm,ˆη⟩=⟨uϵ⋅∇v,eix⋅ξqϵm,ˆη⟩ (as ∇⋅uϵ=0). | (68) |
Using the Taylor expansion of
qϵm,ˆη(x;δ)=ϵ−1qm,ˆη(xϵ;ϵ(ξ⋅ˆη))=ϵ−1qm,ˆη(xϵ;0)+(ξ⋅ˆη)q′m,ˆη(xϵ;0)+O(ϵ(ξ⋅ˆη)2), | (69) |
(prime denotes the derivative with respect to the second variable), with the properties that (cf. Theorem 2.1)
qm,ˆη(xϵ;0)=0 and q′m,ˆη(xϵ;0)⇀MTd(q′m,ˆη(y;0))=0 weakly in L2(Rd); (as q′m,ˆη(y;0)∈L20(Td)) | (70) |
where,
Then by using
−⟨vuϵ,∇(qϵm,ˆηeix⋅ξ)⟩→⟨u⋅∇v,eix⋅ξMTd(q′m,ˆη)⟩=0 in L2loc(Rdξ) strongly. | (71) |
It is also used that, the error term
3rd term of (64) : We use the Taylor expression of
qϵ0,m,ˆη(δ)=ϵ−2q0,m,ˆη(ϵ(ξ⋅ˆη))=12q′′0,m,ˆη(0)+O(ϵ(ξ⋅ˆη)2). | (72) |
So,
−⟨vuϵ,qϵ0,m,ˆη(δ)eix⋅ξˆη⟩→−⟨vu,12q′′0,m,ˆη(0)(ξ⋅ˆη)2ˆηeix⋅ξ⟩ in L2loc(Rdξ) strongly.=−12q′′0,m,ˆη(0)(ξ⋅ˆη)2^(vu)⋅ˆη. | (73) |
4th term of (64) : Finally, we consider the remaining fourth term in (64), and doing integration by parts we get
Bϵm,ˆη(v∇pϵ)(ξ)=⟨v∇pϵ,eix⋅ξϕϵm,ˆη⟩=−⟨pϵ,∇v⋅eix⋅ξϕϵm,ˆη⟩(as ∇⋅(eix⋅ξϕϵm,ˆη)=0). | (74) |
We use the Taylor expansion
ϕϵm,ˆη(x;ξ)=ϕm,ˆη(xϵ;0)+ϵ(ξ⋅ˆη)ϕ′m(xϵ;0)+O((ϵ(ξ⋅ˆη))2)=ϕ0m,ˆη+ϵ(ξ⋅ˆη)ϕ′m,ˆη(xϵ;0)+O((ϵ(ξ⋅ˆη))2)→ϕ0m,ˆη in L2loc(Rdξ,(L2(Ω))d) strongly. | (75) |
And from (60) as
pϵ⇀p in L2(Ω). | (76) |
Thus by passing to the limit in the R.H.S. of (74), we get
−⟨pϵ,∇v⋅eix⋅ξϕϵm,ˆη⟩→−⟨p,∇v⋅eix⋅ξϕ0m,ˆη⟩=⟨∇p,veix⋅ξϕ0m,ˆη⟩(as ∇⋅(eix⋅ξϕ0m,ˆη)=0). | (77) |
Thus
χϵ−1TdBϵm,ˆη(v∇pϵ)(ξ)→ϕ0m,ˆη⋅^(v∇p)(ξ) in L2loc(Rdξ) strongly. | (78) |
This property proved for
Summary so far : Combining the previous results, therefore, by taking the Bloch transformation
(A∗)klαβˆηαˆηβ(ξ⋅ˆη)2(ϕ0m,ˆη)l^(vuk)(ξ)+ϕ0m,ˆη⋅^(v∇p)(ξ)in L2loc(Rdξ) strongly. | (79) |
Step 3. Limit of
Bϵm,ˆη(vf)(ξ)+Bϵm,ˆη(gϵ)(ξ)+Bϵm,ˆη(hϵ)(ξ). | (80) |
We treat below each of these terms separately. Passing to the limit in the first term is straightforward (cf. Corollary 1) and we obtain
χϵ−1Td(ξ)Bϵm,ˆη(vf)(ξ)→ϕ0m,ˆη⋅^(vf) in L2loc(Rdξ) strongly. | (81) |
Limit of
χϵ−1Td(ξ)Bϵm,ˆη(σϵlα∂v∂xα)(ξ)⇀^(σlα∂v∂xα)(ξ)(ϕ0m,ˆη)lin L2loc(Rdξ) weakly. | (82) |
Due to the strong convergence of
χϵ−1Td(ξ)Bϵm,ˆη(μϵΔvuϵ)(ξ)⇀MTd(μ(y))^(Δvu)(ξ)⋅ϕ0m,ˆηin L2loc(Rdξ) weakly. | (83) |
Combining the above two convergence results and doing integration by parts, we obtain
χϵ−1Td(ξ)Bϵm,ˆη(gϵ)(ξ)⇀−2^(σ∇v)(ξ)⋅ϕ0m,ˆη−MTd(μ(y))^(Δvu)(ξ)⋅ϕ0m,ˆη in L2loc(Rdξ) weakly. | (84) |
Limit of
Bϵm,ˆη(hϵ)=−Bϵm,ˆη((∇μϵ⋅∇v)uϵ)(ξ)=−⟨(∇μϵ⋅∇v)uϵ,eix⋅ξϕ0m,ˆη⟩−⟨(∇μϵ⋅∇v)uϵ,eix⋅ξϵ(ξ⋅ˆη)ϕ′m(xϵ;0)+O((ϵ(ξ⋅ˆη))2)⟩. |
We start with the second term. By doing integration by parts, it becomes
(ξ⋅ˆη)∫Rde−ix⋅ξ(μϵ∇y¯ϕ′m(xϵ;0)uϵ)⋅∇v dx+O(ϵ(ξ⋅ˆη)). | (85) |
Thanks to the strong convergence of
(ξ⋅ˆη)∫Rde−ix⋅ξ(MTd(μ(y)∇y¯ϕ′m(y;0))u)⋅∇v dx. | (86) |
Next, we consider the first term of the R.H.S. of (85). After doing integration by parts, one has
∫Rde−ix⋅ξ[μϵΔv (uϵ⋅ϕ0m,ˆη)+((μϵ∇uϵ)ϕ0m,ˆη)⋅∇v−iμϵ((ϕ0m,ˆη⊗ξ)uϵ)⋅∇v]dx. | (87) |
In a manner similar to the above arguments, the limit of (87) would be
∫Rde−ix⋅ξ[MTd(μ(y))Δv (u⋅ϕ0m,ˆη)+(σϕ0m,ˆη)⋅∇v]dx−∫Rde−ix⋅ξ[iMTd(μ(y))((ϕ0m,ˆη⊗ξ)u)⋅∇v] dx. | (88) |
Now combining (86) and (88) and using the fact
ϕ′m(y;0)−iˆηβχlβ(y)(ϕ0m,ˆη)l |
is a constant vector of
∇yϕ′m(y;0)=iˆηβ∇yχlβ(y)(ϕ0m,ˆη)l, |
we see that
−i(ξ⋅ˆη)[MTd(μ(y)ˆηβ∇yχlβ(y)(ϕ0m,ˆη)l)]kα^(∂v∂xαuk)(ξ)+MTd(μ(y))^(Δv uk)(ξ)(ϕ0m,ˆη)k+^(σlβ∂v∂xβ)(ξ)(ϕ0m,ˆη)l−iMTd(μ(y))(ϕ0m,ˆη)kξα^(∂v∂xαuk)(ξ). | (89) |
Step 4. Limit of
(A∗)klαβξαξβ^(vuk)(ξ)(ϕ0m,ˆη)l+^(v∂p∂xl)(ξ)(ϕ0m,ˆη)l=^(vfl)(ξ)(ϕ0m,ˆη)l−2^(σlβ∂v∂xβ)(ξ)(ϕ0m,ˆη)l−MTd(μ(y))^(Δv uk)(ξ)(ϕ0m,ˆη)k −i[MTd(μ(y)∇yχlβ(y))]kα(ϕ0m,ˆη)lξβ^(∂v∂xαuk)(ξ)+MTd(μ(y))^(Δv uk)(ξ)(ϕ0m,ˆη)k +^(σlβ∂v∂xβ)(ξ)(ϕ0m,ˆη)l−iMTd(μ(y))δαβδlk(ϕ0m,ˆη)lξβ^(∂v∂xαuk)(ξ). | (90) |
The above equation has to be considered as the localized homogenized equation in the Fourier space. The conclusion of Theorem 1.1 will follow as a consequence of this equation.
Step 5. Passage from Fourier space (
[L(ξ)−R(ξ)]⋅ϕ0m,ˆη=0 for m=1,..,(d−1). |
Observe that, the quantity
[L(ξ)−R(ξ)]=c(ξ)ξ for some scalar c(ξ). |
Therefore, for all test functions
[L(ξ)−R(ξ)]⋅ˆw(ξ)=0. |
Now by using the Plancherel's theorem, we have
∫RdF−1[L(ξ)−R(ξ)](x)⋅ˉw(x) dx=0,∀w∈(L2(Rd))d satisfying divw=0 | (91) |
where
We easily compute
Il(x)=(−(A∗)klαβ∂2(vuk)∂xβ∂xα+v∂p∂xl)−(vfl−σl,β∂v∂xβ−(A∗)klαβ∂∂xβ(∂v∂xαuk)) in Rd, |
which simplifies in
Il=(−(A∗)klαβ∂2uk∂xβ∂xα+∂p∂xl−fl)v−((A∗)klαβ∂uk∂xα−σl,β)∂v∂xβ in Rd. |
We pose
F1l=(−(A∗)klαβ∂2uk∂xβ∂xα+∂p∂xl−fl) and F2lβ=F2βl=−((A∗)klαβ∂uk∂xα−σl,β) | (92) |
to write
Il=F1l v+F2lβ ∂v∂xβ. |
Using (91), it follows from de Rham's theorem that
Step 5A. To show
divψζ,ω=0 in Rd with ψζ,ω=ζe−inx⋅ω in Ω, where ζ⊥ω. | (93) |
The existence of such a function
divψζ,ω=0 in B(0,R0)∖¯Ω,ψζ,ω=0 on ∂B(0,R0),ψζ,ω=ζe−inx⋅ω on ∂Ω. | (94) |
There exists a solution of (94) (see [15,Page No. 24]) since the boundary data satisfies the required compatibility condition (recall that we assume
∫∂Ωζe−inx⋅ω⋅ν dσ=∫Ω(ζ⋅ω)e−inx⋅ω dx=0. |
Then extending
Now using these
∫ΩF1lζl v0 dx+∫ΩF2lβ∂v0∂xβζl dx+n∫ΩF2lβωβζl v0 dx=0 |
and dividing by
∫Ω(F2 ω⋅ζ)v0 dx=0. | (95) |
As
σlβ=qδlβ+(A∗)klαβ∂uk∂xα. | (96) |
Step 5B. To show
∫Ω(F1kv+q∂v∂xk)dx=0 for all v∈D(Ω), |
which implies
Step 5C Using Step 5A and Step 5B in (92), and considering the relation
−∂σlβ∂xβ+∂p∂xl=fl in Ω, l=1,…,d. | (97) |
Step 5D In this step, we prove that
σϵll=μϵ∂uϵl∂xl=0 in Ω. |
Passing to the limit
σll=0 in Ω. |
Using this relation in (96) with
(A∗)klαl∂uk∂xα+qd=0. | (98) |
On the other hand, from (6) and (7) we have
(A∗)klαl=∫Tdμ(y)∇(χkα+yαek):∇(ylel) dy=∫Tdμ(y)∂∂yl(χkα+yαek)l dy. |
Thus for fixed
(A∗)klαl=MTd(μ)δkα. | (99) |
Using (98) and (99), as
q=−1d(A∗)klαl∂uk∂xα=−1dMTd(μ)δkα∂uk∂xα=0. |
Finally, the macro constitutive law follows as a consequence from (96) :
σlβ=(A∗)klαβ∂uk∂xα. |
Step 5E. Since
−(A∗)klαβ∂2uk∂xα∂xβ+∂p∂xl=fl in Ω for l=1,..,d.divu=0 in Ωu=0 on ∂Ω. | (100) |
This completes the proof of Theorem 1.1.
This work has been carried out within a project supported by Indo -French Centre for Applied Maths -UMI, IFCAM. G. A. is a member of the DEFI project at INRIA Saclay Ile-de-France. M.V. is in the current address during the preparation of the revised version.
The authors thank the referees for their useful comments to improve the presentation of this paper.
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