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Research article Special Issues

Jointly learning and training: using style diversification to improve domain generalization for deepfake detection

  • Received: 13 December 2023 Revised: 14 February 2024 Accepted: 22 February 2024 Published: 06 March 2024
  • Most existing deepfake detection methods often fail to maintain their performance when confronting new test domains. To address this issue, we propose a generalizable deepfake detection system to implement style diversification by alternately learning the domain generalization (DG)-based detector and the stylized fake face synthesizer (SFFS). For the DG-based detector, we first adopt instance normalization- and batch normalization-based structures to extract the local and global image statistics as the style and content features, which are then leveraged to obtain the more diverse feature space. Subsequently, contrastive learning is used to emphasize common style features while suppressing domain-specific ones, and adversarial learning is performed to obtain the domain-invariant features. These optimized features help the DG-based detector to learn generalized classification features and also encourage the SFFS to simulate possibly unseen domain data. In return, the samples generated by the SFFS would contribute to the detector's learning of more generalized features from augmented training data. Such a joint learning and training process enhances both the detector's and the synthesizer's feature representation capability for generalizable deepfake detection. Experimental results demonstrate that our method outperforms the state-of-the-art competitors not only in intra-domain tests but particularly in cross-domain tests.

    Citation: Jicheng Li, Beibei Liu, Hao-Tian Wu, Yongjian Hu, Chang-Tsun Li. Jointly learning and training: using style diversification to improve domain generalization for deepfake detection[J]. Electronic Research Archive, 2024, 32(3): 1973-1997. doi: 10.3934/era.2024090

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  • Most existing deepfake detection methods often fail to maintain their performance when confronting new test domains. To address this issue, we propose a generalizable deepfake detection system to implement style diversification by alternately learning the domain generalization (DG)-based detector and the stylized fake face synthesizer (SFFS). For the DG-based detector, we first adopt instance normalization- and batch normalization-based structures to extract the local and global image statistics as the style and content features, which are then leveraged to obtain the more diverse feature space. Subsequently, contrastive learning is used to emphasize common style features while suppressing domain-specific ones, and adversarial learning is performed to obtain the domain-invariant features. These optimized features help the DG-based detector to learn generalized classification features and also encourage the SFFS to simulate possibly unseen domain data. In return, the samples generated by the SFFS would contribute to the detector's learning of more generalized features from augmented training data. Such a joint learning and training process enhances both the detector's and the synthesizer's feature representation capability for generalizable deepfake detection. Experimental results demonstrate that our method outperforms the state-of-the-art competitors not only in intra-domain tests but particularly in cross-domain tests.



    Fluid flows in branching tubes are common in many biological and industrial applications such as physiological branching flows and flows through pipe and duct networks (see, for instance, [1,2,3,4,5,6,7,8]). This subject is extensively studied in both theoretical and practical points of views. A mathematical model of fluid flows in a network of thin tubes has been derived in [9] from the asymptotic expansion of Navier–Stokes equations. Consistent asymptotic analysis of Navier–Stokes equations in thin tube structures, by letting the diameter of the tubes tend to zero, has been recently studied in a series of papers, such as [10] and [11]. The Navier–Stokes equations with pressure boundary conditions in the junctions of thin pipes are considered in [12] and [13], where approximations based on Leray and Poiseuille problems are constructed therein.

    Let h be a positive integer. Let Gh be the pre-fractal polygonal curve obtained after h-iterations of the contractive similarities of the Sierpinski gasket G (see Figure 1). We consider a network of circular cylindrical pipes whose axes are the sides of the polygon Gh. We assume that these pipes are narrow axisymmetric tubes of radius εh very small with respect to the length 2h of each side of Gh. We consider an incompressible fluid flow in the bounded domain Ωh consisting of these pipes connected, after local adjustments near the bifurcation points, through smooth thin regions centered at the vertices of Gh (see Figure 4). We suppose that each pipe is split into two principal regions: junction zones of length εhln(1/εh) 2h linked to the ends of the pipe and the rest of the pipe. We suppose that the fluid flow in Ωh is driven by some volumic forces and governed by Stokes equations with boundary conditions for the velocity and the pressure on the external boundary of Ωh and inner continuity conditions for the normal velocity on the interfaces between the junction zones and the rest of the pipes (see Section 2 for more details). We assume that the flow in the junction zones is controlled by a typical Reynolds number Rej,h.

    Figure 1.  Representation of the Sierpinski gasket G.

    The main focus of this paper is to study the asymptotic behavior of the fluid flowing through the branching pipes as the radius of the tubes tends to zero and the sequence of pre-fractal curves converges in the Hausdorff metric to the Sierpinski gasket G. Using Γ-convergence methods (see, for instance, [14] and [15]), we prove that the effective potential energy of the fluid turns out to be of the form

    F(v)={μπm(Θ)Hd(G)Gv2dHd+2μπm(Θ)3σGv.Zvdν              if vV+         otherwise,  (1.1)

    where v is the fluid velocity, μ is the fluid viscosity, m(Θ) is the average value (see Eq. (6.10)) of the solution Θ of boundary value problem (6.5), 1m(Θ)Hd(G) is the permeability of the Sierpinski gasket G, Hd being the d -dimensional Hausdorff measure on G where

    d=ln3/ln2 (1.2)

    stands for the fractal dimension of G, Z is a random matrix given in Section 4 (see Eq. (4.15)–(4.18)), ν is a singular measure with respect to the Hausdorff measure Hd on G called the Kusuoka measure (see Eq. (4.11)), which, according to [16], is a Gibbs measure of special kind, V is the admissible velocities space (see Definition 23), and

    1σ=limhεhRej,h. (1.3)

    Depending on the values of σ, we obtain different asymptotic problems:

    1. If σ(0,+), then Rej,h=O(εh). In this case, the effective flow is described (see Theorem 3) by the following singular Brinkman equation in the Sierpinski gasket G:

    2μπm(Θ)Hd3σHd(G)ΔG(u)+μπHdm(Θ)Hd(G)u+νZp.n=HdHd(G)f.n in G (1.4)

    where u is the fluid velocity, p is the pressure, ΔG is the Laplace operator on the Sierpinski gasket (see Lemma 4), f is the effective source term, n=(1,0) on the horizontal part of G, n=(1/2,3/2) on the part of G which is perpendicular to the unit vector (3/2,1/2), and n=(1/2,3/2) on the part of G which is perpendicular to the unit vector (3/2,1/2). This equation includes the singular Brinkman viscous resistance term 2μπm(Θ)Hd3σHd(G)ΔG(u), which is due to the viscous behavior of the fluid flow at the junction zones, and the singular Darcy resistance term μπHdm(Θ)Hd(G)u.

    2. If σ=+, then Rej,h =O(1) or Rej,h as h. In this case, the term μπm(Θ)3σGv.Zvdν in (1.1) disappears and the flow is governed by singular Darcy's law in the Sierpinski gasket G.

    3. If σ=0, then Rej,h =O(εαh) with α>1. In this case, the energy F(v) is finite only if Gv.Zvdν=0, which implies that the velocity of the fluid flow is asymptotically constant in the Sierpinski gasket G.

    The study of asymptotic analysis of boundary value problems in domains with fractal boundaries or containing thin inclusions developing a fractal geometry has been recently addressed in a series of papers (see, for instance, [17,18,19,20,21,22,23,24,25,26,27,28,29]). The problems obtained at the limit generally consist of singular forms containing fractal terms. The problem considered in this work is quite different from the previous ones, as we deal here with the determination of the fluid motion through branching tubes having a fractal structure. The overall effect of the pre-fractal branching networks on the fluid flow appears in the singular effective equation (1.4), according to the characteristics of the flow, as the radius of the tubes tends to zero and the sequence of pre-fractal curves converges in the Hausdorff metric to the Sierpinski gasket G. The asymptotic representation of the solution of the original singularly perturbed problem includes local perturbations representing the flow in the boundary layers in the junction zones. These local perturbations are solutions of Leray problems in semi-infinite cylinders representing the rescaled junctions. The main novelty of this paper lies in the construction of these local perturbations as well as the derivation of the effective flow described above by singular Brinkman and Darcy laws on the fractal G with divergence-free velocity in a fractal sense specified in Definition 22 in Section 5.

    The problem considered in this work has some implications for modeling the behavior of fluid flows in various complex geometrical configurations of branching tubes. An important field to which this model is closely related is the behavior of fluid flows in some physiological structures such as lung airways (see, for instance, [1] and [30]) the cardiovascular system and cerebral arteriovenous (see, for instance, [30], [31], and [32]). It has been shown that physiological branching networks exhibit fractal structures for minimal energy dissipation (see, for instance, [33] and [34]). In particular, blood vessels have self-similar structures with optimal transport property of their fractal networks (see, for instance, [35]). Blood has been treated in [31] as a homogeneous, incompressible, Newtonian viscous fluid, making the assumptions that the flow is steady and axisymmetric with sufficiently small Reynolds number so that the flow is laminar. The authors observed that the overall effect of the non-Newtonian characteristics would be small.

    The present investigation on fractal branching flows provides some motivations in the haemodynamics. The blood vessels can be illustrated, under some simplifying assumptions, by the network Ωh of narrow branching tubes with laminar flow far ahead of the bifurcations and boundary layer flow near the bifurcations, where the local Reynolds number is the most effective factor controlling the flow throughout the whole network.

    This paper is organized as follows. The statement of the problem is presented in Section 2, with a subsection reserved for the nomenclature and another devoted to the position of the problem. In Section 3, we formulate the main results of this work. In Section 4, we introduce the energy forms, the Kusuoka measures, and gradients on the Sierpinski gasket. Section 5 is devoted to some a priori estimates and compactness results. Section 6 is consecrated to the proof of the main results. A final conclusion is made in Section 7.

    A1A2A3 equilateral triangle of vertices A1=(0,0), A2=(1,0), A3=(1/2,3/2)
    G Sierpinski gasket built in the triangle A1A2A3
    Gh prefractal polygonal curve obtained after h-iterations of contractive similarities of G
    Vh set of vertices of Gh
    Eh set of edges of Gh
    V set of all vertices of G
    Tkh kth triangle of Gh
    Ei,kh=[ai,kh,bi,kh] ith edge of Tkh
    2h length of Ei,kh
    yi,kh,1, yi,kh,2 local variables on Tkh
    εh small positive number
    Πh,ik ith tube of radius εh and of length 2h2εh surrounding Ei,kh
    Bhk(ai,kh) small smooth branch junction of thickness of order 2εh centered at the vertex ai,kh
    Bhk(bi,kh) small smooth branch junction of thickness of order 2εh centered at the vertex bi,kh
    Σh,ik,1 interface between Bhk(ai,kh) and Πh,ik
    Σh,ik,2 interface between Bhk(bi,kh) and Πh,ik
    Ωh,ik pipe formed with Bhk(ai,kh), Bhk(bi,kh), Πh,ik, and the interfaces Σh,ik,α; α=1,2, between them
    Ωh network of the interconnected pipes Ωh,ik
    Γh external boundary of Ωh
    Jh,+,ik small junction zone of length εhln(1/εh) located in the region yi,kh,1>0
    Jh,,ik small junction zone of length εhln(1/εh) located in the region yi,kh,1<2h
    Jh union of the junction zones Jh,±,ik
    μ fluid viscosity
    Rej,h typical Reynolds number in Jh
    Reh characteristic Reynolds number in Ωh
    Euh characteristic Euler number in Ωh
    Frh characteristic Froude number in Ωh
    5h3h+1 scaling factor associated to the ramification of the network Ωh
    d the fractal dimension of G
    Hd d -dimensional Hausdorff measure on G
    L2Hd(G) space of square integrable L2-functions with respect to the measure Hd
    EG Dirichlet form in L2Hd(G)
    Z random matrix
    div Z divergence operator on G
    ν Kusuoka measure
    J+±,i semi infinite cylinders representing the rescaled junctions

    Let us consider the points of the plane xOy: A1=(0,0), A2=(1,0), and A3=(1/2,3/2). Let us denote {ψi}i=1,2,3 as the family of contractive similitudes defined on R2 by

    ψi(x)=x+Ai2x=(x1,x2)R2. (2.1)

    Let V0={A1A2A3} be the set of vertices of the equilateral triangle A1A2A3. We define inductively

    Vh+1=i=1,2,3ψi(Vh) (2.2)

    for every hN, and set

    V=hNVh. (2.3)

    The Sierpinski gasket, which is denoted here by G, is defined as the closure of the set V

    G=¯V. (2.4)

    We consider the graph Gh=(Vh,Eh), where Eh is the set of edges [ah,bh]; ah,bhVh, such that |ahbh|=2h; |ahbh| being the Euclidean distance between ah and bh (see Figure 2). The graph Gh is then the standard approximation of the Sierpinski gasket, which means that the sequence (Gh)h converges, as h tends to , in the Hausdorff metric, to the Sierpinski gasket G.

    Figure 2.  The graph Gh for h=0,1,2,3.

    We denote Card(Vh) as the number of vertices of Vh. We can easily check that

    Card(Vh)=3h+1+32hN. (2.5)

    Let k{1,2,...,3h}. We denote Tkh as the kth triangle of the graph Gh obtained at the step h. Let nk be the unit normal to Tkh. Then, nk=(3/2,1/2), nk=(3/2,1/2), or nk=(0,1). We denote E1,kh=[a1,kh,b1,kh] as the edge of Tkh, which is normal to nk=(0,1), E2,kh=[a2,kh,b2,kh] as the edge of Tkh, which is normal to nk=(3/2,1/2), and E3,kh=[a3,kh,b3,kh] as the edge of Tkh which is normal to nk=(3/2,1/2) (see Figure 3).

    Figure 3.  Orientation of the segments E1,kh, E2,kh, and E3,kh.

    Let us consider the following rotation matrices:

    {R1=IdR3R2=(1/23/203/21/20001)R3=Rt2 (2.6)

    IdR3 being the 3×3 identity matrix. We also define the change of variables yi,kh,1, yi,kh,2, x3; i=1,2,3, for every hN, every k{1,2,...,3h}, and every x=(x1,x2,x3)[ai,kh,bi,kh]×R, by

    (yi,kh,1(x)yi,kh,2(x)x3)=Ri(x1ai,kh,1x2ai,kh,2x3). (2.7)

    Let S be the unit disk of R2 centred at the origin. Let (εh)hN be a decreasing sequence of positive numbers, such that

    limhεh=limh2hεhln(1/εh)=0. (2.8)

    We define, for hN, k{1,2,...,3h}, and i=1,2,3, the tube Πh,ik by

    Πh,ik={(x1,x2,x3)R3εh<yi,kh,1(x)<2hεh(yi,kh,2(x),x3)εhS}. (2.9)

    We define the interfaces

    {Σh,ik,1={(x1,x2,x3)R3(yi,kh,2(x),x3)εhSyi,kh,1(x)=εh }Σh,ik,2={(x1,x2,x3)R3(yi,kh,2(x),x3)εhSyi,kh,1(x)=2hεh}Σh,ik=Σh,ik,1Σh,ik,2. (2.10)

    We then set

    {Πh=3hk=1i=1,2,3Πh,ikΣhα=3hk=1i=1,2,3Σh,ik,αα=1,2Σh=Σh1Σh2. (2.11)

    We now define thin, smooth regions which ensure the junctions between the tubes Πh,ik. Let Bhk(ai,kh) and Bhk(bi,kh) be bounded open sets of thickness of order 2εh and centered at the points (ai,kh,1,ai,kh,2,0) and (bi,kh,1,bi,kh,2,0), respectively, such that Bhk(ai,kh) and Bhk(bi,kh) are C2-surfaces with

    {Bhk(ai,kh)Πh=εhSBhk(bi,kh)Πh=εhS (2.12)

    (see Figure 4).

    Figure 4.  Smooth, thin zones Bhk(ai,kh) and Bhk(bi,kh), which ensure the junctions between the tubes Πh,ik.

    We set

    Bh=3hk=1i=1,2,3Bhk(ai,kh)Bhk(bi,kh). (2.13)

    Let us define the pipe Ωh,ik; hN, k{1,2,...,3h}, and i=1,2,3, by

    Ωh,ik=Πh,ikΣh,ikBhk(ai,kh)Bhk(bi,kh). (2.14)

    We consider the network Ωh of interconnected pipes and its external boundary Γh defined by

    Ωh=Σh3hk=1i=1,2,3Ωh,ikΓh=Ωh. (2.15)

    We consider a viscous incompressible fluid flow in Ωh. We suppose that this flow is essentially laminar except in the set Jh of the junction zones, where the main characteristics of the flow and their influence on the fluid motion will be analyzed. On the basis of works [12] and [13], we define the set Jh as

    Jh=3hk=1i=1,2,3Jh,+,ikJh,,ik (2.16)

    where, for every k{1,2,...,3h} and i=1,2,3,

    Jh,+,ik={x=(x1,x2,x3)Ωh0<yi,kh,1(x)<εhln(1/εh)}Jh,,ik={x=(x1,x2,x3)Ωh2hεhln(1/εh)<yi,kh,1(x)<2h}. (2.17)

    Taking into account the typical scales in ΩhJh, we suppose that the characteristic Reynolds number in these regions is of order 2hμ. The characteristic Reynolds number in Ωh can be then defined as

    Reh={Rej,hin Jh2hμin ΩhJh (2.18)

    where Rej,h is assumed to be a typical Reynolds number of the flow in the region Jh. According to [36], the product EuhReh of the characteristic Euler number Euh and the characteristic Reynolds number Reh is the ratio between the caracteristic pressure and viscosity. Then, assuming that the characteristic pressure is the ratio between a constant normal force and the surface of the disk εhS, we may write

    RehEuh=1μπε2h. (2.19)

    According to the above equality, we suppose that the characteristic Euler number Euh in the network Ωh takes the form

    Euh=2hπε2h. (2.20)

    On the other hand, as the diameter of any tube of the network Ωh is 2εh, we deduce, according to [37, page 98], that the ratio of the characteristic Froude number Frh to the characteristic Reynolds number Reh is of order ε2h. Accordingly, we suppose that the characteristic Froude number in Ωh has the following scaling:

    Frh=2hπε2h. (2.21)

    Since the characteristic Reynolds number is small in ΩhJh, we suppose that the inertia effects are negligible in the whole Ωh and the flow is governed by the following Stokes equations:

    { 1Reh5h3h+1Δuh+Euh 5h3h+1ph=1Frh5h3h+1fhin Ωh                                    div uh=0in Ωh (2.22)

    where 5h3h+1 is a scaling factor, which is associated to the ramification of the pre-fractal network Ωh and determined by the decimation principle (see [38] for more details on scaling exponents governing some physical phenomena in fractal media), the source term fh is the solution of the following problem posed in each tube Ωh,ik; k{1,2,...,3h} and i=1,2,3,

    {div fh=ghin Ωh,ik   fh.n=0on Ωh,ik (2.23)

    where n is the outward unit normal on Ωh,ik and gh is a L2(Ωh) function such that

    {                Ωh,ikghdx=0 suph 5h|Ωh|Ωhg2hdx<+ (2.24)

    |A| being the Lebesgue measure of the measurable and bounded subset A of R3. The boundary conditions (2.25) are given, for every i=1,2,3, by

    {             uh=0on ΓhuhΣh1.Rie1=uhΣh2.Rie1on Σh          phn=0on Γh (2.25)

    where, in accordance with the divergence free of the velocity, the condition (2.25)2 ensures that the outward normal velocities are the same on the two interfaces Σh,ik,1 and Σh,ik,2, e1=(1,0,0), and phn is the normal derivative of the pressure on Γh; n being the outward unit normal on Γh.

    Remark 1. The homogeneous Neumann boundary condition (2.25)3 on Γh is justified as follows. According to [39, Chapter II], thin boundary layers are concentrated in the immediate neighborhood of the wall Γh due to the homogeneous Dirichlet boundary condition for the velocity on Γh. The characteristic Reynolds number in these boundary layers, denoted here by Rew,h, is sufficiently large so that the viscous term 1Rew,hΔuh is negligible when one gets too close to the wall Γh. We deduce, according to [40, Remarks page 1119], that the boundary condition

    Euhphn=1Rew,hΔuh.n onΓh

    obtained by taking into account equation (2.22)1 and the fact that fh.n=0 on Γh, can ostensibly be approximated by Euhphn=0 on Γh, which implies that phn=0 on Γh.

    Let us introduce the space Vh defined by

    Vh={vH1(Ωh,R3)vΣh1.Rie1=vΣh2.Rie1i=1,2,3div v=0 in Ωhv=0 on Γh}. (2.26)

    We state here a result of existence and uniqueness of a solution for problem (2.22) with boundary conditions (2.25).

    Lemma 1. Problem (2.22)–(2.25) has a unique velocity solution uhVh and pressure solution phH1(Ωh), which is unique up to an additive constant.

    Proof. Applying the divergence operator to the first equation of problem (2.22), using (2.23)–(2.24)1 and the boundary condition (2.25)3, we deduce that the pressure verifies the Neumann boundary value problem

    { Δph=gh in Ωhphn=0 on Γh. (2.27)

    This problem has a solution phH1(Ωh), which is unique up to an additive constant. On the other hand, as

    Euh 5h3h+1Ωhv.ph=0 (2.28)

    for every vVh, the weak formulation of problem (2.22) can be written as, for every vVh,

    5h3h+1RehΩhuh.vdx=1Frh5h3h+1Ωhfh.vdx. (2.29)

    Using the Poincaré inequality, we have

    |Ωhfh.vdx|Ch{Ωh|v|2dx}1/2 , 

    where Ch is a positive constant. Then, according to the Lax–Milgram theorem, we infer that problem (2.29) has a unique solution uhVh.

    Let us consider the functional Fh defined by

    Fh(v)={5h3h+1RehΩh|v|2dxif vVh+otherwise. (2.30)

    The velocity uh, solution of problem (2.29), is then the solution of the minimization problem

    minvVh{Fh(v)21Frh5h3h+1Ωhfh.vdx}. (2.31)

    One of the main purposes of this paper is to prove the Γ-convergence of the sequence of functionals (Fh)h to the functional F defined in (1.1).

    In this section we state our main results in this work. Let M(R3) be the space of Borel regular measures on R3. According to Proposition 8 in Section 5, we introduce the following topology τ:

    Definition 1. We say that a sequence (vh)h; vhVh, τ-converges to (v,v,v) if

    5hvhπ1Ωh(x)3|Ωh|dxh(v,v,v)dHd(s)δ0(x3)Hd(G)inM(R3)

    where the symbol stands for the weak*-convergence of measures.

    We formulate our result on the Γ-convergence of the sequence of functionals (Fh)h in the following

    Theorem 2. We suppose that σ(0,+). Then

    1. (limsup inequality) For every vV, there exists a sequence (vh)h, with vhVh and (vh)h τ-converges to (v,v,v), where v=0, v=0 on the part of G which is perpendicular to (0,1), v=v3 on the part of G which is perpendicular to (3/2,1/2), and v=v3 on the part of G which is perpendicular to (3/2,1/2), such that

    limsuphFh(vh)F(v)

    where V is defined in Definition 23 of Section 5 and F is the functional energy defined in (1.1),

    2. (liminf inequality) For every sequence (vh)h, such that vhVh and (vh)h τ -converges to (v,v,v), we have vV, v=0 on G, v=0 on the part of G which is perpendicular to (0,1), v=v3 on the part of G which is perpendicular to (3/2,1/2), v=v3 on the part of G which is perpendicular to (3/2,1/2), and

    liminfhFh(vh)F(v).

    We are now in a position to formulate the asymptotic problem.

    Theorem 3. Let (uh,ph) be the solution of problem (2.22) with boundary conditions (2.25). Under the hypothesis of Theorem 2, we have

    1. The sequence (uh)h τ-converges to (u,u,0), with uV, u=0 on the part of G which is perpendicular to (0,1), u=u3 on the part of G which is perpendicular to (3/2,1/2), and u=u3 on the part of G which is perpendicular to (3/2,1/2). There exists pHZ(G); HZ(G) being the space defined in Definition 2 1 of Section 5, and f=(f1,f2,0)L2Hd(G,R3), such that

    {5hˆphπ1Ωh(x)3|Ωh|dxhpdHdδ0(x3)Hd(G)inM(R3)5hfhπ1Ωh(x)3|Ωh|dxhfdHdδ0(x3)Hd(G)inM(R3)limh5h|Ωh|Ωhuh.ph=GuZp.ndν=0

    where n=(1,0) on the horizontal part G1 of G, n=(1/2,3/2) on the part G2 of G which is perpendicular to (3/2,1/2), and n=(1/2,3/2) on the part G3 of G which is perpendicular to (3/2,1/2),

    2. The couple (u,p) is the solution of equation (1.4) stated in the Introduction.

    In this subsection we introduce the notion of Dirichlet forms on the Sierpinski gasket. For the definition and properties of Dirichlet forms, we refer to [41] and [42].

    For any function w:VR, we define

    EhG(w)=(53)hr,sVh|rs|=2h(w(r)w(s))2. (4.1)

    We then define the energy EG on G by

    EG(w)=limhEhG(w) (4.2)

    with domain D={w:VR:EG(w)<}. According to [42, Theorem 2.2.6], every function wD can be uniquely extended to be an element of C(G) still denoted by w. Let us set

    D={wC(G):EG(w)<} (4.3)

    where EG(w)=EG(wV). Then, DC(G)L2Hd(G). We define the space DE as

    DE=¯D.DE (4.4)

    where .DE is the intrinsic norm

    wDE={EG(w)+w2L2Hd(G)}1/2. (4.5)

    We denote EG(.,.) as the bilinear form defined on DE×DE by

    EG(w,z)=12(EG(w+z)EG(w)EG(z))w,zDE (4.6)

    from which we deduce, according to (4.2), that

    EG(w,z)=limhEhG(w,z) (4.7)

    where

    EhG(w,z)=(53)hr,sVh|rs|=2h(w(r)w(s))(z(r)z(s)). (4.8)

    The form EG(.,.) is a closed Dirichlet form in the Hilbert space L2Hd(G) and, according to [43, Theorem 4.1], EG(.,.) is a local regular Dirichlet form in L2Hd(G). This means that

    1. (local property) w,zDE with supp[w] and supp[z] are disjoint compact sets EG(w,z)=0,

    2. (regularity) DEC0(G) is dense both in C0(G) (the space of functions of C(G) with compact support) with respect to the uniform norm and in DE with respect to the intrinsic norm (4.5).

    We deduce that DE is injected in L2Hd(G) and is a Hilbert space with the scalar product associated to the norm (4.5). The second property implies that DE is not trivial (that is, DE is not made by only the constant functions). Moreover every function of DE possesses a continuous representative. Indeed, according to [44, Theorem 6.3. and example 71], the space DE is continuously embedded in the space Cβ(G) of Hölder continuous functions with β=ln53/ln4.

    Now, applying [45, Chap. 6], we have the following result:

    Lemma 4. There exists a unique self-adjoint nonpositive operator ΔG on L2Hd(G) with domain

    DΔG={wL2Hd(G)ΔGwL2Hd(G)}DE

    dense in L2Hd(G), such that, for every wDΔG and zDE,

    EG(w,z)=G(ΔGw)zdHdHd(G).

    In this subsection we define the Kusuoka measure and the gradient on the Sierpinski gasket G. For the definitions and properties of Kusuoka measures and gradients on fractals, we refer to [46,47,48,49].

    Let ϱ:VR. Then, according to [42, Proposition 3.2.1], there exists a unique hD such that h|V0=ϱ and

    EG(h)=inf{EG(v)vDv|V0=ϱ}

    where h is called the harmonic function in G with boundary value h|V0=ϱ. On each Vh, h N, a harmonic function h verifies

    (hψi1...ih)|V0=Ti1...ih(h|V0)i1,...,ih{1,2,3} (4.9)

    (see [42, Proposition 3.2.1]), where ψi1...ih=ψi1ψih and Ti1...ih=Ti1...Tih with

    T1=15[500221212]T2=15[221050122]T3=15[212122005].

    Let M0={(x1,x2,x3)R3x1+x2+x3=0}. Kigami [46] introduced the map Φ:GM0 defined by

    Φ(x)=12((h1(x)h2(x)h3(x))13(111))

    with hi(Aj)=δij for AjV0, where δij is, for i,j=1,2,3, the Kronecker delta symbol. We have the following.

    Proposition 5. [47, Proposition 4.4] If GH=Φ(G), then Φ is a homeomorphism between G and GH. Moreover, define Hi:M0M0; i=1,2,3, by

    Hi(x)=Tti(xΦ(Ai))+Φ(Ai)

    then GH=i=1,2,3Hi(GH) and Φψi=HiΦ for any i=1,2,3.

    GH is called the harmonic Sierpinski gasket, which is the self-similar set associated with the collection of contractions {H1,H2,H3} on M0. Let P be the projection from R3 into M0 defined, for every x=(x1,x2,x3)R3, by

    Px=(x1x2x3)(x1+x2+x3)3(111). (4.10)

    According to [48], the Kusuoka measure ν on G is the unique Borel probability measure defined by

    ν(Gi1...ih)=12(53)htr(Tti1...ihPTi1...ih) (4.11)

    where Gi1...ih=ψi1...ih(G). Let us define

    I={ω=i1i2...in{1,2,3} for any nN} (4.12)

    and π:IG such that ψjπ(ω)=π(jω), for j=1,2,3. For any ωI, there exists a unique xG such that

    {x}=hNGi1...ih and π(ω)=x. (4.13)

    We now define, by abuse of notation, the Kusuoka measure ν on I (see, for instance, [49]) as the pullback of the Kusuoka measure ν on G under the projection map π, that is

    ν(π1(.))=ν(.). (4.14)

    Let us set

    Z(i1...ih)=Tti1...ihPTi1...ihtr(Tti1...ihPTi1...ih). (4.15)

    Then, according to [48], for ν-almost all ω, there exists a limit

    Z(ω)=limhZ(i1...ih). (4.16)

    Let Z(x)Z(π1(x)). Then, Z(x) is well defined on V (see for instance [47]). Indeed, according to [49, Theorem 3.6], for νalmost all xG,

    Z(π1(x))=Z(ω)=limhZh(i1...ih) (4.17)

    where

    Zh(i1...ih)=12(53)hTti1...ihPTi1...ihν(Gi1...ih). (4.18)

    Let U be an open subset of M0 containing GH. Let us define

    C1(G)={u;u=(vGH)Φv C1(U) }. (4.19)

    According to [47], if we fix an orthonormal basis of M0 and regard M0 as R2, then, for any uC1(G),

    u=(ux1ux2). (4.20)

    We have the following.

    Theorem 6. [47, Theorem 4.8] C1(G) is a dense subset of DE under the norm

    u=EG(u,u)+u

    and, for any u,vC1(G),

    EG(u,v)=Gu.Zvdν.

    In this section, we establish some a priori estimates and compactness results which will be useful for the proof of the main results.

    Lemma 7. Let vhVh, such that suphFh(vh)<. If σ(0,+) then

    suph5h|Ωh|Ωh|vh|2dx<+.

    Proof. The proof follows from the Poincaré inequality in a bounded domain with the Dirichlet boundary condition on a part of the boundary and a scaling argument. Let us define, for every k{1,2,...,3h} and i=1,2,3,

    Uh,ik={(yi,kh,1,y,z)R3yi,kh,1(εhln(1/εh),2hεhln(1/εh))(y,z)S}.

    Let φC1(Uh,ik), such that φ=0 on Uh,ikS. Using the Poincaré inequality, we infer that, for every yi,kh,1(εhln(1/εh),2hεhln(1/εh)),

    Sφ2(yi,kh,1,y,z)dydzCS|y,zφ(yi,kh,1,y,z)|2dydz

    where C is a positive constant independent of h and

    y,zφ(yi,kh,1,y,z)=(φy(yi,kh,1,y,z)φz(yi,kh,1,y,z)).

    Now, introducing the scaling yi,kh,2=εhy, x3=εhz, and integrating with respect to yi,kh,1 between εhln(1/εh) and 2hεhln(1/εh), we get

    2hεhln(1/εh)εhln(1/εh)εhSφ2dyi,kh,1dyi,kh,2dx3Cε2h2hεhln(1/εh)εhln(1/εh)εhS|φ|2dyi,kh,1dyi,kh,2dx3

    from which we deduce, using the change of variables (2.7), that, for every vhVh,

    Ωh,ikJh,+,ikJh,,ik|vh|2dxCε2hΩh,ikJh,+,ikJh,,ik|vh|2dx. (5.1)

    We can use the same method in Jh,+,ikJh,,ik to obtain

    Jh,+,ikJh,,ik|vh|2dxCε2hJh,+,ikJh,,ik|vh|2dx. (5.2)

    The combination of (5.1) and (5.2) implies that

    Ωh,ik|vh|2dxCε2hΩh,ik|vh|2dx. (5.3)

    Then, summing over i and k in (5.3), we obtain that

    5h3h+1ε2hRehΩh|vh|2dxC5h3h+1RehΩh|vh|2dx. (5.4)

    As σ(0,+), we have that

    Rej,hCεhC2h

    from which we deduce that 2hC 1Rej,h in Jh. Thus, using (5.4),

    5h3h+1ε2h2hΩh|vh|2dx5h3h+1ε2hRehΩh|vh|2dxCFh(vh). (5.5)

    Observing that 3hε2h2h|Ωh|π, we conclude that

    suph5h|Ωh|Ωh|vh|2dxCsuphFh(vh)<+. (5.6)

    We have the following result:

    Proposition 8. Let 1Ωh be the characteristic function of the set Ωh. Let vhVh, such that suphFh(vh)<+. If σ(0,+), then there exists a subsequence of (vh)h, still denoted as (vh)h, such that

    5hvhπ1Ωh(x)3|Ωh|dxhvdHd(s)δ0(x3)Hd(G) in M(R3)

    where v=(v1,v2,v3)L2Hd(G,R3) with v3=0 on G, v2=0 on the part of G which is perpendicular to (0,1), v2=v13 on the part of G which is perpendicular to (3/2,1/2), and v2=v13 on the part of G which is perpendicular to (3/2,1/2).

    Proof. Let us consider the sequence of measures (ϑh)h on R3 defined by

    ϑh=π1Ωh(x)3|Ωh|dx.

    Using an ergodicity argument (see, for instance, [50, Theorem 6.1]), we deduce that, for every φC0(R3),

    limhR3φ(x)dϑh=limh3hk=1i=1,2,313h+1φ(ai,kh+bi,kh2,0)=1Hd(G)Gφ(s,0)dHd(s)

    from which we deduce that

    ϑhhϑ=1G(s)dHd(s)δ0(x3)Hd(G).

    Let vhL2(Ωh,R3), such that suphFh(vh)<+. If σ(0,+) then, according to Lemma 7,

    suph5h|Ωh|Ωh|vh|2dx<+. (5.7)

    Observing that, for some positive constant C independent of h,

    |R35hvhdϑh|2C5hR3|vh|2dϑhC5h|Ωh|Ωh|vh|2dx

    and, by taking into account (5.7), we deduce that the sequence (5hvhϑh)h is uniformly bounded in variation, hence -weakly relatively compact. Possibly passing to a subsequence, we can suppose that the sequence (5hvhϑh)h -weakly converges to some χ. Let φC0(R3,R3). By using Fenchel's inequality, we have

    liminfh12R3|5hvh|2dϑhliminfh(R35hvh.φdϑh12R3|φ|2dϑh)χ,φ12R3|φ|2dϑ.

    As the left-hand side of this inequality is bounded, we deduce that

    sup{χ,φφC0(R3,R3)G|φ|2(s,0)dHd(s)1}<+

    from which we deduce, according to Riesz' representation Theorem, that there exists v such that v(s,0)L2Hd(G,R3) and χ=v(s,x3)ϑ.

    Let us introduce the function vh,i; i=1,2,3, related to vh by

    vh,i(yi,kh,1,yi,kh,2,x3)=RivhRti((yi,kh,1yi,kh,2x3)+Ri(ai,kh,1ai,kh,20)) (5.8)

    where yi,kh,1,yi,kh,2,x3 are the variables defined in (2.7). We can easily prove, after some computations that for every i=1,2,3,

    div yvh,i=div vh (5.9)

    where div y is the divergence operator in the variables yi,kh,1,yi,kh,2,x3. On the other hand, as Πh,ik is a cylinder of revolution, we can introduce the cylindrical coordinates yi,kh,1yi,kh,1, yi,kh,2=rcosθ, x3=rsinθ, and the polar components of vh,i defined by

    {vh,i1(yi,kh,1,r,θ)=vh,i1(yi,kh,1,rcosθ,rsinθ)vh,ir(yi,kh,1,r,θ)=(vh,i2cosθ+vh,i3sinθ)(yi,kh,1,rcosθ,rsinθ)vh,iθ(yi,kh,1,r,θ)=(vh,i2sinθ+vh,i3cosθ)(yi,kh,1,rcosθ,rsinθ). (5.10)

    Let ˜vh,i=(vh,i1,vh,ir,vh,iθ). The divergence of ˜vh,i in cylindrical coordinates is given by

    div r(˜vh,i)=vh,i1yi,kh,1+vh,irr+vh,irr+1rvh,iθθ. (5.11)

    Since div vh=0, we deduce from (5.9) and (5.11) that

    div yvh,i=div r(˜vh,i)=0. (5.12)

    Using the boundary condition (2.25)2, we have, for every hN,

    vh,i1(εh,r,θ)vh,i1(2hεh,r,θ)=0 (5.13)

    from which we deduce, using Green's formula, that, for ψCc(0,2π) and φ(θ)=θ0ψ(ξ)dξ with φ(2π)=0,

    3hk=1i=1,2,32hεhεhεh02π0vh,i1yi,kh,1φ(θ)rdyi,kh,1drdθ=εh02π03hk=1i=1,2,3(vh,i1(εh,r,θ)vh,i1(2hεh,r,θ))φ(θ)rdrdθ=0. (5.14)

    Since div r(˜vh,i)=0, we deduce from formula (5.11), according to (5.14), that

    2h5h3h+1εh3hk=1i=1,2,32hεhεhεh02π0vh,irφ(θ)dyi,kh,1drdθ+2h5h3h+1εh3hk=1i=1,2,32hεhεhεh02π0vh,irrφ(θ)rdyi,kh,1drdθ+2h5h3h+1εh3hk=1i=1,2,32hεhεhεh02π0vh,iθθφ(θ)dyi,kh,1drdθ=0. (5.15)

    Using Green's formula, we deduce that

    2hεhεhεh02π0vh,irrφ(θ)rdyi,kh,1drdθ=2hεhεhεh02π0vh,irφ(θ)dyi,kh,1drdθ (5.16)

    and

    2hεhεhεh02π0vh,iθθφ(θ)dyi,kh,1drdθ=2hεhεhεh02π0vh,iθψ(θ)dyi,kh,1drdθ. (5.17)

    Combining with (5.15), we deduce that

    2h5h3h+1εh3hk=1i=1,2,32hεhεhεh02π0vh,iθψ(θ)dyi,kh,1drdθ=0 . (5.18)

    Recalling that vh,iθ=vh,i2sinθ+vh,i3cosθ and vh,i3=vh3, and using the first part of this Lemma, we obtain that

    limh2h5h3h+1εh3hk=1i=1,2,32hεhεhεh02π0vh,iθψ(θ)dyi,kh,1drdθ=1Hd(G)G2π0(w(s)sinθ+v3(s)cosθ)ψ(θ)dsdθ=0 (5.19)

    where

    w(s)={v2(s)on G1v1(s)3+v2(s)on G2v1(s)3+v2(s)on G3 (5.20)

    where G1 is the part of G which is perpendicular to (0,1), G2 is the part of G which is perpendicular to (3/2,1/2), and G3 is the part of G which is perpendicular to (3/2,1/2). We deduce from (5.19) that w(s)sinθ+v3(s)cosθ=0 for every θ(0,2π), thus w=v3=0 on G. Therefore, combining with (5.20), v2=0 on G1, v2=v13 on G2, and v2=v13 on G3.

    Proposition 9. We suppose that σ(0,+). Let vhVhH2(Ωh,R3), such that suphFh(vh)<+. Then, for every sequence (φh)h, such that φhH1(Ωh) and

    suph5h|Ωh|Ωh|φh|2dx<+ , 5hφhπ1Ωh(x)3|Ωh|dxhφdHd(s)δ0(x3)Hd(G)inM(R3) (5.21)

    we have

    1. φ(s,0)DE and Gφ.Zφdν<+,

    2. there exists a subsequence of (vh)h, still denoted as (vh)h, and vL2Hd(G), such that

    limh5h|Ωh|Ωhvh.φhdx=Gvn.Zφdν=0

    where n=(1,0) on the horizontal part of G, n=(1/2,3/2) on the part of G which is perpendicular to (3/2,1/2), and n=(1/2,3/2) on the part of G which is perpendicular to (3/2,1/2).

    Proof. 1. Let us define, for every k{1,2,...,3h} and i=1,2,3,

    φih(yi,kh,1,yi,kh,2,x3)=φhRti((yi,kh,1yi,kh,2x3)+Ri(ai,kh,1ai,kh,20)) (5.22)

    and

     ˜φih(yi,kh,1)=1πε2hεhSφih(yi,kh,1,yi,kh,2,x3)dyi,kh,2dx3=1πε2hεhSφh(Rti(yi,kh,1yi,kh,2x3)+(ai,kh,1ai,kh,20))dyi,kh,2dx3 (5.23)

    where yi,kh,1,yi,kh,2,x3 are the change of variables defined in (2.7). Then

    5h|Ωh|Ωh|φh|2dx=5h|Ωh|3hk=1i=1,2,3Ωh,ik|φh|2dx5h3h+13hk=1i=1,2,32h02hπε2hεhS(φihyi,kh,1)2dyi,kh,1dyi,kh,2dx3 5h3h+13hk=1i=1,2,31πε2hεhS(2h0φihyi,kh,1dyi,kh,1)2dyi,kh,2dx3=5h3h+13hk=1i=1,2,31πε2hεhS(φih(2h,yi,kh,2,x3)φih(0,yi,kh,2,x3))2dyi,kh,2dx35h3h+13hk=1i=1,2,3(1πε2hεhS(φih(2h,yi,kh,2,x3)φih(0,yi,kh,2,x3))dyi,kh,2dx3)2=5h3h+13hk=1i=1,2,3(˜φih(2h)˜φih(0))2=5h3h+13hk=1i=1,2,3(˜φh(ai,kh)˜φh(bi,kh))2=EG(˜φh) (5.24)

    where for . We now introduce the harmonic extension of obtained by the so-called decimation procedure (see, for instance, [51, Corollary1]). We define the function as the unique minimizer of the problem

    (5.25)

    Then . For , we define the function from into by

    We have, for every , and

    (5.26)

    We define now, for fixed , the function on as follows. For , we choose such that and set

    (5.27)

    As , we have, according to (5.24), (5.26), and (5.27),

    (5.28)

    from which we deduce, using Section , that has a unique continuous extension on , still denoted as , and that the sequence is bounded in . Therefore, there exists a subsequence, still denoted as , weakly converging in the Hilbert space to some , such that

    (5.29)

    On the other hand, using the hypothesis (5.21), we have that

    (5.30)

    where ; being the triangle obtained at the step in the construction of the fractal . We deduce from this that, for every ,

    (5.31)

    where is the sequence of measures defined by

    (5.32)

    being the Dirac measure at the point . Thus, , , and, according to (5.24) and (5.29),

    (5.33)

    from which we deduce, using Theorem 6, that

    (5.34)

    2. As , we can write

    (5.35)

    where . Since as , using the proof of Lemma 7 and the hypothesis (5.21), we have that

    Thus, passing to the limit in (5.35), we get

    (5.36)

    As , using some density argument, we may suppose that . As , we may write

    (5.37)

    where . On the other hand, there exists a function such that . Indeed, as , is a solution of the equation in with some boundary conditions on . Using the smoothness of , we infer that

    (5.38)

    Then, replacing in (5.37), taking into account the fact that and the estimates on given in Lemma 7, we obtain that

    (5.39)

    As for the fractal , we can construct, according to Proposition 5, a graph approximation of the harmonic Sierpinski gasket and a sequence of thin branching tubes whose axes are iterated curves of the graph . As , there exists , such that . Similarly, there exists , being an open subset of containing , such that . Let us set, for ,

    (5.40)

    where . Then, observing that, there exist such that , using (5.40), the fact that , , and [46, Lemma 3.2], we deduce that

    (5.41)

    Using Lemma 8, there exists a subsequence of , still denoted as , and , such that

    where on the part of which is perpendicular to , on the part of which is perpendicular to , and on the part of which is perpendicular to . The corresponding subsequence of gradients converges to the same limit. Thus, using the limits (5.36)–(5.37), the relations (5.38)–(5.41), and the smoothness of and , we obtain that

    (5.42)

    where we have used the fact that .

    According to the above proposition, we introduce the following

    Definition 2. 1. We define the space by

    (5.43)

    2. Let on the horizontal part of , on the part of which is perpendicular to , and on the part of which is perpendicular to . Let . We define the divergence of on by the relation

    for every .

    3. We define the space by

    (5.44)

    We introduce the following useful result which is due to Bogovskiĭ [52]:

    Lemma 10. Let be a bounded domain with Lipschitz continuous boundary . There exists a linear operator , such that, for every satisfying ,

    where is a constant which only depends on .

    Let us define . As a consequence, we have the following result:

    Lemma 11. Let . There exists a linear operator , such that, for every with ,

    where is a constant which still only depends on .

    Proof. For every satisfying , we define

    Then, since , we can apply Lemma 10 in to obtain

    (5.45)

    Let us define, for every ,

    (5.46)

    Then

    (5.47)

    On the other hand, observing that

    where

    we deduce that

    (5.48)

    Last, according to (5.45), we have

    (5.49)

    Therefore, combining (5.48) and (5.49), we infer that

    (5.50)

    Let be the solution of problem (2.22) with boundary conditions (2.25). Let us define, for every , , and , the zero average-value pressure by

    (5.51)

    and the pressure by

    (5.52)

    The following estimates hold true:

    Lemma 12. If then

    1. , ,

    2. , .

    Proof. 1. Applying Lemma 11 for the solution of problem (2.23), we deduce that, for every and ,

    (5.53)

    Additionally, using the inequality (5.3), we have

    (5.54)

    We deduce from (5.53) and (5.54), that

    (5.55)

    then, using the hypothesis (2.24), we conclude that

    (5.56)

    Multiplying (2.22) by and integrating by parts, we obtain that

    (5.57)

    from which we deduce, in virtue of the fact that , by using inequality (5.6) and estimate (5.56),

    (5.58)

    and, as , according to Lemma 7,

    (5.59)

    2. According to Lemma 11, there exists such that

    (5.60)

    and

    (5.61)

    Let us define on by on each , for every and . Then, according to inequality (5.61), we have that

    (5.62)

    Multiplying (2.22) by and integrating by parts, we deduce that

    (5.63)

    Using the fact that , inequality (5.62), and the uniform boundedness (5.56) and (5.58), we deduce that

    (5.64)

    which implies that

    (5.65)

    On the other hand, multiplying (2.27) by , integrating by parts, and, using the hypothesis (2.24), we get

    (5.66)

    from which we deduce by using (2.24) and the uniform boundedness (5.65):

    Let us define new orthonormal basis systems ; , by

    (6.1)

    where . We define the rescaled junctions and , for , by

    (6.2)

    We consider the following Leray problems:

    (6.3)

    and

    (6.4)

    where is the solution of the auxiliary problem

    (6.5)

    We define, for every and , the sequence of functions by

    (6.6)

    where the sets and are defined in (2.17) and the coordinates , , ; , are related to the variable through the relations (2.7). Let us define, for every and , the intermediate tubes

    (6.7)

    and their upper and lower bases, respectively,

    (6.8)

    Let . Let . Then, ; , for every and every . Let . We define the sequence of vector functions by

    (6.9)

    where

    (6.10)

    and

    (6.11)

    with

    (6.12)

    We introduce the function defined by

    (6.13)

    and the function defined by

    (6.14)

    Let be the solution of the problem

    (6.15)

    We define the sequence of test-functions ; , by

    (6.16)

    We then define the test function in by

    (6.17)

    We have the following results:

    Proposition 13. We have

    1. for small enough,

    2. -converges to , where on the part of which is perpendicular to , on the part of which is perpendicular to , and on the part of which is perpendicular to ,

    3. if , then

    Proof. 1. Introducing the variables and , we have, for small enough, that

    (6.18)

    and

    (6.19)

    This implies that problem (6.15) is solvable. On the other hand, using [53, Theorem Ⅵ.1.2], there exists such that, for any and every ,

    (6.20)

    from which we deduce that

    (6.21)

    which implies that

    (6.22)

    and, using [54, Lemma 9],

    (6.23)

    Since , , for every , and is independent of , we have

    Therefore, for small enough, .

    2. Let . We have

    (6.24)

    and

    (6.25)

    Then, using the estimate (6.20) for , the estimates (6.21)–(6.22) for , the estimate (6.23) for , and the inequality (5.3) applied to , we deduce that

    (6.26)

    3. Let us suppose that . Then, in virtue of the estimates (6.20)–(6.23), we have that

    (6.27)

    where . Then, as

    (6.28)

    and , we deduce that

    (6.29)

    After some computations, we infer that

    (6.30)

    and, for the last limit in (6.27),

    (6.31)

    Using once again the estimate (6.20), we deduce that

    (6.32)

    and

    (6.33)

    Now, combining (6.27)–(6.33), we get the result.

    Proposition 14. If , then for every , there exists a sequence , with and -converges to , where , on the part of which is perpendicular to , on the part of which is perpendicular to , and on the part of which is perpendicular to , such that

    Proof. Let . Let such that with respect to the norm (4.5). We define the sequence by replacing in (6.9), (6.16), and (6.17) by . Then, according to Proposition 13, the sequence -converges to , where on the part of which is perpendicular to , on the part of which is perpendicular to , on the part of which is perpendicular to , and

    The continuity of implies that . The topology being metrizable, we deduce, using a diagonalization argument (see [14, Corollary 1.18]), that the sequence ; , -converges to , with on the part of which is perpendicular to , on the part of which is perpendicular to , on the part of which is perpendicular to , and

    Proposition 15. If , then for every sequence , such that and -converges to , we have , on , on the part of which is perpendicular to , on the part of which is perpendicular to , on the part of which is perpendicular to , and

    Proof. Observe that if , then the lim inf inequality is trivial. We suppose that and, using some regularity argument, we may suppose that . Then, according to Proposition 8, we have that , on , on the part of which is perpendicular to , on the part of which is perpendicular to , on the part of which is perpendicular to , and, according to Proposition 5.9,

    (6.34)

    where is the space defined in Definition 2. Let such that with respect to the norm -strong. We define the sequence by replacing by in test-functions (6.9), (6.16), and (6.17). We deduce from the definition of the subdifferentiability of convex functionals that

    (6.35)

    We then compute

    (6.36)

    Using the estimate (6.23), we deduce that

    (6.37)

    from which we deduce that

    (6.38)

    On the other hand, using the fact that in and according to the problem (6.5) of which is the solution, we deduce that

    (6.39)

    Using the limits (6.24) and (6.30), and the fact that

    (6.40)

    we deduce that

    (6.41)

    Analogously, using the estimate (6.20), the equations (6.5), the expression (6.12) of , and the estimate (6.40), we get

    (6.42)

    and, similarly,

    (6.43)

    As , we have

    (6.44)

    In addition, owing to Proposition 13, we have

    (6.45)

    Thus, combining (6.35)–(6.45), we deduce that

    (6.46)

    Then, letting tend to , we obtain

    and, as a consequence, . Thus, and, taking into account (6.34), we have that .

    Proof. 1. Let be a solution of problem (2.22) with boundary conditions (2.25). According to Lemma 12 and Proposition 8 there exists a subsequence of , still denoted as , such that

    (6.47)

    with on the part of which is perpendicular to , on the part of which is perpendicular to , and on the part of which is perpendicular to . As the boundary is , the velocity is at least in . Thus, according to Proposition 5.9, we have that

    (6.48)

    On the other hand, since is the unique velocity solution of problem (2.31), we deduce from Theorem 2 and [15, Theorem 7.8], that the whole sequence verifies the convergence (6.47),

    (6.49)

    and, taking into account (6.48), we deduce that . In addition, using Lemma 12 and the proof of Proposition 8, we have that

    (6.50)

    with , and, using the uniform boundedness (5.56),

    (6.51)

    with . Using Proposition 5.9 and Lemma 12, we deduce that, for every ,

    (6.52)

    where on the horizontal part of , on the part of which is perpendicular to , and on the part of which is perpendicular to .

    2. According to Theorem 2 and [15, Theorem 7.8], is the solution of the problem

    (6.53)

    Then, using Lemma 4 and the fact that and , for every , we deduce from (6.53) that, for every ,

    (6.54)

    where, by abuse of notation, . Therefore, is the solution (with up to an additive constant) of the following problem:

    which completes the proof of Theorem 3.

    In this paper, we considered the motion of a viscous incompressible fluid in a varying bounded domain consisting of branching cylindrical pipes whose axes are line segments that form a network of pre-fractal polygonal curves obtained after -iterations of the contractive similarities of the standard Sierpinski gasket. We assumed that these pipes are narrow axisymmetric tubes of radius very small with respect to the length of each side of . We supposed that the fluid flow is driven by some volumic forces and governed by Stokes equations with continuity of the velocity at the interfaces separating the junction zones from the rest of the pipes, homogeneous Dirichlet boundary condition for the velocity, and homogeneous Neumann boundary condition for the pressure on the wall of the tubes. The flow in each pipe is split into two streams: boundary layers flow in junction zones of length and laminar flow in the rest of the pipe. We assumed that the flow in the junction zones is controlled by a typical Reynolds number . Using -convergence methods, we studied the asymptotic behavior of the fluid flowing in the branching tubes as the radius of the tubes tends to zero and the sequence of the pre-fractal curves converges in the Hausdorff metric to the Sierpinski gasket. According to critical values taken by , we derived three uncommon effective models of fluid flows in the Sierpinski gasket:

    1. a singular Brinkman equation if ,

    2. a singular Darcy flow if or as ,

    3. a flow with constant velocity if with .

    As far as the modeling is concerned, fractal branching pipe networks have to be considered to describe fluid flows in various complex geometrical configurations. An important field to which this model is closely related is the behavior of fluid flows in some physiological structures such as the blood circulation through arterial networks. Our model may serve as a starting point for further investigations in this area.

    Haifa El Jarroudi: Writing-original draft, Writing-review and editing, Methodology, Formal Analysis; Mustapha El Jarroudi: Writing-original draft, Writing-review and editing, Methodology, Supervision.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors wish to express their gratitude to the anonymous referee for giving a number of valuable comments and helpful suggestions, which improve the presentation of sentation of the manuscript significantly.

    The authors declare there is no conflict of interest.



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