
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 2−h 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)≪ 2−h 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.
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)3σ∫G∇v.Z∇vdν if v∈V∞, +∞ 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=limh→∞εhRej,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+νZ∇p.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)3σ∫G∇v.Z∇vdν 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 ∫G∇v.Z∇vdν=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 |
2−h | 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 2−h−2εh 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<2−h |
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+Ai2, ∀x=(x1,x2)∈R2. | (2.1) |
Let V0={A1, A2, A3} be the set of vertices of the equilateral triangle A1A2A3. We define inductively
Vh+1=∪i=1,2,3ψi(Vh), | (2.2) |
for every h∈N, and set
V∞=∪h∈NVh. | (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,bh∈Vh, such that |ah−bh|=2−h; |ah−bh| 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.
We denote Card(Vh) as the number of vertices of Vh. We can easily check that
Card(Vh)=3h+1+32, ∀h∈N. | (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).
Let us consider the following rotation matrices:
{R1=IdR3, R2=(1/2√3/20−√3/21/20001), 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 h∈N, 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(x1−ai,kh,1x2−ai,kh,2x3). | (2.7) |
Let S be the unit disk of R2 centred at the origin. Let (εh)h∈N be a decreasing sequence of positive numbers, such that
limh→∞εh=limh→∞2hεhln(1/εh)=0. | (2.8) |
We define, for h∈N, k∈{1,2,...,3h}, and i=1,2,3, the tube Πh,ik by
Πh,ik={(x1,x2,x3)∈R3; εh<yi,kh,1(x)<2−h−εh, (yi,kh,2(x),x3)∈εhS}. | (2.9) |
We define the interfaces
{Σh,ik,1={(x1,x2,x3)∈R3; (yi,kh,2(x),x3)∈εhS, yi,kh,1(x)=εh }, Σh,ik,2={(x1,x2,x3)∈R3; (yi,kh,2(x),x3)∈εhS, yi,kh,1(x)=2−h−εh}, Σh,ik=Σh,ik,1∪Σh,ik,2. | (2.10) |
We then set
{Πh=3h∪k=1i=1,2,3Πh,ik, Σhα=3h∪k=1i=1,2,3Σh,ik,α; α=1,2, Σh=Σh1∪Σh2. | (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=εhS, ∂Bhk(bi,kh)∩∂Πh=εhS, | (2.12) |
(see Figure 4).
We set
Bh=3h∪k=1i=1,2,3Bhk(ai,kh)∪Bhk(bi,kh). | (2.13) |
Let us define the pipe Ωh,ik; h∈N, k∈{1,2,...,3h}, and i=1,2,3, by
Ωh,ik=Πh,ik∪Σh,ik∪Bhk(ai,kh)∪Bhk(bi,kh). | (2.14) |
We consider the network Ωh of interconnected pipes and its external boundary Γh defined by
Ωh=Σh∪3h∪k=1i=1,2,3Ωh,ik, Γh=∂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=3h∪k=1i=1,2,3Jh,+,ik∪Jh,−,ik, | (2.16) |
where, for every k∈{1,2,...,3h} and i=1,2,3,
Jh,+,ik={x=(x1,x2,x3)∈Ωh; 0<yi,kh,1(x)<εhln(1/εh)}, Jh,−,ik={x=(x1,x2,x3)∈Ωh; 2−h−εhln(1/εh)<yi,kh,1(x)<2−h}. | (2.17) |
Taking into account the typical scales in Ωh∖Jh, we suppose that the characteristic Reynolds number in these regions is of order 2−hμ. The characteristic Reynolds number in Ωh can be then defined as
Reh={Rej,hin Jh, 2−hμin Ωh∖Jh, | (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=2−hπε2h. | (2.21) |
Since the characteristic Reynolds number is small in Ωh∖Jh, 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+1∇ph=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 Γh, uh∣Σh1.Rie1=uh∣Σh2.Rie1on Σh, ∂ph∂n=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 ∂ph∂n 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
Euh∂ph∂n=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 Euh∂ph∂n=0 on Γh, which implies that ∂ph∂n=0 on Γh.
Let us introduce the space Vh defined by
Vh={v∈H1(Ωh,R3); v∣Σh1.Rie1=v∣Σh2.Rie1; i=1,2,3, div v=0 in Ωh, v=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 uh∈Vh and pressure solution ph∈H1(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 Ωh, ∂ph∂n=0 on Γh. | (2.27) |
This problem has a solution ph∈H1(Ωh), which is unique up to an additive constant. On the other hand, as
Euh 5h3h+1∫Ωhv.∇ph=0, | (2.28) |
for every v∈Vh, the weak formulation of problem (2.22) can be written as, for every v∈Vh,
5h3h+1Reh∫Ωh∇uh.∇vdx=1Frh5h3h+1∫Ωhfh.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 uh∈Vh.
Let us consider the functional Fh defined by
Fh(v)={5h3h+1Reh∫Ωh|∇v|2dxif v∈Vh, +∞otherwise. | (2.30) |
The velocity uh, solution of problem (2.29), is then the solution of the minimization problem
minv∈Vh{Fh(v)−21Frh5h3h+1∫Ωhfh.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; vh∈Vh, τ-converges to (v,v∗,v∗∗) if
√5hvhπ1Ωh(x)3|Ωh|dx∗⇀h→∞(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 v∈V∞, there exists a sequence (vh)h, with vh∈Vh 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∗=v√3 on the part of G which is perpendicular to (−√3/2,1/2), and v∗=−v√3 on the part of G which is perpendicular to (√3/2,1/2), such that
limsuph→∞Fh(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 vh∈Vh and (vh)h τ -converges to (v,v∗,v∗∗), we have v∈V∞, v∗∗=0 on G, v∗=0 on the part of G which is perpendicular to (0,1), v∗=v√3 on the part of G which is perpendicular to (−√3/2,1/2), v∗=−v√3 on the part of G which is perpendicular to (√3/2,1/2), and
liminfh→∞Fh(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 u∈V∞, u∗=0 on the part of G which is perpendicular to (0,1), u∗=u√3 on the part of G which is perpendicular to (−√3/2,1/2), and u∗=−u√3 on the part of G which is perpendicular to (√3/2,1/2). There exists p∈HZ(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|dx∗⇀h→∞pdHd⊗δ0(x3)Hd(G)inM(R3), √5hfhπ1Ωh(x)3|Ωh|dx∗⇀h→∞fdHd⊗δ0(x3)Hd(G)inM(R3), limh→∞5h|Ωh|∫Ωhuh.∇ph=∫GuZ∇p.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:V∞⟶R, we define
EhG(w)=(53)h∑r,s∈Vh|r−s|=2−h(w(r)−w(s))2. | (4.1) |
We then define the energy EG on G by
EG(w)=limh→∞EhG(w), | (4.2) |
with domain D∞={w:V∞⟶R:EG(w)<∞}. According to [42, Theorem 2.2.6], every function w∈D∞ can be uniquely extended to be an element of C(G) still denoted by w. Let us set
D={w∈C(G):EG(w)<∞}, | (4.3) |
where EG(w)=EG(w∣V∞). Then, D⊂C(G)⊂L2Hd(G). We define the space DE as
DE=¯D‖.‖DE, | (4.4) |
where ‖.‖DE is the intrinsic norm
‖w‖DE={EG(w)+‖w‖2L2Hd(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,z∈DE, | (4.6) |
from which we deduce, according to (4.2), that
EG(w,z)=limh→∞EhG(w,z), | (4.7) |
where
EhG(w,z)=(53)h∑r,s∈Vh|r−s|=2−h(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,z∈DE with supp[w] and supp[z] are disjoint compact sets ⟹ EG(w,z)=0,
2. (regularity) DE∩C0(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={w∈L2Hd(G); ΔGw∈L2Hd(G)}⊂DE |
dense in L2Hd(G), such that, for every w∈DΔG and z∈DE,
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:V∞⟶R. Then, according to [42, Proposition 3.2.1], there exists a unique h∈D∞ such that h|V0=ϱ and
EG(h)=inf{EG(v); v∈D∞, v|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
(h∘ψi1...ih)|V0=Ti1...ih(h|V0); i1,...,ih∈{1,2,3}, | (4.9) |
(see [42, Proposition 3.2.1]), where ψi1...ih=ψi1∘⋯∘ψih and Ti1...ih=Ti1...Tih with
T1=15[500221212], T2=15[221050122], T3=15[212122005]. |
Let M0={(x1,x2,x3)∈R3; x1+x2+x3=0}. Kigami [46] introduced the map Φ:G⟶M0 defined by
Φ(x)=1√2((h1(x)h2(x)h3(x))−13(111)), |
with hi(Aj)=δij for Aj∈V0, 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:M0⟶M0; i=1,2,3, by
Hi(x)=Tti(x−Φ(Ai))+Φ(Ai), |
then GH=∪i=1,2,3Hi(GH) and Φ∘ψi=Hi∘Φ 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 n∈N∗}, | (4.12) |
and π:I⟶G such that ψj∘π(ω)=π(jω), for j=1,2,3. For any ω∈I, there exists a unique x∈G such that
{x}=∩h∈N∗Gi1...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(π−1(.))=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)=limh→∞Z(i1...ih). | (4.16) |
Let Z(x)≡Z(π−1(x)). Then, Z(x) is well defined on V∞ (see for instance [47]). Indeed, according to [49, Theorem 3.6], for ν−almost all x∈G,
Z(π−1(x))=Z(ω)=limh→∞Zh(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=(v∣GH)∘Φ, v ∈C1(U) }. | (4.19) |
According to [47], if we fix an orthonormal basis of M0 and regard M0 as R2, then, for any u∈C1(G),
∇u=(∂u∂x1∂u∂x2). | (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,v∈C1(G),
EG(u,v)=∫G∇u.Z∇vdν. |
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 vh∈Vh, 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)∈R3; yi,kh,1∈(εhln(1/εh),2−h−εhln(1/εh))(y,z)∈S}. |
Let φ∈C1(Uh,ik), such that φ=0 on ∂Uh,ik∩∂S. Using the Poincaré inequality, we infer that, for every yi,kh,1∈(εhln(1/εh),2−h−εhln(1/εh)),
∫Sφ2(yi,kh,1,y,z)dydz≤C∫S|∇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)=(∂φ∂y(yi,kh,1,y,z)∂φ∂z(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 2−h−εhln(1/εh), we get
∫2−h−εhln(1/εh)εhln(1/εh)∫εhSφ2dyi,kh,1dyi,kh,2dx3≤Cε2h∫2−h−εhln(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 vh∈Vh,
∫Ωh,ik∖Jh,+,ik∪Jh,−,ik|vh|2dx≤Cε2h∫Ωh,ik∖Jh,+,ik∪Jh,−,ik|∇vh|2dx. | (5.1) |
We can use the same method in Jh,+,ik∪Jh,−,ik to obtain
∫Jh,+,ik∪Jh,−,ik|vh|2dx≤Cε2h∫Jh,+,ik∪Jh,−,ik|∇vh|2dx. | (5.2) |
The combination of (5.1) and (5.2) implies that
∫Ωh,ik|vh|2dx≤Cε2h∫Ωh,ik|∇vh|2dx. | (5.3) |
Then, summing over i and k in (5.3), we obtain that
5h3h+1ε2hReh∫Ωh|vh|2dx≤C5h3h+1Reh∫Ωh|∇vh|2dx. | (5.4) |
As σ∈(0,+∞), we have that
Rej,h≤Cεh≤C2−h, |
from which we deduce that 2h≤C 1Rej,h in Jh. Thus, using (5.4),
5h3h+1ε2h2−h∫Ωh|vh|2dx≤5h3h+1ε2hReh∫Ωh|vh|2dx≤CFh(vh). | (5.5) |
Observing that 3hε2h2−h≈|Ωh|π, we conclude that
suph5h|Ωh|∫Ωh|vh|2dx≤CsuphFh(vh)<+∞. | (5.6) |
We have the following result:
Proposition 8. Let 1Ωh be the characteristic function of the set Ωh. Let vh∈Vh, 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|dx∗⇀h→∞vdHd(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=v1√3 on the part of G which is perpendicular to (−√3/2,1/2), and v2=−v1√3 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 φ∈C0(R3),
limh→∞∫R3φ(x)dϑh=limh→∞3h∑k=1i=1,2,313h+1φ(ai,kh+bi,kh2,0)=1Hd(G)∫Gφ(s,0)dHd(s), |
from which we deduce that
ϑh∗⇀h→∞ϑ=1G(s)dHd(s)⊗δ0(x3)Hd(G). |
Let vh∈L2(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,
|∫R3√5hvhdϑh|2≤C5h∫R3|vh|2dϑh≤C5h|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 φ∈C0(R3,R3). By using Fenchel's inequality, we have
liminfh→∞12∫R3|√5hvh|2dϑh≥liminfh→∞(∫R3√5hvh.φdϑh−12∫R3|φ|2dϑh)≥⟨χ,φ⟩−12∫R3|φ|2dϑ. |
As the left-hand side of this inequality is bounded, we deduce that
sup{⟨χ,φ⟩; φ∈C0(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)=Rivh∘Rti((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,1≡yi,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,i1∂yi,kh,1+vh,irr+∂vh,ir∂r+1r∂vh,iθ∂θ. | (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 h∈N,
vh,i1(εh,r,θ)−vh,i1(2−h−εh,r,θ)=0, | (5.13) |
from which we deduce, using Green's formula, that, for ψ∈C∞c(0,2π) and φ(θ)=∫θ0ψ(ξ)dξ with φ(2π)=0,
3h∑k=1i=1,2,3∫2−h−εhεh∫εh0∫2π0∂vh,i1∂yi,kh,1φ(θ)rdyi,kh,1drdθ=−∫εh0∫2π0∑3hk=1i=1,2,3(vh,i1(εh,r,θ)−vh,i1(2−h−εh,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
2h√5h3h+1εh3h∑k=1i=1,2,3∫2−h−εhεh∫εh0∫2π0vh,irφ(θ)dyi,kh,1drdθ+2h√5h3h+1εh3h∑k=1i=1,2,3∫2−h−εhεh∫εh0∫2π0∂vh,ir∂rφ(θ)rdyi,kh,1drdθ+2h√5h3h+1εh3h∑k=1i=1,2,3∫2−h−εhεh∫εh0∫2π0∂vh,iθ∂θφ(θ)dyi,kh,1drdθ=0. | (5.15) |
Using Green's formula, we deduce that
∫2−h−εhεh∫εh0∫2π0∂vh,ir∂rφ(θ)rdyi,kh,1drdθ=−∫2−h−εhεh∫εh0∫2π0vh,irφ(θ)dyi,kh,1drdθ, | (5.16) |
and
∫2−h−εhεh∫εh0∫2π0∂vh,iθ∂θφ(θ)dyi,kh,1drdθ=−∫2−h−εhεh∫εh0∫2π0vh,iθψ(θ)dyi,kh,1drdθ. | (5.17) |
Combining with (5.15), we deduce that
2h√5h3h+1εh3h∑k=1i=1,2,3∫2−h−εhεh∫εh0∫2π0vh,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
limh→∞2h√5h3h+1εh3h∑k=1i=1,2,3∫2−h−εhεh∫εh0∫2π0vh,iθψ(θ)dyi,kh,1drdθ=1Hd(G)∫G∫2π0(−w(s)sinθ+v3(s)cosθ)ψ(θ)dsdθ=0, | (5.19) |
where
w(s)={v2(s)on G1, −v1(s)√3+v2(s)on G2, v1(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=v1√3 on G2, and v2=−v1√3 on G3.
Proposition 9. We suppose that σ∈(0,+∞). Let vh∈Vh∩H2(Ωh,R3), such that suphFh(vh)<+∞. Then, for every sequence (φh)h, such that φh∈H1(Ωh) and
suph5h|Ωh|∫Ωh|∇φh|2dx<+∞ , √5hφhπ1Ωh(x)3|Ωh|dx∗⇀h→∞φdHd(s)⊗δ0(x3)Hd(G)inM(R3), | (5.21) |
we have
1. φ(s,0)∈DE and ∫G∇φ.Z∇φdν<+∞,
2. there exists a subsequence of (vh)h, still denoted as (vh)h, and v∈L2Hd(G), such that
limh→∞5h|Ωh|∫Ωhvh.∇φhdx=∫Gvn.Z∇φdν=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)=φh∘Rti((yi,kh,1yi,kh,2x3)+Ri(ai,kh,1ai,kh,20)), | (5.22) |
and
˜φih(yi,kh,1)=1πε2h∫εhSφih(yi,kh,1,yi,kh,2,x3)dyi,kh,2dx3=1πε2h∫εhSφh(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|3h∑k=1i=1,2,3∫Ωh,ik|∇φh|2dx≥5h3h+13h∑k=1i=1,2,3∫2−h02hπε2h∫εhS(∂φih∂yi,kh,1)2dyi,kh,1dyi,kh,2dx3 ≥5h3h+13h∑k=1i=1,2,31πε2h∫εhS(∫2−h0∂φih∂yi,kh,1dyi,kh,1)2dyi,kh,2dx3=5h3h+13h∑k=1i=1,2,31πε2h∫εhS(φih(2−h,yi,kh,2,x3)−φih(0,yi,kh,2,x3))2dyi,kh,2dx3≥5h3h+13h∑k=1i=1,2,3(1πε2h∫εhS(φih(2−h,yi,kh,2,x3)−φih(0,yi,kh,2,x3))dyi,kh,2dx3)2=5h3h+13h∑k=1i=1,2,3(˜φih(2−h)−˜φih(0))2=5h3h+13h∑k=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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