Asymptotic approximation for the solution to a semi-linear elliptic problem in a thin aneurysm-type domain

Taras A. Mel’nyk 1  and Arsen V. Klevtsovskiy 1
  • 1 Department of Mathematical Physics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
Taras A. Mel’nyk
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  • Department of Mathematical Physics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
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and Arsen V. Klevtsovskiy
  • Department of Mathematical Physics, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
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Abstract

A semi-linear boundary-value problem with nonlinear Robin boundary conditions is considered in a thin 3D aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter 𝓞(ε). Using the multi-scale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε → 0. Namely, we derive the limit problem (ε = 0) in the corresponding graph, define other terms of the asymptotic approximation and prove energetic and uniform pointwise estimates. These estimates allow us to observe the impact of the aneurysm on some properties of the solution.

1 Introduction

Investigations of various physical and biological processes in channels, junctions and networks are urgent for numerous fields of natural sciences (see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and the references therein). Especial interest is the investigation of the influence of a local geometrical heterogeneity in vessels on the blood flow. This is both an aneurysm (a pathological extension of an artery like bulge) and a stenosis (a pathological restriction of an artery). The understanding of the impact of a local geometric irregularity on properties of solutions to boundary-value problems in such domains can have useful applications in medicine and many areas of applied science. In [15] the authors classified 12 different aneurysms and proposed computational approach for this study. The aneurysm models have been meshed with 800,000–1,200,000 tetrahedral cells containing three boundary layers. It was showed that the geometric aneurysm form essentially impacts on the hemodynamics of the blood flow. However, as was noted by the authors, the question how to model blood flow with sufficient accuracy is still open.

This question was the main motivation for us to develop a new approach (asymptotic one) for the study of boundary-value problems in domains of such type, since numerical methods do not give good approximations through the presence of a local geometric irregularity. It is clear that such domains are prototypes of many other biological and engineering structures, but we prefer to call them thin aneurysm-type domains as more comprehensive and concise.

There are several asymptotic approaches to study such problems (see [1, 3, 10, 11, 12, 13, 16, 17, 18]) with special assumptions, namely: the uniform boundary conditions on the lateral surfaces of the thin cylinders, the right-hand sides depend only on the longitudinal variable in the direction of the corresponding cylinder and they are constant in neighbourhoods of the nodes and vertices, the right-hand sides satisfy especial orthogonality conditions (for more detail see [19, 20]). These assumptions significantly narrow the class of problems that can be studied by such methods.

In the present paper, we continue to develop the asymptotic method proposed in [19], where the complete asymptotic expansion was constructed for the solution to a linear boundary-value problem for the Poisson equation with a nonuniform Neumann boundary conditions in a thin 2D aneurysm-type domain, and in [20], where similar results were obtained for the Poisson equation in a thin 3D aneurysm-type domain, which does not need the above mentioned assumptions. Here, we have adapted this method to semi-linear elliptic problems with nonlinear perturbed Robin boundary conditions in thin aneurysm-type domains. These results were presented on the conference [21].

1.1 Statement of the problem

The model thin aneurysm-type domain Ωε consists of three thin curvilinear cylinders

Ωε(i)={x=(x1,x2,x3)R3:ε<xi<1,j=13(1δij)xj2<ε2hi2(xi)},i=1,2,3,

that are joined through a domain Ωε(0) (referred in the sequel "aneurysm"). Here ε is a small parameter; (0,13); the positive functions {hi}i=13 belong to the space C1([0,1]) and they are equal to some constants in neighborhoods at the points x = 0 and xi = 1, i = 1, 2, 3; the symbol δij is the Kroneker delta, i.e., δii = 1 and δij = 0 if ij.

The aneurysm Ωε(0) (see Fig. 1) is formed by the homothetic transformation with coefficient ε from a bounded domain Ξ(0) ∈ ℝ3, i.e., Ωε(0) = εΞ(0). In addition, we assume that its boundary contains the disks

Υε(i)(ε)={xR3:xi=ε,j=13(1δij)xj2<ε2hi2(ε)},i=1,2,3,

and denote Γε(0):=Ωε(0){Υε(1)(ε)¯Υε(2)(ε)¯Υε(3)(ε)¯}.

Fig. 1
Fig. 1

The aneurysm Ωε(0)

Citation: Open Mathematics 15, 1; 10.1515/math-2017-0114

Thus the model thin aneurysm-type domain Ωε (see Fig. 2) is the interior of the union k=03Ωε(k)¯ and we assume that it has the Lipschitz boundary.

Fig. 2
Fig. 2

The model thin aneurysm-type domain Ωε

Citation: Open Mathematics 15, 1; 10.1515/math-2017-0114

Remark 1.1

We can consider more general thin aneurysm-type domains with arbitrary orientation of thin cylinders (their number can be also arbitrary). But to avoid technical and huge calculations and to demonstrate the main steps of the proposed asymptotic approach we consider a such kind of the thin aneurysm-type domain, when the cylinders are placed on the coordinate axes.

In Ωε, we consider the following semi-linear elliptic problem:

Δuε(x)+κ0(uε(x))=f(x),xΩε,νuε(x)=0,xΓε(0),νuε(x)εκi(uε(x))=φε(x),xΓε(i),i=1,2,3,uε(x)=0,xΥε(i)(1),i=1,2,3,

where Γε(j)=Ωε(i){xR3:ε<xi<1},ν is the outward normal derivative. The the given functions satisfy the following assumptions:

  1. the functions {κj}j=03 belong to the space C1(ℝ) and there exist positive constants κ > 0 and κ+ > 0 such that
    κκj(s)κ+ for sR,j=0,1,2,3;
  2. φε(x):=εφ(j)(xi,x¯iε),xΓε(i), i = 1, 2, 3, where
    x¯i=(x2,x3),i=1,(x1,x3),i=2,(x1,x2),i=3,
    and φ(i)C(Ωε^0(i)¯), i = 1,2,3;
  3. the function fC(Ωε^0¯) and its restrictions on the curvilinear cylinders Ωε^0(i) belong to the spaces C1xi(Ωε^0(i)¯), i = 1, 2, 3, respectively.Here ε̂0 is a fixed positive number and in what follows all values of the small parameter ε belong to the interval (0,ε̂0).Recall that a function uε from the Sobolev space HE={uH1(Ωε):u|Υε(i)(1)=0,i=1,2,3} is called a weak solution to the problem (1) if it satisfies the integral identity
    Ωεuεvdx+Ωεκ0(uε)vdx+εi=13Γε(i)κi(uε)vdσx=Ωεfvdxi=13Γε(i)φεvdσx
    for any function v ∈ 𝓗ε.

The aim of the present paper is to

  1. construct the asymptotic approximation for the solution to the problem (1) as the parameter ε → 0;
  2. derive the corresponding limit problem (ε = 0);
  3. prove the corresponding asymptotic estimates from which the influence of the aneurysm will be observed.

1.2 Existence and uniqueness of the weak solution

In order to obtain operator statement for the problem (1) we introduce the new norm ∥⋅∥ε in 𝓗ε, which is generated by the scalar product

(u,v)ε=Ωεuvdx,u,vHε.

Due to the uniform Dirichlet condition on Υε(i)(1), i = 1, 2, 3, the norm ∥⋅∥ε and the ordinary norm ∥⋅∥H1ε) are uniformly equivalent, i. e., there exist constants C1 > 0 and ε0 > 0 such that for all ε ∈ (0, ε0) and for all u ∈ 𝓗ε the following estimates hold:

uεuH1(Ωε)C1uε.

Remark 1.2

Here and in what follows all constants {Ci} and {ci} in inequalities are independent of the parameter ε.

In the following we will often use the identities (see [22])

εΓε(i)v2dσxC1(ε2Ωε(i)|x¯iv|2dx+Ωε(i)v2dx),vH1(Ωε(i)),i=1,2,3;

and inequalities

κs2+κj(0)sκj(s)sκ+s2+κj(0)ssR,j=0,1,2,3,

that can be deduced from the conditions (2) (see [22]).

Denote by Hε the dual space to 𝓗ε and define a nonlinear operator 𝓐ε : 𝓗εHε through the relation

Aε(u),vε=Ωεuvdx+Ωεκ0(u)vdx+εi=13Γε(i)κi(u)vdσxu,vHε,

where 〈⋅, ⋅〉ε is the duality pairing of Hε and 𝓗ε.

In this case the integral identity (3) can be rewritten as follows

Aε(uε),vε=Fε,vεvHε,

where FεHε is defined by

Fε,vε=Ωεfvdxi=13Γε(i)φεvdσxvHε.

To prove the well-posedness result, we verify some properties of the operator 𝓐ε.

  1. With the help of (6) and Cauchy’s inequality with δ(abδa2+b24δ,a,b>0), we obtain
    Aε(v),vεΩε|v|2dx+Ωεκ|v|2dx+Ωεκ0(0)|v|dx+εi=13Γε(i)κ|v|2dσx+Γε(i)κi(0)|v|dσxvε2δ|κ0(0)|Ωεv2dx+εi=13|κi(0)|Γε(i)v2dσx14δ|κ0(0)||Ωε|+εi=13|κi(0)||Γε(i)|.
    Then using (5), we can select appropriate δ such that
    Aε(v),vεC2vε2C3vHε.
    This inequality means that the operator 𝓐ε is coercive.
  2. Let us show that it is monotone. Taking into account (2), we get
    Aε(u1)Aε(u2),u1u1εΩε|u1u2|2dx+κΩε|u1u2|2dx+κεi=13Γε(i)|u1u2|2dσxu1u2ε2.
  3. The operator 𝓐ε is hemicontinuous. Ineed, the real valued function
    [0,1]τAε(u1+τv),u2ε
    is continuous on [0, 1 ] for all fixed u1, u2, v ∈ 𝓗ε due to the continuity of the functions κj, j = 0, 1, 2, 3 and Lebesque’s dominated convergence theorem.
  4. Let us prove that operator 𝓐ε is bounded. Using Cauchy-Bunyakovsky integral inequality, (4) and (6), we deduce the following inequality:
    Aε(u),vεΩεuvdx+Ωε(κ+|u|+|κ0(0)|)|v|dx+εi=13Γε(i)(κ+|u|+|κi(0)|)|v|dσxuεvε+κ+uL2(Ωε)vL2(Ωε)+|κ0(0)||Ωε|vL2(Ωε)+εi=13(κ+uL2(Γε(i))vL2(Γε(i))+|κi(0)||Γε(i)|vL2(Γε(i)))
    Now with the help of (5), we obtain
    Aε(u),vεC5(1+uε)vεu,vHε.

Thus, the existence and uniqueness of the weak solution for every fixed value ε follow directly from Theorem 2.1 (see [23, Section 2]).

2 Formal asymptotic approximation

In this section we assume that the functions f and φε are smooth enough. Following the approach of [19, 20], we propose ansatzes of the asymptotic approximation for the solution to the problem (1) in the following form:

  1. the regular part of the approximation
    ω0(i)(xi)+εω1(i)(xi)+ε2u2(i)(xi,x¯iε)+ε3u3(i)(xi,x¯iε)
    is located inside each thin cylinder Ωε(i) and their terms depend both on the corresponding longitudinal variable xi and so-called “fast variables” x¯iε(i=1,2,3);
  2. and the inner part of the approximation
    N0(xε)+εN1(xε)+ε2N2(xε)
    is located in a neighborhood of the aneurysm Ωε(0).

2.1 Regular part

Substituting the representation (8) for each fixed index i ∈{1, 2, 3 } into the differential equation of the problem (1), using Taylor’s formula for the function κ0 at s=ω0(i)(xi) and the function f at the point xi = (0,0), and collecting coefficients at ε0, we obtain

Δξ¯iu2(i)(xi,ξ¯i)=d2ω0(i)dxi2(xi)κ0(ω0(i)(xi))+f0(i)(xi),

where ξ¯i=x¯iε and f0(i)(xi):=f(x)|x¯i=(0,0).

It is easy to calculate the outer unit normal to Γε(i):

νi(xi,ξi¯)=11+ε2|hi(xi)|2(εhi(xi)),ν¯i(ξi¯))={(εh1(x1),ν2(1)(ξ¯1),ν3(1)(ξ¯1))1+ε2|h1(x1)|2,i=1(ν1(2)(ξ¯2),εh2(x2),ν3(2)(ξ¯2))1+ε2|h2(x2)|2,i=2,(ν1(3)(ξ¯3),ν2(3)(ξ¯3),εh3(x3))1+ε2|h3(x3)|2,i=3,

where ν¯i(x¯iε) is the outward normal for the disk Υε(i)(xi):={ξ¯iR2:|ξ¯i|<hi(xi)}.

Taking the view of the outer unit normal into account and putting the sum (8) into the third relation of the problem (1), we get with the help of Taylor’s formula for the function κi at s=ω0(i)(xi) the following relation:

εν¯i(ξ¯i)u2(i)(xi,ξ¯i)=hi(xi)εdω0(i)dxi(xi)+ε(κi(ω0(i)(xi))+φ(i)(xi,ξ¯i)).

Relations (10) and (11) form the linear inhomogeneous Neumann boundary-value problem

Δξ¯iu2(i)(xi,ξ¯i)=d2ω0(i)dxi2(xi)κ0(ω0(i)(xi))+f0(i)(xi),ξ¯iΥi(xi),νξ¯iu2(i)(xi,ξ¯i)=hi(xi)dω0(i)dxi(xi)+κi(ω0(i)(xi))+φ(i)(xi,ξ¯i),ξ¯iΥi(xi),u2(i)(xi,)Υi(xi)=0,

to define u2(i). Here u(xi,)Υi(xi):=Υi(xi)u(xi,ξ¯i)dξ¯i, the variable xi is regarded as a parameter from the interval Iε(i) : = {x : xi ∈(εℓ, 1), xi = (0,0 We add the third relation in (12) for the uniqueness of a solution.

Writing down the necessary and sufficient conditions for the solvability of the problem (12), we derive the differential equation

πddxi(hi2(xi)dω0(i)dxi(xi))+πhi2(xi)κ0(ω0(i)(xi))+2πhi(xi)κi(ω0(i)(xi))=πhi2(xi)f0(i)(xi)Υi(xi)φ(i)(xi,ξ¯i)dlξ¯i,xiIε(i),

to define ω0(i) (i ∈{1,2,3}).

Let ω0(i) be a solution of the differential equation (13) (the existence will be proved in the subsection 2.2.1). Thus, there exists a unique solution to the problem (12) for each i ∈{1, 2, 3 }.

For determination of the coefficients u3(i), i = 1, 2, 3, we similarly obtain the following problems:

Δξ¯iu3(i)(xi,ξ¯i)=d2ω1(i)dxi2(xi)κ0(ω0(i)(xi))ω1(i)(xi)+f1(i)(xi,ξ¯i),ξ¯iΥi(xi),νξ¯iu3(i)(xi,ξ¯i)=hi(xi)dω1(i)dxi(xi)+κi(ω0(i)(xi))ω1(i)(xi),ξ¯iΥi(xi),u3(i)(xi,)Υi(xi)=0,

for each i ∈{1, 2, 3 }. Here

f1(i)(xi,ξ¯i)=j=13(1δij)ξjxjf(x)|x¯i=(0,0).

Repeating the previous reasoning, we find that the coefficients {ω1(i)}i=13 have to be solutions to the respective linear ordinary differential equation

πddxi(hi2(xi)dω1(i)dxi(xi))+πhi2(xi)κ0(ω0(i)(xi))ω1(i)(xi)+2πhi(xi)κi(ω0(i)(xi))ω1(i)(xi)=Υi(xi)f1(i)(xi,ξ¯i)dξ¯i,xiIε(i)(i{1,2,3}).

2.2 Inner part

To obtain conditions for the functions {ωk(i)}, i = 1, 2, 3, k ∈{0, 1 } at the point (0,0,0), we introduce the inner part of the asymptotic approximation (9) in a neighborhood of the aneurysm Ωε(0). If we pass to the “fast variables” ξ=xε and tend ε to 0, the domain Ωε is transformed into the unbounded domain Ξ that is the union of the domain Ξ(0) and three semibounded cylinders

Ξ(i)={ξ=(ξ1,ξ2,ξ3)R3:<ξi<+,|ξ¯i|<hi(0)},i=1,2,3,

i.e., Ξ is the interior of i=03 Ξ(i) (see Fig. 3).

Fig. 3
Fig. 3

Domain Ξ

Citation: Open Mathematics 15, 1; 10.1515/math-2017-0114

Let us introduce the following notation for parts of the boundary of the domain Ξ:

  1. Γi = {ξ ∈ ℝ3: < ξi < + ∞, |ξi| = hi(0)}, i = 1, 2, 3,
  2. Γ0 = Ξ∖ ( i=13Γi).

Substituting (9) into the problem (1) and equating coefficients at the same powers of ε, we derive the following relations for Nk, (k ∈{0,1,2}) :

ΔξNk(ξ)=Fk(ξ),ξΞ,νξNk(ξ)=0,ξΓ0,ν¯ξ¯iNk(ξ)=Bk(i)(ξ),ξΓi,i=1,2,3,Nk(ξ)ωk(i)(0)+Ψk(i)(ξ),ξi+,ξ¯iΥi(0),i=1,2,3.

Here

F0F10,F2(ξ)=κ0(N0)+f(0),ξΞ,B0(i)B1(i)0,B2(i)(ξ)=κi(N0)+φ(i)(0,ξ¯i),ξΓi,i=1,2,3.

The right hand sides in the differential equation and boundary conditions on {Γi} of the problem (17) are obtained with the help of the Taylor decomposition of the functions f and φ(i) at the points x = 0 and xi = 0, i = 1, 2, 3, respectively.

The fourth condition in (17) appears by matching the regular and inner asymptotics in a neighborhood of the aneurysm, namely the asymptotics of the terms {Nk} as ξi →+ ∞ have to coincide with the corresponding asymptotics of the terms {ωk(i)} as xi = εξi →+0, i = 1, 2, 3, respectively. Expanding formally each term of the regular asymptotics in the Taylor series at the points xi = 0 and collecting the coefficients of the same powers of ε, we get

Ψ0(i)0,Ψ1(i)(ξ)=ξidω0(i)dxi(0),i=1,2,3,Ψ2(i)(ξ)=ξi22d2ω0(i)dxi2(0)+ξidω1(i)dxi(0)+u2(i)(0,ξ¯i),i=1,2,3.

A solution of the problem (17) at k = 1, 2 is sought in the form

Nk(ξ)=i=13Ψk(i)(ξ)χi(ξi)+N~k(ξ),

where χiC(ℝ+), 0 ≤ χi ≤ 1 and

χi(ξi)=0, if ξi1+,1, if ξi2+,i=1,2,3.

Then k has to be a solution of the problem

ΔξN~k(ξ)=F~k(ξ),ξΞ,νξN~k(ξ)=0,ξΓ0,νξ¯iN~k(ξ)=B~k(i)(ξ),ξΓi,i=1,2,3,

where

F~1(ξ)=i=13(ξidω0(i)dxi(0)χi(ξi)+2dω0(i)dxi(0)χi(ξi)),F~2(ξ)=i=13[(ξi22d2ω0(i)dxi2(0)+ξidω1(i)dxi(0)+u2(i)(0,ξ¯i))χi(ξi)+2(ξid2ω0(i)dxi2(0)+dω1(i)dxi(0))χi(ξi)]+(1i=13χi(ξi))f(0)κ0(N0)+i=13κ0(ω0(i)(0))χi(ξi)

and

B~1(i)0,B~2(i)(ξ)=(1χi(ξi))φ(i)(0,ξ¯i)+κi(N0)i=13κi(ω0(i)(0))χi(ξi),i=1,2,3.

In addition, we demand that k satisfies the following stabilization conditions:

N~k(ξ)ωk(i)(0) as ξi+,ξ¯iΥi(0),i=1,2,3.

The existence of a solution to the problem (20) in the corresponding energetic space can be obtained from general results about the asymptotic behavior of solutions to elliptic problems in domains with different exits to infinity (see e.g. [8, 24]). We will use approach proposed in [6, 8].

Let C0,ξ(Ξ¯) be a space of functions infinitely differentiable in -Ξ and finite with respect to ξ, i.e.,

vC0,ξ(Ξ¯)R>0ξΞ¯ξiR,i=1,2,3:v(ξ)=0.

We now define a space H:=(C0,ξ(Ξ¯),H)¯, where

vH=Ξ|v(ξ)|2dξ+Ξ|v(ξ)|2|ρ(ξ)|2dξ,

and the weight function ρC(ℝ3), 0 ≤ ρ ≤ 1 and

ρ(ξ)=1, if ξΞ(0),|ξi|1, if ξi+1,ξΞ(i),i=1,2,3.

Definition 2.1

A function k from the space 𝓗 is called a weak solution of the problem (20) if the identity

ΞN~kvdξ=ΞF~kvdξi=13ΓiB~k(i)vdσξ.

holds for all v ∈ 𝓗.

Similarly as in [6], we prove the following proposition.

Proposition 2.2

Let ρ1F~kL2(Ξ),ρ1B~k(i)L2(Γi), i = 1, 2, 3. Then there exists a weak solution of problem (20) if and only if

ΞF~kdξ=i=13ΓiB~k(i)dσξ.

This solution is defined up to an additive constant. The additive constant can be chosen to guarantee the existence and uniqueness of a weak solution of problem (20) with the following differentiable asymptotics:

N^k(ξ)=O(exp(γ1ξ1))asξ1+,δk(2)+O(exp(γ2ξ2))asξ2+,δk(3)+O(exp(γ3ξ3))asξ3+,

where γi, i = 1, 2, 3 are positive constants.

The constants δk(2) and δk(3) in (24) are defined as follows:

δk(i)=ΞNiF~k(ξ)dξj=13ΓiNiB~k(j)(ξ)dσξ,i=2,3,k{0,1,2},

where 𝔑2 and 𝔑3 are special solutions to the corresponding homogeneous problem

ΔξN=0 in Ξ,νN=0 on Ξ,

for the problem (20).

Proposition 2.3

The problem (26) has two linearly independent solutions 𝔑2 and 𝔑3 that do not belong to the space 𝓗 and they have the following differentiable asymptotics:

N2(ξ)=ξ1πh12(0)+O(exp(γ1ξ1))asξ1+,C2(2)+ξ2πh22(0)+O(exp(γ2ξ2))asξ2+,C2(3)+O(exp(γ3ξ3))asξ3+,
N3(ξ)=ξ1πh12(0)+O(exp(γ1ξ1))asξ1+,C3(2)+O(exp(γ2ξ2))asξ2+,C3(3)+ξ3πh32(0)+O(exp(γ3ξ3))asξ3+,

Any other solution to the homogeneous problem, which has polynomial growth at infinity, can be presented as a linear combination α1 + α2 𝔑2 + α3 𝔑3.

Proof

The solution 𝔑2 is sought in the form of a sum

N2(ξ)=ξ1πh12(0)χ1(ξ1)+ξ2πh22(0)χ2(ξ2)+N~2(ξ),

where N~2 ∈ 𝓗 and N~2 is the solution to the problem (20) with right-hand sides

F~2(ξ)=1πh12(0)((ξ1χ1(ξ1))+χ1(ξ1)),ξΞ(1),1πh22(0)((ξ2χ2(ξ2))+χ2(ξ2)),ξΞ(2),0,ξΞ(0)Ξ(3).

It is easy to verify that the solvability condition (23) is satisfied. Thus, by virtue of Proposition 2.1 there exist a unique solution N~2 ∈ 𝓗 that has the asymptotics

N~2(ξ)=(1δ1j)C2(j)+O(exp(γjξj)) as ξj+,j=1,2,3.

Similarly we can prove the existence of the solution 𝔑3 with the asymptotics (28).

Obviously, 𝔑2 and 𝔑3 are linearly independent and any other solution to the homogeneous problem, which has polynomial growth at infinity, can be presented as α1 + α2𝔑2 + α3𝔑3.□

Remark 2.4

To obtain formulas (25) for the constants δk(2)andδk(3), it is necessary to substitute the functions N^k,N2andN^k,N3 in the second Green-Ostrogradsky formula

ΞR(N^ΔξNNΔξN^)dξ=ΞR(N^νξNNνξN^)dσξ

respectively, and then pass to the limit as R → + ∞. Here ΞR = Ξ ∩ {ξ : |ξi| < R, i = 1, 2, 3}.

2.2.1 Limit problem

The problem (17) at k = 0 is as follows:

ΔξN0(ξ)=0,ξΞ,νξN0(ξ)=0,ξΓ0,νξ¯iN0(ξ)=0,ξΓi,i=1,2,3,N0(ξ)ω0(i)(0),ξi+,ξ¯iΥi(0),i=1,2,3,

It is easy to verify that δ0(2)=δ0(3)=0 and N^00. Thus, this problem has a solution in 𝓗 if and only if

ω0(1)(0)=ω0(2)(0)=ω0(3)(0);

in this case N0N~0ω0(1)(0).

In the problem (20) at k = 1 the solvability condition (23) reads as follows:

πh12(0)dω0(1)dx1(0)+πh22(0)dω0(2)dx2(0)+πh32(0)dω0(3)dx3(0)=0.

Substituting (8) into the fourth condition in (1) and neglecting terms of order of 𝓞(ε), we arrive at the following boundary conditions:

ω0(i)(1)=0,i=1,2,3.

Thus, taking into account (13), (30), (31) and (32), we obtain for {ω0(i)}i=13 the following semi-linear problem:

πddxi(hi2(xi)dω0(i)dxi(xi))+πhi2(xi)κ0(ω0(i)(xi))+2πhi(xi)κi(ω0(i)(xi))=F^0(i)(xi),xiIi,i=1,2,3,ω0(i)(1)=0,i=1,2,3,ω0(1)(0)=ω0(2)(0)=ω0(3)(0),i=13πhi2(0)dω0(i)dxi(0)=0,

where Ii := {x : xi ∈ (0, 1), xi = (0, 0)} and

F^0(i)(xi):=πhi2(xi)f(x)|x¯i=(0,0)Υi(xi)φ(i)(xi,ξ¯i)dlξ¯j,xIi.

The problem (33) is called limit problem for problem (1).

For functions

ϕ~(x)=ϕ(1)(x1), if x1I1,ϕ(2)(x2), if x2I2,ϕ(3)(x3), if x3I3,

defined on the graph I1I2I3, we introduce the Sobolev space

H0:={ϕ~:ϕ(i)H1(Ii),ϕ(i)(1)=0,i=1,2,3, and ϕ(1)(0)=ϕ(2)(0)=ϕ(3)(0)}

with the scalar product

(ϕ~,ψ~)0:=i=13π01hi2(xi)dϕ(i)dxidψ(i)dxidxi,ϕ~,ψ~H0.

Definition 2.5

A function ω͠ ∈ 𝓗0 is called a weak solution to the problem (33) if it satisfies the integral identity

(ω~,ψ~)0+i=13(π01hi2(xi)κ0(ω(i)(xi))ψ(i)(xi)dxi+2π01hi(xi)κi(ω(i)(xi))ψ(i)(xi)dxi)=i=1301F^0(i)(xi)ψ(i)(xi)dxiψ~H0.

Similarly as was done in Section 1.2, the integral identity (35) can be rewritten as follows

A0(ω~),ψ~0=F0,ψ~0ψ~H0.

where the nonlinear operator 𝓐0 : 𝓗0H0 is defined through the relation

A0(ϕ(i)),ψ(i)0=(ϕ~,ψ~)0+i=13(π01hi2κ0(ϕ(i))ψ(i)dxi+2π01hiκi(ϕ(i))ψ(i)dxi)ϕ~,ψ~H0,

and F0H0 is defined by

F0,ψ~0=i=1301F^0(i)ψ(i)dxiψ~H0,

where 〈⋅,⋅〉0 is the duality pairing of the dual space H0 and 𝓗0.

Using (2) and (6), we can prove that the operator 𝓐0 is bounded, strongly monotone, hemicontinuous and coercive. As a result, the existence and uniqueness of the weak solution to the problem (33) follow directly from Theorem 2.1 (see [23, Section 2]).

2.2.2 Problem for {ω1}

Let us verify the solvability condition (23) for the problem (20) at k = 2. Knowing that N0ω0(1)(0) and taking into account the third relation in problem (12), the equality (23) can be re-written as follows:

i=13[πhi2(0)+1+2(ξid2ω0(i)dxi2(0)+dω1(i)dxi(0))χi(ξi)dξi++2(1χi(ξi))Υi(0)(f(0)κ0(ω0(i)(0)))dξ¯idξi+2(1χi(ξi))Υi(0)(φ(i)(0,ξ¯i)+κi(ω0(i)(0)))dlξ¯idξi]+Ξ(0)(f(0)κ0(ω0(i)(0)))dξ=0.

Whence, integrating by parts in the first integrals with regard to (13), we obtain the following relations for {ω1(i)}:

i=13πhi2(0)dω1(i)dxi3(0)=d1,

where

d1=i=13πhi2(0)(f(0)κ0(ω0(i)(0)))2πhi(0)κi(ω0(i)(0))Υi(0)φ(i)(0,ξ¯i)dlξ¯i|Ξ(0)|(f(0)κ0(ω0(i)(0))).

Hence, if the functions {ω1(i)}i=13 satisfy (37), then there exists a weak solution 2 of the problem (20). According to Proposition 2.2, it can be chosen in a unique way to guarantee the asymptotics (24).

It remains to satisfy the stabilization conditions (21) at k = 1. For this, we represent a weak solution of the problem (20) in the following form:

N~1=ω1(1)(0)+N^1.

Taking into account the asymptotics (24), we have to put

ω1(1)(0)=ω1(2)(0)δ1(2)=ω1(3)(0)δ1(3).

As a result, we get the solution of the problem (17) with the following asymptotics:

N1(ξ)=ω1(i)(0)+Ψ1(i)(ξ)+O(exp(γiξi))as ξi+,i=1,2,3.

Let us denote by

G1(ξ):=ω1(i)(0)+Ψ1(i)(ξ),ξΞ(i),i=1,2,3.

Remark 2.6

Due to (40), the function N1G1 is exponentially decrease as ξi → + ∞, i = 1, 2, 3.

Relations (39) and (37) are the first and second transmission conditions for {ω1(i)}i=13 at x = 0. Thus, the coefficients {ω1(1),ω1(2),ω1(3)} are determined from the linear problem

πddxi(hi2(xi)dω1(i)dxi(xi))+πhi2(xi)κ0(ω0(i)(xi))ω1(i)(xi)+2πhi(xi)κi(ω0(i)(xi))ω1(i)(xi)=F^1(i)(xi),xiIi,i=1,2,3,ω1(i)(1)=0,i=1,2,3,ω1(1)(0)=ω1(2)(0)δk(2)=ω1(3)(0)δk(3),i=13πhi2(0)dω1(i)dxi(0)=d1,

where

F^1(i)(xi)=Υi(xi)f1(i)(xi,ξ¯i)dξ¯i,xIi,i=1,2,3.

The constants δ1(2) and δ1(3) are uniquely determined (see Remark 2.4) by formula

δ1(i)=ΞNij=13(ξjdω0(j)dxj(0)χj(ξj)+2dω0(j)dxj(0)χj(ξj))dξ,i=2,3.

With the help of the substitutions

ϕ1(1)(x1)=ω1(1)(x1),ϕ1(2)(x2)=ω1(2)(x2)δ1(2)(1x2),ϕ1(3)(x3)=ω1(3)(x3)δ1(3)(1x3),

we reduce the problem (41) to the respective integral identity in the space 𝓗0 and then the existence and uniqueness of a solution of this identity (and hence the problem (41)) follows from the Riesz representation theorem.

3 Justification

With the help of the coefficients {ω0(i)},{ω1(i)},N1 and smooth cut-off functions defined by formulas

χ(i)(xi)=1, if xi3,0, if xi2,i=1,2,3,

we construct the following asymptotic approximation:

Uε(1)(x)=i=13χ(i)(xiεα)(ω0(i)(xi)+εω1(i)(xi))+(1i=13χ(i)(xiεα))(ω0(1)(0)+εN1(xε)),xΩε,

where α is a fixed number from the interval (23,1).

Theorem 3.1

Let assumptions made in the statement of the problem (1) be satisfied. Then the sum (44) is the asymptotic approximation for the solution uε to the boundary-value problem (1) in the Sobolev space H1ε), i. e.,

C0>0ε0>0ε(0,ε0):uεUε(1)H1(Ωε)<_C0ε1+α2.

Proof

Substituting Uε(0) in the equations and the boundary conditions of problem (1), we find

ΔUε(1)+κ0(Uε(1))f=R^ε in Ωε,vUε(1)εκi(Uε(1))φε=R˘ε,(i) on Γε(i),i=1,2,3,Uε(1)=0on Υε(i)(1),i=1,2,3,νUε(1)=0on Γε(0),

where

R^ε(x)=i=13(2εαdχ(i)dζi(ζi)|ζi=xiεα(dω0(i)dxi(xi)dω0(i)dxi(0)+εdω1(i)dxi(xi)(N1ξi(ξ)G1ξi(ξ))|ξ=xε)+ε2αd2χ(i)dζi2(ζi)|ζi=xiεα(ω0(i)(xi)ω0(i)(0)xidω0(i)dxi(0)+εω1(i)(xi)εω1(i)(0)εN1(xε)+εG1(xε))+χ(i)(xiεα)(d2ω0(i)dxi2(xi)+εd2ω1(i)dxi2(xi)))+κ0(Uε(1)(x))f(x),

and

R˘ε,(i)(x)=εhi(xi)1+ε2|hi(xi)|2χ(i)(xiεα)(dω0(i)dxi(xi)+εdω1(i)dxi(xi))εκi(Uε(1)(x))φε(x).

From (46) we derive the following integral relation:

ΩεUε(1)vdx+Ωεκ0(Uε(1))vdx+εi=13Γε(i)κi(Uε(1))vdσxΩεfvdx+i=13Γε(i)φεvdσx=Rε(v)vHε,

where

Rε(v)=ΩεR^εvdxi=13Γε(i)R˘ε,(i)vdσx.

From (12) and (14) we deduce that integral identities

Υi(xi)d2ω0(i)dxi2ηdξ¯i=Υi(xi)Tξ¯iu2(i).ξ¯iηdξ¯iΥi(xi)hidω0(i)dxiηdlξ¯i+Υi(xi)κ0(ω0(i))ηdξ¯i+Υi(xi)κi(ω0(i))ηdlξ¯iΥi(xi)f0(i)ηdξ¯i+Υi(xi)φ(i)ηdlξ¯i

and

Υi(xi)d2ω1(i)dxi2ηdξ¯i=Υi(xi)ξ¯iu3(i)ξ¯iηdξ¯iΥi(xi)hidω1(i)dxiηdlξ¯i+Υi(xi)κ0(ω0(i))ω1(i)ηdξ¯i+Υi(xi)κi(ω0(i))ω1(i)ηdlξ¯iΥi(xi)f1(i)ηdξ¯i

hold for all ηH1i(xi)) and for all xiIε(i), i = 1, 2, 3.

Using (50) and (51), we rewrite Rε in the form

Rε(v)=j=110Rε,j(v),

where

Rε,1(v)=Ωε(κ0(Uε(1)(x))i=13χ(i)(xiεα)(κ0(ω0(i)(xi))+εκ0(ω0(i)(xi))ω1(i)(xi)))v(x)dx,Rε,2(v)=εi=13Γε(i)(κi(Uε(1)(x))χ(i)(xiεα)(κi(ω0(i)(xi))+εκi(ω0(i)(xi))ω1(i)(xi)))v(x)dσx,Rε,3(v)=Ωε(f(x)i=13χ(i)(xiεα)(f0(i)(xi)+εf1(i)(xi,x¯iε)))v(x)dx,Rε,4(v)=i=13Γε(i)(1χ(i)(xiεα))ε(x)v(x)dσx,Rε,5(v)=εi=13Γε(i)hi(xi)(dω0(i)dxi(xi)+εdω1(i)dxi(xj))111+ε2|hi(xi)|2χp(i)(xiεα)v(x)dσx,Rε,6(v)=2εαi=13Ωεdχ(i)dζi(ζi)|ζi=xiεα(dω0(i)dxi(xi)dω0(i)dxi(0)+εdω1(i)dxi(xi))v(x)dx,Rε,7(v)=ε2αi=13Ωεd2χ(i)dζi2(ζi)|ζi=xiεα(ω0(i)(xi)ω0(i)(0)xidω0(i)dxi(0)+εω1(i)(xi)εω1(i)(0))v(x)dx,Rε,8(v)=ε2i=13Iε(i)Υi(xi)χ(i)(xiεα)ξ¯iu2(i)(xi,ξ¯i)ξ¯iv(x)dξ¯idxi,Rε,9(v)=ε3i=13Iε(i)Υi(xi)χ(i)(xiεα)ξ¯iu3(i)(xi,ξ¯i)ξ¯iv(x)dξ¯idxi,Rε,10(v)=i=13Ωε(2εαdχ(i)dζi(ζi)(N1ξi(ξ)G1ξi(ξ))+ε12αd2χ(i)dζi2(ζi)(N1(ξ)G1(ξ)))ζi=xiεα,ξ=xεv(x)dx.

Let us estimate the value Rε. Using (5) and (6), we deduce the following estimates:

|Rε,1(v)|Cˇ|Ξ(0)|+3πi=13hi2(0)ε1+α2vL2(Ωε),
|Rε,j(v)|Cˇi=136πhi(0)ε1+α2vH1(Ωε),j=2,4,
|Rε,3(v)|Cˇi=13πmaxxiIihi2(xi)ε2+|Ξ(0)|+2πi=13hi2(0)ε1+α2vL2(Ωε),
|Rε,5(v)|Cˇi=132πmaxxiIihi(xi)ε3vH1(Ωε),
|Rε,j(v)|Cˇi=13πhi2(0)ε1+α2vL2(Ωε),j=6,7,
|Rε,8(v)|Cˇε2xvL2(Ωε),|Rε,9(v)|Cˇε3xvL2(Ωε).

Due to the exponential decreasing of function N1G1 (see Remark 2.6) and the fact that the support of the derivative of χ(i) belongs to the set {xi : 2ℓεαxi ≤ 3 ℓεα}, we arrive at

|Rε,10(v)|Cˇi=13πhi2(0)ε1α2exp(2ε1αmini=1,2,3γi)vL2(Ωε).

Subtracting the integral identity (3) from (49), we obtain

Ωε(Uε(1)uε)vdx+Ωε(κ0(Uε(1))κ0(uε))vdx+εi=13Γε(i)(κi(Uε(1))κi(uε))vdσx=Rε(v)vHε.

Now set v = Uε(1)uε in (59). Then, taking into account (2) and (52)(58), we arrive at the inequality

Ωε|(Uε(1)uε)|2dxCε1+α2Uε(1)uεH1(Ωε),

whence thanks to (4) it follows (45).□

Corollary 3.2

The differences between the solution uε of problem (1) and the sum

Uε(0)(x)=i=13χ(i)(xiεα)ω0(i)(xi)+(1i=13χ(i)(xiεα))ω0(1)(0),xΩε

admit the following asymptotic estimates:

uεUε(0)H1(Ωε)C~0ε1+α2,uεUε(0)L2(Ωε)C~0ε1+α2,

where α is a fixed number from the interval (23,1).

In thin cylinders Ωε,α(i):=Ωε(i){xR3:xiIε,α(i):=(3εα,1)},i=1,2,3, the following estimates hold:

uεω0(i)H1(Ωε,α(i))C~1ε1+α2,i=1,2,3,

where {ω0(i)}i=13 is the solution of the limit problem (33).

In the neighbourhood Ωε,(0) := Ωε ∩ {x : xi < 2ℓε, i = 1, 2, 3} of the aneurysm Ωε(0), we get estimates

xuεξN1L2(Ωε,(0))uεω0(i)(0)εN1H1(Ωε,(0))C~4ε1+α2,

Proof

Denote by χ,α,ε(i)():=χ(i)(εα) (the function χ(i) is determined in (43)). Using the smoothness of the functions {ω1(i)} and the exponential decay of the functions {N1G1}, i = 1, 2, 3, at infinity, we deduce the inequalities (61) from estimate (45):

uεUε(0)H1(Ωε)uεUε(1)H1(Ωε)+εi=13χ,α,ε(i)ω1(i)+(1i=13χ,α,ε(i))N1H1(Ωε)C1ε1+α2+εi=13(χ,α,ε(i)ω1(i)+(1χ,α,ε(i))N1)H1(Ωε(i))+εN1H1(Ωε(0))C1ε1+α2+i=13(1χ,α,ε(i))xidω0(i)dxi(0)H1(Ωε(i))+εi=13(1χ,α,ε(i))(ω1(i)(0)ω1(i))H1(Ωε(i))+εi=13ω1(i)H1(Ωε(i))+εi=13(1χ,α,ε(i))(N1G1)H1(Ωε(i))+ε32N1H1(Ξ(0))C~0ε1+α2.

With the help of estimate (45), we deduce

uεω0(i)H1(Ωε,α(i))uεUε(1)H1(Ωε)+εω1(i)H1(Ωε,α(i))C~2ε1+α2,

whence we get (62).

The energetic estimate (63) in a neighbourhood of the aneurysm Ωε(0) follows directly from (45).□

Using the Cauchy-Buniakovskii-Schwarz inequality and the continuously embedding of the space H1(Iε,α(i)) in C(Iε,α(i)¯), from (62) we get the following corollary.

Corollary 3.3

If hi(xi) ≡ hiconst, (i = 1, 2, 3), then

Eε(i)(uε)ω0(i)H1(Iε,α(i))C~2εα2,
maxxiIε,α(i)¯|Eε(i)(uε)(xi)ω0(i)(xi)|C~3εα2,i=1,2,3,

where

(Eε(i)uε)(xi)=1πε2hi2Υε(i)(0)uε(x)dx¯i,i=1,2,3.

4 Conclusions

  1. An important problem of existing multi-scale methods is their stability and accuracy. The proof of the error estimate between the constructed approximation and the exact solution is a general principle that has been applied to the analysis of the efficiency of a multi-scale method. In our paper, we have done this for the solution to the problem (1).The results of Theorem 3.1 and Corollary 3.2 showed the possibility to replace the complex boundary-value problem (1) with the corresponding one-dimensional boundary-value problem (33) in the graph I=i=13Ii with sufficient accuracy measured by the parameter ε characterizing the thickness and the local geometrical irregularity. In this regard, the uniform pointwise estimates (65), which are important for applied problems, also confirm this conclusion.
  2. In [16], the authors considered the boundary-value problem
    Δuε(x)=f(x1)in Qε,νuε(x)=0on the lateral side of Qε,uε(x)=±t as x1=±a,
    where Qε is a thin 2D rod with a small local geometric irregularity in the middle.The energetic estimate (61) partly confirms the first formal result of [16] (see p. 296) that the local geometric irregularity of the analyzed structure does not significantly affect the global-level properties of the framework, which are described by the limit problem (33) and its solution {ω0(i)}i=13 (the first terms of the asymptotics). But thanks to estimates (45) and (63) it has become possible to identify the impact of the geometric irregularity and material characteristics of the aneurysm on the global level through the second terms {ω1(i)}i=13 of the regular asymptotics (8). They depend on the constants d1,δ1(2) and δ1(3) that take into account all those factors (see (38) and (42)). This conclusion does not coincide with the second main result of [16] (see p. 296) thatthe joints of normal type manifest themselves on the local level only”.In addition, in [16] the authors stated that the main idea of their approach “is to use a local perturbation corrector of the form εN(x/ε)du0dx1 with the condition that the function N(y) is localized near the joint”. e., N(y) → 0 as |y| → + ∞, and the main assumption of this approach is that ∇yNL1(Q).As shown the coefficients {Nk} of the inner asymptotics (9) behave as polynomials at infinity and do not decrease exponentially (see (40)). Therefore, they influence directly the terms of the regular asymptotics beginning with the second terms. Thus, the main assumption made in [16] is not correct.
  3. From the first estimate in (61) it follows that the gradient ∇uε is equivalent to {dω0(i)dxi}i=13 in the L2-norm over whole junction Ωε as ε → 0. Obviously, this estimate is not informative in the neighbourhood Ωε,(0) of the aneurysm Ωε(0).Thanks to estimates (45) and (63), we get the approximation of the gradient (flux) of the solution both in the curvilinear cylinders Ωε,α(i),i=1,2,3:
    uε(x)dω0(i)dxi(xi)+εdω1(i)dxi(xi)asε0
    and in the neighbourhood Ωε,(0) of the aneurysm:
    uε(x)ξ(N1(ξ))|ξ=xεasε0.
  4. We hope that this asymptotic approach can be applied to the study of the blood flow in vessels with a local geometric heterogeneity what we are going to do in our further studies. Nevertheless, the results obtained in this article can be considered as the first steps in this direction, since it is known that for the incompressible flow it is possible in some cases to couple pressure and velocity through the Poisson equation (κ0 ≡ 0) for pressure. Also the pressure Poisson equation with Neumann boundary conditions is encountered in the time-discretization of the incompressible Navier-Stokes equations.

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If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Bakhvalov N.S., Panasenko G.P., Homogenization: Averaging processes in periodic media, Kluwer, Dordrecht, Boston, London, 1989.

  • [2]

    Borisyuk A.O., Experimental study of wall pressure fluctuations in rigid and elastic pipes behind an axisymmetric narrowing, Journal of Fluids and Structures, 2010, 26, 658-674.

  • [3]

    Cardone G., Corbo-Esposito A., Panasenko G., Asymptotic partial decomposition for diffusion with sorption in thin structures, Nonlinear Analysis, 2006, 65, 79-106.

  • [4]

    Chechkin G.A., Jikov V.V., Lukkassen D., Piatnitski A. L., On homogenization of networks and junctions, Asymptotic Analysis, 2002, 30, 61-80.

  • [5]

    Cioranescu D., Saint Jean Paulin J., Homogenization of Reticulated Structures, volume 139 of Applied Mathematical Sciences, Springer-Verlag, New York, 1999.

  • [6]

    Meľnyk T.A., Homogenization of the Poisson equation in a thick periodic junction, Zeitschrift für Analysis und ihre Anwendungen, 1999, 18:4, 953-975.

  • [7]

    Meľnyk T.A., Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure, J. Math.Anal.Appl., 2015, 424, 1237-1260.

  • [8]

    Nazarov S.A., Junctions of singularly degenerating domains with different limit dimensions, J. Math. Sci., 1996, 80:5, 1989-2034.

  • [9]

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