Abstract

We study the nonsteady Stokes flow in a thin tube structure composed by two thin rectangles with lateral elastic boundaries which are connected by a domain with rigid boundaries. After a variational approach of the problem which gives us existence, uniqueness, regularity results, and some a priori estimates, we construct an asymptotic solution. The existence of a junction region between the two rectangles imposes to consider, as part of the asymptotic solution, some boundary layer correctors that correspond to this region. We present and solve the problems for all the terms of the asymptotic expansion. For two different cases, we describe the order of steps of the algorithm of solving the problem and we construct the main term of the asymptotic expansion. By means of the a priori estimates, we justify our asymptotic construction, by obtaining a small error between the exact and the asymptotic solutions.

1. Introduction

A few years ago we began to publish several papers dealing with an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure. Problems involving the interaction between a fluid and a deformable structure have been studied extensively in the last years due to their applications in many areas such as: engineering, biomechanics, biology, and hydroelasticity.

In order to model the blood flow through a vessel we considered in [1] a non steady viscous flow in a thin rectangle with elastic walls, when at the ends of the flow domain periodicity conditions are imposed. The asymptotic approach was then extended for the nonperiodic case in [2], when the inflow and outflow velocities are given functions. Generalizations of the previous cited papers were obtained in [3โ€“5]. In the first two papers we extended the results of [1] to a-three dimensional case, by considering a periodic, axisymmetric flow in a thin cylinder with elastic lateral boundary, while in [4] we studied two different cases: when the inertial term of the equation for the wall displacement is much smaller than the stress term and when these two terms are of the same order. Moreover, [5] deals with the asymptotic analysis for a fluid with variable viscosity.

In all these papers, the fluid flow was described by the Stokes equations and the behavior of the elastic structure was simulated either by the Sophie Germainโ€™s or by the Koiterโ€™s equation. The fluid-structure interaction was mathematically expressed by the equality of the fluid velocity at the boundary and the time derivative of the wall displacement. We constructed an asymptotic solution and we proved that it represents a good approximation for the exact solution, by obtaining a small error between them.

In this paper we consider the nonsteady Stokes flow in a thin tube structure. In two-dimensional case, a tube structure is some connected union of thin rectangles, having a dimension much smaller than the other one. The main difference between this paper and our previous works [2, 5] is that now the boundary of the flow domain contains elastic parts and rigid parts, as well.

The flow domain consists of two thin rectangles with elastic boundary. The junction between the rectangles is realized by means of rigid boundaries. The interaction between the viscous fluid and the elastic boundaries produces normal displacements. The elastic boundaries behavior is described by the Sophie Germainโ€™s equation.

This domain models a vessel structure where a stent was placed. In fact, for treating arterial stenoses or occlusions, percutaneous angioplasty is indicated. It uses small inflatable balloons, single-use, that help dilate the artery at the site of narrowing. During angioplasty, a small wire mesh tube called a stent may be permanently placed in the newly opened artery or vein to help it remain open.

There are two types of stents: bare stents (wire mesh) and covered stents (also commonly called stent grafts). The first are in use as part of the expansion of retrecissements vessels, the second impermeable and can be used to prevent or treat hemorrhage. Stents are used for arteries in the heart, the kidney, the arm, or the leg. They are also used for the aorta in the abdomen or chest (see Figures 1 and 2).

We assimilate then the part of the vessel where the stent is placed to a domain with rigid boundaries.

We suppose that the viscous fluid has a variable viscosity depending on a longitudinal variable for each rectangle. This situation models a blood flow in a vessel structure where the viscosity depends on the concentration of some substances diluted in blood or some blood cells. Indeed, the asymptotic analysis of the convection-diffusion equation set in such domains [6, 7] shows that in the case of the Neumann (impermeability) condition at the lateral boundary and small Reynolds numbers, the concentration is asymptotically close to the one-dimensional description, that is, the convection-diffusion equation set on the graph. The solution of the problem on the graph is the leading term of the asymptotic expansion, and it evidently depends on the longitudinal variable. On the other hand, the viscosity often depends on the concentration of the diluted substances or distributed cells, and so, it depends on the longitudinal variable. Of course, the fluid motion equation is coupled with the diffusion-convection equation in this case. However, if the velocity is small (in our case, it is of order ๐œ€2), then neglecting the convection, in comparison with the diffusion term or iterating with respect to the small term,we get the steady state diffusion equation; in absence of the source term in the right-hand side, it has a piecewise-linear asymptotic solution on the graph for the concentration. So, in this simplified situation, the diffusion equation can be solved before the fluid motion equation, and we obtain for the flow, the Stokes or Navier-Stokes equation with a variable viscosity depending (via concentration) on the longitudinal variable.

There are, of course, many other practical problems involving fluids with variable viscosity. For example, the presence of bacteria in suspension (see [8]) may change locally the viscosity.

The outline of the paper is as follows. In Section 2 we give a description of the flow domain and of the coupled system which models our problem. The next section presents the variational formulation of the problem. The literature contains an important number of papers dealing with a variational approach of fluid-structure interaction problems. For instance, results concerning the existence of weak or strong solutions when the fluid domain is either fixed or depending on time can be found in [9โ€“12]. We establish in this section results such as: existence, uniqueness, regularity and a priori estimates. In Section 4 we construct the asymptotic solution. As in our previous papers [1โ€“5], the problem depends on two small parameters. The first small parameter, ๐œ€, is defined as the ratio of the dimensions of the thin rectangles; the second one, ๐›ฟ, corresponds to the softness of the wall. For various values of the small parameters ๐œ€ and ๐›ฟ, an asymptotic expansion of the solution is constructed; the parameter ๐›ฟ is taken of the form ๐›ฟ=๐œ€๐›พ, with ๐›พโˆˆโ„•, ๐›พโ‰ฅ3. The asymptotic expansion is different for the cases: ๐›พ>3 and ๐›พ=3. The asymptotic solution contains three types of terms: the regular part, defined as in [1], the boundary layer correctors corresponding to the boundary conditions and the boundary layer correctors which realize the junction between the motion in the two rectangles. The first two types of correctors have already been introduced in [1, 2]. The third type is characteristic for structures with junction regions. The asymptotic solution is more complicated in this case since it contains also some truncation functions introduced in order to restrict the influence of the boundary layer correctors to the regions to which they correspond. We present and solve the problems for all the components of the asymptotic solution. For the two cases ๐›พ>3 and ๐›พ=3 the order of solving the problems is presented and the leading term of the asymptotic expansion is described. In the last section, we establish the error between the exact solution and the asymptotic one, by means of the a priori estimates obtained in Section 3. The small error between the two solutions justifies our asymptotic expansion.

2. The Physical Problem

We consider an incompressible, viscous fluid, with variable viscosity, occupying a thin domain, โ„ฌ๐œ€. The flow domain is a thin tube structure, composed by two thin rectangles with lateral elastic boundaries, connected by a region with rigid boundaries. We introduce the first small parameter of our problem, ๐œ€, ๐œ€=1/๐‘ž, ๐‘žโˆˆโ„• in connection with the ratio of the two dimensions of the rectangles, as below. The thin rectangles are given by:๐ท1๐œ€=๐‘ฅ๎€ฝ๎€ท1,๐‘ฅ2๎€ธโˆˆโ„2โˆถ0<๐‘ฅ1<1,โˆ’๐œ€<๐‘ฅ2๎€พ,๐ท<๐œ€2๐œ€=๐‘ฅ๎€ฝ๎€ท1,๐‘ฅ2๎€ธโˆˆโ„2โˆถโˆ’๐œ€<๐‘ฅ1<๐œ€,0<๐‘ฅ2๎€พ,<1(2.1) and the junction region is๐ท๐‘Ÿ๐œ€=๐‘ฅ๎€ฝ๎€ท1,๐‘ฅ2๎€ธโˆˆโ„2โˆถโˆ’๐œ€<๐‘ฅ1<2๐œ€,โˆ’๐œ€<๐‘ฅ2๎€พโงต๐‘ฅ<2๐œ€๎€ท๎€ฝ๎€ท1,๐‘ฅ2๎€ธโˆˆโ„2โˆถโˆ’๐œ€<๐‘ฅ1<0,โˆ’๐œ€<๐‘ฅ2<0,๐‘ฅ21+๐‘ฅ22โ‰ฅ๐œ€2๎€พโˆช๎‚†๎€ท๐‘ฅ1,๐‘ฅ2๎€ธโˆˆโ„2โˆถ๐œ€<๐‘ฅ1<2๐œ€,๐œ€<๐‘ฅ2๎€ท๐‘ฅ<2๐œ€,1๎€ธโˆ’2๐œ€2+๎€ท๐‘ฅ2๎€ธโˆ’2๐œ€2โ‰ค๐œ€2.๎‚‡๎‚(2.2) The flow domain โ„ฌ๐œ€โŠ‚โ„2 is given by (๐ท1๐œ€โˆฉ{๐‘ฅ1โ‰ฅ2๐œ€})โˆช(๐ท2๐œ€โˆฉ{๐‘ฅ2โ‰ฅ2๐œ€})โˆช๐ท๐‘Ÿ๐œ€, as shown in Figure 3.

The main difference between this paper and our previous works [2, 5] is that now the boundary of the flow domain โ„ฌ๐œ€ contains elastic parts and rigid parts as well. Let โŒข๐ด๐œ€+๐ต๐œ€+ and โŒข๐ด๐œ€โˆ’๐ต๐œ€โˆ’ be the rigid parts of the boundary of โ„ฌ๐œ€ defined as follows:โŒข๐ด๐œ€+๐ต๐œ€+=๎‚†๎€ท๐‘ฅ1,๐‘ฅ2๎€ธโˆˆโ„2โˆถ๐œ€<๐‘ฅ1<2๐œ€,๐œ€<๐‘ฅ2๎€ท๐‘ฅ<2๐œ€,1๎€ธโˆ’2๐œ€2+๎€ท๐‘ฅ2๎€ธโˆ’2๐œ€2=๐œ€2๎‚‡,โŒข๐ด๐œ€โˆ’๐ต๐œ€โˆ’=๐‘ฅ๎€ฝ๎€ท1,๐‘ฅ2๎€ธโˆˆโ„2โˆถ๐‘ฅ1=โˆ’๐œ€,0โ‰ค๐‘ฅ2๎€พโˆช๐‘ฅ<2๐œ€๎€ฝ๎€ท1,๐‘ฅ2๎€ธโˆˆโ„2โˆถ0โ‰ค๐‘ฅ1<2๐œ€,๐‘ฅ2๎€พโˆช๐‘ฅ=โˆ’๐œ€๎€ฝ๎€ท1,๐‘ฅ2๎€ธโˆˆโ„2โˆถโˆ’๐œ€<๐‘ฅ1<0,โˆ’๐œ€<๐‘ฅ2<0,๐‘ฅ21+๐‘ฅ22=๐œ€2๎€พ;(2.3) the elastic parts of ๐œ•โ„ฌ๐œ€ are given by:ฮ“๐œ€ยฑ=๎€ฝ๎€ทยฑ๐œ€,๐‘ฅ2๎€ธโˆถ2๐œ€<๐‘ฅ2๎€พโˆช๐‘ฅ<1๎€ฝ๎€ท1๎€ธ,ยฑ๐œ€โˆถ2๐œ€<๐‘ฅ1๎€พ<1.(2.4) Let ๐น๐œ€1={(1,๐‘ฅ2)โˆถโˆ’๐œ€<๐‘ฅ2<๐œ€} and ๐น๐œ€2={(๐‘ฅ1,1)โˆถโˆ’๐œ€<๐‘ฅ1<๐œ€} be the inflow and outflow parts of the boundary of โ„ฌ๐œ€. Denoting ๐น๐œ€ยฑ=ฮ“๐œ€ยฑโˆชโŒข๐ด๐œ€ยฑ๐ต๐œ€ยฑ, we can write ๐œ•โ„ฌ๐œ€=๐น๐œ€+โˆช๐น๐œ€โˆ’โˆช๐น๐œ€1โˆช๐น๐œ€2.

We study the nonsteady, slow flow of the viscous fluid in the domain โ„ฌ๐œ€ previously described, when the fluid interacts with the elastic boundaries, ฮ“๐œ€ยฑ. The interaction between the fluid and the elastic boundaries produces the normal displacements ๐‘‘ยฑ=๐‘‘ยฑ(๐‘ฅ1,๐‘ฅ2,๐‘ก). We neglect the tangential displacements and we consider that the elastic boundaries are clamped. We study the problem for ๐‘กโˆˆ(0,๐‘‡), with ๐‘‡ an arbitrary positive constant independent on ๐œ€ and we assume that the membranes are not very elastic so that the displacement of the boundaries is small enough. Consequently, at each time ๐‘ก, we approximate the position of the elastic membranes by its initial position and, hence, the fluid flow equations are considered in the initial configuration. We suppose that the displacements ๐‘‘ยฑโˆถ๐น๐œ€ยฑโ†ฆโ„ have the following form:๐‘‘ยฑ๎€ท๐‘ฅ1,๐‘ฅ2๎€ธ=โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘‘,๐‘กยฑ๎€ท๐‘ฅ1๎€ธ,๐‘กon๐ท1๐œ€โˆฉฮ“๐œ€ยฑ,๐‘‘ยฑ๎€ท๐‘ฅ2๎€ธ,๐‘กon๐ท2๐œ€โˆฉฮ“๐œ€ยฑ,0on๐น๐œ€ยฑโงตฮ“๐œ€ยฑ.(2.5)

The problem described above, with nonhomogeneous boundary conditions for the velocity, is modeled by the following coupled system:๐œŒ๐‘“๐œ•๐ฎ๐œ•๐‘กโˆ’2div๐‘ฅ๎€ท๐œˆ(๐‘ฅ)๐ท๐‘ฅ๐ฎ๎€ธ+โˆ‡๐‘ฅ๐‘=๐Ÿinโ„ฌ๐œ€ร—(0,๐‘‡),div๐‘ฅ๐ฎ=0inโ„ฌ๐œ€๐œ•ร—(0,๐‘‡),๐œŒโ„Ž2๐‘‘ยฑ๐œ•๐‘ก2+โ„Ž3๐ธ๐œ•124๐‘‘ยฑ๐œ•๐‘ฅ4๐‘–๐œ•+๐œ‡5๐‘‘ยฑ๐œ•๐‘ฅ4๐‘–๐œ•๐‘ก=๐‘”ยฑยฑ๐‘|๐ท๐‘–๐œ€๐œ€ยฑโˆฉฮ“๎‚€on๐ท๐‘–๐œ€โˆฉฮ“๐œ€ยฑ๎‚๐‘‘ร—(0,๐‘‡),๐‘–=1,2,+=0onโŒข๐ด๐œ€+๐ต๐œ€+ร—๐‘‘(0,๐‘‡),โˆ’=0onโŒข๐ด๐œ€โˆ’๐ต๐œ€โˆ’ร—(0,๐‘‡),๐ฎ=๐๐œ€๎€ทon๐œ•โ„ฌ๐œ€โงต๎€ทฮ“๐œ€+โˆชฮ“๐œ€โˆ’๐‘‘๎€ธ๎€ธร—(0,๐‘‡),ยฑ(1,ยฑ๐œ€,๐‘ก)=๐œ•๐‘‘ยฑ๐œ•๐‘ฅ1๐‘‘(1,ยฑ๐œ€,๐‘ก)=0in(0,๐‘‡),ยฑ(2๐œ€,ยฑ๐œ€,๐‘ก)=๐œ•๐‘‘ยฑ๐œ•๐‘ฅ1(๐‘‘2๐œ€,ยฑ๐œ€,๐‘ก)=0in(0,๐‘‡),ยฑ(ยฑ๐œ€,1,๐‘ก)=๐œ•๐‘‘ยฑ๐œ•๐‘ฅ2๐‘‘(ยฑ๐œ€,1,๐‘ก)=0in(0,๐‘‡),ยฑ(ยฑ๐œ€,2๐œ€,๐‘ก)=๐œ•๐‘‘ยฑ๐œ•๐‘ฅ2(ยฑ๐œ€,2๐œ€,๐‘ก)=0in(0,๐‘‡),๐ฎโ‹…๐ง=ยฑ๐œ•๐‘‘ยฑ๐œ•๐‘ก,๐ฎโ‹…๐‰=0onฮ“๐œ€ยฑร—(0,๐‘‡),๐ฎ(๐‘ฅ,0)=๐ŸŽinโ„ฌ๐œ€,๐‘‘ยฑ(๐‘ฅ,0)=๐œ•๐‘‘ยฑ๐œ•๐‘ก(๐‘ฅ,0)=0onฮ“๐œ€ยฑ,(2.6) with ๐ง the outer unit normal on the boundary of โ„ฌ๐œ€ and ๐‰ the unit tangent vector to ๐œ•โ„ฌ๐œ€. The given data contained by the previous system are: some material constants and some given functions. ๐œŒ๐‘“,๐œŒ,๐œ‡,๐ธ are positive given constants in connection with the properties of materials representing the density of the fluid, the density of the elastic walls, a viscosity coefficient, and the two-dimensional Youngโ€™s modulus (The two dimensional Youngโ€™s modulus ๐ธ is defined as ๐ธ(3)/(1โˆ’ฬ‚๐œˆ2) where ๐ธ(3) is the common three dimensional Young modulus and ฬ‚๐œˆ is the Poisson ratio.), respectively, and the positive constant โ„Ž stands for the thickness of the elastic walls.

The given functions are: ๐œˆ, the variable viscosity of the fluid, which satisfies ๐œˆโˆˆ๐ถ1(โ„ฌ๐œ€),๐œˆ(๐‘ฅ)โ‰ฅ๐›ผ>0 for all ๐‘ฅโˆˆโ„ฌ๐œ€, ๐Ÿ, the exterior force applied to the fluid, ๐‘”ยฑ, the exterior forces applied on the elastic walls, with ๐‘”ยฑโˆถ๐น๐œ€ยฑโ†ฆโ„,๐‘”ยฑ๎€ท๐‘ฅ1,๐‘ฅ2๎€ธ=โŽงโŽชโŽชโŽจโŽชโŽชโŽฉ๐‘”,๐‘กยฑ๎€ท๐‘ฅ1๎€ธ,๐‘กon๐ท1๐œ€โˆฉฮ“๐œ€ยฑ,๐‘”ยฑ๎€ท๐‘ฅ2๎€ธ,๐‘กon๐ท2๐œ€โˆฉฮ“๐œ€ยฑ,0on๐น๐œ€ยฑโงตฮ“๐œ€ยฑ(2.7) and a small inflow-outflow velocity defined by means of the function ๐œ“๐œ€ defined as follows:๐๐œ€๎€ท๐‘ฅ1,๐‘ฅ2๎€ธ,๐‘ก=๐œ€2๐๎€ท๐œ‰1,๐œ‰2๎€ธ,๐‘ก,โˆ€๐‘ฅโˆˆโ„ฌ๐œ€,(2.8) with (๐œ‰1,๐œ‰2)=(๐‘ฅ1/๐œ€,๐‘ฅ2/๐œ€). The function ๐ is the trace of a function denoted also by ๐ with the following properties:๐๎€ท๐œ‰1,๐œ‰2๎€ธ๎€ท๐œ‰,๐‘ก=๐œ“2๎€ธ๐ž,๐‘ก1๎€ท๐œ‰,โˆ€1,๐œ‰2๎€ธโˆˆ๐๎€ท๐œ‰(2,โˆž)ร—(โˆ’1,1),1,๐œ‰2๎€ธ๎€ท๐œ‰,๐‘ก=๐œ“1๎€ธ๐ž,๐‘ก2๎€ท๐œ‰,โˆ€1,๐œ‰2๎€ธโˆˆ(โˆ’1,1)ร—(2,โˆž),div๐œ‰๐=0,in๐ท๐‘Ÿ,๐=๐ŸŽon๐น+โˆช๐นโˆ’,๐๎€ท๐œ‰1,๐œ‰2๎€ธ๎€œ,0=๐ŸŽ,1โˆ’1๐œ“๎€ท๐œ‰2๎€ธ,๐‘กd๐œ‰2+๎€œ1โˆ’1๐œ“๎€ท๐œ‰1๎€ธ,๐‘กd๐œ‰1=0,(2.9) where ๐ท๐‘Ÿ=(1/๐œ€)๐ท๐‘Ÿ๐œ€,๐นยฑ=๐น๐œ€ยฑ. The unknowns of the system (2.6) are the velocity of the fluid, ๐ฎ, the pressure of the fluid, ๐‘, and the normal displacements of the elastic walls, ๐‘‘ยฑ.

The fluid flow is described by the nonsteady Stokes equations. For the normal displacements we consider the Sophie-Germainโ€™s equation. A โ€œviscousโ€ type term, ๐œ‡(๐œ•5๐‘‘๐‘–ยฑ/๐œ•๐‘ฅ4๐‘–๐œ•๐‘ก), is added to the usual forth-order equation for the normal displacements. It corresponds to the viscoelastic behavior of the wall (the so called Kelvin-Voigt model). Usually, the Youngโ€™s modulus, ๐ธ, has a value of 104โˆ’106โ€‰Pa. On the other hand, we assume that the characteristic longitudinal space scale for vessels is of order of cm and the time scale is of order of seconds. Let us use the S.I. system of units. This leads us to the necessity of scaling of every derivative is ๐‘ฅ๐‘– by the factor 102; that is, the fourth derivative will contain the additional factor 108. If โ„Ž is of order 10โˆ’3โ€‰m or 10โˆ’2โ€‰m, then the coefficient ๐œŒโ„Ž can be taken in the further analysis as a value of order of 1. The coefficient โ„Ž3๐ธ/12 in (2.6)3 will be replaced (after scaling in ๐‘ฅ๐‘–) by a great coefficient ๐›ฟโˆ’1 with the value ๐›ฟ of order of 10โˆ’7 to 10โˆ’4. If the ratio of thickness and the length of the vessel ๐œ€ are of order 10โˆ’2, then ๐›ฟ is of order from ๐œ€2 to ๐œ€4. We assume that the โ€œviscousโ€ term is much smaller than the term with the coefficient ๐›ฟโˆ’1 and hence the new coefficient denoted also by ๐œ‡, obtained after scaling in ๐‘ฅ๐‘–, is ๐‘‚(1). More details concerning (2.6) can be found, for instance in [1].

Due to the properties of the function ๐๐œ€, the compatibility condition for the coupled system which describes the physical problem is๎€œ0=๐œ•โ„ฌ๐œ€d๐ฎ(๐‘ฅ,๐‘ก)โ‹…๐งd๐‘ =๎‚ต๎€œd๐‘ก12๐œ€๎€ท๐‘‘+๎€ท๐‘ฅ1๎€ธ,๐‘กโˆ’๐‘‘โˆ’๎€ท๐‘ฅ1,๐‘ก๎€ธ๎€ธd๐‘ฅ1+๎€œ12๐œ€๎€ท๐‘‘+๎€ท๐‘ฅ2๎€ธ,๐‘กโˆ’๐‘‘โˆ’๎€ท๐‘ฅ2,๐‘ก๎€ธ๎€ธd๐‘ฅ2๎‚ถ.(2.10) Using next the initial condition for the displacements, condition for the above coupled system becomes๎€œ12๐œ€๎€ท๐‘‘+๎€ท๐‘ฅ1๎€ธ,๐‘กโˆ’๐‘‘โˆ’๎€ท๐‘ฅ1,๐‘ก๎€ธ๎€ธd๐‘ฅ1+๎€œ12๐œ€๎€ท๐‘‘+๎€ท๐‘ฅ2๎€ธ,๐‘กโˆ’๐‘‘โˆ’๎€ท๐‘ฅ2,๐‘ก๎€ธ๎€ธd๐‘ฅ2=0.(2.11) This condition states that the global area of the flow domain is preserved.

3. Variational Formulation: Existence, Uniqueness, Regularity, and A Priori Estimates

In order to obtain the above properties for the solution of the physical problem, we introduce the variational framework.

To simplify the computations, we consider first (2.6) with homogeneous boundary conditions on ๐น๐œ€1 and ๐น๐œ€2, that is, the problem for ๐๐œ€=๐ŸŽ. Then, the same properties for the solution of (2.6) follow with the usual technique for nonhomogeneous problems.

Taking into account the conditions for the velocity and for the displacements and the condition (2.11) we introduce the following spaces:๐‘‰๐œ€=๎‚†๎€ท๐ป๐ฏโˆˆ1๎€ทโ„ฌ๐œ€๎€ธ๎€ธ2โˆถdiv๐ฏ=0inโ„ฌ๐œ€,๐ฏ=๐ŸŽon๐œ•โ„ฌ๐œ€โงต๎€ทฮ“๐œ€+โˆชฮ“๐œ€โˆ’๎€ธ,๐ฏโ‹…๐‰=0onฮ“๐œ€ยฑ๎‚‡,๐‘Š๐œ€=๎‚ป๎€ท๐›ฝ+,๐›ฝโˆ’๎€ธโˆˆ๐ป1๎€ท๐น๐œ€+๎€ธร—๐ป1๎€ท๐น๐œ€โˆ’๎€ธโˆถ๎€ท๐›ฝ+,๐›ฝโˆ’๎€ธโˆˆ๐ป20๎€ทฮ“๐œ€+๎€ธร—๐ป20๎€ทฮ“๐œ€โˆ’๎€ธ,๎€ท๐›ฝ+,๐›ฝโˆ’๎€ธ=(0,0)onโŒข๐ด๐œ€+๐ต๐œ€+ร—โŒข๐ด๐œ€โˆ’๐ต๐œ€โˆ’,๎€œ12๐œ€๎€ท๐›ฝ1+๎€ท๐‘ฅ1๎€ธโˆ’๐›ฝ1โˆ’๎€ท๐‘ฅ1๎€ธ๎€ธd๐‘ฅ1+๎€œ12๐œ€๎€ท๐›ฝ2+๎€ท๐‘ฅ2๎€ธโˆ’๐›ฝ2โˆ’๎€ท๐‘ฅ2๎€ธ๎€ธd๐‘ฅ2๎‚ผ.=0(3.1) Choosing for the data the regularity ๐Ÿโˆˆ๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2) and ๐ =(๐‘”+,๐‘”โˆ’)โˆˆ๐ฟ2(0,๐‘‡;๐น๐œ€+)ร—๐ฟ2(0,๐‘‡;๐น๐œ€โˆ’) we consider the following variational problem:Find(๐ฎ,๐)โˆˆ๐ฟ2(0,๐‘‡;๐‘‰๐œ€)ร—๐ป1(0,๐‘‡;๐‘Š๐œ€๎€ทฬ‡ฬˆ๐๎€ธ),with๐ฎ,โˆˆ๐ฟ2๎€ท0,๐‘‡;(๐‘‰๐œ€)๎…ž๎€ธร—๐ฟ2๎€ท0,๐‘‡;(๐‘Š๐œ€)๎…ž๎€ธ๐œŒwhichsatis๏ฌesa.e.in(0,๐‘‡)โˆถ๐‘“d๎€œd๐‘กโ„ฌ๐œ€๎€œ๐ฎโ‹…๐‹d๐‘ฅ+2โ„ฌ๐œ€+๐œˆD๐ฎโˆถD๐‹d๐‘ฅ2๎“๐‘–=1๎ƒฏd๐œŒโ„Ž๎€œd๐‘ก12๐œ€๐œ•๐๐œ•๐‘กโ‹…๐œทd๐‘ฅ๐‘–+โ„Ž3๐ธ๎€œ1212๐œ€๐œ•2๐๐œ•๐‘ฅ2๐‘–โ‹…๐œ•2๐œท๐œ•๐‘ฅ2๐‘–d๐‘ฅ๐‘–๎€œ+๐œ‡12๐œ€๐œ•3๐๐œ•๐‘ฅ2๐‘–โ‹…๐œ•๐œ•๐‘ก2๐œท๐œ•๐‘ฅ2๐‘–d๐‘ฅ๐‘–๎ƒฐ=๎€œโ„ฌ๐œ€๐Ÿโ‹…๐‹d๐‘ฅ+2๎“๐‘–=1๎€œ12๐œ€๐ โ‹…๐œทd๐‘ฅ๐‘–โˆ€๐‹โˆˆ๐‘‰๐œ€,โˆ€๐œทโˆˆ๐‘Š๐œ€,with๐‹โ‹…๐ง=ยฑ๐›ฝยฑonฮ“๐œ€ยฑ,๐ฎโ‹…๐ง=ยฑ๐œ•๐‘‘ยฑ๐œ•๐‘กonฮ“๐œ€ยฑ,ฬ‡๐ฎ(0)=๐ŸŽ,๐(0)=๐(0)=๐ŸŽ.(3.2) Here and below ๐=(๐‘‘+,๐‘‘โˆ’) and ๐œท=(๐›ฝ+,๐›ฝโˆ’).

For the nonhomogeneous boundary conditions we still obtain the variational formulation (3.2) with ๐ฎ replaced by ๐ฎโˆ’๐๐œ€ and ๐Ÿ replaced by ๐Ÿโˆ’๐œŒ๐‘“(๐œ•๐๐œ€/๐œ•๐‘ก)+2div(๐œˆ๐ท๐๐œ€).

Theorem 3.1. The variational problem (3.2) has the unique solution (๐ฎ,๐) with (ฬ‡ฬˆ๐ฎ,๐)โˆˆ๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)ร—๐ฟ2(0,๐‘‡;๐ฟ2(๐น๐œ€+)ร—๐ฟ2(๐น๐œ€โˆ’)).

Proof. Let us start with the proof of the uniqueness for the solution of (3.2). Consider (๐ฎ1,๐1) and (๐ฎ2,๐2) two solutions of (3.2) and define (๐ฎ,๐)=(๐ฎ1โˆ’๐ฎ2,๐1โˆ’๐2). Subtracting the two relations (3.2)1 and taking as test function (๐‹,๐œท)=(๐ฎ,๐) we get: ๐œŒ๐‘“2d๎€œd๐‘กโ„ฌ๐œ€๐ฎ2๎€œd๐‘ฅ+2โ„ฌ๐œ€+๐œˆD๐ฎโˆถD๐ฎd๐‘ฅ2๎“๐‘–=1โŽ›โŽœโŽœโŽ๐œŒโ„Ž2d๎€œd๐‘ก12๐œ€๎‚€๐œ•๐๎‚๐œ•๐‘ก2d๐‘ฅ๐‘–+โ„Ž3๐ธd24๎€œd๐‘ก12๐œ€๎ƒฉ๐œ•2๐๐œ•๐‘ฅ2๐‘–๎ƒช2d๐‘ฅ๐‘–๎€œ+๐œ‡12๐œ€๎ƒฉ๐œ•3๐๐œ•๐‘ฅ2๐‘–๎ƒช๐œ•๐‘ก2d๐‘ฅ๐‘–โŽžโŽŸโŽŸโŽ =0.(3.3) Integrating from 0 to ๐‘ก this equality and taking into account the initial conditions, we obtain: ๐ฎ=๐ŸŽ a.e. in (0,๐‘‡) and ๐=๐ŸŽ a.e. in (0,๐‘‡). Hence the problem (3.2) has a unique solution.
For proving the existence and the regularity of the functions ๐ฎ and ๐, we will use the Galerkinโ€™s method.
We begin with the construction of a basis for the space ๐‘Š๐œ€. Let {๐œ๐‘—}๐‘—โˆˆโ„• be a basis of ๐ป20(2๐œ€,1) chosen by considering the eigenfunctions of the following problem: ๐œ๐‘—(๐‘–๐‘ฃ)=๐›ผ๐‘—๐œ๐‘—๐œin(2๐œ€,1),๐‘—(2๐œ€)=๐œ๐‘—๐œ(1)=0,๎…ž๐‘—(2๐œ€)=๐œ๎…ž๐‘—(1)=0,(3.4) where ๐œ๐‘—(๐‘–๐‘ฃ) is the fourth derivative of ๐œ๐‘— and ๐›ผ๐‘—>0, for all ๐‘—โˆˆโ„•. We define {๐œท๐‘—}๐‘—โˆˆโ„• as follows: ๐œท๐‘—=(๐›ฝ+๐‘—,๐›ฝโˆ’๐‘—) where ๐›ฝยฑ๐‘—๎€ท๐‘ฅ1,๐‘ฅ2๎€ธ=โŽงโŽชโŽจโŽชโŽฉ๐œ๐‘—๎€ท๐‘ฅ1๎€ธ๎€ท๐‘ฅfor1,๐‘ฅ2๎€ธโˆˆ๎€ท๐‘ฅ(2๐œ€,1)ร—{ยฑ๐œ€},0for1,๐‘ฅ2๎€ธโˆˆโŒข๐ด๐œ€ยฑ๐ต๐œ€ยฑ,๐œ๐‘—๎€ท๐‘ฅ2๎€ธ๎€ท๐‘ฅfor1,๐‘ฅ2๎€ธโˆˆ{ยฑ๐œ€}ร—(2๐œ€,1).(3.5) It is easy to check that {๐œท๐‘—}๐‘—โˆˆโ„• is a basis for ๐‘Š๐œ€. We choose the functions of the basis {๐œ๐‘—}๐‘—โˆˆโ„• such that ๎€œโ„ฌ๐œ€๐œท๐‘–โ‹…๐œท๐‘˜d๐‘ฅ=๐›ฟ๐‘–๐‘˜.(3.6) As a consequence of the previous relation we also get: ๎€œโ„ฌ๐œ€๐œท๐‘–๎…ž๎…žโ‹…๐œท๐‘˜๎…ž๎…ž=๐›ผ๐‘˜๐›ฟ๐‘–๐‘˜.(3.7) We consider now {๐๐‘–}๐‘–โˆˆโ„• a basis of the space ๐‘‰๐œ€0={๐ฎโˆˆ(๐ป10(โ„ฌ๐œ€))2โˆถdiv๐ฎ=0inโ„ฌ๐œ€}, constructed with the eigenfunctions of the following Stokes problem: ๎€ทโˆ’2div๐œˆD๐๐‘–๎€ธ+โˆ‡๐‘ž๐‘–=๐œ†๐‘–๐๐‘–inโ„ฌ๐œ€,div๐๐‘–=0inโ„ฌ๐œ€,๐๐‘–=0on๐œ•โ„ฌ๐œ€,(3.8) with ๐œ†๐‘–>0 for all ๐‘–โˆˆโ„•. The functions ๐๐‘–,๐‘–โˆˆโ„• are uniquely determined from the condition ๎€œโ„ฌ๐œ€๐๐‘—โ‹…๐๐‘˜d๐‘ฅ=๐›ฟ๐‘—๐‘˜,(3.9) which implies 2๎€œโ„ฌ๐œ€๐œˆD๐๐‘–โˆถD๐๐‘˜d๐‘ฅ=๐œ†๐‘˜๐›ฟ๐‘—๐‘˜.(3.10) Next, for any ๐œท๐‘— we consider the following problem: ๎€ท๐‹Find๐‘—,๐‘๐‘—๎€ธโˆˆ๐‘‰๐œ€ร—๐ฟ2๎€ทโ„ฌ๐œ€๎€ธ๎€ท๎€ท๐‹suchthatโˆ’2div๐œˆD๐‘—๎€ธ๎€ธ+โˆ‡๐‘๐‘—๎€ท๐ป=0inโˆ’1๎€ทโ„ฌ๐œ€๎€ธ๎€ธ2,๐‹๐‘—โ‹…๐ง=ยฑ๐›ฝยฑ๐‘—onฮ“๐œ€ยฑ.(3.11) Following the classical results of [13] for nonhomogeneous Stokes problems, we obtain a unique ๐‹๐‘— and a function ๐‘๐‘— unique up to an additive constant. Moreover, for any ๐‘—,๐‘˜โˆˆโ„•๎€œโ„ฌ๐œ€๐œˆD๐‹๐‘—โˆถD๐๐‘˜d๐‘ฅ=0.(3.12) By means of the functions {๐œท๐‘—}๐‘—โˆˆโ„•,{๐๐‘–}๐‘–โˆˆโ„•,{๐‹๐‘—}๐‘—โˆˆโ„•, we are now in a position to define, for each ๐‘›,๐‘šโˆˆโ„•, an approximate solution (๐ฎ๐‘š๐‘›,๐๐‘›) of (3.2) as follows: ๐ฎ๐‘š๐‘›(๐‘ฅ,๐‘ก)=๐‘š๎“๐‘–=1๐‘Ž๐‘–(๐‘ก)๐๐‘–(๐‘ฅ)+๐‘›๎“๐‘—=1ฬ‡๐‘๐‘—(๐‘ก)๐‹๐‘—๐(๐‘ฅ),๐‘›(๐‘ฅ,๐‘ก)=๐‘›๎“๐‘—=1๐‘๐‘—(๐‘ก)๐œท๐‘—(๐‘ฅ),(3.13) with ๐‘Ž๐‘–,๐‘๐‘—โˆถ[0,๐‘‡]โ†ฆโ„,๐‘–=1,โ€ฆ,๐‘š,๐‘—=1,โ€ฆ,๐‘› scalar unknown functions. These functions are determined below from the condition that (๐ฎ๐‘š๐‘›,๐๐‘›) is the solution for the problem: ๐œŒ๐‘“๎€œโ„ฌ๐œ€๐œ•๐ฎ๐‘š๐‘›๐œ•๐‘กโ‹…๐๐‘–๎€œd๐‘ฅ+2โ„ฌ๐œ€๐œˆD๐ฎ๐‘š๐‘›โˆถD๐๐‘–๎€œd๐‘ฅ=โ„ฌ๐œ€๐Ÿโ‹…๐๐‘–๐œŒd๐‘ฅ,for๐‘–โˆˆ{1,โ€ฆ,๐‘š},๐‘“๎€œโ„ฌ๐œ€๐œ•๐ฎ๐‘š๐‘›๐œ•๐‘กโ‹…๐‹๐‘—๎€œd๐‘ฅ+2โ„ฌ๐œ€๐œˆD๐ฎ๐‘š๐‘›โˆถD๐‹๐‘—+d๐‘ฅ2๎“๐‘™=1๎ƒฉ๎€œ๐œŒโ„Ž12๐œ€๐œ•2๐๐‘›๐œ•๐‘ก2โ‹…๐œท๐‘—d๐‘ฅ๐‘™+โ„Ž3๐ธ๎€œ1212๐œ€๐œ•2๐๐‘›๐œ•๐‘ฅ2๐‘™โ‹…๐œ•2๐œท๐‘—๐œ•๐‘ฅ2๐‘™d๐‘ฅ๐‘™๎€œ+๐œ‡12๐œ€๐œ•3๐๐‘›๐œ•๐‘ฅ2๐‘™โ‹…๐œ•๐œ•๐‘ก2๐œท๐ฃ๐œ•๐‘ฅ2๐‘™d๐‘ฅ๐‘™๎ƒช=๎€œโ„ฌ๐œ€๐Ÿโ‹…๐‹๐‘—d๐‘ฅ+2๎“๐‘™=1๎€œ12๐œ€๐ โ‹…๐œท๐‘—d๐‘ฅ๐‘™๐ฎ,for๐‘—โˆˆ{1,โ€ฆ,๐‘›},๐‘š๐‘›โ‹…๐ง=ยฑ๐œ•๐‘‘ยฑ๐‘›๐œ•๐‘กonฮ“๐œ€ยฑ,๐ฎ๐‘š๐‘›(0)=๐ŸŽ,๐๐‘›ฬ‡๐(0)=๐‘›(0)=๐ŸŽ.(3.14) We introduce the notations: ๐‘๐‘–๐‘˜=๎€œโ„ฌ๐œ€๐‹๐‘˜โ‹…๐๐‘–d๐‘ฅ,๐‘ž๐‘˜๐‘—=๎€œโ„ฌ๐œ€๐‹๐‘˜โ‹…๐‹๐‘—d๐‘ฅ,๐‘Ÿ๐‘–๐‘˜๎€œ=2โ„ฌ๐œ€๐œˆD๐‹๐‘–โˆถD๐‹๐‘˜d๐‘ฅ.(3.15) Taking into account the previous notations and the relations (3.6), (3.7), (3.9), (3.10), (3.12) we get from (3.14) the following ๐‘š+๐‘› linear differential system for the unknown functions ๐‘Ž๐‘–,๐‘๐‘—,๐‘–=1,โ€ฆ,๐‘š,๐‘—=1,โ€ฆ,๐‘›: ๐œŒ๐‘“ฬ‡๐‘Ž๐‘–(๐‘ก)+๐œ†๐‘–๐‘Ž๐‘–(๐‘ก)+๐‘›๎“๐‘˜=1๐‘๐‘–๐‘˜ฬˆ๐‘๐‘˜๎€œ(๐‘ก)=โ„ฌ๐œ€๐Ÿโ‹…๐๐‘–๐œŒd๐‘ฅ,๐‘“๐‘š๎“๐‘˜=1๐‘๐‘˜๐‘—ฬ‡๐‘Ž๐‘˜(๐‘ก)+๐œŒ๐‘“๐‘š๎“๐‘˜=1๐‘ž๐‘˜๐‘—ฬˆ๐‘๐‘˜(๐‘ก)+๐‘›๎“๐‘˜=1๐‘Ÿ๐‘˜๐‘—ฬ‡๐‘๐‘˜ฬˆ๐‘(๐‘ก)+๐œŒโ„Ž๐‘—โ„Ž(๐‘ก)+3๐ธ๐›ผ12๐‘—๐‘๐‘—(๐‘ก)+๐œ‡๐›ผ๐‘—ฬ‡๐‘๐‘—=๎€œ(๐‘ก)โ„ฌ๐œ€๐Ÿโ‹…๐‹๐‘—d๐‘ฅ+2๎“๐‘™=1๎€œ12๐œ€๐ โ‹…๐œท๐‘—d๐‘ฅ๐‘™,๐‘Ž๐‘–(0)=๐‘๐‘—ฬ‡๐‘(0)=๐‘—(0)=0,๐‘–=1,โ€ฆ,๐‘š,๐‘—=1,โ€ฆ,๐‘›.(3.16) The previous system uniquely determines the unknown functions ๐‘Ž๐‘–,๐‘๐‘—,๐‘–=1,โ€ฆ,๐‘š,๐‘—=1,โ€ฆ,๐‘›. For more details see [1, Sectionโ€‰โ€‰3].
In the sequel we obtain some a priori estimates which give the regularity of the solution for (3.2). Computing โˆ‘๐‘š๐‘–=1๐‘Ž๐‘–(๐‘ก)โ‹…(3.14)1+โˆ‘๐‘›๐‘—=1ฬ‡๐‘๐‘—(๐‘ก)โ‹…(3.14)2 and using (3.13) we get:๐œŒ๐‘“2d๎€œd๐‘กโ„ฌ๐œ€๎€ท๐ฎ๐‘š๐‘›๎€ธ2๎€œd๐‘ฅ+โ„ฌ๐œ€๐œˆ๎€ทD๐ฎ๐‘š๐‘›๎€ธ2+2๎“๐‘–=1โŽ›โŽœโŽœโŽ๐œŒโ„Ž2d๎€œd๐‘ก12๐œ€๎‚ต๐œ•๐๐‘›๎‚ถ๐œ•๐‘ก2d๐‘ฅ๐‘–+โ„Ž3๐ธd24๎€œd๐‘ก12๐œ€๎ƒฉ๐œ•2๐๐‘›๐œ•๐‘ฅ2๐‘–๎ƒช2d๐‘ฅ๐‘–๎€œ+๐œ‡12๐œ€๎ƒฉ๐œ•3๐๐‘›๐œ•๐‘ฅ2๐‘–๎ƒช๐œ•๐‘ก2d๐‘ฅ๐‘–โŽžโŽŸโŽŸโŽ =๎€œโ„ฌ๐œ€๐Ÿโ‹…๐ฎ๐‘š๐‘›d๐‘ฅ+2๎“๐‘–=1๎€œ12๐œ€๐ โ‹…๐œ•๐๐‘›๐œ•๐‘กd๐‘ฅ๐‘–.(3.17) Integrating from 0 to ๐‘ก, using the property of ๐œˆ and the initial conditions we obtain, as in [1], the first estimates: โ€–โ€–๐ฎ๐‘›๐‘šโ€–โ€–๐ฟโˆž(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)โ€–โ€–โ‰ค๐ถ(๐Ÿ,๐ ),D๐ฎ๐‘›๐‘šโ€–โ€–๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))4)โ€–โ€–โ€–โ‰ค๐ถ(๐Ÿ,๐ ),๐œ•๐‘‘ยฑ๐‘›โ€–โ€–โ€–๐œ•๐‘ก๐ฟโˆž(0,๐‘‡;๐ฟ2(ฮ“๐œ€ยฑ))โ€–โ€–โ€–๐œ•โ‰ค๐ถ(๐Ÿ,๐ ),2๐‘‘ยฑ๐‘›๐œ•๐‘ 2โ€–โ€–โ€–๐ฟโˆž(0,๐‘‡;๐ฟ2(ฮ“๐œ€ยฑ))โ€–โ€–โ€–๐œ•โ‰ค๐ถ(๐Ÿ,๐ ),3๐‘‘ยฑ๐‘›๐œ•๐‘ 2โ€–โ€–โ€–๐œ•๐‘ก๐ฟ2(ฮ“๐œ€ยฑร—(0,๐‘‡))โ‰ค๐ถ(๐Ÿ,๐ ),(3.18) with ๐ถ(๐Ÿ,๐ )=๐ถ(๐‘‡,๐œ‡,๐œŒ๐‘“,๐œŒ,๐ธ,โ„Ž)(โ€–๐Ÿโ€–๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)+โ€–๐ โ€–๐ฟ2((0,๐‘‡);๐ฟ2(๐น๐œ€+ร—๐น๐œ€โˆ’))) and ๐‘  is the variable of the parametrisation for ฮ“๐œ€+ or on ฮ“๐œ€โˆ’.
The second estimates are obtained computing โˆ‘๐‘š๐‘–=1ฬ‡๐‘Ž๐‘–(๐‘ก)โ‹…(3.14)1+โˆ‘๐‘›๐‘—=1ฬˆ๐‘๐‘—(๐‘ก)โ‹…(3.14)2:โ€–โ€–โ€–๐œ•๐ฎ๐‘›๐‘šโ€–โ€–โ€–๐œ•๐‘ก๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)โ‰ค๎‚ตโ„Ž3๐ธ๎‚ถ121/2โ€–โ€–๐ถ(๐Ÿ,๐ ),D๐ฎ๐‘›๐‘šโ€–โ€–๐ฟโˆž((0,๐‘‡);(๐ฟ2(โ„ฌ๐œ€))4)โ‰ค๎‚ตโ„Ž3๐ธ๎‚ถ121/2โ€–โ€–โ€–๐œ•๐ถ(๐Ÿ,๐ ),2๐‘‘ยฑ๐‘›๐œ•๐‘ก2โ€–โ€–โ€–๐ฟ2(ฮ“๐œ€ยฑร—(0,๐‘‡))โ‰ค๎‚ตโ„Ž3๐ธ๎‚ถ121/2โ€–โ€–โ€–๐œ•๐ถ(๐Ÿ,๐ ),3๐‘‘ยฑ๐‘›๐œ•๐‘ 2โ€–โ€–โ€–๐œ•๐‘ก๐ฟโˆž(0,๐‘‡;๐ฟ2(ฮ“๐œ€ยฑ))โ‰ค๎‚ตโ„Ž3๐ธ๎‚ถ121/2๐ถ(๐Ÿ,๐ ).(3.19) From (3.18)1,2 and (3.19)1,2 we get the boundedness of {๐ฎ๐‘š๐‘›}๐‘š,๐‘›โˆˆโ„• in ๐ฟโˆž(0,๐‘‡;๐‘‰๐œ€)โˆฉ๐ป1(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2). Consequently, we can pass to the limit in (3.14) which yields the existence result of the theorem. To achieve the proof, we notice that the regularity stated in (3.2) follows from the estimates (3.18) and (3.19).

As a consequence of Theorem 3.1 we introduce the pressure which appears in (2.6)1,3.

Corollary 3.2. There exists a unique function ๐‘โˆˆ๐ฟ2(0,๐‘‡;๐ป1(โ„ฌ๐œ€)) such that (๐ฎ,๐‘,๐) satisfies (2.6)1,3 a.e. in โ„ฌ๐œ€ร—(0,๐‘‡) and on (๐ท๐‘–๐œ€โˆฉฮ“๐œ€ยฑ)ร—(0,๐‘‡), respectively.

To obtain this result we follow the ideas of [1, Sectionโ€‰โ€‰3].

The last result of this section presents the estimates for (๐ฎ,๐‘,๐), the unique solution of (2.6).

Corollary 3.3. Let (๐ฎ,๐‘,๐) be the solution of the problem (2.6) corresponding to the data ๐Ÿ,๐ , with ๐Ÿโˆˆ๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2) and ๐ โˆˆ๐ฟ2(0,๐‘‡;๐น๐œ€+)ร—๐ฟ2(0,๐‘‡;๐น๐œ€โˆ’). Then the following estimates hold: โ€–๐ฎโ€–๐ฟโˆž(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)โ‰ค๐ถ(๐Ÿ,๐ ),โ€–D๐ฎโ€–๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))4)โ€–โ€–โ€–โ‰ค๐ถ(๐Ÿ,๐ ),๐œ•๐ฎโ€–โ€–โ€–๐œ•๐‘ก๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)โ‰ค๎‚ตโ„Ž3๐ธ๎‚ถ121/2โ€–โ€–โ€–๐ถ(๐Ÿ,๐ ),๐œ•๐‘‘ยฑโ€–โ€–โ€–๐œ•๐‘ก๐ฟโˆž(0,๐‘‡;๐ฟ2(ฮ“๐œ€ยฑ))โ€–โ€–โ€–๐œ•โ‰ค๐ถ(๐Ÿ,๐ ),2๐‘‘ยฑ๐œ•๐‘ 2โ€–โ€–โ€–๐ฟโˆž(0,๐‘‡;๐ฟ2(ฮ“๐œ€ยฑ))โ€–โ€–โ€–๐œ•โ‰ค๐ถ(๐Ÿ,๐ ),2๐‘‘ยฑ๐œ•๐‘ก2โ€–โ€–โ€–๐ฟ2(ฮ“๐œ€ยฑร—(0,๐‘‡))โ‰ค๎‚ตโ„Ž3๐ธ๎‚ถ121/2โ€–โ€–โ€–๐œ•๐ถ(๐Ÿ,๐ ),3๐‘‘ยฑ๐œ•๐‘ 2โ€–โ€–โ€–๐œ•๐‘ก๐ฟโˆž(0,๐‘‡;๐ฟ2(ฮ“๐œ€ยฑ))โ‰ค๎‚ตโ„Ž3๐ธ๎‚ถ121/2๐ถ(๐Ÿ,๐ ),โ€–โˆ‡๐‘โ€–๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)โ‰ค๎‚ตโ„Ž3๐ธ๎‚ถ121/2๐ถ(๐Ÿ,๐ ).(3.20)

Remark 3.4. If we consider the nonhomogeneous problem (which corresponds to ๐๐œ€) we obtain for the corresponding solution the same estimates (3.20) with a different ๐Ÿ, but which depends in the same way on ๐œ€.

4. Asymptotic Analysis

In order to approximate the solution of (2.6), we consider more regular data than in Section 3. We suppose that:๐๐œ€โˆˆ๎€ท๐’žโˆž๎€ทโ„ฌ๐œ€ร—(0,๐‘‡)๎€ธ๎€ธ2,๎€ท๐’ž๐Ÿโˆˆโˆž๎€ทโ„ฌ๐œ€ร—(0,๐‘‡)๎€ธ๎€ธ2,๐‘”ยฑโˆˆ๐’žโˆž๎€ท๐น๐œ€ยฑ๎€ธ,ร—(0,๐‘‡)๐œˆโˆˆ๐’žโˆž๎€ทโ„ฌ๐œ€๎€ธ,โˆƒ๐‘กโˆ—<๐‘‡suchthat๐Ÿ(๐‘ฅ,๐‘ก)=๐‘”ยฑ(๐‘ฅ,๐‘ก)=๐๐œ€(๐‘ฅ,๐‘ก)=0โˆ€(๐‘ฅ,๐‘ก)โˆˆโ„ฌ๐œ€ร—๎€ท0,๐‘กโˆ—๎€ธ,(4.1) and ๐Ÿ, ๐‘”ยฑ and ๐œˆ are chosen as follows:๐Ÿ๎€ท๐‘ฅ1,๐‘ฅ2๎€ธ=โŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉ๐‘“๎€ท๐‘ฅ,๐‘ก1๎€ธ๐ž,๐‘ก1๎‚€โ„ฌin๐œ€โˆฉ๎‚†๐‘ฅ1>13๐‘“๎€ท๐‘ฅ๎‚‡๎‚ร—(0,๐‘‡),2๎€ธ๐ž,๐‘ก2๎‚€โ„ฌin๐œ€โˆฉ๎‚†๐‘ฅ2>13ร—๎‚€โ„ฌ๎‚‡๎‚(0,๐‘‡),๐ŸŽin๐œ€โˆฉ๎‚†๐‘ฅ1โ‰ค13,๐‘ฅ2โ‰ค13๐‘”๎‚‡๎‚ร—(0,๐‘‡),ยฑ๎€ท๐‘ฅ1,๐‘ฅ2๎€ธ=โŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉ๐‘”,๐‘กยฑ๎€ท๐‘ฅ1๎€ธ๎‚€๐น,๐‘กon๐œ€ยฑโˆฉ๎‚†๐‘ฅ1>13๐‘”๎‚‡๎‚ร—(0,๐‘‡),ยฑ๎€ท๐‘ฅ2๎€ธ๎‚€๐น,๐‘กon๐œ€ยฑโˆฉ๎‚†๐‘ฅ2>130๎‚€๐น๎‚‡๎‚ร—(0,๐‘‡),on๐œ€ยฑโˆฉ๎‚†๐‘ฅ1โ‰ค13,๐‘ฅ2โ‰ค13ร—๐œˆ๎€ท๐‘ฅ๎‚‡๎‚(0,๐‘‡),1,๐‘ฅ2๎€ธ=โŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉ๐œˆ๎€ท๐‘ฅ1๎€ธinโ„ฌ๐œ€โˆฉ๎‚†13<๐‘ฅ1<23๎‚‡,๐œˆ๎€ท๐‘ฅ2๎€ธinโ„ฌ๐œ€โˆฉ๎‚†13<๐‘ฅ2<23๎‚‡,๐œˆ0inโ„ฌ๐œ€โˆฉ๐‘ฅ๎‚€๎‚†1โ‰ค13,๐‘ฅ2โ‰ค13๎‚‡โˆช๎‚†๐‘ฅ1โ‰ฅ23๎‚‡โˆช๎‚†๐‘ฅ2โ‰ฅ23,๎‚‡๎‚(4.2) with ๐œˆ0 constant.

4.1. Construction of the Asymptotic Solution

In the sequel we introduce the second small parameter ๐›ฟ=12/โ„Ž3๐ธ and we take ๐›ฟ=๐œ€๐›พ, with ๐›พโˆˆโ„•, ๐›พโ‰ฅ3. The asymptotic solution approximating the periodic flow in an infinite rectangle (see [1, Sectionโ€‰โ€‰5]) is modified by using two types of correctors: the first type corresponds to the boundary conditions on ๐‘ฅ1=1 and ๐‘ฅ2=1 and the second one represents the boundary layer functions in a neighborhood of (๐‘ฅ1,๐‘ฅ2)=๐ŸŽ. So, the asymptotic solution contains the regular part of the solution (which has two terms, due to the form of the flow domain), two boundary layer functions corresponding to ๐‘ฅ1=1 and to ๐‘ฅ2=1, respectively, and the correctors in (๐‘ฅ1,๐‘ฅ2)=๐ŸŽ. Since the terms of the asymptotic solution should vanish in different parts of the flow domain, we multiply them with the truncation functions defined as follows: ๐œ’, ๐œ‚โˆถโ„โ†ฆโ„, ๐œ’,๐œ‚โˆˆ๐’žโˆž(โ„),โŽงโŽชโŽจโŽชโŽฉโŽงโŽชโŽจโŽชโŽฉ3๐œ’(๐œ)=0,|๐œ|<2,1,|๐œ|>3,๐œ‚(๐œ)=0,|๐œ|<4,71,|๐œ|>8.(4.3)

We construct the asymptotic solution as below:ฬ‚๐ฎ๐‘Ž(๐‘˜)๎€ท๐‘ฅ1,๐‘ฅ2๎€ธ,๐‘ก=๐ฎ1(๐‘˜)๎‚€๐‘ฅ1,๐‘ฅ2๐œ€๎‚๐œ’๎‚€๐‘ฅ,๐‘ก1๐œ€๎‚+๐ฎ2(๐‘˜)๎‚€๐‘ฅ1๐œ€,๐‘ฅ2๎‚๐œ’๎‚€๐‘ฅ,๐‘ก2๐œ€๎‚+๐ฎ(๐‘˜)1๐‘๐‘™๎‚ต๐‘ฅ1โˆ’1๐œ€,๐‘ฅ2๐œ€๎‚ถ๐œ‚๎€ท๐‘ฅ,๐‘ก1๎€ธ+๐ฎ(๐‘˜)2๐‘๐‘™๎‚ต๐‘ฅ1๐œ€,๐‘ฅ2โˆ’1๐œ€๎‚ถ๐œ‚๎€ท๐‘ฅ,๐‘ก2๎€ธ+๐ฎ(๐‘˜)0๐‘๐‘™๎‚€๐‘ฅ1๐œ€,๐‘ฅ2๐œ€๎‚๐œ‚๎€ท,๐‘ก1โˆ’๐‘ฅ1๎€ธ๐œ‚๎€ท1โˆ’๐‘ฅ2๎€ธ,ฬ‚๐‘๐‘Ž(๐‘˜)๎€ท๐‘ฅ1,๐‘ฅ2๎€ธ,๐‘ก=๐‘1(๐‘˜)๎‚€๐‘ฅ1,๐‘ฅ2๐œ€๎‚๐œ’๎‚€๐‘ฅ,๐‘ก1๐œ€๎‚+๐‘2(๐‘˜)๎‚€๐‘ฅ1๐œ€,๐‘ฅ2๎‚๐œ’๎‚€๐‘ฅ,๐‘ก2๐œ€๎‚+๐‘(๐‘˜)1๐‘๐‘™๎‚ต๐‘ฅ1โˆ’1๐œ€,๐‘ฅ2๐œ€๎‚ถ๐œ‚๎€ท๐‘ฅ,๐‘ก1๎€ธ+๐‘(๐‘˜)2๐‘๐‘™๎‚ต๐‘ฅ1๐œ€,๐‘ฅ2โˆ’1๐œ€๎‚ถ๐œ‚๎€ท๐‘ฅ,๐‘ก2๎€ธ+๐‘(๐‘˜)0๐‘๐‘™๎‚€๐‘ฅ1๐œ€,๐‘ฅ2๐œ€๎‚๐œ‚๎€ท,๐‘ก1โˆ’๐‘ฅ1๎€ธ๐œ‚๎€ท1โˆ’๐‘ฅ2๎€ธ,๎๐‘‘(๐‘˜)ยฑ๐‘Ž๎€ท๐‘ฅ1,๐‘ฅ2๎€ธ,๐‘ก=๐‘‘(๐‘˜)ยฑ1๎€ท๐‘ฅ1๎€ธ๐œ’๎‚€๐‘ฅ,๐‘ก1๐œ€๎‚+๐‘‘(๐‘˜)ยฑ2๎€ท๐‘ฅ2๎€ธ๐œ’๎‚€๐‘ฅ,๐‘ก2๐œ€๎‚+๐‘‘(๐‘˜)1ยฑ๐‘๐‘™๎‚ต๐‘ฅ1โˆ’1๐œ€๎‚ถ๐œ‚๎€ท๐‘ฅ,๐‘ก1๎€ธ+๐‘‘(๐‘˜)2ยฑ๐‘๐‘™๎‚ต๐‘ฅ2โˆ’1๐œ€๎‚ถ๐œ‚๎€ท๐‘ฅ,๐‘ก2๎€ธ+๐‘‘(๐‘˜)0ยฑ๐‘๐‘™๎‚€๐‘ฅ1๐œ€,๐‘ฅ2๐œ€๎‚๐œ‚๎€ท,๐‘ก1โˆ’๐‘ฅ1๎€ธ๐œ‚๎€ท1โˆ’๐‘ฅ2๎€ธ.(4.4)

Due to the definition of the truncation functions, we notice that in โ„ฌ๐œ€โˆฉ{1/4โ‰ค๐‘ฅ๐‘–โ‰ค3/4}, that is, in a neighborhood of the region of variable viscosity, the asymptotic solution reduces to its regular part, (๐ฎ๐‘–(๐‘˜),๐‘๐‘–(๐‘˜),๐‘‘(๐‘˜)ยฑ๐‘–), in a neighborhood of {๐‘ฅ๐‘–=1} it reduces to (๐ฎ๐‘–(๐‘˜)+๐ฎ(๐‘˜)๐‘–๐‘๐‘™,๐‘๐‘–(๐‘˜)+๐‘(๐‘˜)๐‘–๐‘๐‘™,๐‘‘(๐‘˜)ยฑ๐‘–+๐‘‘(๐‘˜)๐‘–ยฑ๐‘๐‘™), ๐‘–=1,2, while in ๐ท๐‘Ÿ๐œ€ the asymptotic solution is equal to the corrector in ๐‘ฅ=๐ŸŽ,(๐ฎ(๐‘˜)0๐‘๐‘™,๐‘(๐‘˜)0๐‘๐‘™,๐‘‘(๐‘˜)0ยฑ๐‘๐‘™). This means that the regular part of the asymptotic solution (which can be computed and has a very simple expression) represents an approximation for the exact solution in all the flow domain except some neighborhoods of ๐‘ฅ1=1, ๐‘ฅ2=1 and ๐‘ฅ=๐ŸŽ.

We give next the expressions of the three components of the asymptotic solution and we explain their role in this construction.

4.1.1. The Regular Part of the Asymptotic Solution

For each rectangle ๐ท1๐œ€ and ๐ท2๐œ€ we define the corresponding regular part of the asymptotic expansion as in [1]. The regular part corresponding to ๐ท1๐œ€ has the expression:๐ฎ1(๐‘˜)๎‚€๐‘ฅ1,๐‘ฅ2๐œ€๎‚=,๐‘ก๐‘˜๎“๐‘—=0๐œ€๐‘—+2๐‘ข11,๐‘—๎‚€๐‘ฅ1,๐‘ฅ2๐œ€๎‚๐ž,๐‘ก1+๐‘˜๎“๐‘—=0๐œ€๐‘—+3๐‘ข12,๐‘—๎‚€๐‘ฅ1,๐‘ฅ2๐œ€๎‚๐ž,๐‘ก2,๐‘1(๐‘˜)๎‚€๐‘ฅ1,๐‘ฅ2๐œ€๎‚=,๐‘ก๐‘˜๎“๐‘—=0๐œ€๐‘—+1๐‘1๐‘—๎‚€๐‘ฅ1,๐‘ฅ2๐œ€๎‚+,๐‘ก๐‘˜๎“๐‘—=0๐œ€๐‘—๐‘ž1๐‘—๎€ท๐‘ฅ1๎€ธ,๐‘‘,๐‘ก(๐‘˜)ยฑ1๎€ท๐‘ฅ1๎€ธ=,๐‘ก๐‘˜๎“๐‘—=0๐œ€๐‘—+๐›พ๐‘‘1(ยฑ)๐‘—๎€ท๐‘ฅ1๎€ธ.,๐‘ก(4.5) It represents the solution of the problem (2.6)1,2,3,11 set in the infinite rectangle in ๐‘‚๐‘ฅ1 direction, (โˆ’โˆž,โˆž)ร—(โˆ’๐œ€,๐œ€); the functions which appear in (4.5) can be computed explicitly and represent a good approximation of the exact solution of (2.6) in a neighborhood of the region with variable viscosity of ๐ท1๐œ€ (as we will prove in the last section).

In a similar way we introduce the regular part of the asymptotic solution corresponding to ๐ท2๐œ€.

Hence, the sum of the first two terms of the asymptotic solution represents a good approximation of the exact solution of (2.6) in some neighborhood of the region of variable viscosity of the flow domain, but it is not close to it on the inflow/outflow boundaries and in ๐ท๐‘Ÿ๐œ€. Since the purpose of the asymptotic construction is to approximate the exact solution with a small error between the exact and asymptotic solution, we modify and complete the regular part by two types of boundary layer correctors.

4.1.2. The Boundary Layer Correctors for ๐‘ฅ1=1 and ๐‘ฅ2=1

These boundary layer functions are introduced in order to repair the traces of the regular part of the asymptotic solution on ๐‘ฅ1=1 and ๐‘ฅ2=1. They are given by:๐ฎ(๐‘˜)๐‘–๐‘๐‘™๎‚€๐‘ฅ๐œ€๎‚=,๐‘ก๐‘˜๎“๐‘—=0๐œ€๐‘—+2๐ฎ๐‘—(๐‘–)๎‚€๐‘ฅ๐œ€๎‚,๐‘,๐‘ก(๐‘˜)๐‘–๐‘๐‘™๎‚€๐‘ฅ๐œ€๎‚=,๐‘ก๐‘˜๎“๐‘—=0๐œ€๐‘—+1๐‘๐‘—(๐‘–)๎‚€๐‘ฅ๐œ€๎‚,๐‘‘,๐‘ก(๐‘˜)๐‘–ยฑ๐‘๐‘™๎‚€๐‘ฅ๐‘–๐œ€๎‚=,๐‘ก๐‘˜๎“๐‘—=0๐œ€๐‘—+๐›พ๐‘‘๐‘—(๐‘–)๎‚€๐‘ฅ๐‘–๐œ€๎‚,๐‘ก,๐‘–โˆˆ{1,2}.(4.6) The corrector with ๐‘–=1 corresponds to the end ๐‘ฅ1=1 and that with ๐‘–=2 corresponds to ๐‘ฅ2=1.

From the definition of the truncation functions and of the asymptotic solution we notice that the influence of each corrector defined above is significant only near the corresponding end of the flow domain.

4.1.3. The Boundary Layer Corrector in ๐‘ฅ=0

This corrector is necessary in order to realize the junction between the two parts of the asymptotic solution, corresponding to the two branches of the flow domain and to obtain the conditions on the rigid boundaries of the domain. The expressions of the correctors in ๐‘ฅ=๐ŸŽ are given by:๐ฎ(๐‘˜)0๐‘๐‘™๎‚€๐‘ฅ๐œ€๎‚=,๐‘ก๐‘˜๎“๐‘—=โˆ’1๐œ€๐‘—+2๐ฎ๐‘—(0)๎‚€๐‘ฅ๐œ€๎‚,๐‘,๐‘ก(๐‘˜)0๐‘๐‘™๎‚€๐‘ฅ๐œ€๎‚=,๐‘ก๐‘˜๎“๐‘—=โˆ’1๐œ€๐‘—+1๐‘๐‘—(0)๎‚€๐‘ฅ๐œ€๎‚,๐‘‘,๐‘ก(๐‘˜)0ยฑ๐‘๐‘™๎‚€๐‘ฅ๐œ€๎‚=,๐‘ก๐‘˜๎“๐‘—=0๐œ€๐‘—+๐›พ๐‘‘(0)(ยฑ)๐‘—๎‚€๐‘ฅ๐œ€๎‚.,๐‘ก(4.7) From the definition of the truncation function ๐œ‚ it follows that the corrector in ๐‘ฅ=๐ŸŽ appears in the expression of the asymptotic solution (4.4) only in โ„ฌ๐œ€โˆฉ{๐‘ฅ1โ‰ค1/4,๐‘ฅ2โ‰ค1/4}, which represents a neighborhood of ๐ท๐‘Ÿ๐œ€ with constant viscosity and where the forces are equal to zero.

Remark 4.1. The boundary layer method is close to the well-known method of matching of asymptotic expansions (see [14]). However there is a difference between these two methods. In the boundary layer method the regular expansion is reexpanded with respect to the fast variable (as in the matching method), but then the new expansion is multiplied by a cutting function. This product being inserted into the equation gives a discrepancy in the right-hand side which is then compensated by an appropriate boundary layer corrector. So, we get finally a unique expression for the asymptotic approximation in each point of the domain. In the matching approach the reexpanded regular ansatz (outer expansion) coexists with the so-called inner expansion defined near the boundary, so that there are several overlapping expressions for an asymptotic approximation in different parts of the domain. We emphasize that this difference is not too important and corresponds more to the form of presentation of the result.

4.2. The Determination of the Asymptotic Solution

This subsection is devoted to the resolution of the problems satisfied by the three different components of the asymptotic solution presented in the previous subsection. Since some computations are different with respect to the values of ๐›พ, we will analyse the problems and the order of solving them for ๐›พ>3 and for ๐›พ=3. Moreover, in each case we will specify the leading term of the asymptotic expansion.

We begin the approach with the problems for the boundary layer correctors corresponding to ๐‘ฅ1=1 and ๐‘ฅ2=1 since the study of these problems is the same both for ๐›พ>3 and for ๐›พ=3. To fix the ideas, we obtain in the sequel the problems for the corrector corresponding to the end ๐‘ฅ1=1. As we noticed before, the term containing this corrector is not equal to zero only in a neighborhood of the boundary ๐‘ฅ1=1. So, the problems and the other relations corresponding to this corrector are obtained substituting the asymptotic solution in (2.6)1,2,3,6,7,11. We obtain two separate problems: one for the velocity-pressure correctors and the other one for the displacements correctors.

Since in this neighborhood the viscosity is constant, the problem for (๐ฎ๐‘—(1),๐‘๐‘—(1)) has constant coefficients. Denoting by ฮ 1 the semi-infinite rectangle (โˆ’โˆž,0)ร—(โˆ’1,1) and imposing for the velocity and for the pressure the condition of decay at โˆ’โˆž, we obtain for (๐ฎ๐‘—(1),๐‘๐‘—(1)) the problem:โˆ’๐œˆ0ฮ”๐œ‰๐œ‰๐ฎ๐‘—(1)+โˆ‡๐œ‰๐‘๐‘—(1)=โˆ’๐œŒ๐‘“๐œ•๐ฎ(1)๐‘—โˆ’2๐œ•๐‘กinฮ 1ร—(0,๐‘‡),div๐œ‰๐ฎ๐‘—(1)=0inฮ 1ร—๐ฎ(0,๐‘‡),๐‘—(1)๎€ท๐œ‰1๎€ธ=,ยฑ1,๐‘ก๐œ•๐‘‘(1)(ยฑ)๐‘—+2โˆ’๐›พ๎€ท๐œ‰๐œ•๐‘ก1๎€ธ๐ž,๐‘ก2๐ฎin(โˆ’โˆž,0)ร—(0,๐‘‡),๐‘—(1)๎€ท0,๐œ‰2๎€ธ๎€ท๐œ‰,๐‘ก=๐œ“2๎€ธ๐›ฟ,๐‘ก๐‘—0๐ž1โˆ’๐‘ข11,๐‘—๎€ท1,๐œ‰2๎€ธ๐ž,๐‘ก1โˆ’๐‘ข12,๐‘—โˆ’1๎€ท1,๐œ‰2๎€ธ๐ž,๐‘ก2๐ฎin(โˆ’1,1)ร—(0,๐‘‡),๐‘—(1)โŸถ๐ŸŽ,๐‘๐‘—(1)โŸถ0,uniformlywhen๐œ‰1โŸถโˆ’โˆž.(4.8) The compatibility condition for (4.8) reads:๎€œ1โˆ’1๐‘ข11,๐‘—๎€ท1,๐œ‰2๎€ธ,๐‘กd๐œ‰2=d๎€œd๐‘ก0โˆ’โˆž๎‚€๐‘‘(1)(+)๐‘—+2โˆ’๐›พโˆ’๐‘‘(1)(โˆ’)๐‘—+2โˆ’๐›พ๎‚๎€ท๐œ‰1๎€ธ,๐‘กd๐œ‰1+๐›ฟ๐‘—0๎€œ1โˆ’1๐œ“๎€ท๐œ‰2๎€ธ,๐‘กd๐œ‰2.(4.9) For any ๐›พโ‰ฅ3 the right-hand side of (4.8)3 is known; so the boundary layer correctors for the velocity and for the pressure corresponding to ๐‘ฅ1=1 are uniquely determined from (4.8) (see [2, Sectionโ€‰โ€‰4]). The condition (4.9) represents a relation for the regular part of the asymptotic solution.

The boundary layer correctors for the displacements exponentially stabilizing to zero at โˆ’โˆž are obtained as the unique solution of the following problems:๐œ•4๐‘‘(1)(ยฑ)๐‘—๐œ•๐œ‰41๐œ•=โˆ’๐œŒโ„Ž2๐‘‘(1)(ยฑ)๐‘—โˆ’4โˆ’๐›พ๐œ•๐‘ก2๐œ•โˆ’๐œ‡5๐‘‘(1)(ยฑ)๐‘—โˆ’๐›พ๐œ•๐œ‰41๐œ•๐‘กยฑ๐‘|(1)๐‘—โˆ’5๐œ‰2=ยฑ1๐œ•in(โˆ’โˆž,0)ร—(0,๐‘‡),๐‘Ž๐‘‘(1)(ยฑ)๐‘—๐œ•๐œ‰๐‘Ž1โŸถ0,uniformly,whenonly๐œ‰1โŸถโˆ’โˆž,๐‘Žโˆˆ{0,1,2,3}.(4.10)

Since at the step ๐‘— the problem (4.10) gives both ๐‘‘(1)(ยฑ)๐‘— and ๐‘‘(1)(ยฑ)๐‘—+1, introducing the asymptotic solution into (2.6)7 we obtain two boundary conditions for the regular part of the asymptotic solution for the displacements:๐‘‘1(ยฑ)๐‘—(1,๐‘ก)=โˆ’๐‘‘(1)(ยฑ)๐‘—(0,๐‘ก),๐œ•๐‘‘1(ยฑ)๐‘—๐œ•๐‘ฅ1(1,๐‘ก)=โˆ’๐œ•๐‘‘(1)(ยฑ)๐‘—+1๐œ•๐œ‰1(0,๐‘ก).(4.11)

In a similar way, we obtain the boundary layer correctors corresponding to the end ๐‘ฅ2=1. The boundary layers for the velocity-pressure are defined on ฮ 2ร—(0,๐‘‡), with ฮ 2=(โˆ’1,1)ร—(โˆ’โˆž,0), and the boundary layers for the displacements are defined also on (โˆ’โˆž,0)ร—(0,๐‘‡).

We study next the problems for the regular parts of the asymptotic solution. The results are obtained for the regular part corresponding to ๐ท1๐œ€; the regular part corresponding to ๐ท2๐œ€ may be obtained from the previous with some obvious changes.

Introducing (4.5) into (2.6)1,2,3,11 and collecting together the terms of the same order with respect to ๐œ€ we are leaded to consider the following problem for (๐‘ข11,๐‘—,๐‘ข12,๐‘—,๐‘1๐‘—,๐‘ž1๐‘—,๐‘‘1(ยฑ)๐‘—):๎€ท๐‘ฅโˆ’๐œˆ1๎€ธ๐œ•2๐‘ข11,๐‘—๐œ•๐œ‰22+๐œ•๐‘ž1๐‘—๐œ•๐‘ฅ1=๐‘“๐›ฟ๐‘—0โˆ’๐œŒ๐‘“๐œ•๐‘ข11,๐‘—โˆ’2๐œ•๐œ•๐‘ก+2๐œ•๐‘ฅ1๎ƒฉ๐œˆ๎€ท๐‘ฅ1๎€ธ๐œ•๐‘ข11,๐‘—โˆ’2๐œ•๐‘ฅ1๎ƒช๎€ท๐‘ฅ+๐œˆ1๎€ธ๐œ•2๐‘ข12,๐‘—โˆ’2๐œ•๐œ‰2๐œ•๐‘ฅ1โˆ’๐œ•๐‘1๐‘—โˆ’1๐œ•๐‘ฅ1,๐œ•๐‘1๐‘—๐œ•๐œ‰2=โˆ’๐œŒ๐‘“๐œ•๐‘ข12,๐‘—โˆ’3+๐œ•๐œ•๐‘ก๐œ•๐‘ฅ1๎ƒฉ๐œˆ๎€ท๐‘ฅ1๎€ธ๐œ•๐‘ข12,๐‘—โˆ’3๐œ•๐‘ฅ1๎ƒช+๐œ•๐œ•๐‘ฅ1๎ƒฉ๐œˆ๎€ท๐‘ฅ1๎€ธ๐œ•๐‘ข11,๐‘—โˆ’1๐œ•๐œ‰2๎ƒช๎€ท๐‘ฅ+2๐œˆ1๎€ธ๐œ•2๐‘ข12,๐‘—โˆ’1๐œ•๐œ‰22,๐œ•๐‘ข11,๐‘—๐œ•๐‘ฅ1+๐œ•๐‘ข12,๐‘—๐œ•๐œ‰2๐œ•=0in(0,1)ร—(โˆ’1,1)ร—(0,๐‘‡),4๐‘‘1(+)๐‘—๐œ•๐‘ฅ41โˆ’๐‘ž1๐‘—=๐‘”+๐›ฟ๐‘—0+๐‘1|๐‘—โˆ’1๐œ‰2=1๐œ•โˆ’๐œŒโ„Ž2๐‘‘1(+)๐‘—โˆ’๐›พ๐œ•๐‘ก2๐œ•โˆ’๐œ‡5๐‘‘1(+)๐‘—โˆ’๐›พ๐œ•๐‘ฅ41๐œ•๐œ•๐‘กin(0,1)ร—{โˆ’1}ร—(0,๐‘‡),4๐‘‘1(โˆ’)๐‘—๐œ•๐‘ฅ41+๐‘ž1๐‘—=๐‘”โˆ’๐›ฟ๐‘—0โˆ’๐‘1|๐‘—โˆ’1๐œ‰2=โˆ’1๐œ•โˆ’๐œŒโ„Ž2๐‘‘1(โˆ’)๐‘—โˆ’๐›พ๐œ•๐‘ก2๐œ•โˆ’๐œ‡5๐‘‘1(โˆ’)๐‘—โˆ’๐›พ๐œ•๐‘ฅ41๐ฎ๐œ•๐‘กin(0,1)ร—{1}ร—(0,๐‘‡),1๐‘—๎€ท๐‘ฅ1๎€ธ=,ยฑ1,๐‘ก๐œ•๐‘‘1(ยฑ)๐‘—โˆ’๐›พ+3๎€ท๐‘ฅ๐œ•๐‘ก1๎€ธ๐ž,๐‘ก2in(0,1)ร—{ยฑ1}ร—(0,๐‘‡).(4.12) The two cases, ๐›พ>3 and ๐›พ=3, appear because of the last relation of the previous system. We can see, indeed, that for ๐›พ>3 the unknown of this relation is ๐ฎ1๐‘—, while for ๐›พ=3, (4.12)6 contains two unknowns. From this point, the computations are different with respect to the values of ๐›พ.

We introduce the functions:๐‘1๎€ท๐œ‰2๎€ธ=12๎€ท๐œ‰22๎€ธโˆ’1,๐‘2๎€ท๐œ‰2๎€ธ=๎€œ๐œ‰2โˆ’1๐‘1(๐œ)d๐œ,(4.13) with the properties: ๐‘1๎…ž๎…ž=1,๐‘1(ยฑ1)=0 and ๐‘2(1)=โˆ’2/3.

We also use the notations:๐‘ˆ1๐‘—โˆ’1๎€ท๐‘ฅ1,๐œ‰2๎€ธ,๐‘ก=๐œŒ๐‘“๐œ•๐‘ข11,๐‘—โˆ’2๐œ•๐œ•๐‘กโˆ’2๐œ•๐‘ฅ1๎ƒฉ๐œˆ๎€ท๐‘ฅ1๎€ธ๐œ•๐‘ข11,๐‘—โˆ’2๐œ•๐‘ฅ1๎ƒช๎€ท๐‘ฅโˆ’๐œˆ1๎€ธ๐œ•2๐‘ข12,๐‘—โˆ’2๐œ•๐œ‰2๐œ•๐‘ฅ1+๐œ•๐‘1๐‘—โˆ’1๐œ•๐‘ฅ1,๐’Ÿ1(ยฑ)๐‘—โˆ’1๎€ท๐‘ฅ1๎€ธ,๐‘ก=ยฑ๐‘1|๐‘—โˆ’1๐œ‰2=1๐œ•โˆ’๐œŒโ„Ž2๐‘‘1(ยฑ)๐‘—โˆ’๐›พ๐œ•๐‘ก2๐œ•โˆ’๐œ‡5๐‘‘1(ยฑ)๐‘—โˆ’๐›พ๐œ•๐‘ฅ41,๐ท๐œ•๐‘กโˆ’1๎€œโˆถ๐นโŸถ๐œ‰2โˆ’1๐น๎€ท๐‘ฅ1๎€ธ๐ท,๐œd๐œ,โˆ’2๎€œโˆถ๐นโŸถ๐œ‰2โˆ’1๎€œ๐œƒโˆ’1๐น๎€ท๐‘ฅ1๎€ธ1,๐œd๐œd๐œƒโˆ’2๎€ท๐œ‰2๎€ธ๎€œ+11โˆ’1๎€œ๐œƒโˆ’1๐น๎€ท๐‘ฅ1๎€ธ๐ผ,๐œd๐œd๐œƒ,๐‘ฅโˆ’๐‘˜1๎€œโˆถ๐นโŸถ๐‘ฅ10๎€œ๐‘Ž๐‘˜โˆ’10โ‹ฏ๎€œ๐‘Ž10๐น๎€ท๐‘ ,๐œ‰2๎€ธ๐ฝ,๐‘กd๐‘ ,๐‘ฅ1โˆถ๐นโŸถ๐ผ๐‘ฅโˆ’41(๐น)+๐‘ฅ31๎‚ต2๎€œ10๐ผ๐‘ฅโˆ’31๎€œ(๐น)โˆ’10๐ผ๐‘ฅโˆ’21๎‚ถ(๐น)โˆ’๐‘ฅ2๎‚ต3๎€œ10๐ผ๐‘ฅโˆ’31๎€œ(๐น)โˆ’10๐ผ๐‘ฅโˆ’21๎‚ถ.(๐น)(4.14)

4.2.1. The Order of Solving the Problems for ๐›พ>3

The regular part corresponding to ๐ท1๐œ€ is computed by integrating (4.12), as stated below.

Proposition 4.2. The unknowns ๐ฎ1๐‘—,๐‘ž1๐‘—,๐‘1๐‘—,๐‘‘1(ยฑ)๐‘— are determined from (4.12), up to nine functions of ๐‘ก.

Proof. Integrating twice (4.12)1 from โˆ’1 to ๐œ‰2 and using the boundary conditions (4.12)6 we get: ๐‘ข11,๐‘—๎€ท๐‘ฅ1,๐œ‰2๎€ธ=1,๐‘ก๐œˆ๎€ท๐‘ฅ1๎€ธ๐ทโˆ’2๎‚€๐‘ˆ1๐‘—โˆ’1๎‚+1๐œˆ๎€ท๐‘ฅ1๎€ธ๎ƒฉ๐œ•๐‘ž1๐‘—๐œ•๐‘ฅ1โˆ’๐‘“๐›ฟ๐‘—0๎ƒช๐‘1๎€ท๐œ‰2๎€ธ,(4.15) which contains as unknowns ๐‘ข11,๐‘— and ๐‘ž1๐‘—. The other functions contained by this relation are either known from previous computations or equal to zero.
We integrate next the incompressibility condition (4.12)3 with respect to ๐œ‰2 with the boundary condition (4.12)6 for ๐œ‰2=โˆ’1 and we obtain๐‘ข12,๐‘—๎€ท๐‘ฅ1,๐œ‰2๎€ธ=,๐‘ก๐œ•๐‘‘1(โˆ’)๐‘—โˆ’๐›พ+3โˆ’๐œ•๐œ•๐‘ก๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๎ƒฉ๐œ•๐‘ž1๐‘—๐œ•๐‘ฅ1โˆ’๐‘“๐›ฟ๐‘—0๐‘๎ƒช๎ƒช2๎€ท๐œ‰2๎€ธโˆ’๐ทโˆ’1๐ทโˆ’2๎ƒฉ๐œ•๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๐‘ˆ1๐‘—โˆ’1.๎ƒช๎ƒช(4.16) The previous two relations give the components of the velocity ๐ฎ1๐‘— with respect to ๐‘ž1๐‘—.
The pressure approximations are determined from (4.12)2, supposing that the integration functions, depending on ๐‘ฅ1, ๐‘ก are equal to zero, since we consider that any function depending only on ๐‘ฅ1, ๐‘ก could be contained in ๐‘ž1๐‘—+1. ๐‘1๐‘—=๐ทโˆ’1๎ƒฉโˆ’๐œŒ๐‘“๐œ•๐‘ข12,๐‘—โˆ’3+๐œ•๐œ•๐‘ก๐œ•๐‘ฅ1๎ƒฉ๐œˆ๎€ท๐‘ฅ1๎€ธ๐œ•๐‘ข12,๐‘—โˆ’3๐œ•๐‘ฅ1๎ƒช+๐œ•๐œ•๐‘ฅ1๎ƒฉ๐œˆ๎€ท๐‘ฅ1๎€ธ๐œ•๐‘ข11,๐‘—โˆ’1๐œ•๐œ‰2๎ƒช๎€ท๐‘ฅ+2๐œˆ1๎€ธ๐œ•2๐‘ข12,๐‘—โˆ’1๐œ•๐œ‰22๎ƒช.(4.17) Taking ๐œ‰2=1 in (4.16) and using the boundary condition (4.12)6 for ๐œ‰2=1 we obtain the following second order differential equation for the function ๐‘ž1๐‘—: ๐œ•๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๎ƒฉ๐œ•๐‘ž1๐‘—๐œ•๐‘ฅ1โˆ’๐‘“๐›ฟ๐‘—0=3๎ƒช๎ƒช2๐œ•๎‚€๐‘‘๐œ•๐‘ก1(+)๐‘—โˆ’๐›พ+3โˆ’๐‘‘1(โˆ’)๐‘—โˆ’๐›พ+3๎‚+32๎€œ1โˆ’1๐ทโˆ’2๎ƒฉ๐œ•๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๐‘ˆ1๐‘—โˆ’1๎ƒช๎ƒชd๐œ‰2.(4.18) Integrating (4.18) from ๐‘ฅ1 to 1, we express ๐œ•๐‘ž1๐‘—/๐œ•๐‘ฅ1 by means of (๐œ•๐‘ž1๐‘—/๐œ•๐‘ฅ1)(1,๐‘ก), which represents the only unknown of this expression. This function of ๐‘ก is obtained as follows: we take ๐‘ฅ1=1 in (4.15) and we introduce the result into (4.9). Hence, we determined the expression of ๐œ•๐‘ž1๐‘—/๐œ•๐‘ฅ1 in (0,1)ร—(0,๐‘‡), which is: 1๐œˆ๎€ท๐‘ฅ1๎€ธ๎ƒฉ๐œ•๐‘ž1๐‘—๐œ•๐‘ฅ1โˆ’๐‘“๐›ฟ๐‘—0๎ƒช3=โˆ’2๐›ฟ๐‘—0๎€œ1โˆ’1๐œ“๎€ท๐œ‰2๎€ธ,๐‘กd๐œ‰2โˆ’32d๎€œd๐‘ก0โˆ’โˆž๎‚€๐‘‘(1)(+)๐‘—+2โˆ’๐›พโˆ’๐‘‘(1)(โˆ’)๐‘—+2โˆ’๐›พ๎‚๎€ท๐œ‰1๎€ธ,๐‘กd๐œ‰1โˆ’32๐œ•๎€œ๐œ•๐‘ก1๐‘ฅ1๎‚€๐‘‘1(+)๐‘—โˆ’๐›พ+3โˆ’๐‘‘1(โˆ’)๐‘—โˆ’๐›พ+3๎‚3(๐‘ ,๐‘ก)d๐‘ +2๎€œ1โˆ’1๐ทโˆ’2๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๐‘ˆ1๐‘—โˆ’1๎ƒชd๐œ‰2.(4.19) Introducing (4.19) into (4.15) and (4.18) into (4.16) we determine ๐ฎ1๐‘— in (0,1)ร—(โˆ’1,1)ร—(0,๐‘‡).
We integrate next ๐œˆ(๐‘ฅ1)โ€‰ยทโ€‰(4.19) from 0 to ๐‘ฅ1 and we get ๐‘ž1๐‘— determined up to the function of ๐‘ก, ๐‘ž1๐‘—(0,๐‘ก): ๐‘ž1๐‘—๎€ท๐‘ฅ1๎€ธ,๐‘ก=๐›ฟ๐‘—0๎€œ๐‘ฅ103๐‘“(๐‘ ,๐‘ก)d๐‘ โˆ’2๎€œ๐‘ฅ10๎€œ๐œˆ(๐œƒ)1๐œƒ๐œ•๎‚€๐‘‘๐œ•๐‘ก1(+)๐‘—โˆ’๐›พ+3โˆ’๐‘‘1(โˆ’)๐‘—โˆ’๐›พ+3๎‚โˆ’3(๐‘ ,๐‘ก)d๐‘ d๐œƒ2๎‚ตd๎€œd๐‘ก0โˆ’โˆž๎‚€๐‘‘(1)(+)๐‘—+2โˆ’๐›พโˆ’๐‘‘(1)(โˆ’)๐‘—+2โˆ’๐›พ๎‚๎€ท๐œ‰1๎€ธ,๐‘กd๐œ‰1+๐›ฟ๐‘—0๎€œ1โˆ’1๐œ“๎€ท๐œ‰2๎€ธ,๐‘กd๐œ‰2๎€œ๎‚ถ๎‚ต๐‘ฅ10๐œˆ๎‚ถ+3(๐‘ )d๐‘ 2๎€œ๐‘ฅ10๎€œ1โˆ’1๐ทโˆ’2๎‚€๐‘ˆ1๐‘—โˆ’1๎€ท๐‘ ,๐œ‰2๎€ธ๎‚,๐‘กd๐œ‰2d๐‘ +๐‘ž1๐‘—(0,๐‘ก).(4.20) The functions ๐‘‘1(ยฑ)๐‘— satisfy the fourth-order differential equations: ๐œ•4๐‘‘1(ยฑ)๐‘—๐œ•๐‘ฅ41=ยฑ๐‘ž1๐‘—+๐‘”ยฑ๐›ฟ๐‘—0+๐’Ÿ1(ยฑ)๐‘—โˆ’1,(4.21) with ๐‘ž1๐‘— given by (4.20). Writting ๐‘ž1๐‘— as ๐‘„1๐‘—(๐‘ฅ1,๐‘ก)+๐‘ž1๐‘—(0,๐‘ก) and integrating four times (4.21) with respect to ๐‘ฅ1 we obtain the following expressions for ๐‘‘1(ยฑ)๐‘—: ๐‘‘1(+)๐‘—๎€ท๐‘ฅ1๎€ธ,๐‘ก=๐‘‘1(+)๐‘—๎€ท(0,๐‘ก)1โˆ’3๐‘ฅ12+2๐‘ฅ13๎€ธ+๐œ•๐‘‘1(+)๐‘—๐œ•๐‘ฅ1๎€ท๐‘ฅ(0,๐‘ก)1โˆ’2๐‘ฅ12+๐‘ฅ13๎€ธ+๐‘‘1(+)๐‘—๎€ท(1,๐‘ก)3๐‘ฅ12โˆ’2๐‘ฅ13๎€ธ+๐œ•๐‘‘1(+)๐‘—๐œ•๐‘ฅ1๎€ท(1,๐‘ก)โˆ’๐‘ฅ12+๐‘ฅ13๎€ธ+๐‘ž1๐‘—๎‚ต๐‘ฅ(0,๐‘ก)12โˆ’๐‘ฅ2413+๐‘ฅ1214๎‚ถ24+๐ฝ๐‘ฅ1๎€ท๐‘„1๐‘—๎€ธ+๐ฝ๐‘ฅ1๎€ท๐‘”+๎€ธ๐›ฟ๐‘—0+๐ฝ๐‘ฅ1๎‚€๐’Ÿ1(+)๐‘—โˆ’1๎‚,๐‘‘1(โˆ’)๐‘—๎€ท๐‘ฅ1๎€ธ,๐‘ก=๐‘‘1(โˆ’)๐‘—๎€ท(0,๐‘ก)1โˆ’3๐‘ฅ12+2๐‘ฅ13๎€ธ+๐œ•๐‘‘1(โˆ’)๐‘—๐œ•๐‘ฅ1๎€ท๐‘ฅ(0,๐‘ก)1โˆ’2๐‘ฅ12+๐‘ฅ13๎€ธ+๐‘‘1(โˆ’)๐‘—๎€ท(1,๐‘ก)3๐‘ฅ12โˆ’2๐‘ฅ13๎€ธ+๐œ•๐‘‘1(โˆ’)๐‘—๐œ•๐‘ฅ1๎€ท(1,๐‘ก)โˆ’๐‘ฅ12+๐‘ฅ13๎€ธโˆ’๐‘ž1๐‘—๎‚ต๐‘ฅ(0,๐‘ก)12โˆ’๐‘ฅ2413+๐‘ฅ1214๎‚ถ24โˆ’๐ฝ๐‘ฅ1๎€ท๐‘„1๐‘—๎€ธ+๐ฝ๐‘ฅ1๎€ท๐‘”โˆ’๎€ธ๐›ฟ๐‘—0+๐ฝ๐‘ฅ1๎‚€๐’Ÿ1(โˆ’)๐‘—โˆ’1๎‚.(4.22) Hence, the regular part of the asymptotic solution corresponding to ๐ท1๐œ€ is determined up to the functions ๐‘ž1๐‘—(0,๐‘ก), ๐‘‘1(ยฑ)๐‘—(0,๐‘ก), (๐œ•๐‘‘1(ยฑ)๐‘—/๐œ•๐‘ฅ1)(0,๐‘ก), ๐‘‘1(ยฑ)๐‘—(1,๐‘ก), (๐œ•๐‘‘1(ยฑ)๐‘—/๐œ•๐‘ฅ1)(1,๐‘ก), which achieves the proof.

In a similar way we express the regular part of the asymptotic solution corresponding to ๐ท2๐œ€ depending on 9 unknown functions of ๐‘ก.

We continue the construction with the problems for the corrector in ๐‘ฅ=๐ŸŽ. As we previously said, the term of the asymptotic solution containing this corrector is not equal to zero only in a neighborhood of ๐ท๐‘Ÿ๐œ€. In this neighborhood, the expression of the asymptotic solution reduces to (๐ฎ1(๐‘˜)(๐‘ฅ1,๐œ‰2,๐‘ก)๐œ’(๐œ‰1)+๐ฎ2(๐‘˜)(๐œ‰1,๐‘ฅ2,๐‘ก)๐œ’(๐œ‰2)+๐ฎ(๐‘˜)0๐‘๐‘™(๐œ‰,๐‘ก),๐‘1(๐‘˜)(๐‘ฅ1,๐œ‰2,๐‘ก)๐œ’(๐œ‰1)+๐‘2(๐‘˜)(๐œ‰1,๐‘ฅ2,๐‘ก)๐œ’(๐œ‰2๐‘)+(๐‘˜)0๐‘๐‘™(๐œ‰,๐‘ก),๐‘‘(๐‘˜)ยฑ1(๐‘ฅ1,๐‘ก)๐œ’(๐œ‰1)+๐‘‘(๐‘˜)ยฑ2(๐‘ฅ2,๐‘ก)๐œ’(๐œ‰2)+๐‘‘(๐‘˜)0ยฑ๐‘๐‘™(๐œ‰,๐‘ก)). For obtaining the problems satisfied by the correctors in ๐‘ฅ=๐ŸŽ, we introduce the previous expression of the asymptotic solution in (2.6)1,2,3,4,5,6,11, with ๐œˆ=๐œˆ0 and ๐Ÿ=๐ =๐ŸŽ in (2.6)1,3; for derivating the terms which contain the two types of variables ๐‘ฅ๐‘– and ๐œ‰๐‘– we proceed as follows: we replace ๐œ•/๐œ•๐‘ฅ๐‘– by (1/๐œ€)(๐œ•/๐œ•๐œ‰๐‘–), we replace ๐‘ฅ๐‘– by ๐œ€๐œ‰๐‘– and we expand the functions as a Taylor expansion with respect to ๐‘ฅ๐‘–=๐œ€๐œ‰๐‘–. We introduce the notations โ„ฌ=๐ท๐‘Ÿโˆช{(๐œ‰1,๐œ‰2)โˆถ๐œ‰1โ‰ฅ2,๐œ‰2โˆˆ(โˆ’1,1)}โˆช{(๐œ‰1,๐œ‰2)โˆถ๐œ‰1โˆˆ(โˆ’1,1),๐œ‰2โ‰ฅ2},โŒข๐ดยฑ๐ตยฑ=(1/๐œ€)โŒข๐ด๐œ€ยฑ๐ต๐œ€ยฑ and we obtain for (๐ฎ๐‘—(0),๐‘๐‘—(0)) the following nondivergence free problem:โˆ’๐œˆ0ฮ”๐œ‰๐œ‰๐ฎ๐‘—(0)+โˆ‡๐œ‰๐‘๐‘—(0)=โˆ’๐œŒ๐‘“๐œ•๐ฎ(0)๐‘—โˆ’2๐œ•๐‘ก+๐…๐‘—+๎€ท๎€ท๐œ‰1โˆ’๐œ’1๎€ธ๎€ธ๎…ž๐œ‰1๐‘ž1๐‘—+1(0,๐‘ก)๐ž1+๎€ท๎€ท๐œ‰1โˆ’๐œ’2๎€ธ๎€ธ๎…ž๐œ‰2๐‘ž2๐‘—+1(0,๐‘ก)๐ž2inโ„ฌ,div๐ฎ๐‘—(0)=ฬƒ๐œƒ๐‘—๐ฎinโ„ฌ,๐‘—(0)=๐ŸŽonโŒข๐ด+๐ต+โˆชโŒข๐ดโˆ’๐ตโˆ’,๐ฎ๐‘—(0)๎€ท๐œ‰1๎€ธ=,ยฑ1,๐‘ก๐œ•๐‘‘(0)(ยฑ)๐‘—+2โˆ’๐›พ๎€ท๐œ‰๐œ•๐‘ก1๎€ธ๐ž,ยฑ1,๐‘ก2for๐œ‰1๐ฎโ‰ฅ2,๐‘—(0)๎€ทยฑ1,๐œ‰2๎€ธ=,๐‘ก๐œ•๐‘‘(0)(ยฑ)๐‘—+2โˆ’๐›พ๎€ท๐œ•๐‘กยฑ1,๐œ‰2๎€ธ๐ž,๐‘ก1for๐œ‰2๐ฎโ‰ฅ2,๐‘—(0)โŸถ๐ŸŽuniformlywhen๐œ‰1๐ฎโŸถโˆž,๐‘—(0)โŸถ๐ŸŽuniformlywhen๐œ‰2โŸถโˆž,(4.23) for ๐‘—โ‰ฅ0 andโˆ’๐œˆ0ฮ”๐œ‰๐œ‰๐ฎ(0)โˆ’1+โˆ‡๐œ‰๐‘(0)โˆ’1=๎€ท๎€ท๐œ‰1โˆ’๐œ’1๎€ธ๎€ธ๎…ž๐œ‰1๐‘ž10(0,๐‘ก)๐ž1+๎€ท๎€ท๐œ‰1โˆ’๐œ’2๎€ธ๎€ธ๎…ž๐œ‰2๐‘ž20(0,๐‘ก)๐ž2inโ„ฌ,div๐ฎ(0)โˆ’1๐ฎ=0inโ„ฌ,(0)โˆ’1=๐ŸŽonโŒข๐ด+๐ต+โˆชโŒข๐ดโˆ’๐ตโˆ’,๐ฎ(0)โˆ’1๎€ท๐œ‰1๎€ธ,ยฑ1,๐‘ก=๐ŸŽfor๐œ‰1๐ฎโ‰ฅ2,(0)โˆ’1๎€ทยฑ1,๐œ‰2๎€ธ,๐‘ก=๐ŸŽfor๐œ‰2๐ฎโ‰ฅ2,(0)โˆ’1โŸถ๐ŸŽuniformlywhen๐œ‰1๐ฎโŸถโˆž,(0)โˆ’1โŸถ๐ŸŽuniformlywhen๐œ‰2โŸถโˆž.(4.24) Unlike the problem (4.8) which give the correctors for ๐‘ฅ1=1, the problems (4.23), (4.24) have unknown right hand sides. The functions ๐…๐‘— are known, but ๐‘ž1๐‘—+1(0,๐‘ก) and ๐‘ž2๐‘—+1(0,๐‘ก) are elements from the next approximation, so they are unknown. The function ฬƒ๐œƒ๐‘— is given by:ฬƒ๐œƒ๐‘—=โˆ’div๐œ‰๎ƒฉ๐œ’๎€ท๐œ‰1๎€ธ๐‘—๎“๐‘™=0๐œ‰๐‘™1๎ƒฉ๐œ•๐‘™!๐‘™๐‘ข11,๐‘—โˆ’๐‘™๐œ•๐‘ฅ1๐‘™๎€ท0,๐œ‰2๎€ธ๐ž,๐‘ก1+๐œ•๐‘™๐‘ข12,๐‘—โˆ’1โˆ’๐‘™๐œ•๐‘ฅ1๐‘™๎€ท0,๐œ‰2๎€ธ๐ž,๐‘ก2๎ƒช๎€ท๐œ‰+๐œ’2๎€ธ๐‘—๎“๐‘™=0๐œ‰๐‘™2๎ƒฉ๐œ•๐‘™!๐‘™๐‘ข21,๐‘—โˆ’1โˆ’๐‘™๐œ•๐‘ฅ2๐‘™๎€ท๐œ‰1๎€ธ๐ž,0,๐‘ก1+๐œ•๐‘™๐‘ข22,๐‘—โˆ’๐‘™๐œ•๐‘ฅ2๐‘™๎€ท๐œ‰1๎€ธ๐ž,0,๐‘ก2,๎ƒช๎ƒช(4.25) and it has to satisfy a compatibility condition.

The problem (4.24) has the unique solution (with the pressure unique up to an additive function of ๐‘ก)๐ฎ(0)โˆ’1๐‘=๐ŸŽ,(0)โˆ’1=โŽงโŽชโŽจโŽชโŽฉ๎€ท๎€ท๐œ‰1โˆ’๐œ’1๐‘ž๎€ธ๎€ธ10(0,๐‘ก)for๐œ‰1๐‘žโ‰ฅ2,10(0,๐‘ก)=๐‘ž20(0,๐‘ก)in๐ท๐‘Ÿ,๎€ท๎€ท๐œ‰1โˆ’๐œ’2๐‘ž๎€ธ๎€ธ20(0,๐‘ก)for๐œ‰2โ‰ฅ2.(4.26) We obtain next the problems for the correctors in ๐‘ฅ=๐ŸŽ corresponding to the displacements. We first notice that๐‘‘(0)(ยฑ)๐‘—=0onโŒข๐ดยฑ๐ตยฑ.(4.27) Let us denote by ๎๐‘‘(0)(ยฑ)๐‘—โˆ’1 the unique solution of๐œ•4๎๐‘‘(0)(ยฑ)๐‘—โˆ’1๐œ•๐œ‰41๎€ท๎€ท๐œ‰=๐œŒโ„Ž1โˆ’๐œ’1+๎€ธ๎€ธ๐‘—๎“๐‘™=0๐œ‰๐‘™1๐œ•๐‘™!๐‘™+2๐‘‘1(ยฑ)๐‘—โˆ’๐›พโˆ’4โˆ’๐‘™๐œ•๐‘ก2๐œ•๐‘ฅ๐‘™1+(0,๐‘ก)๐‘—๎“๐‘™=2๐œ•4๐œ‰41๎ƒฉ๎€ท๎€ท๐œ‰1โˆ’๐œ’1๐œ‰๎€ธ๎€ธ๐‘™1๎ƒช๐œ•๐‘™!๐‘™๐‘‘1(ยฑ)๐‘—โˆ’๐‘™๐œ•๐‘ฅ๐‘™1(0,๐‘ก)+๐œ‡๐‘—๎“๐‘™=0๐œ•4๐œ‰41๎ƒฉ๎€ท๎€ท๐œ‰1โˆ’๐œ’1๐œ‰๎€ธ๎€ธ๐‘™1๎ƒช๐œ•๐‘™!๐‘™+1๐‘‘1(ยฑ)๐‘—โˆ’๐›พโˆ’๐‘™๐œ•๐‘ก๐œ•๐‘ฅ๐‘™1โˆ’๎€ท๎€ท๐œ‰(0,๐‘ก)1โˆ’๐œ’1๎€ธ๎€ธ๐‘—๎“๐‘™=0๐œ‰๐‘™1๐œ•๐‘™!๐‘™๐‘1๐‘—โˆ’5โˆ’๐‘™๐œ•๐‘ฅ๐‘™1โˆ’๎€ท๎€ท๐œ‰(0,ยฑ1,๐‘ก)1โˆ’๐œ’1๎€ธ๎€ธ๐‘—๎“๐‘™=0๐œ‰๐‘™1๐œ•๐‘™!๐‘™๐‘ž1๐‘—โˆ’4โˆ’๐‘™๐œ•๐‘ฅ๐‘™1๐œ•(0,๐‘ก),๐‘Ž๎๐‘‘(0)(ยฑ)๐‘—โˆ’1๐œ•๐œ‰๐‘Ž1โŸถ0,uniformly,when๐œ‰1โŸถโˆ’โˆž,๐‘Žโˆˆ{0,1,2,3},(4.28) and by ๎‚๐‘‘(0)(ยฑ)๐‘—โˆ’1 the unique solution of๐œ•4๎‚๐‘‘(0)(ยฑ)๐‘—โˆ’1๐œ•๐œ‰41=๐œ•4๎๐‘‘(0)(ยฑ)๐‘—โˆ’1๐œ•๐œ‰41๐œ•โˆ’๐œŒโ„Ž2๐‘‘(0)(ยฑ)๐‘—โˆ’๐›พโˆ’4๐œ•๐‘ก2๐œ•โˆ’๐œ‡5๐‘‘(0)(ยฑ)๐‘—โˆ’๐›พ๐œ•๐œ‰41๐œ•๐‘ก+๐‘(0)๐‘—โˆ’52/๐œ‰=ยฑ1,๐œ•๐‘Ž๎‚๐‘‘(0)(ยฑ)๐‘—โˆ’1๐œ•๐œ‰๐‘Ž1โŸถ0,uniformly,when๐œ‰1โŸถโˆ’โˆž,๐‘Žโˆˆ{0,1,2,3}.(4.29) Introducing the asymptotic solution into (2.6)3 and identifying the coefficients of the power of ๐œ€ we obtain:๐‘‘(0)(ยฑ)๐‘—๎€ท๐œ‰1๎€ธ=๎‚๐‘‘,ยฑ1,๐‘ก(0)(ยฑ)๐‘—โˆ’1๎€ท๐œ‰1๎€ธโˆ’๎€ท๎€ท๐œ‰,ยฑ1,๐‘ก1โˆ’๐œ’1ร—โŽ›โŽœโŽœโŽ๎‚๐‘‘๎€ธ๎€ธ(0)(ยฑ)๐‘—โˆ’1(๎€ท2,ยฑ1,๐‘ก)โˆ’2โˆ’๐œ‰1๎€ธ๐œ•๎‚๐‘‘(0)(ยฑ)๐‘—โˆ’1๐œ•๐œ‰1(โŽžโŽŸโŽŸโŽ ๐œ‰2,ยฑ1,๐‘ก)1๐‘‘>2,(4.30)1(ยฑ)๐‘—๐œ•๎‚๐‘‘(0,๐‘ก)=2(0)(ยฑ)๐‘—โˆ’1๐œ•๐œ‰1๎‚๐‘‘(2,ยฑ1,๐‘ก)โˆ’(0)(ยฑ)๐‘—โˆ’1(2,ยฑ1,๐‘ก),๐œ•๐‘‘1(ยฑ)๐‘—๐œ•๐‘ฅ1๐œ•๎‚๐‘‘(0,๐‘ก)=โˆ’(0)(ยฑ)๐‘—๐œ•๐œ‰1(2,ยฑ1,๐‘ก).(4.31) The relation (4.30) gives the corrector in ๐‘ฅ=๐ŸŽ for the displacements corresponding to one branch of the flow domain.

We notice that the right-hand side of (4.31) is known, since the problems (4.28) and (4.29) can be solved also if we replace ๐‘—โˆ’1 by ๐‘—.

We are now in a position to return to the regular part of the asymptotic solution. For each regular part of the asymptotic solution, we need 9 relations, in order to determine the 9 unknown functions of integration. For the regular part corresponding to ๐ท1๐œ€, (4.11) and (4.31) give 8 functions: ๐‘‘1(ยฑ)๐‘—(0,๐‘ก), (๐œ•๐‘‘1(ยฑ)๐‘—/๐œ•๐‘ฅ1)(0,๐‘ก), ๐‘‘1(ยฑ)๐‘—(1,๐‘ก), (๐œ•๐‘‘1(ยฑ)๐‘—/๐œ•๐‘ฅ1)(1,๐‘ก). It remains undetermined ๐‘ž1๐‘—(0,๐‘ก). We show next that at the jth step this function is already determined at the previous step. For this purpose, we return to the problem (4.23). As usual, we should construct boundary layer functions stabilizing to zero at infinity. As we imposed this condition for the velocity in (4.23)6,7, it is known that the pressure stabilizes to some functions of ๐‘ก. At the jth step, we determine ๐‘ž1๐‘—+1(0,๐‘ก) and ๐‘ž2๐‘—+1(0,๐‘ก) from the condition of the exponential decay of ๐‘๐‘—(0) at infinity. To this end, we consider instead of (4.23) and (4.24), the following problem with known right-hand side for ๐‘—โ‰ฅโˆ’1:โˆ’๐œˆ0ฮ”๐œ‰๐œ‰๐ฎ๐‘—(0)+โˆ‡๐œ‰๐‘๐‘—(0)=โˆ’๐œŒ๐‘“๐œ•๐ฎ(0)๐‘—โˆ’2๐œ•๐‘ก+๐…๐‘—inโ„ฌ,div๐ฎ๐‘—(0)=ฬƒ๐œƒ๐‘—๐ฎinโ„ฌ,๐‘—(0)=๐ŸŽonโŒข๐ด+๐ต+โˆชโŒข๐ดโˆ’๐ตโˆ’,๐ฎ๐‘—(0)๎€ท๐œ‰1๎€ธ=,ยฑ1,๐‘ก๐œ•๐‘‘(0)(ยฑ)๐‘—+2โˆ’๐›พ๎€ท๐œ‰๐œ•๐‘ก1๎€ธ๐ž,ยฑ1,๐‘ก2for๐œ‰1๐ฎโ‰ฅ2,๐‘—(0)๎€ทยฑ1,๐œ‰2๎€ธ=,๐‘ก๐œ•๐‘‘(0)(ยฑ)๐‘—+2โˆ’๐›พ๎€ท๐œ•๐‘กยฑ1,๐œ‰2๎€ธ๐ž,๐‘ก1for๐œ‰2๐ฎโ‰ฅ2,๐‘—(0)โŸถ๐ŸŽ,๐‘๐‘—(0)โŸถ๐›ผ1๐‘—(๐‘ก)uniformlywhen๐œ‰1๐ฎโŸถโˆž,๐‘—(0)โŸถ๐ŸŽ,๐‘๐‘—(0)โŸถ๐›ผ2๐‘—(๐‘ก)uniformlywhen๐œ‰2โŸถโˆž(4.32) with ๐›ผ1๐‘—(๐‘ก),๐›ผ2๐‘—(๐‘ก) unknown. Supposing for the time being that the function ฬƒ๐œƒ๐‘— satisfies the compatibility condition, it follows that (4.32) has a unique solution (the uniqueness for the pressure being understood up to an additive function of ๐‘ก). This means that the function ๐›ผ๐‘—(๐‘ก), with ๐›ผ๐‘—(๐‘ก)=๐›ผ1๐‘—(๐‘ก)โˆ’๐›ผ2j(๐‘ก) is fixed. We define for ๐‘—โ‰ฅโˆ’1๐‘๐‘—(0)=๐‘๐‘—(0)๎€ท๐œ‰+๐œ’1๎€ธ๐‘ž1๐‘—+1๎€ท๐œ‰(0,๐‘ก)+๐œ’2๎€ธ๐‘ž2๐‘—+1(0,๐‘ก).(4.33) Standard computations show that (๐ฎ๐‘—(0),๐‘๐‘—(0)) is solution for (4.32) iff (๐ฎ๐‘—(0),๐‘๐‘—(0)) satisfies (4.23) or (4.24) for ๐‘—=โˆ’1 with ๐‘๐‘—(0)โ†’๐›ผ๐‘–๐‘—(๐‘ก)+๐‘ž๐‘–๐‘—+1(0,๐‘ก) uniformly when ๐œ‰๐‘–โ†’โˆž,๐‘–=1,2. To obtain the desired behavior for ๐‘๐‘—(0) when ๐œ‰๐‘–โ†’โˆž,๐‘–=1,2, it suffices to take ๐›ผ๐‘–๐‘—(๐‘ก)=โˆ’๐‘ž๐‘–๐‘—+1(0,๐‘ก).

A first equation for the unknowns ๐‘ž1๐‘—+1(0,๐‘ก) and ๐‘ž2๐‘—+1(0,๐‘ก) is๐‘ž1๐‘—+1(0,๐‘ก)โˆ’๐‘ž2๐‘—+1(0,๐‘ก)=๐›ผ๐‘—(๐‘ก),(4.34) with ๐›ผโˆ’1(๐‘ก)=0. The other equation is obtained introducing the asymptotic solution into the compatibility condition (2.11):๎€œ10๎‚€๐‘‘1(+)๐‘—+1๎€ท๐‘ฅ1๎€ธ,๐‘กโˆ’๐‘‘1(โˆ’)๐‘—+1๎€ท๐‘ฅ1๎€ธ๎‚,๐‘กd๐‘ฅ1+๎€œ10๎‚€๐‘‘2(+)๐‘—+1๎€ท๐‘ฅ2๎€ธ,๐‘กโˆ’๐‘‘2(โˆ’)๐‘—+1๎€ท๐‘ฅ2๎€ธ๎‚,๐‘กd๐‘ฅ2=โˆ’๐‘—๎“๐‘™=0โŽ›โŽœโŽœโŽ๐œ•๐‘™๎‚€๐‘‘1(+)๐‘—โˆ’๐‘™โˆ’๐‘‘1(โˆ’)๐‘—โˆ’๐‘™๎‚๐œ•๐‘ฅ๐‘™1๐œ•(0,๐‘ก)+๐‘™๎‚€๐‘‘2(+)๐‘—โˆ’๐‘™โˆ’๐‘‘2(โˆ’)๐‘—โˆ’๐‘™๎‚๐œ•๐‘ฅ๐‘™2โŽžโŽŸโŽŸโŽ ๎€œ(0,๐‘ก)30๐œ๐‘™โˆ’๎€œ๐‘™!(๐œ’(๐œ)โˆ’1)d๐œ0โˆ’โˆž๎‚€๐‘‘(1)(+)๐‘—๎€ท๐œ‰1๎€ธ,1,๐‘กโˆ’๐‘‘(1)(+)๐‘—๎€ท๐œ‰1๎€ธ๎‚,โˆ’1,๐‘กd๐œ‰1โˆ’๎€œ0โˆ’โˆž๎‚€๐‘‘(2)(+)๐‘—๎€ท1,๐œ‰2๎€ธ,๐‘กโˆ’๐‘‘(2)(+)๐‘—๎€ทโˆ’1,๐œ‰2๎€ธ๎‚,๐‘กd๐œ‰2โˆ’๎€œโˆž2๎‚€๐‘‘(0)(+)๐‘—๎€ท๐œ‰1๎€ธ,1,๐‘กโˆ’๐‘‘(0)(+)๐‘—๎€ท๐œ‰1๎€ธ๎‚,โˆ’1,๐‘กd๐œ‰1โˆ’๎€œโˆž2๎‚€๐‘‘(0)(+)๐‘—๎€ท1,๐œ‰2๎€ธ,๐‘กโˆ’๐‘‘(0)(+)๐‘—๎€ทโˆ’1,๐œ‰2๎€ธ๎‚,๐‘กd๐œ‰2.(4.35) The right-hand side of (4.35) is known from the jth approximation. Replacing in the left hand side of (4.35) ๐‘‘1(ยฑ)๐‘—+1 given by (4.22) (for ๐‘— replaced by ๐‘—+1) and ๐‘‘2(ยฑ)๐‘—+1 given by a similar relation, we are leaded to the second equation for the unknowns ๐‘ž1๐‘—+1(0,๐‘ก) and ๐‘ž2๐‘—+1(0,๐‘ก):๐‘ž1๐‘—+1(0,๐‘ก)+๐‘ž2๐‘—+1(0,๐‘ก)=๐›ฝ๐‘—(๐‘ก),(4.36) with ๐›ฝ๐‘— a known function for ๐‘—โ‰ฅโˆ’1, with๐›ฝโˆ’1๎‚ต๎€œ(๐‘ก)=โˆ’6!10๐ผ๐‘ฅโˆ’11(๐‘“)d๐‘ฅ1+๎€œ10๐ผ๐‘ฅโˆ’12(๐‘“)d๐‘ฅ2๎‚ถ+3ร—6!2๎‚ต๎€œ1โˆ’1๐œ“๎€ท๐œ‰2๎€ธ,๐‘กd๐œ‰2๎€œ10๐ผ๐‘ฅโˆ’11(๐œˆ)d๐‘ฅ1+๎€œ1โˆ’1๐œ“๎€ท๐œ‰1๎€ธ,๐‘กd๐œ‰1๎€œ10๐ผ๐‘ฅโˆ’12(๐œˆ)d๐‘ฅ2๎‚ถโˆ’6!2๎‚ต๎€œ10๐ผ๐‘ฅโˆ’41๎€ท๐‘”+โˆ’๐‘”โˆ’๎€ธd๐‘ฅ1+๎€œ10๐ผ๐‘ฅโˆ’42๎€ท๐‘”+โˆ’๐‘”โˆ’๎€ธd๐‘ฅ2๎‚ถ+6!4๎‚ต๎€œ10๐ผ๐‘ฅโˆ’31๎€ท๐‘”+โˆ’๐‘”โˆ’๎€ธd๐‘ฅ1+๎€œ10๐ผ๐‘ฅโˆ’32๎€ท๐‘”+โˆ’๐‘”โˆ’๎€ธd๐‘ฅ2๎‚ถโˆ’6!๎‚ต๎€œ4!10๐ผ๐‘ฅโˆ’21๎€ท๐‘”+โˆ’๐‘”โˆ’๎€ธd๐‘ฅ1+๎€œ10๐ผ๐‘ฅโˆ’22๎€ท๐‘”+โˆ’๐‘”โˆ’๎€ธd๐‘ฅ2๎‚ถ.(4.37) From (4.34) and (4.36) we determine ๐‘ž1๐‘—+1(0,๐‘ก) and ๐‘ž2๐‘—+1(0,๐‘ก), which means that the right-hand side of (4.23) is now known, that allows to solve (4.23), if we show that the problem (4.23) is well posed, that is, the compatibility condition is fulfilled. Moreover, this justifies the previous assertion that ๐‘ž๐‘–๐‘—(0,๐‘ก),๐‘–=1,2 are computed at the (๐‘—โˆ’1)th step. We can say that the entire ๐‘—th approximation was obtained for ๐›พ>3 if we prove.

Proposition 4.3. For any ๐‘—โˆˆโ„•, the function ฬƒ๐œƒ๐‘— satisfies the compatibility condition: ๎€œโ„ฌฬƒ๐œƒ๐‘—๎€ท๐œ‰1,๐œ‰2๎€ธ๎€œ,๐‘กd๐œ‰=๐œ•โ„ฌ๐ฎ๐‘—(0)โ‹…๐งd๐‘ .(4.38)

Proof. From (4.23) and (4.35) it follows: ๎€œ๐œ•โ„ฌ๐ฎ๐‘—(0)dโ‹…๐งd๐‘ =โˆ’๎‚ต๎€œd๐‘ก0โˆ’โˆž๎‚€๐‘‘(1)(+)๐‘—+2โˆ’๐›พ๎€ท๐œ‰1๎€ธ,๐‘กโˆ’๐‘‘(1)(โˆ’)๐‘—+2โˆ’๐›พ๎€ท๐œ‰1๎€ธ๎‚,๐‘กd๐œ‰1โˆ’๎€œ0โˆ’โˆž๎‚€๐‘‘(2)(+)๐‘—+2โˆ’๐›พ๎€ท๐œ‰2๎€ธ,๐‘กโˆ’๐‘‘(2)(โˆ’)๐‘—+2โˆ’๐›พ๎€ท๐œ‰2๎€ธ๎‚,๐‘กd๐œ‰2๎‚ถโˆ’dd๐‘ก๐‘—+2โˆ’๐›พ๎“๐‘™=0โŽ›โŽœโŽœโŽ๐œ•๐‘™๎‚€๐‘‘1(+)๐‘—+2โˆ’๐›พโˆ’๐‘™โˆ’๐‘‘1(โˆ’)๐‘—+2โˆ’๐›พโˆ’๐‘™๎‚๐œ•๐‘ฅ๐‘™1(+๐œ•0,๐‘ก)๐‘™๎‚€๐‘‘2(+)๐‘—+2โˆ’๐›พโˆ’๐‘™โˆ’๐‘‘2(โˆ’)๐‘—+2โˆ’๐›พโˆ’๐‘™๎‚๐œ•๐‘ฅ๐‘™2โŽžโŽŸโŽŸโŽ ๎€œ(0,๐‘ก)30๐‘ ๐‘™โˆ’d๐‘™!(๐œ’(๐‘ )โˆ’1)d๐‘ ๎‚ต๎€œd๐‘ก10๎‚€๐‘‘1(+)๐‘—+3โˆ’๐›พ๎€ท๐‘ฅ1๎€ธ,๐‘กโˆ’๐‘‘1(โˆ’)๐‘—+3โˆ’๐›พ๎€ท๐‘ฅ1๎€ธ๎‚,๐‘กd๐‘ฅ1+๎€œ10๎‚€๐‘‘2(+)๐‘—+3โˆ’๐›พ๎€ท๐‘ฅ2๎€ธ,๐‘กโˆ’๐‘‘2(โˆ’)๐‘—+3โˆ’๐›พ๎€ท๐‘ฅ2๎€ธ๎‚,๐‘กd๐‘ฅ2๎‚ถ.(4.39) We express next the three terms of the right-hand side of the previous equality with respect to the components of the velocity corresponding to the regular parts of the asymptotic solution.
From (4.9), the corresponding relation for ๐‘ฅ2=1 and (2.9)6 the first term of the right-hand side is equal to๐‘‡1๎€œ=โˆ’1โˆ’1๐‘ข11,๐‘—๎€ท1,๐œ‰2๎€ธ,๐‘กd๐œ‰2โˆ’๎€œ1โˆ’1๐‘ข22,๐‘—๎€ท๐œ‰1๎€ธ,1,๐‘กd๐œ‰1.(4.40) The third term is expressed by means of (4.12)6 and its analogous for ๐ฎ2๐‘— and it is equal to ๐‘‡3๎€œ=โˆ’10๎‚€๐‘ข12,๐‘—๎€ท๐‘ฅ1๎€ธ,1,๐‘กโˆ’๐‘ข12,๐‘—๎€ท๐‘ฅ1๎€ธ๎‚,โˆ’1,๐‘กd๐‘ฅ1โˆ’๎€œ10๎‚€๐‘ข21,๐‘—๎€ท1,๐‘ฅ2๎€ธ,๐‘กโˆ’๐‘ข21,๐‘—๎€ทโˆ’1,๐‘ฅ2๎€ธ๎‚,๐‘กd๐‘ฅ2.(4.41) Computing now ๐‘‡1+๐‘‡3 and using (4.12)3 and the similar one for ๐ฎ2๐‘— we get ๐‘‡1+๐‘‡3๎€œ=โˆ’1โˆ’1๐‘ข11,๐‘—๎€ท0,๐œ‰2๎€ธ,๐‘กd๐œ‰2โˆ’๎€œ1โˆ’1๐‘ข22,๐‘—๎€ท๐œ‰1๎€ธ,0,๐‘กd๐œ‰1.(4.42) The second term is given by ๐‘‡2=๐‘—๎“๐‘™=0โŽ›โŽœโŽœโŽ๐œ•๐‘™๎‚€๐‘ข12,๐‘—โˆ’1โˆ’๐‘™(0,1,๐‘ก)โˆ’๐‘ข12,๐‘—โˆ’1โˆ’๐‘™๎‚(0,โˆ’1,๐‘ก)๐œ•๐‘ฅ๐‘™1+๐œ•๐‘™๎‚€๐‘ข21,๐‘—โˆ’1โˆ’๐‘™(1,0,๐‘ก)โˆ’๐‘ข21,๐‘—โˆ’1โˆ’๐‘™๎‚(โˆ’1,0,๐‘ก)๐œ•๐‘ฅ๐‘™2โŽžโŽŸโŽŸโŽ ๎€œ30s๐‘™(=๐‘™!๐œ’(๐‘ )โˆ’1)d๐‘ ๐‘—๎“๐‘™=0๎€œ(0,3)ร—(โˆ’1,1)div๐œ‰๎ƒฉ๎€ท๎€ท๐œ‰1โˆ’๐œ’1๐œ‰๎€ธ๎€ธ๐‘™1๎ƒฉ๐œ•๐‘™!๐‘™๐‘ข11,๐‘—โˆ’๐‘™๎€ท0,๐œ‰2๎€ธ,๐‘ก๐œ•๐‘ฅ๐‘™1๐ž1โˆ’๐œ•๐‘™๐‘ข12,๐‘—โˆ’1โˆ’๐‘™๎€ท0,๐œ‰2๎€ธ,๐‘ก๐œ•๐‘ฅ๐‘™1๐ž2+๎ƒช๎ƒชd๐œ‰๐‘—๎“๐‘™=0๎€œ(โˆ’1,1)ร—(0,3)div๐œ‰๎ƒฉ๎€ท๎€ท๐œ‰1โˆ’๐œ’2๐œ‰๎€ธ๎€ธ๐‘™2๎ƒฉ๐œ•๐‘™!๐‘™๐‘ข21,๐‘—โˆ’1โˆ’๐‘™๎€ท๐œ‰1๎€ธ,0,๐‘ก๐œ•๐‘ฅ๐‘™2๐ž1โˆ’๐œ•๐‘™๐‘ข22,๐‘—โˆ’๐‘™๎€ท๐œ‰2๎€ธ,0,๐‘ก๐œ•๐‘ฅ๐‘™2๐ž2+๎€œ๎ƒช๎ƒชd๐œ‰1โˆ’1๐‘ข11,๐‘—๎€ท0,๐œ‰2๎€ธ,๐‘กd๐œ‰2+๎€œ1โˆ’1๐‘ข22,๐‘—๎€ท๐œ‰1๎€ธ,0,๐‘กd๐œ‰2.(4.43) Finally, computing ๐‘‡1+๐‘‡2+๐‘‡3, we obtain the following expression of the right-hand side of (4.38): ๎€œ๐œ•โ„ฌ๐ฎ๐‘—(0)=โ‹…๐งd๐‘ ๐‘—๎“๐‘™=0๎€œ(0,3)ร—(โˆ’1,1)div๐œ‰๎ƒฉ๎€ท๎€ท๐œ‰1โˆ’๐œ’1๐œ‰๎€ธ๎€ธ๐‘™1๎ƒฉ๐œ•๐‘™!๐‘™๐‘ข11,๐‘—โˆ’๐‘™๎€ท0,๐œ‰2๎€ธ,๐‘ก๐œ•๐‘ฅ๐‘™1๐ž1โˆ’๐œ•๐‘™๐‘ข12,๐‘—โˆ’1โˆ’๐‘™๎€ท0,๐œ‰2๎€ธ,๐‘ก๐œ•๐‘ฅ๐‘™1๐ž2+๎ƒช๎ƒชd๐œ‰๐‘—๎“๐‘™=0๎€œ(โˆ’1,1)ร—(0,3)div๐œ‰๎ƒฉ๎€ท๎€ท๐œ‰1โˆ’๐œ’2๐œ‰๎€ธ๎€ธ๐‘™2๎ƒฉ๐œ•๐‘™!๐‘™๐‘ข21,๐‘—โˆ’1โˆ’๐‘™๎€ท๐œ‰1๎€ธ,0,๐‘ก๐œ•๐‘ฅ๐‘™2๐ž1โˆ’๐œ•๐‘™๐‘ข22,๐‘—โˆ’๐‘™๎€ท๐œ‰1๎€ธ,0,๐‘ก๐œ•๐‘ฅ๐‘™2๐ž2๎ƒช๎ƒชd๐œ‰.(4.44) We compute next the left-hand side of (4.38). Since ๐œ’(๐œ‰1)=๐œ’(๐œ‰2)=0 on โ„ฌโˆฉ{๐œ‰1<2,๐œ‰2<2}, we can write: ๎€œโ„ฌฬƒ๐œƒ๐‘—๎€ท๐œ‰1,๐œ‰2๎€ธ๎€œ,๐‘กd๐œ‰=(2,3)ร—(โˆ’1,1)ฬƒ๐œƒ๐‘—๎€ท๐œ‰1,๐œ‰2๎€ธ๎€œ,๐‘กd๐œ‰+(โˆ’1,1)ร—(2,3)ฬƒ๐œƒ๐‘—๎€ท๐œ‰1,๐œ‰2๎€ธ,๐‘กd๐œ‰.(4.45) We express the first term of the right-hand side of the previous relation taking into account that for (๐œ‰1,๐œ‰2)โˆˆ(2,3)ร—(โˆ’1,1), we have ๐œ’(๐œ‰2)=0: ๎€œ(2,3)ร—(โˆ’1,1)ฬƒ๐œƒ๐‘—๎€ท๐œ‰1,๐œ‰2๎€ธ=๎€œ,๐‘กd๐œ‰(0,3)ร—(โˆ’1,1)div๐œ‰๎ƒฉ๎€ท๎€ท๐œ‰1โˆ’๐œ’1๎€ธ๎€ธ๐‘—๎“๐‘™=0๐œ‰๐‘™1๎ƒฉ๐œ•๐‘™!๐‘™๐‘ข11,๐‘—โˆ’๐‘™๐œ•๐‘ฅ1๐‘™๎€ท0,๐œ‰2๎€ธ๐ž,๐‘ก1+๐œ•๐‘™๐‘ข12,๐‘—โˆ’1โˆ’๐‘™๐œ•๐‘ฅ1๐‘™๎€ท0,๐œ‰2๎€ธ๐ž,๐‘ก2๎ƒช๎ƒชd๐œ‰.(4.46) For obtaining the above equality we also used the property: div๐œ‰๎ƒฉ๐‘—๎“๐‘™=0๐œ‰๐‘™1๎ƒฉ๐œ•๐‘™!๐‘™๐‘ข11,๐‘—โˆ’๐‘™๐œ•๐‘ฅ1๐‘™๎€ท0,๐œ‰2๎€ธ๐ž,๐‘ก1+๐œ•๐‘™๐‘ข12,๐‘—โˆ’1โˆ’๐‘™๐œ•๐‘ฅ1๐‘™๎€ท0,๐œ‰2๎€ธ๐ž,๐‘ก2๎ƒช๎ƒช=0.(4.47) A similar result is obtained for the second term of the right-hand side and, comparing the expression of โˆซโ„ฌฬƒ๐œƒ๐‘—(๐œ‰1,๐œ‰2,๐‘ก)d๐œ‰ with the right-hand side of (4.44), the proof is complete.

The last result of this subsection gives the leading term of the regular part of the asymptotic solution corresponding to ๐ท1๐œ€.

Proposition 4.4. For ๐‘—=0 the regular part of the asymptotic solution corresponding to ๐ท1๐œ€ is given by ๐‘ข11,0๎€ท๐‘ฅ1,๐œ‰2๎€ธ3,๐‘ก=โˆ’2๎‚ต๎€œ1โˆ’1๐œ“๎€ท๐œ‰2๎€ธ,๐‘กd๐œ‰2๎‚ถ๐‘1๎€ท๐œ‰2๎€ธ,๐‘ข12,0๎€ท๐‘ฅ1,๐œ‰2๎€ธ๐‘,๐‘ก=0,10๎€ท๐‘ฅ1,๐œ‰2๎€ธ๐‘ž,๐‘ก=0,10๎€ท๐‘ฅ1๎€ธ=๎€œ,๐‘ก๐‘ฅ103๐‘“(๐‘ ,๐‘ก)d๐‘ โˆ’2๎‚ต๎€œ1โˆ’1๐œ“๎€ท๐œ‰2๎€ธ,๐‘กd๐œ‰2๎‚ถ๎€œ๐‘ฅ101๐œˆ(๐‘ )d๐‘ +2๐›ฝโˆ’1(๐‘ก),(4.48) with ๐›ฝโˆ’1(๐‘ก) defined by (4.37) and ๐œ•4๐‘‘1(ยฑ)0๐œ•๐‘ฅ41=๐‘”ยฑยฑ๐‘ž10,๐‘‘1(ยฑ)0(0,๐‘ก)=๐‘‘1(ยฑ)0(1,๐‘ก)=๐œ•๐‘‘1(ยฑ)0๐œ•๐‘ฅ1(0,๐‘ก)=๐œ•๐‘‘1(ยฑ)0๐œ•๐‘ฅ1(1,๐‘ก)=0.(4.49)

4.2.2. The Solution of the Problems for ๐›พ=3

Unlike in the previous case, for ๐›พ=3 we cannot solve the problems corresponding to ๐ท1๐œ€ and to ๐ท2๐œ€ separately. We begin with the problems corresponding to the functions with the index 1. The relations (4.15)โ€“(4.17) are still true. Hence, ๐‘1๐‘— is also given by (4.17). The difference with respect to the previous case is that now we cannot express the other unknowns only in function of ๐‘ž1๐‘—, since in the right-hand side of (4.16) ๐‘‘1(โˆ’)๐‘— is also unknown.

We introduce the notations:๐‘‘1๐‘—๎€ท๐‘ฅ1๎€ธ,๐‘ก=๐‘‘1(+)๐‘—๎€ท๐‘ฅ1๎€ธ,๐‘ก+๐‘‘1(โˆ’)๐‘—๎€ท๐‘ฅ1๎€ธ,๐’Ÿ,๐‘ก1๐‘—โˆ’1=๐’Ÿ1(+)๐‘—โˆ’1+๐’Ÿ1(โˆ’)๐‘—โˆ’1,(4.50) with ๐’Ÿ1(ยฑ)๐‘—โˆ’1 given by (4.14)2. The function ๐‘‘1๐‘— is obtained as the unique solution of a fourth-order differential problem:๐œ•4๐‘‘1๐‘—๐œ•๐‘ฅ41=๎€ท๐‘”++๐‘”โˆ’๎€ธ๐›ฟ๐‘—0+๐’Ÿ1๐‘—โˆ’1,๐‘‘1๐‘—๎‚€๐‘‘(1,๐‘ก)=โˆ’(1)(+)๐‘—(0,๐‘ก)+๐‘‘(1)(โˆ’)๐‘—๎‚,(0,๐‘ก)๐œ•๐‘‘1๐‘—๐œ•๐‘ฅ1๎ƒฉ(1,๐‘ก)=โˆ’๐œ•๐‘‘(1)(+)๐‘—+1๐œ•๐œ‰1(0,๐‘ก)+๐œ•๐‘‘(1)(โˆ’)๐‘—+1๐œ•๐œ‰1๎ƒช,๐‘‘(0,๐‘ก)1๐‘—โŽ›โŽœโŽœโŽ๐œ•๎‚๐‘‘(0,๐‘ก)=2(0)(+)๐‘—โˆ’1๐œ•๐œ‰1๐œ•๎‚๐‘‘(2,1,๐‘ก)+(0)(โˆ’)๐‘—โˆ’1๐œ•๐œ‰1โŽžโŽŸโŽŸโŽ โˆ’๎‚€๎‚๐‘‘(2,โˆ’1,๐‘ก)(0)(+)๐‘—โˆ’1๎‚๐‘‘(2,1,๐‘ก)+(0)(โˆ’)๐‘—โˆ’1๎‚,(2,โˆ’1,๐‘ก)๐œ•๐‘‘1๐‘—๐œ•๐‘ฅ1โŽ›โŽœโŽœโŽ๐œ•๎‚๐‘‘(0,๐‘ก)=โˆ’(0)(+)๐‘—๐œ•๐œ‰1๐œ•๎‚๐‘‘(2,1,๐‘ก)+(0)(โˆ’)๐‘—๐œ•๐œ‰1โŽžโŽŸโŽŸโŽ ,(2,โˆ’1,๐‘ก)(4.51) the boundary conditions being obtained by means of (4.11) and (4.31). This means that ๐‘‘1(โˆ’)๐‘— can also be expressed via ๐‘‘1(+)๐‘—. We show that all the unknowns of the regular part of the asymptotic solution can be expressed in function of ๐‘‘1(+)๐‘—; hence, the regular part of the asymptotic solution corresponding to ๐ท1๐œ€ is determined if we obtain and solve a problem for ๐‘‘1(+)๐‘—. As we cannot obtain all the necessary conditions for ๐‘‘1(+)๐‘—, we have to consider the problem for the couple of unknowns, ๐‘‘1(+)๐‘—, ๐‘‘2(+)๐‘—.

Theorem 4.5. The approximations of the displacement, ๐‘‘1(+)๐‘—, ๐‘‘2(+)๐‘—, are obtained as the unique solution of the following system of two sixth-order parabolic equations: ๐œ•๐‘‘1(+)๐‘—๎€ท๐‘ฅ๐œ•๐‘ก1๎€ธโˆ’1,๐‘ก3๐œ•๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๐œ•5๐‘‘1(+)๐‘—๐œ•๐‘ฅ51๎€ท๐‘ฅ1๎€ธ๎ƒช,๐‘ก=๐บ1๐‘—๎€ท๐‘ฅ1๎€ธโˆ’,๐‘ก๐œ•๐ป1๐œ•๐‘ฅ1๎€ท๐‘ฅ1๎€ธ๐›ฟ,๐‘ก๐‘—0,๐œ•๐‘‘2(+)๐‘—๎€ท๐‘ฅ๐œ•๐‘ก2๎€ธโˆ’1,๐‘ก3๐œ•๐œ•๐‘ฅ2๎ƒฉ1๐œˆ๎€ท๐‘ฅ2๎€ธ๐œ•5๐‘‘2(+)๐‘—๐œ•๐‘ฅ52๎€ท๐‘ฅ2๎€ธ๎ƒช,๐‘ก=๐บ2๐‘—๎€ท๐‘ฅ2๎€ธโˆ’,๐‘ก๐œ•๐ป2๐œ•๐‘ฅ2๎€ท๐‘ฅ2๎€ธ๐›ฟ,๐‘ก๐‘—0,๐‘‘1(+)๐‘—(1,๐‘ก)=โˆ’๐‘‘(1)(+)๐‘—(0,๐‘ก),๐œ•๐‘‘1(+)๐‘—๐œ•๐‘ฅ1(1,๐‘ก)=โˆ’๐œ•๐‘‘(1)(+)๐‘—+1๐œ•๐œ‰1(๐‘‘0,๐‘ก),1(+)๐‘—๐œ•๎‚๐‘‘(0,๐‘ก)=2(0)(+)๐‘—โˆ’1๐œ•๐œ‰1๎‚๐‘‘(2,1,๐‘ก)โˆ’(0)(+)๐‘—(2,1,๐‘ก),๐œ•๐‘‘1(+)๐‘—๐œ•๐‘ฅ1๐œ•๎‚๐‘‘(0,๐‘ก)=โˆ’(0)(+)๐‘—๐œ•๐œ‰1๐‘‘(2,1,๐‘ก),2(+)๐‘—(1,๐‘ก)=โˆ’๐‘‘(2)(+)๐‘—(0,๐‘ก),๐œ•๐‘‘2(+)๐‘—๐œ•๐‘ฅ2(1,๐‘ก)=โˆ’๐œ•๐‘‘(2)(+)๐‘—+1๐œ•๐œ‰2๐‘‘(0,๐‘ก),2(+)๐‘—๐œ•๎‚๐‘‘(0,๐‘ก)=2(0)(+)๐‘—โˆ’1๐œ•๐œ‰2๎‚๐‘‘(1,2,๐‘ก)โˆ’(0)(+)๐‘—(1,2,๐‘ก),๐œ•๐‘‘2(+)๐‘—๐œ•๐‘ฅ2๐œ•๎‚๐‘‘(0,๐‘ก)=โˆ’(0)(+)๐‘—๐œ•๐œ‰2๐œ•(1,2,๐‘ก),5๐‘‘1(+)๐‘—๐œ•๐‘ฅ51(1,๐‘ก)=๐ฟ1๐‘—๐œ•(๐‘ก),5๐‘‘2(+)๐‘—๐œ•๐‘ฅ52(1,๐‘ก)=๐ฟ2๐‘—๐œ•(๐‘ก),4๐‘‘1(+)๐‘—๐œ•๐‘ฅ41๐œ•(0,๐‘ก)โˆ’4๐‘‘2(+)๐‘—๐œ•๐‘ฅ42(0,๐‘ก)=๐‘€๐‘—๐œ•(๐‘ก),5๐‘‘1(+)๐‘—๐œ•๐‘ฅ51๐œ•(0,๐‘ก)+5๐‘‘2(+)๐‘—๐œ•๐‘ฅ52(0,๐‘ก)=๐‘๐‘—๐‘‘(๐‘ก),1(+)๐‘—๎€ท๐‘ฅ1๎€ธ,0=๐‘‘2(+)๐‘—๎€ท๐‘ฅ1๎€ธ,0=0,(4.52) where ๐ป๐‘–, ๐บ๐‘–๐‘—, ๐ฟ๐‘–๐‘—(๐‘ก), ๐‘€๐‘—(๐‘ก), and ๐‘๐‘—(๐‘ก) are known functions, given below.

Proof. The first relation between the unknowns ๐‘‘1(+)๐‘— and ๐‘ž1๐‘— is given by (4.12)4, that is, ๐œ•4๐‘‘1(+)๐‘—๐œ•๐‘ฅ41โˆ’๐‘ž1๐‘—=๐‘”+๐›ฟ๐‘—0+๐’Ÿ1(+)๐‘—โˆ’1.(4.53) The second one is obtained from (4.16) for ๐›พ=3 and ๐œ‰2=1, (4.12)6 for ๐œ‰2=1 and (4.50)1: 2๐œ•๐‘‘1(+)๐‘—=๐œ•๐‘ก๐œ•๐‘‘1๐‘—+2๐œ•๐‘ก3๐œ•๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๎ƒฉ๐œ•๐‘ž1๐‘—๐œ•๐‘ฅ1โˆ’๐‘“๐›ฟ๐‘—0โˆ’๎€œ๎ƒช๎ƒช1โˆ’1๐ทโˆ’2๎ƒฉ๐œ•๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๐‘ˆ1๐‘—โˆ’1๎ƒชd๐œ‰2.(4.54) Eliminating ๐‘ž1๐‘— from (4.53) and (4.54) we get (4.52)1 with ๐ป1๎€ท๐‘ฅ1๎€ธ=1,๐‘ก๎€ท๐‘ฅ3๐œˆ1๎€ธ๎‚ต๐œ•๐œ•๐‘ฅ1๐‘”+๎‚ถ,๐บ+๐‘“1๐‘—๎€ท๐‘ฅ1๎€ธ=1,๐‘ก2๎ƒฉ๐œ•๐‘‘1๐‘—โˆ’๎€œ๐œ•๐‘ก1โˆ’1๐ทโˆ’2๎ƒฉ๐œ•๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๐‘ˆ1๐‘—โˆ’1๎ƒช๎ƒชd๐œ‰2๎ƒชโˆ’13๐œ•๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๐œ•๐œ•๐‘ฅ1๐’Ÿ1(+)๐‘—โˆ’1๎ƒช.(4.55)
As we can see, ๐ป1 depends only on the data and ๐บ1๐‘— depends on some functions determined at the previous approximations. Similar expressions are obtained for ๐ป2 and ๐บ2๐‘—.
Relations (4.52)3,4 are in fact (4.11) and (4.31), respectively, and the same remark holds for (4.52)5,6. We obtain next (4.52)7 as follows: we derivate (4.53) with respect to ๐‘ฅ1 and we take ๐‘ฅ1=1; we also take ๐‘ฅ1=1 in (4.15) and we eliminate (๐œ•๐‘ž1๐‘—/๐œ•๐‘ฅ1)(1,๐‘ก) from these two relations, by means of (4.9). The right-hand side of (4.52)7 is๐ฟ1๐‘—๎‚ต๐œ•(๐‘ก)=๐œ•๐‘ฅ1๐‘”+3(1,๐‘ก)+๐‘“(1,๐‘ก)โˆ’2๐œˆ0๎€œ1โˆ’1๐œ“d๐œ‰2๎‚ถ๐›ฟ๐‘—0+32๎€œ1โˆ’1๐ทโˆ’2๎‚€๐‘ˆ1๐‘—โˆ’1๎‚๎€ท1,๐œ‰2๎€ธ,๐‘กd๐œ‰2+๐œ•๐œ•๐‘ฅ1๐’Ÿ1(+)๐‘—โˆ’1โˆ’3๐œˆ02d๎€œd๐‘ก0โˆž๎‚€๐‘‘(1)(+)๐‘—โˆ’1โˆ’๐‘‘(1)(โˆ’)๐‘—โˆ’1๎‚๎€ท๐œ‰1๎€ธ,๐‘กd๐œ‰1(4.56) and a similar expression we obtain for ๐ฟ2๐‘—(๐‘ก). Relation (4.52)9 is a consequence of (4.34) for ๐‘—+1 replaced by ๐‘—, of (4.53) for ๐‘ฅ1=0 and of the corresponding relation for the index 2. The right hand side of (4.52)9 is ๐‘€1๐‘—(0,๐‘ก)=๐›ผ๐‘—โˆ’1(๐‘ก)+๐’Ÿ1(+)๐‘—โˆ’1(0,๐‘ก)โˆ’๐’Ÿ2(+)๐‘—โˆ’1(0,๐‘ก).(4.57) Finally, (4.52)10 is a consequence of the compatibility condition for (4.32), obtained as follows: (4.25) can be written as ฬƒ๐œƒ๐‘—=ฬ‚๐œƒ๐‘—โˆ’1โˆ’div๐œ‰๎‚€๐œ’๎€ท๐œ‰1๎€ธ๐‘ข11,๐‘—๎€ท๐œ‰1๎€ธ๐ž,0,๐‘ก1๎€ท๐œ‰+๐œ’2๎€ธ๐‘ข22,๐‘—๎€ท0,๐œ‰2๎€ธ๐ž,๐‘ก2๎‚,(4.58) where ฬ‚๐œƒ๐‘—โˆ’1=โˆ’div๐œ‰๎ƒฉ๐œ’๎€ท๐œ‰1๎€ธ๐‘—๎“๐‘™=1๐œ‰1๐‘™โˆ’1(๎ƒฉ๐œ‰๐‘™โˆ’1)!1๐‘™๐œ•๐‘™๐‘ข11,๐‘—โˆ’๐‘™๐œ•๐‘ฅ1๐‘™๎€ท0,๐œ‰2๎€ธ๐ž,๐‘ก1+๐œ•๐‘™โˆ’1๐‘ข12,๐‘—โˆ’๐‘™๐œ•๐‘ฅ1๐‘™โˆ’1๎€ท0,๐œ‰2๎€ธ๐ž,๐‘ก2๎ƒช๎€ท๐œ‰+๐œ’2๎€ธ๐‘—๎“๐‘™=1๐œ‰2๐‘™โˆ’1๎ƒฉ๐œ•(๐‘™โˆ’1)!๐‘™๐‘ข21,๐‘—โˆ’๐‘™๐œ•๐‘ฅ2๐‘™๎€ท๐œ‰1๎€ธ๐ž,0,๐‘ก1+๐œ‰2๐‘™๐œ•๐‘™๐‘ข22,๐‘—โˆ’๐‘™๐œ•๐‘ฅ2๐‘™๎€ท๐œ‰1๎€ธ๐ž,0,๐‘ก2๎ƒช๎ƒช(4.59) from (4.32) we get ๐›พ๐‘—(๐‘ก)โˆถ=๐œ•๐‘ž1๐‘—๐œ•๐‘ฅ1(0,๐‘ก)+๐œ•๐‘ž2๐‘—๐œ•๐‘ฅ2(0,๐‘ก)=โˆ’3๐œˆ02๎€œโ„ฌฬ‚๐œƒ๐‘—โˆ’1+3๐œˆ02๎€œโˆž2๎‚€๐‘‘(0)(+)๐‘—โˆ’1๎€ท๐œ‰1๎€ธ,1,๐‘กโˆ’๐‘‘(0)(โˆ’)๐‘—โˆ’1๎€ท๐œ‰1๎€ธ๎‚,โˆ’1,๐‘กd๐œ‰1+3๐œˆ02๎€œโˆž2๎‚€๐‘‘(0)(+)๐‘—โˆ’1๎€ท1,๐œ‰2๎€ธ,๐‘กโˆ’๐‘‘(0)(โˆ’)๐‘—โˆ’1๎€ทโˆ’1,๐œ‰2๎€ธ๎‚,๐‘กd๐œ‰2+32๎€œ1โˆ’1๐ทโˆ’2๎‚€๐‘ˆ1๐‘—โˆ’1๎‚๎€ท0,๐œ‰2๎€ธ,๐‘กd๐œ‰2+32๎€œ1โˆ’1๐ทโˆ’2๎‚€๐‘ˆ2๐‘—โˆ’1๎‚๎€ท๐œ‰1๎€ธ,0,๐‘กd๐œ‰1.(4.60)
Next, we derivate (4.53), we take ๐‘ฅ1=0 and we add with the same computation for the indexโ€‰โ€‰2. Replacing the previous relation, we obtain (4.52)10, with๐‘๐‘—(๐‘ก)=๐›พ๐‘—๐œ•(๐‘ก)+๐œ•๐‘ฅ1๎€ท๐’Ÿ(+)๐‘—โˆ’1(0,๐‘ก)+๐’Ÿ(โˆ’)๐‘—โˆ’1๎€ธ(0,๐‘ก)(4.61) which completes the proof.

The other components of the regular part of the asymptotic solution corresponding to ๐ท1๐œ€ have the following expressions:๐‘‘1(โˆ’)๐‘—๎€ท๐‘ฅ1๎€ธ,๐‘ก=๐‘‘1๐‘—๎€ท๐‘ฅ1๎€ธ,๐‘กโˆ’๐‘‘1(+)๐‘—๎€ท๐‘ฅ1๎€ธ,๐‘ž,๐‘ก1๐‘—๎€ท๐‘ฅ1๎€ธ=๐œ•,๐‘ก4๐‘‘1(+)๐‘—๐œ•๐‘ฅ41๎€ท๐‘ฅ1๎€ธ,๐‘กโˆ’๐‘”+๎€ท๐‘ฅ1๎€ธ๐›ฟ,๐‘ก๐‘—0+๐’Ÿ1(+)๐‘—โˆ’1๎€ท๐‘ฅ1๎€ธ,๐‘ข,๐‘ก11,๐‘—๎€ท๐‘ฅ1,๐œ‰2๎€ธ=1,๐‘ก๐œˆ๎€ท๐‘ฅ1๎€ธ๐ทโˆ’2๎‚€๐‘ˆ1๐‘—โˆ’1๎‚+1๐œˆ๎€ท๐‘ฅ1๎€ธ๎ƒฉ๐œ•๐‘ž1๐‘—๐œ•๐‘ฅ1โˆ’๐‘“๐›ฟ๐‘—0๎ƒช๐‘1๎€ท๐œ‰2๎€ธ,๐‘ข12,๐‘—๎€ท๐‘ฅ1,๐œ‰2๎€ธ=,๐‘ก๐œ•๐‘‘1(โˆ’)๐‘—โˆ’๐›พ+3โˆ’๐œ•๐œ•๐‘ก๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๎ƒฉ๐œ•๐‘ž1๐‘—๐œ•๐‘ฅ1โˆ’๐‘“๐›ฟ๐‘—0๐‘๎ƒช๎ƒช2๎€ท๐œ‰2๎€ธโˆ’๐ทโˆ’1๐ทโˆ’2๎ƒฉ๐œ•๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๐‘ˆ1๐‘—โˆ’1,๐‘๎ƒช๎ƒช1๐‘—=๐ทโˆ’1๎ƒฉโˆ’๐œŒ๐‘“๐œ•๐‘ข12,๐‘—โˆ’3+๐œ•๐œ•๐‘ก๐œ•๐‘ฅ1๎ƒฉ๐œˆ๎€ท๐‘ฅ1๎€ธ๐œ•๐‘ข12,๐‘—โˆ’3๐œ•๐‘ฅ1๎ƒช+๐œ•๐œ•๐‘ฅ1๎ƒฉ๐œˆ๎€ท๐‘ฅ1๎€ธ๐œ•๐‘ข11,๐‘—โˆ’1๐œ•๐œ‰2๎ƒช๎€ท๐‘ฅ+2๐œˆ1๎€ธ๐œ•2๐‘ข12,๐‘—โˆ’1๐œ•๐œ‰22๎ƒช.(4.62) As in the previous subsection, we discuss the expression of the leading term of the asymptotic solution.

Proposition 4.6. For ๐‘—=0 the regular part of the asymptotic solution corresponding to ๐ท1๐œ€ is given by ๐‘ข11,0๎€ท๐‘ฅ1,๐œ‰2๎€ธ=1,๐‘ก๐œˆ๎€ท๐‘ฅ1๎€ธ๎ƒฉ๐œ•5๐‘‘1(+)0๐œ•๐‘ฅ51๎€ท๐‘ฅ1๎€ธโˆ’๐œ•,๐‘ก๐œ•๐‘ฅ1๐‘”+๎€ท๐‘ฅ1๎€ธ๎€ท๐‘ฅ,๐‘กโˆ’๐‘“1๎€ธ๎ƒช๐‘,๐‘ก1๎€ท๐œ‰2๎€ธ,๐‘ข12,0๎€ท๐‘ฅ1,๐œ‰2๎€ธ๐œ•,๐‘ก=โˆ’๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๎ƒฉ๐œ•5๐‘‘1(+)0๐œ•๐‘ฅ51๎€ท๐‘ฅ1๎€ธโˆ’๐œ•,๐‘ก๐œ•๐‘ฅ1๐‘”+๎€ท๐‘ฅ1๎€ธ๎€ท๐‘ฅ,๐‘กโˆ’๐‘“1๎€ธ๐‘,๐‘ก๎ƒช๎ƒช2๎€ท๐œ‰2๎€ธ,๐‘ž10๎€ท๐‘ฅ1๎€ธ=๐œ•,๐‘ก4๐‘‘1(+)0๐œ•๐‘ฅ41๎€ท๐‘ฅ1๎€ธ,๐‘กโˆ’๐‘”+๎€ท๐‘ฅ1๎€ธ,๐‘,๐‘ก10๎€ท๐‘ฅ1,๐œ‰2๎€ธ๐‘‘,๐‘ก=0,1(โˆ’)0๎€ท๐‘ฅ1๎€ธ,๐‘ก=๐‘‘10๎€ท๐‘ฅ1๎€ธ,๐‘กโˆ’๐‘‘1(+)0๎€ท๐‘ฅ1๎€ธ,,๐‘ก(4.63) with ๐‘‘10,๐‘‘1(+)0 satisfying the problems presented in the proof, (4.64) and (4.65).

Proof. The relations (4.63)1,2,3,4 are consequences of (4.15)โ€“(4.17) and of (4.53) for ๐‘—=0. It remains to show that ๐‘‘10,๐‘‘1(+)0,๐‘‘2(+)0 can be determined as unique solutions of some problems.
For ๐‘—=0 the problem for ๐‘‘10 becomes:๐œ•4๐‘‘10๐œ•๐‘ฅ41๎€ท๐‘ฅ1๎€ธ,๐‘ก=๐‘”+(๐‘ฅ,๐‘ก)+๐‘”โˆ’๎€ท๐‘ฅ1๎€ธ,๐‘‘,๐‘ก10(1,๐‘ก)=๐œ•๐‘‘10๐œ•๐‘ฅ1๐‘‘(1,๐‘ก)=0,10(0,๐‘ก)=๐œ•๐‘‘10๐œ•๐‘ฅ1(0,๐‘ก)=0.(4.64) Finally, the problem for ๐‘‘1(+)0,๐‘‘2(+)0 is the following: ๐œ•๐‘‘1(+)0๎€ท๐‘ฅ๐œ•๐‘ก1๎€ธโˆ’1,๐‘ก3๐œ•๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๐œ•5๐‘‘1(+)0๐œ•๐‘ฅ51๎€ท๐‘ฅ1๎€ธ๎ƒช=1,๐‘ก2๐œ•๐‘‘10โˆ’1๐œ•๐‘ก3๐œ•๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๎‚ต๐œ•๐œ•๐‘ฅ1๐‘”+๎‚ถ๎ƒช,+๐‘“๐œ•๐‘‘2(+)0๎€ท๐‘ฅ๐œ•๐‘ก2๎€ธโˆ’1,๐‘ก3๐œ•๎ƒฉ1๐œ•๐‘ก๐œˆ๎€ท๐‘ฅ2๎€ธ๐œ•5๐‘‘2(+)0๐œ•๐‘ฅ52๎€ท๐‘ฅ2๎€ธ๎ƒช=1,๐‘ก2๐œ•๐‘‘20๐œ•๐‘ฅ2โˆ’13๐œ•๐œ•๐‘ฅ2๎ƒฉ1๐œˆ๎€ท๐‘ฅ2๎€ธ๎‚ต๐œ•๐œ•๐‘ฅ2๐‘”+๎‚ถ๎ƒช,๐‘‘+๐‘“1(+)0(1,๐‘ก)=๐œ•๐‘‘1(+)0๐œ•๐‘ฅ1(1,๐‘ก)=๐‘‘1(+)0(0,๐‘ก)=๐œ•๐‘‘1(+)0๐œ•๐‘ฅ1๐‘‘(0,๐‘ก)=0,2(+)0(1,๐‘ก)=๐œ•๐‘‘2(+)0๐œ•๐‘ฅ2(1,๐‘ก)=๐‘‘2(+)0(0,๐‘ก)=๐œ•๐‘‘2(+)0๐œ•๐‘ฅ2๐œ•(0,๐‘ก)=0,5๐‘‘1(+)0๐œ•๐‘ฅ51(1,๐‘ก)=โˆ’3๐œˆ02๎€œ1โˆ’1๐œ“๎€ท๐œ‰2๎€ธ,๐‘กd๐œ‰2๐œ•+๐‘“(1,๐‘ก)+๐œ•๐‘ฅ1๐‘”+๐œ•(1,๐‘ก),5๐‘‘2(+)0๐œ•๐‘ฅ52(1,๐‘ก)=โˆ’3๐œˆ02๎€œ1โˆ’1๐œ“๎€ท๐œ‰1๎€ธ,๐‘กd๐œ‰1๐œ•+๐‘“(1,๐‘ก)+๐œ•๐‘ฅ2๐‘”+๐œ•(1,๐‘ก),4๐‘‘1(+)0๐œ•๐‘ฅ41๐œ•(0,๐‘ก)โˆ’4๐‘‘2(+)0๐œ•๐‘ฅ42๐œ•(0,๐‘ก)=0,5๐‘‘1(+)0๐œ•๐‘ฅ51๐œ•(0,๐‘ก)+5๐‘‘2(+)0๐œ•๐‘ฅ52๐‘‘(0,๐‘ก)=0,1(+)0๎€ท๐‘ฅ1๎€ธ,0=๐‘‘2(+)0๎€ท๐‘ฅ2๎€ธ,0=0.(4.65)

At the end of this subsection we notice that the system (4.52) can be solved also for ๐‘— replaced by ๐‘—+1 which gives, by means of (4.53), ๐‘ž1๐‘—+1(0,๐‘ก), ๐‘ž2๐‘—+1(0,๐‘ก). Hence, the system (4.23) can be solved, and all the ๐‘—th approximation is determined for ๐›พ=3.

4.2.3. Conclusions

In the previous two subsections we determined the ๐‘—th approximation of the asymptotic solution for two different cases: ๐›พ>3 and ๐›พ=3. For ๐›พ>3 we can separate the problems for the two branches of the elastic tube and we obtain the regular part of the asymptotic solution (for ๐ท๐œ€1) as follows:(i)the macroscopic variable ๐‘ž1๐‘— is the unique solution of the problem: ๐œ•๐œ•๐‘ฅ1๎ƒฉ1๐œˆ๎€ท๐‘ฅ1๎€ธ๎ƒฉ๐œ•๐‘ž1๐‘—๐œ•๐‘ฅ1๎ƒช๎ƒช=๐‘Ž๐‘ž๐‘—๎€ท๐‘ฅ1๎€ธ,,๐‘ก๐œ•๐‘ž1๐‘—๐œ•๐‘ฅ1(1,๐‘ก)=๐‘๐‘ž๐‘—๐‘ž(๐‘ก),1๐‘—(0,๐‘ก)=๐‘๐‘ž๐‘—(๐‘ก),(4.66) with ๐‘Ž๐‘ž๐‘—,๐‘๐‘ž๐‘—,๐‘๐‘ž๐‘— known functions;(ii)the macroscopic variables ๐‘‘1(ยฑ)๐‘— are the unique solutions of the fourth-order differential problems: ๐œ•4๐‘‘1(+)๐‘—๐œ•๐‘ฅ41=๐‘Ž๐‘‘๐‘—๎€ท๐‘ฅ1๎€ธ,๐‘‘,๐‘ก1(+)๐‘—(0,๐‘ก),๐‘‘1(+)๐‘—(1,๐‘ก),๐œ•๐‘‘1(+)๐‘—๐œ•๐‘ฅ1(0,๐‘ก),๐œ•๐‘‘1(+)๐‘—๐œ•๐‘ฅ1(1,๐‘ก)given,(4.67) and a similar one for ๐‘‘1(โˆ’)๐‘—;(iii)๐‘ข11,๐‘—,๐‘ข12,๐‘—,๐‘1๐‘— given by (4.15)โ€“(4.17). The main term of the regular part of the asymptotic solution is presented in (4.48)-(4.49). Unlike in the case ๐›พ>3, for ๐›พ=3 the problems must be solved for the two branches together, since they cannot be uncoupled;(iv)we first introduce the auxiliary functions ๐‘‘1๐‘—,๐‘‘2๐‘—, with ๐‘‘1๐‘— the unique solution of (4.51); then the regular part of the asymptotic solution is obtained as follows;(v)the macroscopic variables ๐‘‘1(+)๐‘—,๐‘‘2(+)๐‘— are obtained as the unique solution of the sixth-order parabolic system (4.52);(vi)the macroscopic variables ๐‘ž1๐‘—,๐‘ž2๐‘— are given by (4.53) and the similar equation for the index 2;(vii)๐‘ข11,๐‘—,๐‘ข12,๐‘—,๐‘1๐‘— given also by (4.15)โ€“(4.17), but we obtain different expressions.

The main term of the regular part of the asymptotic solution is given by (4.63) and (4.65).

5. Error Estimates

In the last section, we estimate the error between the exact solution and the asymptotic one, in order to justify the asymptotic expansion. Introducing the asymptotic solution of order ๐‘˜, (ฬ‚๐ฎ๐‘Ž(๐‘˜),ฬ‚๐‘๐‘Ž(๐‘˜),๎๐‘‘(๐‘˜)ยฑ๐‘Ž), into (2.6) we want to obtain for it a problem of the same type as (2.6).

The next computations hold for any integer ๐›พโ‰ฅ3. We begin with the continuity equation satisfied by the asymptotic velocity of order ๐‘˜. Standard computations givediv๐‘ฅฬ‚๐ฎ๐‘Ž(๐‘˜)=๐‘‘๐œ€(๐‘˜)inโ„ฌ๐œ€ร—(0,๐‘‡),(5.1) with๐‘‘๐œ€(๐‘˜)(โŽงโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽฉ๐‘ฅ,๐‘ก)=0inโ„ฌ๐œ€โˆฉ๎‚†๐‘ฅ1<18๎‚‡โˆช๎‚†๐‘ฅ2<18๎‚‡,โˆ’๐œ‚๎…ž๎€ท1โˆ’๐‘ฅ๐‘–๎€ธ๐‘˜๎“๐‘—=0๐œ€๐‘—+2๐‘ข(0)1,๐‘—๎‚€๐‘ฅ๐œ€๎‚,๐‘กinโ„ฌ๐œ€โˆฉ๎‚†18<๐‘ฅ๐‘–<14๎‚‡๐œ‚,๐‘–=1,2,๎…ž๎€ท๐‘ฅ1๎€ธ๐‘˜๎“๐‘—=0๐œ€๐‘—+2๐‘ข(1)1,๐‘—๎‚ต๐‘ฅ1โˆ’1๐œ€,๐‘ฅ2๐œ€๎‚ถ,๐‘กinโ„ฌ๐œ€โˆฉ๎‚†14<๐‘ฅ1๎‚‡,๐œ‚<1๎…ž๎€ท๐‘ฅ2๎€ธ๐‘˜๎“๐‘—=0๐œ€๐‘—+2๐‘ข(2)2,๐‘—๎‚ต๐‘ฅ1๐œ€,๐‘ฅ2โˆ’1๐œ€๎‚ถ,๐‘กinโ„ฌ๐œ€โˆฉ๎‚†14<๐‘ฅ2๎‚‡.<1(5.2) We notice that the asymptotic velocity of order ๐‘˜ is not a divergence free function.

We present next the equations for the asymptotic solution, which correspond to (2.6)1 and (2.6)3:๐œŒ๐‘“๐œ•ฬ‚๐ฎ๐‘Ž(๐‘˜)๐œ•๐‘กโˆ’2div๐‘ฅ๎‚€๐œˆ(๐‘ฅ)๐ท๐‘ฅฬ‚๐ฎ๐‘Ž(๐‘˜)๎‚+โˆ‡๐‘ฅฬ‚๐‘๐‘Ž(๐‘˜)=๐Ÿ+๐…๐œ€(๐‘˜)inโ„ฌ๐œ€๐œ•ร—(0,๐‘‡),๐œŒโ„Ž2๎๐‘‘(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ก2+1๐œ€๐›พ๐œ•4๎๐‘‘(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ฅ4๐‘–๐œ•+๐œ‡5๎๐‘‘(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ฅ4๐‘–๐œ•๐‘ก=๐‘”ยฑยฑฬ‚๐‘(๐‘˜)๐‘Ž|๐ท๐œ€๐œ€ยฑโˆฉฮ“+๐บ(๐‘˜)ยฑ๐œ€๎‚€on๐ท๐‘–๐œ€โˆฉฮ“๐œ€ยฑ๎‚ร—(0,๐‘‡),๐‘–=1,2,(5.3) with ๐…๐œ€(๐‘˜),๐บ(๐‘˜)ยฑ๐œ€ residual known functions. Since the expressions of these residuals are rather complicated, but not too technical to obtain, let us give below only the estimates for their norms in the corresponding spaces: โ€–๐…๐œ€(๐‘˜)โ€–๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)=๐‘‚(๐œ€๐‘˜+3/2), โ€–๐บ(๐‘˜)ยฑ๐œ€โ€–๐ฟ2((๐ท๐‘–๐œ€โˆฉฮ“๐œ€ยฑ)ร—(0,๐‘‡))=๎‚ป๐‘‚๎€ท๐œ€3/2๎€ธ๐‘‚๎€ท๐œ€for๐‘˜โˆˆ{0,1,2,3},๐‘š๐‘–๐‘›{๐‘˜โˆ’5/2,๐‘˜+4โˆ’๐›พ}๎€ธfor๐‘˜โ‰ฅ4.(5.4) These estimates will be used to derive finally the error estimate between the exact and the asymptotic solution. More details concerning these estimates can be found in [2, Theoremโ€‰โ€‰5.1].

We derive in the sequel the conditions on the elastic boundaries. Introducing the asymptotic solution of order ๐‘˜ into (2.6)11 we getฬ‚๐ฎ๐‘Ž(๐‘˜)๐œ•๎๐‘‘โ‹…๐ง=ยฑ(๐‘˜)ยฑ๐‘Ž๐œ•๐‘กยฑ๐ด๐‘˜ยฑ๐œ€,ฬ‚๐ฎ๐‘Ž(๐‘˜)โ‹…๐‰=0onฮ“๐œ€ยฑร—(0,๐‘‡),(5.5) where the expression of ๐ด๐‘˜ยฑ๐œ€ on ฮ“๐œ€ยฑโˆฉ๐œ•๐ท1๐œ€ is given by:๐ด๐‘˜ยฑ๐œ€๎€ท๐‘ฅ1๎€ธ=โŽงโŽชโŽชโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽชโŽชโŽฉโˆ’๎ƒฉ๐œ€,ยฑ1,๐‘ก๐‘˜+4๐œ•๐‘‘1ยฑ๐‘˜+4โˆ’๐›พ๐œ•๐‘ก+โ‹ฏ+๐œ€๐‘˜+๐›พ๐œ•๐‘‘1ยฑ๐‘˜๎ƒช๎€ท๐‘ฅ๐œ•๐‘ก1๎€ธ๎€ท๐‘ฅ,๐‘กโˆ’๐œ‚1๎€ธ๎ƒฉ๐œ€๐‘˜+3๐œ•๐‘‘(1)ยฑ๐‘˜+3โˆ’๐›พ๐œ•๐‘ก+โ‹ฏ+๐œ€๐‘˜+๐›พ๐œ•๐‘‘(1)ยฑ๐‘˜๎ƒช๎‚ต๐‘ฅ๐œ•๐‘ก1โˆ’1๐œ€๎‚ถ,ยฑ1,๐‘กonฮ“๐œ€ยฑโˆฉ๎‚†๐‘ฅ1>14๎‚‡,๎‚€๐‘ฅโˆ’๐œ’1๐œ€๎‚๎ƒฉ๐œ€๐‘˜+4๐œ•๐‘‘1ยฑ๐‘˜+4โˆ’๐›พ๐œ•๐‘ก+โ‹ฏ+๐œ€๐‘˜+๐›พ๐œ•๐‘‘1ยฑ๐‘˜๎ƒช๎€ท๐‘ฅ๐œ•๐‘ก1๎€ธ๎€ท,๐‘กโˆ’๐œ‚1โˆ’๐‘ฅ1๎€ธ๎ƒฉ๐œ€๐‘˜+3๐œ•๐‘‘(0)ยฑ๐‘˜+3โˆ’๐›พ๐œ•๐‘ก+โ‹ฏ+๐œ€๐‘˜+๐›พ๐œ•๐‘‘(0)ยฑ๐‘˜๎ƒช๎‚€๐‘ฅ๐œ•๐‘ก1๐œ€๎‚,ยฑ1,๐‘กonฮ“๐œ€ยฑโˆฉ๎‚†2๐œ€<๐‘ฅ1<14๎‚‡,(5.6) and an analogous expression can be written on ฮ“๐œ€ยฑโˆฉ๐œ•๐ท2๐œ€. We notice that for ๐›พ=3 the first terms in the above definition are zero.

We complete the problem (5.1), (5.3), (5.5) with the following boundary conditions:ฬ‚๐ฎ๐‘Ž(๐‘˜)๎๐‘‘=๐ŸŽ,(๐‘˜)ยฑ๐‘Ž๎‚ต=0onโŒข๐ด๐œ€+๐ต๐œ€+โˆชโŒข๐ด๐œ€โˆ’๐ต๐œ€โˆ’๎‚ถฬ‚๐ฎร—(0,๐‘‡),๐‘Ž(๐‘˜)๎€ท1,๐‘ฅ2๎€ธ,๐‘ก=๐œ€2๐œ“๐œ€๎€ท๐‘ฅ2๎€ธ๐ž,๐‘ก1+๐œ€๐‘˜+3๐‘ข12,๐‘˜๎€ท1,๐œ‰2๎€ธ๐ž,๐‘ก2on๐น๐œ€1ฬ‚๐ฎร—(0,๐‘‡),๐‘Ž(๐‘˜)๎€ท๐‘ฅ1๎€ธ,1,๐‘ก=๐œ€2๐œ“๐œ€๎€ท๐‘ฅ1๎€ธ๐ž,๐‘ก2+๐œ€๐‘˜+3๐‘ข21,๐‘˜๎€ท๐œ‰1๎€ธ๐ž,1,๐‘ก1on๐น๐œ€2๎๐‘‘ร—(0,๐‘‡),(๐‘˜)ยฑ๐‘Ž(๐œ•๎๐‘‘1,ยฑ๐œ€,๐‘ก)=0,(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ฅ1(1,ยฑ๐œ€,๐‘ก)=๐œ€๐‘˜+๐›พ๐œ•๐‘‘1ยฑ๐‘˜๐œ•๐‘ฅ1(๎๐‘‘1,๐‘ก),(๐‘˜)ยฑ๐‘Ž๐œ•๎๐‘‘(2๐œ€,ยฑ๐œ€,๐‘ก)=(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ฅ1๎๐‘‘(2๐œ€,ยฑ๐œ€,๐‘ก)=0,(๐‘˜)ยฑ๐‘Ž๐œ•๎๐‘‘(ยฑ๐œ€,1,๐‘ก)=0,(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ฅ1(ยฑ๐œ€,1,๐‘ก)=๐œ€๐‘˜+๐›พ๐œ•๐‘‘2ยฑ๐‘˜๐œ•๐‘ฅ1๎๐‘‘(1,๐‘ก),(๐‘˜)ยฑ๐‘Ž๐œ•๎๐‘‘(ยฑ๐œ€,2๐œ€,๐‘ก)=(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ฅ1(ยฑ๐œ€,2๐œ€,๐‘ก)=0๐‘กโˆˆ(0,๐‘‡),(5.7) and initial conditions:ฬ‚๐ฎ๐‘Ž(๐‘˜)(๐‘ฅ,0)=๐ŸŽinโ„ฌ๐œ€,๎๐‘‘(๐‘˜)ยฑ๐‘Ž๐œ•๎๐‘‘(๐‘ฅ,0)=(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ก(๐‘ฅ,0)=0onฮ“๐œ€ยฑ.(5.8) The compatibility condition for the problem (5.1), (5.3) with the boundary and initial conditions (5.5), (5.7), (5.8) is given by:๎€œโ„ฌ๐œ€๐‘‘๐œ€(๐‘˜)๎€œd๐‘ฅ=12๐œ€๐œ•๎‚€๎๐‘‘๐œ•๐‘ก(๐‘˜)+๐‘Ž๎€ท๐‘ฅ1๎€ธโˆ’๎๐‘‘,๐œ€,๐‘ก(๐‘˜)โˆ’๐‘Ž๎€ท๐‘ฅ1๎€ธ๎‚,โˆ’๐œ€,๐‘กd๐‘ฅ1+๎€œ12๐œ€๐œ•๎‚€๎๐‘‘๐œ•๐‘ก(๐‘˜)+๐‘Ž๎€ท๐œ€,๐‘ฅ2๎€ธโˆ’๎๐‘‘,๐‘ก(๐‘˜)โˆ’๐‘Ž๎€ทโˆ’๐œ€,๐‘ฅ2๎€ธ๎‚,๐‘กd๐‘ฅ2+๎€œ12๐œ€๎€ท๐ด๐‘˜+๐œ€๎€ท๐‘ฅ1๎€ธ,๐œ€,๐‘กโˆ’๐ด๐‘˜โˆ’๐œ€๎€ท๐‘ฅ1,โˆ’๐œ€,๐‘ก๎€ธ๎€ธd๐‘ฅ1+๎€œ12๐œ€๎€ท๐ด๐‘˜โˆ’๐œ€๎€ท๐œ€,๐‘ฅ2๎€ธ,๐‘กโˆ’๐ด๐‘˜+๐œ€๎€ทโˆ’๐œ€,๐‘ฅ2,๐‘ก๎€ธ๎€ธd๐‘ฅ2.(5.9) We notice that the boundary conditions for ฬ‚๐‘ข๐‘Ž(๐‘˜) and ๎๐‘‘(๐‘˜)ยฑ๐‘Ž are of different type from those for ๐ฎ and ๐‘‘ยฑ, respectively; so we cannot apply directly the estimates (3.20) which hold for different right-hand sides in the equations but for the same boundary conditions. Moreover, the asymptotic velocity of order ๐‘˜ is not a divergence free function. To overcome this difficulty, we define new functions ๐”๐‘Ž(๐‘˜) and ๐ท(๐‘˜)ยฑ๐‘Ž in order to obtain for (๐”๐‘Ž(๐‘˜),ฬ‚๐‘(๐‘˜)ยฑ๐‘Ž,๐ท(๐‘˜)ยฑ๐‘Ž) a problem of the same type as (2.6).

Construction of ๐ท(๐‘˜)ยฑ๐‘Ž
For this construction we use some ideas from [5]. We first define ๐ทยฑ(๐‘˜)โˆถ๐น๐œ€ยฑร—(0,๐‘‡)โ†’โ„, satisfying: ๐ทยฑ(๐‘˜)(1,ยฑ๐œ€,๐‘ก)=0,๐œ•๐ทยฑ(๐‘˜)๐œ•๐‘ฅ1(1,ยฑ๐œ€,๐‘ก)=๐œ€๐‘˜+๐›พ๐œ•๐‘‘1ยฑ๐‘˜๐œ•๐‘ฅ1๐ท(1,๐‘ก),ยฑ(๐‘˜)(2๐œ€,ยฑ๐œ€,๐‘ก)=๐œ•๐ทยฑ(๐‘˜)๐œ•๐‘ฅ1๐ท(2๐œ€,ยฑ๐œ€,๐‘ก)=0,ยฑ(๐‘˜)(ยฑ๐œ€,1,๐‘ก)=0,๐œ•๐ทยฑ(๐‘˜)๐œ•๐‘ฅ1(ยฑ๐œ€,1,๐‘ก)=๐œ€๐‘˜+๐›พ๐œ•๐‘‘2ยฑ๐‘˜๐œ•๐‘ฅ2๐ท(1,๐‘ก),ยฑ(๐‘˜)(ยฑ๐œ€,2๐œ€,๐‘ก)=๐œ•๐ทยฑ(๐‘˜)๐œ•๐‘ฅ2๐ท(ยฑ๐œ€,2๐œ€,๐‘ก)=0,ยฑ(๐‘˜)๎€ท๐‘ฅ(๐‘ฅ,๐‘ก)=01,๐‘ฅ2๎€ธโˆˆโŒข๐ด๐œ€ยฑ๐ต๐œ€ยฑ,๐ทยฑ(๐‘˜)(๐‘ฅ,0)=๐œ•๐ทยฑ(๐‘˜)๎€œ๐œ•๐‘ก(๐‘ฅ,0)=0,(5.10)12๐œ€๐œ•๎‚€๐ท๐œ•๐‘ก+(๐‘˜)๎€ท๐‘ฅ1๎€ธ,๐œ€,๐‘กโˆ’๐ทโˆ’(๐‘˜)๎€ท๐‘ฅ1๎€ธ๎‚,โˆ’๐œ€,๐‘กd๐‘ฅ1+๎€œ12๐œ€๐œ•๎‚€๐ท๐œ•๐‘ก+(๐‘˜)๎€ท๐œ€,๐‘ฅ2๎€ธ,๐‘กโˆ’๐ทโˆ’(๐‘˜)๎€ทโˆ’๐œ€,๐‘ฅ2๎€ธ๎‚,๐‘กd๐‘ฅ2=๎€œโ„ฌ๐œ€๐‘‘๐œ€(๐‘˜)๎€œd๐‘ฅโˆ’12๐œ€๎€ท๐ด๐‘˜+๐œ€๎€ท๐‘ฅ1๎€ธ,๐œ€,๐‘กโˆ’๐ด๐‘˜โˆ’๐œ€๎€ท๐‘ฅ1,โˆ’๐œ€,๐‘ก๎€ธ๎€ธd๐‘ฅ1โˆ’๎€œ12๐œ€๎€ท๐ด๐‘˜+๐œ€๎€ท๐œ€,๐‘ฅ2๎€ธ,๐‘กโˆ’๐ด๐‘˜โˆ’๐œ€๎€ทโˆ’๐œ€,๐‘ฅ2,๐‘ก๎€ธ๎€ธd๐‘ฅ2=โˆถ๐‘Š๐‘˜๐œ€(๐‘ก).(5.11) Let us consider ๐ทยฑ(๐‘˜)โŽงโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽฉ๎€ท๐‘ฅ(๐‘ฅ,๐‘ก)=1๐‘ฅโˆ’1๎€ธ๎€ท1๎€ธโˆ’2๐œ€2๎ƒฉ๐‘Žยฑ(๐‘˜)1๎€ท๐‘ฅ(๐‘ก)1๎€ธ+1โˆ’1(1โˆ’2๐œ€)2๐œ€๐‘˜+๐›พ๐œ•๐‘‘1ยฑ๐‘˜๐œ•๐‘ฅ1๎ƒช๎€ท๐‘ฅ(1,๐‘ก)in(2๐œ€,1)ร—{ยฑ๐œ€},2๐‘ฅโˆ’1๎€ธ๎€ท2๎€ธโˆ’2๐œ€2๎ƒฉ๐‘Žยฑ(๐‘˜)2๎€ท๐‘ฅ(๐‘ก)2๎€ธ+1โˆ’1(1โˆ’2๐œ€)2๐œ€๐‘˜+๐›พ๐œ•๐‘‘2ยฑ๐‘˜๐œ•๐‘ฅ2๎ƒช(1,๐‘ก)in{ยฑ๐œ€}ร—(2๐œ€,1),0onโŒข๐ด๐œ€ยฑ๐ต๐œ€ยฑ,(5.12) with ๐‘Žยฑ(๐‘˜)๐‘–,๐‘–=1,2 four functions which are determined by imposing some conditions, as we show below. It is obvious that the functions ๐ทยฑ(๐‘˜) satisfy (5.10)1,2,3,4 without any conditions on ๐‘Žยฑ(๐‘˜)๐‘–.

We introduce next the following notations:๐‘Š๐œ€๐‘˜,1๎€œ(๐‘ก)=๐ท1๐œ€๐‘‘๐œ€(๐‘˜)๎€œd๐‘ฅโˆ’12๐œ€๎€ท๐ด๐‘˜+๐œ€๎€ท๐‘ฅ1๎€ธ,๐œ€,๐‘กโˆ’๐ด๐‘˜โˆ’๐œ€๎€ท๐‘ฅ1,โˆ’๐œ€,๐‘ก๎€ธ๎€ธd๐‘ฅ1,๐‘Š๐œ€๐‘˜,2๎€œ(๐‘ก)=๐ท2๐œ€๐‘‘๐œ€(๐‘˜)๎€œd๐‘ฅโˆ’12๐œ€๎€ท๐ด๐‘˜+๐œ€๎€ท๐œ€,๐‘ฅ2๎€ธ,๐‘กโˆ’๐ด๐‘˜โˆ’๐œ€๎€ทโˆ’๐œ€,๐‘ฅ2,๐‘ก๎€ธ๎€ธd๐‘ฅ2.(5.13) We notice that from (5.2) we obtain ๐‘Š๐‘˜๐œ€(๐‘ก)=๐‘Š๐œ€๐‘˜,1(๐‘ก)+๐‘Š๐œ€๐‘˜,2(๐‘ก). A relation between the functions ๐‘Ž+(๐‘˜)1 and ๐‘Žโˆ’(๐‘˜)1 follows from โˆซ(d/d๐‘ก)12๐œ€(๐ท+(๐‘˜)(๐‘ฅ1,๐œ€,๐‘ก)โˆ’๐ทโˆ’(๐‘˜)(๐‘ฅ1,โˆ’๐œ€,๐‘ก))d๐‘ฅ1=๐‘Š๐œ€๐‘˜,1(๐‘ก), with the initial conditions (5.10)6. In a similar way we establish a relation between ๐‘Ž+(๐‘˜)2 and ๐‘Žโˆ’(๐‘˜)2. So we proved the possibility of construction of functions ๐ทยฑ(๐‘˜) which satisfy (5.10) and (5.11).

We define next๎๐ท(๐‘˜)ยฑ๐‘Ž=๎๐‘‘(๐‘˜)ยฑ๐‘Žโˆ’๐ทยฑ(๐‘˜),(5.14) and we verify that it satisfies the same boundary conditions as ๐‘‘ยฑ.

Construction of ๎๐”๐‘Ž(๐‘˜)
We look for a function ๐”(๐‘˜)โˆถโ„ฌ๐œ€โ†’โ„2 satisfying the following problem, for ๐‘กโˆˆ(0,๐‘‡): ๐”(๐‘˜)๎€ท๐ป(๐‘ก)โˆˆ1(โ„ฌ๐œ€)๎€ธ2,div๐‘ฅ๐”(๐‘˜)(๐‘ก)=๐‘‘๐œ€(๐‘˜)(๐‘ก)inโ„ฌ๐œ€,๐”(๐‘˜)(๐‘ก)=0inโ„ฌ๐œ€โˆฉ๎€ฝ๐‘ฅ1<2๐œ€,๐‘ฅ2๎€พ,๐”<2๐œ€(๐‘˜)๎€ท1,๐‘ฅ2๎€ธ,๐‘ก=๐œ€๐‘˜+3๐‘ข12,๐‘˜๎€ท1,๐œ‰2๎€ธ๐ž,๐‘ก2on๐น๐œ€1,๐”(๐‘˜)๎€ท๐‘ฅ1๎€ธ,1,๐‘ก=๐œ€๐‘˜+3๐‘ข21,๐‘˜๎€ท๐œ‰1๎€ธ๐ž,1,๐‘ก1on๐น๐œ€2,๐”(๐‘˜)๎€ท๐‘ฅ1๎€ธ=๎ƒฉ,ยฑ๐œ€,๐‘ก๐œ•๐ทยฑ(๐‘˜)๐œ•๐‘กยฑ๐ด๐‘˜ยฑ๐œ€๎ƒช๎€ท๐‘ฅ1๎€ธ๐ž,ยฑ๐œ€,๐‘ก2onฮ“๐œ€ยฑโˆฉ๐œ•๐ท1๐œ€,๐”(๐‘˜)๎€ทยฑ๐œ€,๐‘ฅ2๎€ธ=๎ƒฉ,๐‘ก๐œ•๐ทยฑ(๐‘˜)๐œ•๐‘กยฑ๐ด๐‘˜ยฑ๐œ€๎ƒช๎€ทยฑ๐œ€,๐‘ฅ2๎€ธ๐ž,๐‘ก1onฮ“๐œ€ยฑโˆฉ๐œ•๐ท2๐œ€.(5.15)

Remark 5.1. Since ๐”(๐‘˜) vanishes in โ„ฌ๐œ€โˆฉ{๐‘ฅ1<2๐œ€,๐‘ฅ2<2๐œ€}, (5.15) represents in fact two separate problems, one in ๐ท1๐œ€ and the second one on ๐ท2๐œ€. The compatibility condition for each problem is fulfilled due to the construction of ๐ทยฑ(๐‘˜).

We can prove that.

Proposition 5.2. The problem (5.15) has at least one solution, with the property โ€–โ€–๐”(๐‘˜)โ€–โ€–(๐‘ก)(๐ป1(โ„ฌ๐œ€))2๎€ท๐œ€=๐‘‚๐‘˜+3/2๎€ธ.(5.16)

Proof. Due to (5.15)3, we can consider the problem (5.15) as two separate problems: one in ๐ท1๐œ€ and the second in ๐ท2๐œ€, each problem with its compatibility condition. So, we can obtain the estimate (5.16) only for ๐ท1๐œ€. For any (๐‘ฅ1,๐‘ฅ2)โˆˆ๐ท1๐œ€โˆฉ{๐‘ฅ1>2๐œ€} we introduce the new variable (๐‘ฆ1,๐‘ฆ2)=((๐‘ฅ1โˆ’2๐œ€)/(1โˆ’2๐œ€),๐‘ฅ2/๐œ€); obviously (๐‘ฆ1,๐‘ฆ2)โˆˆ(0,1)ร—(โˆ’1,1). We define a new function ๐œผ๐œ€1(๐‘˜)โˆถ(0,1)ร—(โˆ’1,1)ร—(0,๐‘‡)โ†’โ„2, by ๐œผ๐œ€1(๐‘˜)(๐‘ฆ1,๐‘ฆ2,๐‘ก)=(1/(1โˆ’2๐œ€))๐‘ˆ1(๐‘˜)(๐‘ฅ1,๐‘ฅ2,๐‘ก)๐ž1+(1/๐œ€)๐‘ˆ2(๐‘˜)(๐‘ฅ1,๐‘ฅ2,๐‘ก)๐ž2. Obvious computations lead to the following problem for ๐œผ๐œ€1(๐‘˜)(๐‘ก): div๐‘ฆ๐œผ๐œ€1(๐‘˜)(๐‘ก)=๐‘‘๐œ€(๐‘˜)๎€ท(1โˆ’2๐œ€)๐‘ฆ1+2๐œ€,๐œ€๐‘ฆ2๎€ธ๐œผ,๐‘กin(0,1)ร—(โˆ’1,1),๐œ€1(๐‘˜)๎€ท0,๐‘ฆ2๎€ธ๐œผ,๐‘ก=๐ŸŽin(โˆ’1,1),๐œ€1(๐‘˜)๎€ท1,๐‘ฆ2๎€ธ,๐‘ก=๐œ€๐‘˜+2๐‘ข12,๐‘˜๎€ท1,๐‘ฆ2๎€ธ๐ž,๐‘ก2๐œผin(โˆ’1,1),๐œ€1(๐‘˜)๎€ท๐‘ฆ1๎€ธ=1,ยฑ1,๐‘ก๐œ€๎ƒฉ๐œ•๐ทยฑ(๐‘˜)๐œ•๐‘กยฑ๐ด๐‘˜ยฑ๐œ€๎ƒช๎€ท(1โˆ’2๐œ€)๐‘ฆ1๎€ธ๐ž+2๐œ€,ยฑ๐œ€,๐‘ก2in(0,1).(5.17) The right-hand sides of (5.17)1,4 contain either correctors (which are of order ๐‘‚(exp(โˆ’๐œŽ/๐œ€)), with ๐œŽ independent on ๐œ€), or terms of order ๐‘‚(๐œ€๐‘˜+2). Applying a result of [15], Chapter III, page 127 we get: โ€–โ€–๐œผ๐œ€1(๐‘˜)โ€–โ€–(๐‘ก)(๐ป1((0,1)ร—(โˆ’1,1)))2๎€ท๐œ€=๐‘‚๐‘˜+2๎€ธ.(5.18) Expressing next the norm of ๐”(๐‘˜) with respect to the norm of ๐œผ๐œ€1(๐‘˜) we obtain โ€–๐”(๐‘˜)(๐‘ก)โ€–(๐ป1(๐ท1๐œ€โˆฉ{๐‘ฅ1>2๐œ€}))2โ‰ค(1/๐œ€1/2)โ€–๐œผ๐œ€1(๐‘˜)(๐‘ก)โ€–(๐ป1((0,1)ร—(โˆ’1,1)))2, that is, โ€–๐”(๐‘˜)(๐‘ก)โ€–(๐ป1(๐ท1๐œ€โˆฉ{๐‘ฅ1>2๐œ€}))2=๐‘‚(๐œ€๐‘˜+(3/2)). With similar computations we get the same estimate for โ€–๐”(๐‘˜)(๐‘ก)โ€–(๐ป1(๐ท2๐œ€โˆฉ{๐‘ฅ2>2๐œ€}))2, which completes the proof.

We notice that from (5.16) and from the regularity of the right-hand sides of (5.17) it follows that โ€–๐”(๐‘˜)โ€–๐ฟ2(0,๐‘‡;(๐ป1(โ„ฌ๐œ€))2)=๐‘‚(๐œ€๐‘˜+(3/2)).

The function๎๐”๐‘Ž(๐‘˜)=ฬ‚๐ฎ๐‘Ž(๐‘˜)โˆ’๐”(๐‘˜)(5.19) satisfies the same type of boundary conditions as ๐ฎ on ๐œ•โ„ฌ๐œ€. Moreover ๎๐”๐‘Ž(๐‘˜)(๐‘ฅ,0)=0, due to (4.1)4. As a consequence of the previous constructions, we obtain for (๎๐”๐‘Ž(๐‘˜),ฬ‚๐‘๐‘Ž(๐‘˜),๎๐ท(๐‘˜)ยฑ๐‘Ž) the following problem:๐œŒ๐‘“๐œ•๎๐”๐‘Ž(๐‘˜)๐œ•๐‘กโˆ’2div๐‘ฅ๎‚€๐œˆ(๐‘ฅ)๐ท๐‘ฅ๎๐”๐‘Ž(๐‘˜)๎‚+โˆ‡๐‘ฅฬ‚๐‘๐‘Ž(๐‘˜)=๐Ÿ+โ„ฑ๐œ€(๐‘˜)inโ„ฌ๐œ€ร—(0,๐‘‡),div๐‘ฅ๎๐”๐‘Ž(๐‘˜)=0inโ„ฌ๐œ€๐œ•ร—(0,๐‘‡),๐œŒโ„Ž2๎๐ท(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ก2+1๐œ€๐›พ๐œ•4๎๐ท(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ฅ4๐‘–๐œ•+๐œ‡5๎๐ท(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ฅ4๐‘–๐œ•๐‘ก=๐‘”ยฑยฑฬ‚๐‘(๐‘˜)๐‘Ž|๐ท๐‘–๐œ€๐œ€ยฑโˆฉฮ“+๐’ข(๐‘˜)ยฑ๐œ€onฮ“๐œ€ยฑ๎๐ทร—(0,๐‘‡),(๐‘˜)ยฑ๐‘Ž=0onโŒข๐ด๐œ€ยฑ๐ต๐œ€ยฑ๎๐”ร—(0,๐‘‡),๐‘Ž(๐‘˜)=๐๐œ€๎€ทon๐œ•โ„ฌ๐œ€โงต๎€ทฮ“๐œ€+โˆชฮ“๐œ€โˆ’ร—๎๐ท๎€ธ๎€ธ(0,๐‘‡),(๐‘˜)ยฑ๐‘Ž๐œ•๎๐ท(1,ยฑ๐œ€,๐‘ก)=(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ฅ1๎๐ท(1,ยฑ๐œ€,๐‘ก)=0in(0,๐‘‡),(๐‘˜)ยฑ๐‘Ž๐œ•๎๐ท(2๐œ€,ยฑ๐œ€,๐‘ก)=(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ฅ1๎๐ท(2๐œ€,ยฑ๐œ€,๐‘ก)=0in(0,๐‘‡),(๐‘˜)ยฑ๐‘Ž๐œ•๎๐ท(ยฑ๐œ€,1,๐‘ก)=(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ฅ2๎๐ท(ยฑ๐œ€,1,๐‘ก)=0in(0,๐‘‡),(๐‘˜)ยฑ๐‘Ž๐œ•๎๐ท(ยฑ๐œ€,2๐œ€,๐‘ก)=(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ฅ2๎๐”(ยฑ๐œ€,2๐œ€,๐‘ก)=0in(0,๐‘‡),๐‘Ž(๐‘˜)๐œ•๎๐ทโ‹…๐ง=ยฑ(๐‘˜)ยฑ๐‘Ž,๎๐”๐œ•๐‘ก๐‘Ž(๐‘˜)โ‹…๐‰=0onฮ“๐œ€ยฑ๎๐”ร—(0,๐‘‡),๐‘Ž(๐‘˜)(๐‘ฅ,0)=๐ŸŽinโ„ฌ๐œ€,๎๐ท(๐‘˜)ยฑ๐‘Ž๐œ•๎๐ท(๐‘ฅ,0)=(๐‘˜)ยฑ๐‘Ž๐œ•๐‘ก(๐‘ฅ,0)=0onฮ“๐œ€ยฑ,(5.20) with โ„ฑ๐œ€(๐‘˜)=๐…๐œ€(๐‘˜)โˆ’๐œŒ๐‘“(๐œ•๐”(๐‘˜)/๐œ•๐‘ก)+2div๐‘ฅ(๐œˆ(๐‘ฅ)๐ท๐‘ฅ๐”(๐‘˜)),๐’ข(๐‘˜)ยฑ๐œ€=๐บ(๐‘˜)ยฑ๐œ€โˆ’๐œŒโ„Ž(๐œ•2๐ทยฑ(๐‘˜)/๐œ•๐‘ก2)โˆ’(1/๐œ€๐›พ)(๐œ•4๐ทยฑ(๐‘˜)/๐œ•๐‘ฅ4๐‘–)โˆ’๐œ‡(๐œ•5๐ทยฑ(๐‘˜)/๐œ•๐‘ฅ4๐‘–๐œ•๐‘ก).

We are now in a position to apply (3.20) in order to establish the error between the exact solution of (2.6) and its approximation given by (4.4).

Theorem 5.3. Let (ฬ‚๐ฎ๐‘Ž(๐‘˜),ฬ‚๐‘๐‘Ž(๐‘˜),๎๐‘‘(๐‘˜)ยฑ๐‘Ž) be the asymptotic solution of order ๐‘˜ given by (4.4) and (๐ฎ,๐‘,๐‘‘) the exact solution of the physical problem (2.6). Then the following estimates hold: โ€–โ€–ฬ‚๐ฎ๐ฎโˆ’๐‘Ž(๐‘˜)โ€–โ€–๐ฟโˆž(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)=๎‚ป๐‘‚๎€ท๐œ€3/2๎€ธ๐‘‚๎€ท๐œ€for๐พโˆˆ{0,1,2,3},min{๐‘˜โˆ’5/2,๐‘˜+4โˆ’๐›พ}๎€ธโ€–โ€–D๎‚€ฬ‚๐ฎfor๐พโ‰ฅ4,๐ฎโˆ’๐‘Ž(๐‘˜)๎‚โ€–โ€–๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))4)=๎‚ป๐‘‚๎€ท๐œ€3/2๎€ธ๐‘‚๎€ท๐œ€for๐พโˆˆ{0,1,2,3},min{๐‘˜โˆ’5/2,๐‘˜+4โˆ’๐›พ}๎€ธโ€–โ€–โ€–๐œ•for๐พโ‰ฅ4,๎‚€๐‘‘๐œ•๐‘กยฑโˆ’๎๐‘‘(๐พ)ยฑ๐‘Ž๎‚โ€–โ€–โ€–๐ฟโˆž(0,๐‘‡;๐ฟ2(ฮ“๐œ€ยฑ))=๎‚ป๐‘‚๎€ท๐œ€3/2๎€ธ๐‘‚๎€ท๐œ€for๐พโˆˆ{0,1,2,3},min{๐‘˜โˆ’5/2,๐‘˜+4โˆ’๐›พ}๎€ธโ€–โ€–โ€–๐œ•for๐พโ‰ฅ4,2๐œ•๐‘ 2๎‚€๐‘‘ยฑโˆ’๎๐‘‘(๐พ)ยฑ๐‘Ž๎‚โ€–โ€–โ€–๐ฟโˆž(0,๐‘‡;๐ฟ2(ฮ“๐œ€ยฑ))=๎‚ป๐‘‚๎€ท๐œ€3/2+๐›พ/2๎€ธ๐‘‚๎€ท๐œ€for๐พโˆˆ{0,1,2,3},min{๐‘˜โˆ’5/2,๐‘˜+4โˆ’๐›พ}+๐›พ/2๎€ธโ€–โ€–โˆ‡๎‚€for๐พโ‰ฅ4,๐‘โˆ’ฬ‚๐‘๐‘Ž(๐‘˜)๎‚โ€–โ€–๐ฟโˆž(0,๐‘‡;(๐ปโˆ’1(โ„ฌ๐œ€))2)=๎‚ป๐‘‚๎€ท๐œ€5/2โˆ’๐›พ/2๎€ธ๐‘‚๎€ท๐œ€for๐พโˆˆ{0,1,2,3},min{๐‘˜โˆ’5/2,๐‘˜+4โˆ’๐›พ}โˆ’๐›พ/2๎€ธfor๐พโ‰ฅ4.(5.21)

Proof. Let us obtain the estimate (5.21)2. From (5.16) and (5.19) it follows that โ€–โ€–D๎‚€ฬ‚๐ฎ๐ฎโˆ’๐‘Ž(๐‘˜)๎‚โ€–โ€–๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))4)โ‰คโ€–โ€–โ€–D๎‚€๎๐”๐ฎโˆ’๐‘Ž(๐‘˜)๎‚โ€–โ€–โ€–๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))4)๎€ท๐œ€+๐‘‚๐‘˜+3/2๎€ธ.(5.22) It remains to estimate the first term of the right-hand side of the previous inequality. For this we first estimate min(โ€–โ„ฑ๐œ€(๐‘˜)โ€–๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2),โ€–๐’ข(๐‘˜)ยฑ๐œ€โ€–๐ฟ2((๐ท๐‘–๐œ€โˆฉฮ“๐œ€ยฑ)ร—(0,๐‘‡))) and then we apply (3.20)2. This estimate is given either by ๐บ(๐‘˜)ยฑ๐œ€ or by (1/๐œ€๐›พ)(๐œ•4๐ทยฑ(๐‘˜)/๐œ•๐‘ฅ4๐‘–), the other terms having greater order than these. Following the ideas of [2, Theoremโ€‰โ€‰5.1] we obtain (5.21)2; the other estimates of (5.21) are proved with the same technique and the proof is achieved.

In order to improve the estimates (5.21) we analyze the order of the leading term of the asymptotic solution. We prove that.

Proposition 5.4. For the leading term of the asymptotic solution one has the following estimates: โ€–โ€–ฬ‚๐ฎ๐‘Ž(0)โ€–โ€–๐ฟโˆž(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)๎€ท๐œ€=๐‘‚5/2๎€ธ,โ€–โ€–๐ทฬ‚๐ฎ๐‘Ž(0)โ€–โ€–๐ฟ2(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))4)๎€ท๐œ€=๐‘‚3/2๎€ธ,โ€–โ€–โ€–๐œ•๎๐‘‘๐œ•๐‘ก(0)ยฑ๐‘Žโ€–โ€–โ€–๐ฟโˆž(0,๐‘‡;๐ฟ2(ฮ“๐œ€ยฑ))=๐‘‚(๐œ€๐›พโ€–โ€–โ€–๐œ•),2๐œ•๐‘ 2๎๐‘‘(0)ยฑ๐‘Žโ€–โ€–โ€–๐ฟโˆž(0,๐‘‡;๐ฟ2(ฮ“๐œ€ยฑ))=๐‘‚(๐œ€๐›พโ€–โ€–),โˆ‡ฬ‚๐‘๐‘Ž(0)โ€–โ€–๐ฟโˆž((0,๐‘‡);(๐ปโˆ’1(โ„ฌ๐œ€))2)๎€ท๐œ€=๐‘‚1/2๎€ธ.(5.23)

Proof. Let us establish (5.23)1. From (4.48)1,2 (for ๐›พ>3) and from (4.63)1,2 (for ๐›พ=3) we obtain for the regular velocity corresponding to ๐ท1๐œ€โ€–โ€–๐ฎ1(0)๐œ’โ€–โ€–๐ฟโˆž(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)๎€ท๐œ€=๐‘‚5/2๎€ธ.(5.24) This means that the regular part of the asymptotic velocity in the norm of (5.24) is ๐‘‚(๐œ€5/2). It remains to estimate in the same norm the boundary layer functions. For the corrector corresponding to ๐‘ฅ1=1 we have โ€–โ€–๐ฎ(0)1๐‘๐‘™๐œ‚โ€–โ€–(๐ฟ2(โ„ฌ๐œ€))2โ‰ค๐œ€3๎‚ต๎€œ0โˆ’1/4๐œ€๎€œ1โˆ’1๎‚€๐ฎ0(1)(๐œ‰1,๐œ‰2๎‚,๐‘ก)2๎‚ถd๐œ‰1/2โ‰ค๐œ€3๎‚ต๎€œ0โˆ’โˆž๎€œ1โˆ’1๎‚€๐ฎ0(1)(๐œ‰1,๐œ‰2๎‚,๐‘ก)2๎‚ถd๐œ‰1/2๎€ท๐œ€=๐‘‚3๎€ธ,(5.25) which yields โ€–โ€–๐ฎ(0)1๐‘๐‘™๐œ‚โ€–โ€–๐ฟโˆž(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)๎€ท๐œ€=๐‘‚3๎€ธ.(5.26) Finally, for the corrector in ๐‘ฅ=๐ŸŽ we have โ€–โ€–๐ฎ(0)0๐‘๐‘™โ€–โ€–๐œ‚โ‹…๐œ‚(๐ฟ2(โ„ฌ๐œ€))2โ‰ค๐œ€3๎‚ต๎€œโ„ฌโˆฉ{๐œ‰1<1/๐œ€4,๐œ‰2<1/๐œ€4}๎‚€๐ฎ0(0)(๐œ‰1,๐œ‰2๎‚,๐‘ก)2๎‚ถd๐œ‰1/2๎€ท๐œ€=๐‘‚3๎€ธ,(5.27) that leads to an estimate of the same type as (5.26). Hence, we obtain (5.23)1 as a consequence of (5.24) and (5.26). More details concerning this type of computations can be found in [5, Theoremโ€‰โ€‰5.1].

The previous Proposition allows us to improve the estimates given by Theorem 5.3 in the following sense.

Theorem 5.5. Let (ฬ‚๐ฎ๐‘Ž(๐‘—),ฬ‚๐‘๐‘Ž(๐‘—),๎๐‘‘(๐‘—)ยฑ๐‘Ž) be the asymptotic solution of order ๐‘— and (๐ฎ,๐‘,๐‘‘) the exact solution of the physical problem. Then the following estimates hold: โ€–โ€–ฬ‚๐ฎ๐ฎโˆ’๐‘Ž(๐‘—)โ€–โ€–๐ฟโˆž(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)๎€ท๐œ€=๐‘‚๐‘—+7/2๎€ธ,โ€–โ€–D๎‚€ฬ‚๐ฎ๐ฎโˆ’๐‘Ž(๐‘—)๎‚โ€–โ€–๐ฟ2((0,๐‘‡);(๐ฟ2(โ„ฌ๐œ€))4)๎€ท๐œ€=๐‘‚๐‘—+5/2๎€ธ,โ€–โ€–โ€–๐œ•๎‚€๐‘‘๐œ•๐‘กยฑโˆ’๎๐‘‘(๐‘—)ยฑ๐‘Ž๎‚โ€–โ€–โ€–๐ฟโˆž(0,๐‘‡;๐ฟ2(ฮ“๐œ€ยฑ))๎€ท๐œ€=๐‘‚๐‘—+๐›พ+1๎€ธ,โ€–โ€–โ€–๐œ•2๐œ•๐‘ 2๎‚€๐‘‘ยฑโˆ’๎๐‘‘(๐‘—)ยฑ๐‘Ž๎‚โ€–โ€–โ€–๐ฟโˆž(0,๐‘‡;๐ฟ2(ฮ“๐œ€ยฑ))๎€ท๐œ€=๐‘‚๐‘—+๐›พ+1๎€ธ,โ€–โ€–โˆ‡๎‚€๐‘โˆ’ฬ‚๐‘๐‘Ž(๐‘—)๎‚โ€–โ€–๐ฟโˆž((0,๐‘‡);(๐ปโˆ’1(โ„ฌ๐œ€))2)๎€ท๐œ€=๐‘‚๐‘—+3/2๎€ธ.(5.28)

Proof. Let ๐‘—โ‰ฅ0 be a fixed integer and ๐‘˜โ‰ซ๐‘—. Then from (5.21)1 and (5.23)1 we get โ€–โ€–ฬ‚๐ฎ๐ฎโˆ’๐‘Ž(๐‘—)โ€–โ€–๐ฟโˆž(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)โ‰คโ€–โ€–ฬ‚๐ฎ๐ฎโˆ’๐‘Ž(๐‘˜)โ€–โ€–๐ฟโˆž(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)+โ€–โ€–ฬ‚๐ฎ๐‘Ž(๐‘˜)โˆ’ฬ‚๐ฎ๐‘Ž(๐‘—)โ€–โ€–๐ฟโˆž(0,๐‘‡;(๐ฟ2(โ„ฌ๐œ€))2)๎€ท๐œ€=๐‘‚min{๐‘˜โˆ’5/2,๐‘˜+4โˆ’๐›พ}๎€ธ๎€ท๐œ€+๐‘‚๐‘—+7/2๎€ธ๎€ท๐œ€=๐‘‚๐‘—+7/2๎€ธ.(5.29) The other estimates of (5.28) are obtained in a similar way, and the proof is achieved.

Acknowledgments

The authors were partially supported by the following grants: SFR MODMAD of the University of Saint-Etienne, ENISE and the School of Mines of Saint-Etienne, the joint French-Russian PICS CNRS grant โ€œMathematical modeling of blood diseasesโ€ and by the Grant no. 14.740.11.0875 โ€œMultiscale problems: analysis and methodsโ€ of the Ministry of Education and Research of Russian Federation.