Laurent Chupin
email: laurent.chupin@math.univ-bpclermont.fr

Sébastien Martin
email: sebastien.martin@parisdescartes.fr

Blaise Pascal

VISCOELASTIC FLOWS IN A ROUGH CHANNEL: A MULTISCALE ANALYSIS

In this paper, we consider viscoelastic flows in a rough domain (with typical roughness patterns of size ε ≪ 1). We present and rigorously justify an asymptotic expansion with respect to ε, at any order, based upon the definition of elementary problems: Oldroyd-type problems at the global scale defined on a smoothened domain and boundary-layer corrector problems. The resulting analysis guarantees optimality with respect to the truncation error.

Introduction

Many studies investigate the effect of wall roughness on Newtonian flows. In 1827, C. L. Navier [START_REF] Navier | Sur les lois de l'équilibre et du mouvement des corps élastiques[END_REF] was one of the first scientists to note that the roughness could drag a fluid. Since then, numerous studies attempted to prove mathematical results in this direction, see for instance the works of W. Jäger and A. Mikelic [START_REF] Jäger | On the roughness-induced effective boundary conditions for an incompressible viscous flow[END_REF], Y. Amirat and co-authors [START_REF] Amirat | Effect of rugosity on a flow governed by stationary Navier-Stokes equations[END_REF][START_REF] Amirat | Influence de la rugosité en hydrodynamique laminaire[END_REF] and more recently the works of D. Bresch and V. Milisic [START_REF] Bresch | High order multi-scale wall-laws, Part I: the periodic case[END_REF]. Note that all these works formulate the roughness using a periodic function (whose amplitude and period are supposed to be small). In a context of more general "roughness" patterns, there exists similar recent results, see [START_REF] Basson | Wall laws for fluid flows at a boundary with random roughness[END_REF][START_REF] Gérard-Varet | The Navier wall law at a boundary with random roughness[END_REF]. All the previous work deal with a Newtonian flow, for which the Stokes or Navier-Stokes equations are classically considered.

Much literature research has been devoted to non-Newtonian fluids, in both mathematical aspects and applications. It is well known that numerous biological fluids, blood or physiological secretions like tears or synovial fluids, show these non-Newtonian characteristics. In engineering applications people are interested in controlling the flows characteristics to suit various requirements such as maintaining the fluid qualities in a wide range of temperatures and stresses. Introduction of additives lead to non-Newtonian behavior of the modern lubricants for instance. Another application domain is linked to polymers, whose non-Newtonian characteristics appear in a wide range of applications such as the molding or injection processes. Some particular classes of non-Newtonian models have often been considered. This includes the Bingham flow or the quasi-Newtonian fluids (Carreau's law [START_REF] Carreau | An analysis of the viscous behaviour of polymeric solutions[END_REF][START_REF] Kennedy | Flow Analysis of Injection Molds[END_REF], the power law or Williamson's law, in which various stress-velocity relations are chosen [START_REF] Tanner | Rheology: an Historical Perspective, volume 7 of Rheology series[END_REF] or [START_REF] Blavier | On the stationary quasi-Newtonian flow obeying a power-law[END_REF][START_REF] Boughanim | Derivation of the two-dimensional Carreau law for a quasi-Newtonian fluid flow through a thin slab[END_REF] for mathematical aspects) and also micropolar ones [START_REF] Łukaszewicz | Micropolar fluids. Modeling and Simulation in Science[END_REF]. These models, however, consider the fluid as viscous and elasticity effects are neglected. The introduction of such a viscoelastic behavior is primarily described by the Weissenberg number, denoted We which can be viewed as a measure of the elasticity of the fluid and is related to its characteristic relaxation time. One of the laws which seems the most able to describe viscoelastic flows is the Oldroyd model. This model is based on a constitutive equation which is an interpolation between Date: March 11, 2015. purely viscous and purely elastic behaviors, thus introducing a supplementary parameter r which describes the relative proportion of both behaviors (the solvent to solute ratio). Considering the Oldroyd model [START_REF] Oldroyd | On the formulation of rheological equations of state[END_REF], the momentum, continuity and constitutive equations for an incompressible flow of such a non-Newtonian fluid are, respectively, [START_REF] Achdou | Effective boundary conditions for laminar flows over periodic rough boundaries[END_REF] ρ ∂ t u + u • ∇uη(1r)∆u + ∇pdiv σ = 0,

(2) div u = 0,

(3) λ ∂ t σ + u • ∇σ + g a (∇u, σ) + σ -δ∆σ = 2ηrD(u).

In these equations, ρ, η and λ are positive constants which respectively correspond to the fluid density, the fluid viscosity and the relaxation time. It is important to notice the presence of a term δ∆σ, δ > 0 corresponding to a spatial diffusion of the polymeric stresses. Usually (i.e. for the Oldroyd model) this term is deleted, but it can be physically justified: physical effects that can contribute to the diffusive process include hydrodynamic interactions [START_REF] El-Kareh | Existence of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion[END_REF], particle diffusion [START_REF] Bhave | A constitutive equation for liquid-crystalline polymer solutions[END_REF] and semiflexibility of polymer blends [START_REF] Liu | Free energy functionals for semiflexible polymer solutions and blends[END_REF]. Thus, the diffusive Oldroyd models have been the subject of intense studies related to the understanding of shear-banded flows or phase coexistence, see [START_REF] Olmsted | Coexistence and phase separation in sheared complex fluids[END_REF][START_REF] Spenley | Nonmonotonic constitutive laws and the formation of shear-banded flows[END_REF][START_REF] Olmsted | Phase coexistence of complex fluids in shear flow[END_REF][START_REF] Lu | Effects of nonlocal stress on the determination of shear banding flow[END_REF][START_REF] Olmsted | Johnson-Segalman model with a diffusion term in cylindrical Couette flow[END_REF] and, as a consequence, the investigation of the mathematical properties of the diffusive model has gained an increasing interest recently [START_REF] El-Kareh | Existence of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion[END_REF][START_REF] Barrett | Existence of global weak solutions for some polymeric flow models[END_REF][START_REF] Constantin | Note on global regularity for two-dimensional Oldroyd-B fluids with diffusive stress[END_REF][START_REF] Chupin | Stationary Oldroyd model with diffusive stress: mathematical analysis of the model and vanishing diffusion process[END_REF].

Equations ( 1)-( 3) make up a system of 10 scalar equations with 10 unknowns: the lubricant velocity vector u = (u i ) 1≤i≤3 , the pressure p and the extra-stress symmetric tensor σ = (σ i,j ) 1≤i,j≤3 . The bilinear application g a , -1 ≤ a ≤ 1, is defined by

g a (∇u, σ) = σ • W(u) -W(u) • σ -a(σ • D(u) + D(u) • σ)
where D(u) and W(u) are respectively the symmetric and skew-symmetric parts of the velocity gradient ∇u. Usually, D(u) is called the rate of strain tensor and W(u) is called the vorticity tensor. Notice that the parameter a is considered to interpolate between upper convected (a = 1) and lower convective derivatives (a = -1), the case a = 0 being the corotational case [START_REF] Joseph | Fluid Dynamics of Viscoelastic Liquids[END_REF]. Note that taking r = 1 allows us to recover various forms of the generalized Maxwell model. Conversely, a Newtonian flow is described by choosing r = 0.

In this paper, we focus on viscoelastic flows in a rough domain (with typical roughness patterns of size ε ≪ 1). We present and rigorously justify an asymptotic expansion with respect to ε. The development is done at any order, so that we are guaranteed to be optimal with respect to the truncation error. We also highlight the particular effects of roughness.

Several relevant questions are not addressed in this article. First, recent works on random roughness, see [START_REF] Basson | Wall laws for fluid flows at a boundary with random roughness[END_REF][START_REF] Gérard-Varet | The Navier wall law at a boundary with random roughness[END_REF], could make us think that our results can be extended to more general cases of roughness. In fact, the construction of our development strongly depends on the behavior of solutions of the Stokes equation on a halfspace, whose lower boundary is periodic. The behavior of such solutions must be sufficiently decreasing at infinity to justify our development. Unfortunately, it seems that this decrease is only logarithmic in the case of a random boundary (while it is exponential in our periodic case). Second, another task related to the regularity of the roughness patterns is not addressed in this paper: what is the behavior of the solution when the patterns are not Lipschitz continuous? In particular, what is the influence of roughness jump discontinuities over the flow? Finally, the choice to make appear a spatial diffusion (δ > 0) in the Oldroyd model could by argue. From a mathematical point of view, it is clearly an advantage since we know that such diffusion allows to have solutions to the initial problem (1)- [START_REF] Amirat | Effect of rugosity on a flow governed by stationary Navier-Stokes equations[END_REF]. Nevertheless, even if we admit the existence of a smooth solution to the initial problem without diffusion (δ = 0), the development proposed with respect to the roughness parameter ε seems to be unsuitable.

The paper is composed of five sections. In Section 2, we introduce the diffusive Oldroyd model, we precisely describe the roughness geometry and we recall a fundamental result: there exists a solution (u, p, σ) to the model. In Section 3 we introduce the ansatz, which is a formal asymptotic expansion of the solution (u, p, σ) with respect to the roughness parameter ε ≪ 1. By identifying the powers of ε in such a development, we obtain some elementary problems at any order. Section 4 is devoted to the mathematical study of these elementary problems: well-posedness and properties. Section 5 provides a rigorous justification of the asymptotic expansion by analyzing the remainder and deriving error estimates. In Section 6, we show that it is possible to effectively determine all the contributions of the ansatz. We will notice that regarding each elementary problem and their overlaps, this crucial result is not obvious: we prove that the solution of each problem can be built using the only previous elementary solutions.

The diffusive Oldroyd system in a rough channel: statement of the problem

As a matter of fact, the derivation of reduced models is crucial if one aims at performing numerical simulations of the flow. However, roughness patterns lead to a sharp increase of the computational costs because the mesh of the domain has to be built according to the constraints defined by the roughness patterns. In order to avoid such a costly procedure, reduced models can be defined by considering the fluid flow in the smooth domain (therefore avoiding heavy costs in terms of numerical computations by using coarse meshes) and adding a so-called boundary layer correction which takes into account the influence of the roughness pattern. More precisely, for a Newtonian flow, we can proceed as follows (see [START_REF] Navier | Sur les lois de l'équilibre et du mouvement des corps élastiques[END_REF][START_REF] Achdou | Effective boundary conditions for laminar flows over periodic rough boundaries[END_REF][START_REF] Jäger | On the roughness-induced effective boundary conditions for an incompressible viscous flow[END_REF][START_REF] Bresch | High order multi-scale wall-laws, Part I: the periodic case[END_REF], and also [START_REF] Gérard-Varet | The Navier wall law at a boundary with random roughness[END_REF][START_REF] Basson | Wall laws for fluid flows at a boundary with random roughness[END_REF] for random roughness patterns):

• at order 0: assume that u is the velocity fluid associated to the rough domain ω ε and u 0 is the velocity fluid associated to the smooth domain (i.e. the domain has been truncated by considering a flat boundary instead of the oscillating one and imposing the Dirichlet condition on the smooth boundary), see Fig. 1. Then one has

u -u 0 L 2 (ωε) ε and u -u 0 H 1 (ωε) √ ε.
• at order 1: in order to counterbalance the error introduced by the truncation of the domain, it is possible to introduce a corrector term which leads to the justification of wall laws. By considering the smooth domain with a slip condition defined as

       u (1) = α 11 ∂ y u (1) + α 12 ∂ y u (2)
u (2) = α 21 ∂ y u (1) + α 22 ∂ y u (2) u (3) = 0 on the smooth boundary (the reals α i,j being coefficients depending on the roughness shape), a velocity field u is defined, leading to a refinement of the approximation. For instance we have (see [START_REF] Jäger | On the roughness-induced effective boundary conditions for an incompressible viscous flow[END_REF]) For a non-Newtonian flow of the viscoelastic type, we aim at describing the roughness effects extensively in the following sense:

u -u L 2 (ωε) ε 3/2 and u -u H 1 (ωε) ε. 1 ε ε 1 1 1/ε
• build an asymptotic expansion based upon elementary solutions (i.e. solutions of problems defined on smooth domains, thus avoiding complex geometries) ; • prove in a rigorous way that the asymptotic expansion is valid at any order ;

• define an algorithm associated to an efficient numerical procedure. Therefore we consider the Navier-Stokes equations with the Oldroyd model in a rough channel. Consider the domain

ω ε := (x, y) ∈ T d × R, -εH x ε < y < 1 ,
where

T d is the d-dimensional torus, d = 1 or d = 2
, and H a smooth periodic and positive function. The boundary ω ε is denoted γ ε and it is composed of two connex components: The upper smooth boundary γ + = T d × {1} and the upper highly oscillating boundary which is denoted γ - ε , see Fig. 2. Then we consider the following set of equations:

(4)            Re (u • ∇u) -(1 -r)∆u + ∇p = div σ + f, in ω ε , div u = 0, in ω ε , We (u • ∇σ + g a (∇u, σ)) + σ -D ∆σ = 2rD(u), in ω ε , u = 0, on γ ε , D ∂ n σ = 0, on γ ε . 1 ε ε γ + γ - ε Figure 2.
Channel with oscillating boundary.

This system is the non-dimensional version of the system (1)-( 3). We have introduced the Reynolds number Re, the Weissenberg number We, a relaxation parameter r ∈ [0, 1] and the diffusion coefficient D.

In the next sections, we aim at describing the structure of the solution (u, p) with respect to the roughness number ε. Before entering into this description, let us recall the main mathematical results related to the stationary diffusive Oldroyd model. The problem defined in a strong form can be associated to a variational formulation. Then we have (see [START_REF] Chupin | Stationary Oldroyd model with diffusive stress: mathematical analysis of the model and vanishing diffusion process[END_REF] for details):

Theorem 1. Let f ∈ H -1 (ω ε ) 3 . Let Re ≥ 0, We ≥ 0, 0 < r < 1, -1 ≤ a ≤ 1 and D > 0.
Let us introduce the following constants:

C (I) := 8|a|C 2 ωε We f H -1 min(1 -r, D) 2 , C (II) := √ 2r min(1 -r, D) 4|a|C 2 ωε We 1 -1 -C (I) ,
where C ωε is a constant which only depends on the domain ω ε .

• Existence. If C (I) ≤ 1 then, the problem (4) admits a variational solution (u, σ) which satisfies

2r ∇u 2 L 2 + σ 2 H 1 ≤ C 2 (II) .
Moreover there exists a pressure field p ∈ L 2 (Ω) such that (u, p) satisfies the first equation of the problem (4) in the sense of distributions. • Uniqueness. The variational formulation of the problem (4) admits at most one solution if one of the following conditions is satisfied: a) f H -1 is small enough; b) Re and We are small enough. • Regularity. If f is regular then the variational solution is regular and satisfies the problem (4) in a classical sense.

Let us remark that the corotationnal case (namely a = 0) allows us to get rid of the smallness assumption on the data.

Asymptotic expansion

3.1. Main ideas: ansatz. Let us describe the structure of the solution by using a suitable ansatz:

                                             u(x, y) = u 0 (x, y) + εU 1 x, x ε , y ε + εu 1 (x, y) + ε 2 U 2 x, x ε , y ε + • • • = +∞ k=0 ε k u k (x, y) + U k x, x ε , y ε , p(x, y) = p 0 (x, y) + P 1 x, x ε , y ε + εp 1 (x, y) + εP 2 x, x ε , y ε + • • • = +∞ k=0 ε k p k (x, y) + P k+1 x, x ε , y ε , σ(x, y) = σ 0 (x, y) + εΣ 1 x, x ε , y ε + εσ 1 (x, y) + ε 2 Σ 2 x, x ε , y ε + • • • = +∞ k=0 ε k σ k (x, y) + Σ k x, x ε , y ε .
The definition of the asymptotic expansion has to be completed by the description of the problems satisfied by the elementary solutions. Then another task consists in showing that each elementary problem is well posed. The final task consists in showing that the level of truncation in the asymptotic expansion is directly related to the quality of the approximation of the exact solution.

In order to identify the elementary problems satisfied by the elementary solutions, we proceed as follows:

(1) separation of the macroscopic variables (x, y) and the microscopic ones which will be denoted (X, Y ) = ( x ε , y ε ) ; (2) identification of terms with the same order with respect to ε in the equations ;

(3) identification of terms with the same order with respect to ε in the boundary conditions.

Notation 1. We will use lowercase letters to denote elements corresponding to the real physical domain (x, u, σ, ω...) and uppercase letters for all that concerns the microscopic field (X, U, Σ, Ω...).

To complete this subsection, and before getting into the details, let us explain the way the asymptotic expansion has been built: at main order, the fluid flow is governed by a classical viscoelastic model with boundary conditions located at the flat bottom y = 0 (definition of (u 0 , p 0 , σ 0 )). But, of course, the boundary layer has been omitted and, in fact, the boundary condition should have been imposed on the oscillating boundary instead of the smooth one ; it can be shown that the resulting error is of order ε (the size of the boundary layer) and, therefore, a so-called boundary layer correction is introduced in order to counterbalance the mentioned boundary value default (definition of (U 1 , P 1 , Σ 1 )). Now if we analyze the approximate solution defined as (u 0 + εU 1 , p 0 + P 1 , σ 0 + εΣ 1 ), the equations in the domain and the boundary condition at the oscillating boundary are satisfied by means of construction ; unfortunately, the boundary condition on the upper boundary is not satisfied by the approximate solution because of the behavior of the boundary layer corrector at infinity. However the error is of order ε and it is is possible to build a solution on the smooth domain (definition of (u 1 , p 1 , σ 1 )) which counterbalances this boundary value default on the upper boundary. Again, at his step, by considering the smooth domain only (in particular, homogeneous boundary conditions are considered at the lower smooth boundary), the boundary layer has been omitted and, in fact, the boundary condition should have been imposed on the oscillating boundary ; the resulting error is now of order ε 2 . Thus, if we compare

(u ε , p ε , σ ε ) -(u 0 , p 0 , σ 0 ) and (u ε , p ε , σ ε ) -(u 0 + εU 1 + εu 1 , p 0 + P 1 + εp 1 , σ 0 + εΣ 1 + εσ 1 )
, the error has been decreased by an order of magnitude and, besides, the same procedure can be applied by introducing a boundary layer correction which counterbalances the boundary value default of order ε 2 at the oscillating boundary.

In a more general way, the boundary value default introduced at the oscillating boundary can be counterbalanced by a boundary layer correction ; the resulting boundary value default at the upper boundary can be counterbalanced by a viscoelastic flow defined in the smooth domain. Through this procedure, the resulting approximation satisfies the equations in the domain, the boundary condition at the upper boundary and the error on the boundary condition on the oscillating boundary has been decreased by an order of magnitude.

Let us define the two rescaled sub-domains. As a matter of fact, the main flow is defined on the smooth domain whereas, due to the consideration of the roughness patterns, the boundary layer is rescaled by the homothetic transformation (X, Y ) := ( x ε , y ε ). Definition 1. The smooth domain is defined by

ω 0 := (x, y) ∈ T d × R, 0 < y < 1 .
The upper boundary equals γ + and the lower boundary corresponds to γ - 0 . The normal outward unit vector the lower (resp. upper) boundary is n = (0, -1) (resp. (0, 1)) on γ - 0 (resp. γ + ).

Definition 2. The boundary layer domain is defined by

Ω = (X, Y ) ∈ T d × R, -H(X) < Y . The boundary {(X, Y ) ∈ T d × R, Y = -H(X)} is denoted Γ.
We denote by N := -(∇H, 1) the outward vector to the lower boundary. Note that N is not a unit vector.

Notation 2. The usual notation for classical operator of derivation are ∇, div and ∆. The problems considered in this paper make appear two kinds of functions: the first ones, like the velocity u, which only depend on the macroscopic variables (x, y), and the others, like the velocity U which depend on (x, X, Y ). In the first case the classical operators are defined as usual, for instance

∆u = d ℓ=1 ∂ 2 x ℓ u + ∂ 2 y u.
In the case of function depending on (x, X, Y ), the notations are the following:

∆U = d ℓ=1 ∂ 2 X ℓ U + ∂ 2 Y U and ∆ x U = d ℓ=1 ∂ 2 x ℓ U.
Notice that the divergence with respect to the variable x is also defined for a function U : R d+1 → R d+1 by

div x U = d ℓ=1 ∂ x ℓ U (ℓ) .
The normal derivative of a function U defined on Γ is defined by

∂ N U = (N • ∇)U = - d ℓ=1 ∂ x ℓ H ∂ x ℓ U -∂ Y U.
3.2. Elementary problems. Let us define the problem at the main scale: it consists in considering the viscoelastic problem on the smooth domain, i.e. by truncating the rough boundary from the initial domain, associated to homogeneous conditions on both boundaries.

Main order:

pb (0)                        Re (u 0 • ∇u 0 ) -(1 -r)∆u 0 + ∇p 0 -div(σ 0 ) = f 0 + a 0 , on ω 0 div(u 0 ) = 0, on ω 0 We (u 0 • ∇σ 0 + g a (∇u 0 , σ 0 )) + σ 0 -D∆σ 0 = 2rD(u 0 ), on ω 0 u 0 = 0, on γ - 0 ∪ γ + ∂ n σ 0 = 0, on γ + ∂ n σ 0 = b 0 , on γ - 0
where f 0 = f is the source term coming from the modelisation, and where the constants a 0 and b 0 have to be fixed.

Remark 1. The choice of constants a 0 and b 0 will be discussed further. Roughly speaking, they play the role of a degree of freedom which will be fixed in order to ensure the well-posedness of elementary problems and a suitable behavior of elementary solutions to be defined. From a practical point of view, we will set a 0 = b 0 = 0.

Imposing the homogeneous Dirichlet condition on the smooth lower boundary γ - 0 instead of the oscillating one γ - ε is a source of error. Indeed, (u 0 , p 0 , σ 0 ) satisfies all the equations except the boundary condition at the oscillating boundary γ - ε . The value at the oscillating boundary can be determined by a Taylor expansion:

u 0 (x, -εH( x ε )) = +∞ k=0 -εH( x ε ) k k! ∂ k y u 0 (x, 0). Since u 0 (x, 0) = 0, one can check that u 0 | γ - ε is of order ε, namely u 0 (x, -εH( x ε )) = -εH( x ε )∂ y u 0 (x, 0) + O(ε 2
), and this leading term of order ε will be counterbalanced by first the boundary layer corrector.

Remark 2. Note that the other terms (of orders ε 2 , ε 3 etc.) will be treated and counterbalanced in subsequent boundary layer problems.

Remark 3. The same methodology applies for the Neumann condition related to the elastic tensor.

As a consequence of the previous remarks, we define the following boundary layer problem.

Correction with boundary layer n o 1:

PB (1)                    -(1 -r)∆U 1 + ∇P 1 = 0, on Ω div(U 1 ) = 0, on Ω U 1 = H∂ y u 0 | γ - ε , on Γ -D ∆Σ 1 = 0, on Ω ∂ N Σ 1 = (∇H • ∇ x )σ 0 | γ - ε -b 0 on Γ. Remark 4.
Let us present the behavior of the solution (the proofs will be given by proposition 1 later).

(1) When Y → +∞, the velocity U 1 exponentially decreases towards the constant defined as

U ∞ := lim Y →∞ T d U 1 (•, Y ).
(2) For b 0 = 0, the elastic tensor Σ 1 exponentially decreases towards 0.

At this stage, it can be shown that the approximation satisfies the equations in the domain and at the oscillating boundary. At the upper boundary, the Neumann boundary condition for the elastic tensor is satisfied because of the exponential decay of σ 0 at infinity. However the homogeneous Dirichlet condition is not satisfied for the velocity field. This is why a so-called main corrector defined on the smooth domain is defined in order to counterbalance the boundary default introduced by the approximation at the upper boundary.

Defining the linear operators L (A) , L (B) and L (C) by

L (A) (u 1 ) := Re (u 1 • ∇u 0 + u 0 • ∇u 1 ), L (B) (u 1 ) := We (u 1 • ∇σ 0 + g a (∇u 1 , σ 0 )), L (C) (σ 1 ) := We (u 0 • ∇σ 1 + g a (∇u 0 , σ 1 )),
we define the following problems:

Main correction n o 1:

pb (1)                                L (A) (u 1 ) -(1 -r)∆u 1 + ∇p 1 = div(σ 1 ) + a 1 , on ω 0 div(u 1 ) = 0, on ω 0 L (B) (u 1 ) + L (C) (σ 1 ) + σ 1 -D∆σ 1 = 2rD(u 1 ), on ω 0 u 1 = -lim Y →∞ T U 1 , on γ + u 1 = 0, on γ - 0 ∂ n σ 1 = 0, on γ + ∂ n σ 1 = b 1 , on γ - 0 .
Correction with boundary layer n o 2:

PB (2)                            -(1 -r)∆U 2 + ∇P 2 = F 2 -a 0 , on Ω div(U 2 ) = 0, on Ω U 2 = H∂ y u 1 | γ - 0 -1 2 H 2 ∂ 2 y u 0 | γ - 0 on Γ -D ∆Σ 2 = G 2 , on Ω ∂ N Σ 2 = -H∂ 2 y σ 0 | γ - 0 + (∇H • ∇ x )σ 1 | γ - 0 -H(∇H • ∇ x )∂ y σ 0 | γ - 0 +(∇H • ∇ x )Σ 1 -b 1 on Γ.
with

F 2 := div(Σ 1 ) -∇ x P 1 + 2(1 -r)∇ x • ∇U 1 -Re(u 0 • ∇U 1 ), G 2 := 2rD(U 1 ) + 2D ∇ x • ∇Σ 1 -We (u 0 • ∇Σ 1 + g a (∇U 1 , σ 0 )) .
Remark 5. With a suitable choice of a 0 (resp. b 1 ), the problem related to the velocity field (resp. constraint field) is well-posed and satisfies the property with exponential decay towards a constant (resp. 0). Moreover, b 1 does not depend on σ 1 ! Main correction at order k ≥ 2:

pb (k)                            L (A) (u k ) -(1 -r)∆u k + ∇p k = div(σ k ) + f k + a k on ω 0 , div(u k ) = 0 on ω 0 , L (B) (u k ) + L (C) (σ k ) + σ k -D∆σ k = 2rD(u k ) + g k on ω 0 , u k = -lim Y →∞ T U k on γ + , u k = 0 on γ - 0 , ∂ n σ k = 0 on γ + , ∂ n σ k = b k on γ - 0 .
where f k and g k only depend on solutions that were defined previously:

f k := -Re k-1 i=1 u i • ∇u k-i , g k := We k-1 i=1 (u i • ∇σ k-i + g(∇u i , σ k-i )).
Higher order correction terms of the asymptotic expansion are defined as the solutions of the following elementary problems.

Correction with boundary layer of order k ≥ 3:

PB (k)                  -(1 -r)∆U k + ∇P k = F k -a k-2 on Ω, div(U k ) = 0
on Ω,

U k = D k on Γ, -D ∆Σ k = G k on Ω, ∂ N Σ k = N k -b k-1 on Γ,
where F k , D k , G k and N k only depend on solutions that were defined previously.

F k := div(Σ k-1 ) + 2(1 -r)∇ x • ∇U k-1 +(1 -r)∆ x U k-2 -∇ x P k-1 + div x (Σ k-2 ) -Re k-2 i=0 ((u i + U i ) • (∇U k-1-i + ∇ x U k-2-i ) + U i • ∇u k-i-2 ) , D k := - k p=1 (-1) p p! H p ∂ (p) y u k-p | b , G k := 2rD(U k-1 ) + 2rD x (U k-2 ) -Σ k-2 + 2D ∇ x • ∇Σ k-1 + D ∆ x Σ k-2 -We k-2 i=0 u i • (∇Σ k-1-i + ∇ x Σ k-i-2 ) +U i • (∇σ k-i-2 + ∇Σ k-1-i + ∇ x Σ k-i-2 ) +g a (∇u i , Σ k-i-2 ) + g a (∇U i+1 + ∇ x U i , σ k-i-2 + Σ k-i-2 ) , N k := (∇H • ∇ x )σ k-1 | γ - 0 + (∇H • ∇ x )Σ k-1 + k-1 p=1 (-1) p p! H p ∂ (p) y ((∇H • ∇ x )σ k-1-p + ∂ y σ k-1-p ) | γ - 0 .

Analysis of the elementary problems: well-posedness and properties of the solutions

The elementary problems related to the boundary layer correctors with respect to the velocity take the following form:

PB (ref1)      -(1 -r)∆U + ∇P = F on Ω, div(U) = 0 on Ω, U = D on Γ, with F ∈ L 2 (Ω) and D ∈ L 2 (T d ).
Notation 3. We denote F j the following Fourier coefficients of a function F defined on T d × (0, +∞):

F (X, Y ) = j∈Z d F j (Y )e 2πij•X .
Notation 4. All constants depending only on the domain and on physical constants will be considered harmless, there will be denoted C. In the same way, we will use the notation Q(Y ) to denote any polynomial with coefficients depending only on the domain or on physical constant. In particular the quantities C and Q(Y ) do not depend on the variables x or X.

Proposition 1. If the following conditions are satisfied

F 0 (Y ) ≤ Q(Y ) e -Y , for all Y > 0, F j (Y ) ≤ Q(Y ) e -j Y , for all j ∈ Z d \ {0} and Y > 0,
then PB (ref1) admits a unique solution (U, P ) satisfying ∇U ∈ L 2 (Ω) and P ∈ L 2 (Ω).

(1) There exists

U ∞ ∈ R d such that U 0 (Y ) -U ∞ + P 0 (Y ) ≤ Q(Y ) e -Y , for all Y > 0, U 0 ′ (Y ) + P 0 ′ (Y ) ≤ Q(Y ) e -Y ,
for all Y > 0.

(2) We have

U(X, Y ) -U ∞ + |P (X, Y )| ≤ Q(Y ) e -Y , for all X ∈ T d , Y > 0,
and, for all ℓ ≥ 1,

∂ ℓ X U(X, Y ) + |∂ ℓ X P (X, Y )| ≤ Q(Y ) e -Y
, for all X ∈ T d , Y > 0. Remark 6. In this Proposition, the vector U ∞ can be identified as the limit of U 0 (Y ) when Y goes to +∞:

U ∞ := lim Y →∞ T U ∈ R d .
Proof. Let us introduce the following decomposition of the vectors U = (U (1) , U (2) ) ∈ R d × R and F = (F (1) , F (2) ) ∈ R d × R. Then, we pass to the Fourier transform with respect to X. Passing in the Fourier regime, equations satisfied by (U, P ) inside the domain Ω, in problem PB (1) , can be translated into (5)

         j 2 U (1) j -U (1) j ′′ + ij P j = F (1) j on {Y > 0} ∀j ∈ Z d , j 2 U (2) j -U (2) j ′′
+ P j ′ = F (2) j on {Y > 0} ∀j ∈ Z d , ij • U (1) j + U (2) j ′ = 0 on {Y > 0} ∀j ∈ Z d ,
where

U (1) j ′ , U (2) j
′ and P j belong to L 2 (0, +∞). Now we solve the Fourier problem and describe the behavior of the solution of the Stokes problem.

(1) Let us discuss the case j = 0. The system reduces to

         -U (1) 0 ′′ = F (1) 0 with U (1) 0 ′ ∈ L 2 (0, +∞), -U (2) 0 ′′ + P 0 ′ = F (2) 0 with P 0 ∈ L 2 (0, +∞), U (2) 0
′ = 0 with U (2) 0 ′ ∈ L 2 (0, +∞).
By integration, this leads us to the following equalities

U (1) 0 ′ (Y ) = +∞ Y F (1) 0 (ξ) dξ, U (2) 0 ′
(Y ) = 0, U (1)
0 (Y ) = U

0 (Y ) = U (2) 0 (0), P 0 (Y ) = - +∞ Y F (2) 0 (Z) dZ. (2)
By assumption on the source term, we have

+∞ Y +∞ Z F (1) 0 (ζ) dζ dZ ≤ Q(Y ) e -Y +∞ Y F (ℓ) 0 (Z) dZ ≤ Q(Y ) e -Y , ℓ = 1, 2.
Defining

U ∞ := (U (1)
∞ , U (2)
∞ ) ∈ R d × R as U (1) ∞ := U (1) 0 (0) - +∞ 0 +∞ Z F (1) 0 (ξ) dξ dZ, U (2)
∞ := U (2) 0 (0), we obtain, for all Y > 0, | U (1) 0 (Y ) -U (1) ∞ | + | U (2) 0 (Y ) -U (2) ∞ | + | P 0 (Y )| ≤ Q(Y ) e -Y , | U (1) 0 ′ (Y )| + | U (2) 0 ′ (Y )| + | P 0 ′ (Y )| ≤ Q(Y ) e -Y .
(2) Let us discuss the case j = 0. If the source terms F

(1) j and F

(2) j

were written as Q(Y )e -j Y then the estimate is proved in [START_REF] Chupin | Rigorous derivation of the thin film approximation with roughnessinduced correctors[END_REF]Appendix B]. In the present case, the source terms only satisfy

F (1) j (Y ) ≤ Q(Y )e -j Y , ∀Y > 0, | F (2) j (Y )| ≤ Q(Y )e -j Y , ∀Y > 0,
and the proof can be adapted from [START_REF] Chupin | Rigorous derivation of the thin film approximation with roughnessinduced correctors[END_REF] by using a comparison principle. In particular, we can show that

U j (Y ) + P j (Y ) ≤ Q(Y ) e -j Y ,
for all j ∈ Z d \ {0} and Y > 0.

The elementary problems related to the boundary layer correctors with respect to the elastic constraint take the following form:

PB (ref2) -∆Σ = G on Ω, ∂ N Σ = N on Γ, with G ∈ L 2 (Ω) and N ∈ L 2 (T d ).
We note that the problems related to the boundary layer correctors are matricial problems: the unknowns Σ k have matricial values. The elementary problem that we analyze here is a scalar case but can easily be extend component by component.

Proposition 2. Assume that {Y <0} G(X, Y ) dX dY = T d N(X) dX.

If the following conditions are satisfied

G 0 (Y ) ≤ Q(Y ) e -Y ,
for all Y > 0,

G j (Y ) ≤ Q(Y ) e -j Y , for all j ∈ Z d \ {0} and Y > 0,
then PB (ref2) admits a unique solution Σ ∈ L 2 (Ω). Moreover, we have

Σ 0 (Y ) ≤ Q(Y ) e -Y , for all Y > 0, Σ j (Y ) ≤ Q(Y ) e -j Y
, for all j ∈ Z d \ {0} and Y > 0.

In particular, we have

|Σ(X, Y )| ≤ Q(Y ) e -Y , for all X ∈ T d , Y > 0.
Proof. If the source terms satisfy G 0 = 0 and, for j ≥ 1, G j (Y ) = Q(Y )e -j Y , then the estimate is proved in [14, Lemma 2.2]. In the present case, the source terms only satisfy

| G 0 (Y )| ≤ Q(Y ) e -Y and | G j (Y )| ≤ Q(Y )e -j Y .
The estimate on Σ 0 can be obtained by straightforward integration. For j = 0, the proof can be adapted from [START_REF] Chupin | Roughness effect on Neumann boundary condition[END_REF] by using a comparison principle.

Applications: analysis of problems PB (k) . Let us recall that the definition of the boundary layer correction problems PB (k) need to specify the value of a k-2 and b k-1 .

Let us first describe how to determine a k-2 . In order to apply Proposition 1, we need to impose that the source term in the momentum equation of PB (k) , namely

F := F k -a k-2
, satisfies a sharp decrease for each Fourier mode:

• Averaging this source term F with respect to X gives

F 0 (Y ) = T d (F k (X, Y ) -a k-2 ) dX = T d F k (X, Y ) dX -lim Y →+∞ T d F k (X, Y ) dX + lim Y →+∞ T d F k (X, Y ) dX -a k-2 .
On one hand, F k is composed of elementary solutions (U i , Σ i , P i ) 1≤i≤k-1 which, by induction, satisfy the expected decreasing behavior. We have

T d F k (X, Y ) -lim Y →+∞ T d F k (X, Y ) dX ≤ Q(Y ) e -Y .
On the other hand, in order to satisfy the assumption needed to apply Proposition 1, we impose

(6) a k-2 = lim Y →+∞ T d F k (X, Y ) dX.
• By induction on k, the Fourier coefficients F j , for j = 0, of the source term

F = F k -a k-2 satisfy F j (Y ) ≤ Q(Y ) e -j Y , ∀j ∈ Z d \ {0}, ∀Y > 0.
Let us now describe how to determine b k-1 . In order to apply Proposition 2, we need to impose the compatibility condition between the source term in the Laplace equation of PB (k) , namely G := D -1 G k , and the Neumann boundary term

N = N k -b k-1 : (7) b k-1 = T d N k (X) dX -D -1 Y <0 G k (X, Y ) dX dY.
Besides, by induction on k, the Fourier coefficients G j of the source term

G := D -1 G k satisfy G 0 (Y ) ≤ Q(Y ) e -Y , ∀Y > 0, G j (Y ) ≤ Q(Y ) e -j Y , ∀j ∈ Z d \ {0}, ∀Y > 0.
5. Error estimates 5.1. Remainder. The asymptotic expansion truncated at a given order leads us to introduce the so-called remainder (R, Q, S):

u(x) = N j=0 ε j u j (x) + U j x, x ε + ε N R(x), p(x) = N j=0 ε j p j (x) + P j+1 x, x ε + ε N Q(x), σ(x) = N j=0 ε j σ j (x) + Σ j x, x ε + ε N S(x).
We aim at establishing estimates on the remainder, at any order.

Applying the Oldroyd operator to the remainder (R, Q, S) and considering the properties of the elementary solutions, we get the following set of equations

• momentum equation, in ω ε : Re ε N R • ∇R + L (A) ε (R) -(1 -r)∆R + ∇Q = div(S) + F ε , • continuity equation, in ω ε : div(R) = 0, • constitutive equation, in ω ε : We ε N (R • ∇S + g a (∇R, S)) + L (B) ε (R) + L (C) ε (S) + S -D∆S = 2rD(R) + G ε ,
• boundary conditions on the velocity, on γ - ε ∪ γ + : R = D (±) ε , • boundary conditions on the elastic constraint, on γ - ε ∪ γ + :

∂ n S = N (±) ε ,
where operators are defined as follows.

L (A) ε (R) = R • N -1 k=0 ε k (∇u k + ∇ x U k + ∇U k+1 ) +ε N R • (∇u N + ∇ x U N ) + N k=0 ε k (u k + U k ) • ∇R, L (B) ε (R) = R • N -1 k=0 ε k (∇σ k + ∇ x Σ k + ∇Σ k+1 ) +ε N R • (∇σ N + ∇ x Σ N ) + g a ∇R, N k=0 ε k (σ k + Σ k ) , L (C) ε (S) = N k=0 ε k (u k + U k ) • ∇S + ε N g a (∇u N + ∇ x U N , S) +g a N -1 k=0 ε k (∇u k + ∇ x U k + ∇U k+1 ), S .
The source terms are defined by

F ε = -Re N k=1 (u k + U k ) • (∇u N -k + ∇ x U N -k + ∇U N +1-k ) -Re u 0 • (∇u N + ∇ x U N ) + (1 -r)∆ x U N + (1 -r)∆ x U N -∇p N -∇ x P N +1 + div(σ N ) + div x (Σ N ), G ε = 2rD(u N ) + 2rD x (U N ) -σ N -Σ N + D ∆σ N + D ∆ x Σ N -We N k=1 (u k + U k ) • (∇σ N -k + ∇ x Σ N -k + ∇Σ N +1-k ) -We u 0 • (∇σ N + ∇ x Σ N ) -We N -1 k=0 g a (∇u k + ∇ x U k + ∇U k+1 , σ N -k + Σ N -k ) -We g a (∇u N + ∇ x U N , σ 0 ),
and the contributions to the boundary relation are given by

D (+) ε = N k=0 ε k-N lim Y →∞ T U k -U k | Y =1/ε , D (-) ε = -ε -N N k=0 ε k +∞ i=N -k+1 (-εH) i i! ∂ (i) y u k | γ - 0 , N (+) ε = - N -1 k=0 ε k-N ∂ Y Σ k+1 | y=1,Y =1/ε , N (-) ε = -ε -N N k=0 ε k +∞ i=N -k+1 (-εH) i i! ∂ (i) y ((∇H • ∇ x )σ k + ∂ y σ k )| γ - 0 .
Proposition 3. The following estimates hold:

F ε L 2 ≤ C, G ε L 2 ≤ C.
For all ℓ ≥ 0, for all x ∈ T d , we have

|∇ ℓ x D (+) ε (x)| ≤ Q 1 ε e -1 ε , |∇ ℓ x D (-) ε (x)| ≤ C ε 1-ℓ , |∇ ℓ x N (+) ε (x)| ≤ Q 1 ε e -1 ε , |∇ ℓ x N (-) ε (x)| ≤ C ε 1-ℓ .
Proof.

• The estimates for F ε and G ε are obvious.

• By using Proposition 1, we know that for each integer k we have

U k x, x ε , 1 ε -lim Y →∞ T U k ≤ Q 1 ε e -1 ε , for all x ∈ T d ,
and, for all ℓ ≥ 1,

∇ ℓ X U k x, x ε , 1 ε ≤ Q 1 ε e -1 ε , for all x ∈ T d .
We immediately deduce that for all ℓ ≥ 0 and for all x ∈ T d we have

|∇ ℓ x D (+) ε (x)| ≤ Q 1 ε e -1 ε . • We estimate ∇ ℓ x D (-) ε
remarking that, using the Taylor formulae, for each integer k we can write

+∞ i=N -k+1 (-εH) i i! ∂ (i) y u k (x, 0) = (-εH) N -k+1 (N -k + 1)! ∂ (N -k+1) y u k (x, ξ k ), with ξ k ∈ [0, -εH(x/ε)]. That implies D (-) ε (x, -εH(x/ε)) = -ε N k=0 (-H) N -k+1 (N -k + 1)! ∂ (N -k+1) y u k (x, ξ k ). Now, since D (-) ε
is a finite sum, the estimate directly follows from its analysis. • Finally, the estimate on ∇ ℓ

x N (±) ε is based on the same arguments, noticing that the Neumann data, for the remainder, can be reduced to a finite sum of boundary terms, namely

N (N ) ε (x, -εH(x/ε)) = ε N k=0 (-H) N -k+1 (N -k + 1)! ∂ (N -k+1) y ((∇H • ∇ x )σ k + ∂ y σ k )(x, ζ k ).
5.2. Lift procedure. Well-posedness of the set of equations satisfied by the remainder is obtained by means of construction. Let us point out the fact that the remainder satisfies a diffusive Oldroyd-type system. In this step, we aim at modifying the set of equations by using a lift procedure in order to deal with homogeneous boundary conditions and preserve the homogeneous incompressibility condition.

Definition 3. Let τ ∈ C ∞ (R) be such that τ (y) = 0 if y ≤ 0, 1 if y ≥ 1.
We define R bound and S bound as

(8) R bound (x, y) = τ (y) D (+) ε (x) + (1 -τ (y)) D (-) ε (x), S bound (x, y) = τ (y) N (+) ε (x) + (1 -τ (y)) N (-) ε (x)
, and we define R div as a solution of

div(R div ) = -div(R bound ) in ω ε , R div = 0 on γ - ε ∪ γ + .
The lift velocity and constraint fields are thus defined as:

R lift := R bound + R div , S lift := S bound
and the lifted velocity and constraint field are thus defined as:

R := R -R lift , Q := Q, S := S -S lift .
Remark 7. The lifted remainder ( R, S) satisfies a system which is identical to the system satisfied by (R, S), up to some modifications:

• the boundary conditions are homogeneous ;

• the incompressibility condition is (still) homogeneous ;

• the source terms F ε , G ε , and linear operators L (A) , L (B) , L (C) have been (slightly) modified (see further, page

Remark 8. The definition of R div is guaranteed by the following result, due to Bogovskii [START_REF] Bogovskii | Solution of the first boundary value problem for an equation of continuity of an incompressible medium[END_REF] (see also [START_REF] Boyer | Éléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles[END_REF]):

Proposition 4 (Bogovskii). If H ∈ H m (ω ε ), m ≥ 0, is such that Ωε H = 0, then there exists a solution R ∈ H m+1 (ω ε ) of div( R) = H in ω ε , R = 0 on γ - ε ∪ γ + , such that ∇ R H m (ωε) ≤ C H H m (ωε) .
Thus, the existence of such a lift function R relies on the identity Ωε H = 0 with H = -div(R bound ). By the Stokes formula, we have

Ωε div(R bound ) = γ - ε D (-) ε • n + γ + D (+) ε • n = Ωε div(R) = 0.
Proposition 5. The following estimates hold:

R lift H 2 ≤ C ε -1 , S lift H 2 ≤ C ε -1 .
Proof.

• By definition of R bound and S bound (see 8), the estimates

R bound H 2 ≤ C ε -1 , S bound H 2 ≤ C ε -1 .
directly follows from the Proposition 3. • Following the Bogovskii inequality (see Proposition 4) and the Poincaré inequality, we have

R div H 2 ≤ C R bound H 2 .
• Since R lift = R bound + R div and S lift = S bound , the two previous steps imply the result of the Proposition 5.

Now estimates for a homogeneous (w.r.t. boundary conditions and incompressibility condition) system have to be established. This is the purpose of the next subsection.

5.3.

Estimate on the remainder. The lifted remainder ( R, Q, S) satisfies the following system:

• momentum equation, in ω ε :

ε N Re R • ∇ R + L (A) ε ( R) -(1 -r)∆ R + ∇ Q = div( S) + F ε , • constitutive equation, in ω ε : ε N We R • ∇ S + g a (∇ R, S) + L (B) ε ( R) + L (C) ε ( S) + S -D∆ S = 2rD( R) + G ε
• the homogeneous incompressibility condition, homogeneous Dirichlet conditions for the velocity, homogeneous Neumann conditions for the constraint.

The linear operators are given by:

L (A) ε ( R) = L (A) ε ( R) + Re ε N ( R • ∇R lift + R lift • ∇ R), L (B) ε ( R) = L (B) ε ( R) + We ε N ( R • ∇S lift + g a (∇ R, S lift )), L (C) ε ( S) = L (C) ε ( S) + We ε N (R lift • ∇ S + g a (∇R lift , S)).
The source terms are given by:

F ε = F ε -Re ε N R lift • ∇R lift + L (A) ε (R lift ) + (1 -r)∆R lift , G ε = G ε -S lift + D∆S lift -We(L (B) ε (R lift ) + L (C) ε (S lift )) -ε N We (R lift • ∇S lift + g a (∇R lift , S lift )) .
Remark 9. From the Propositions 3 and 5 we can obtain a bound on the new source terms:

F ε L 2 ≤ Cε -1 , G ε L 2 ≤ Cε -1 .
Theorem 2. The remainder satisfies:

∇R L 2 + S H 1 ≤ C ε -1 .
Proof. Due to the relations between (R, S, Q) and ( R, S, Q)

R := R -R lift , Q := Q, S := S -S lift ,
using the Proposition 5, it suffices to analyze the error on ( R, S, Q).

The estimate is then obtained using a classical energy estimate on the system satisfied by ( R, S, Q). More precisely, we first take the scalar product in L 2 (ω ε ) of the momentum equation by 2r R. Next we take the scalar product in L 2 (ω ε ) of the constitutive equation by S. We finally add the results to obtain [START_REF] Bogovskii | Solution of the first boundary value problem for an equation of continuity of an incompressible medium[END_REF] 2r(1r)

ω ∇ R 2 + ω S 2 + D ω ∇ S 2 = RHS.
The term RHS is composed as follow:

RHS = -2r ωε L (A) ε ( R) • + r ωε F ε • R -ε N ω g a (∇ R, S) : S - ωε L (B) ε ( R) • S - ωε L (C) ε ( S) • S + ωε G ε • S.
It is not very difficult to show that the source terms and linear terms of RHS can be controlled by the terms on the left-hand side of the estimate [START_REF] Bogovskii | Solution of the first boundary value problem for an equation of continuity of an incompressible medium[END_REF]. However the quadratic term does not lead in a straightforward way to a suitable estimate. Therefore we have to consider a new argument which is based on a fixed point procedure. Let us consider the following linearized system, denoted (R lin. ):

ε N Re R n • ∇ R n+1 + L (A) ε ( R n+1 ) -(1 -r)∆ R n+1 + ∇ Q n+1 = div( S n+1 ) + F ε , ε N We R n • ∇ S n+1 + g a (∇ R n , S n+1 ) + L (B) ε ( R n+1 ) + L (C) ε ( S n+1 ) + S n+1 -D∆ S n+1 = 2rD( R n+1 ) + G ε , and div R n+1 = 0.
where ( R n , S n ) are given. The idea relies on the following arguments:

(1) we show that ( R n , S n ) n is bounded in H 1 (ω ε ), up to smallness assumptions ;

(2) by the Cauchy criterion, we show that the sequence ( R n , S n ) n converges in H 1 (ω ε ) ; (3) we let n tend to +∞ and show that the limit of ( R n , S n ) n is the solution of the system satisfied by the remainder. The limit still satisfies the estimates of step 1.

Step 1. Using the classical energy estimate, we have

2r(1 -r) ω ∇ R n+1 2 + ω S n+1 2 + D ω ∇ S n+1 2 = RHS (n) .
The term RHS (n) is composed as follow:

RHS (n) = -2r ωε L (A) ε ( R n+1 ) • R n+1 + r ωε F ε • R n+1 -We ε N ω g a (∇ R n , S n+1 ) : S n+1 - ωε L (B) ε ( R n+1 ) • S n+1 - ωε L (C) ε ( S n+1 ) • S n+1 + ωε G ε • S n+1 .
We distinguish three types of terms:

• source terms

r ωε F ε • R n+1 + G ε • S n+1
Using Cauchy-Schwarz, Poincaré and Young inequalities, we have

r ωε F ε • R n+1 ≤ r F ε L 2 (ωε) R n+1 L 2 (ωε) ≤ rC P F ε L 2 (ωε) ∇ R n+1 L 2 (ωε) ≤ r(1 -r) ∇ R n+1 2 L 2 (ωε) + r 4(1 -r) C 2 P F ε 2 L 2 (ωε)
As a matter of fact, r(1-r) ∇ R n+1 2

L 2 (ωε) can be absorbed by the heft-hand side of the energy estimate. The other source term can be treated in a very similar way.

• linear terms

-2r ωε L (A) ε ( R n+1 ) • R n+1 - ωε L (B) ε ( R n+1 ) • S n+1 - ωε L (C) ε ( S n+1 ) • S n+1 .
Conventional arguments are the Hölder inequality, the Sobolev injections like H 1 (ω ε ) ⊂ L 4 (ω ε ), with constant denoted C S , and Young's inequality. For instance, the first term of

L (A) ε ( R n+1 )
, that is R n+1 • ∇u 0 , can be treated as follows:

2r ωε ( R n+1 • ∇u 0 ) • R n+1 ≤ 2r R n+1 2 L 4 (ωε) ∇u 0 L 2 (ωε) ≤ 2r C S ∇ R n+1 2 L 2 (ωε) ∇u 0 L 2 (ωε) .
Under smallness assumptions on u 0 (and therefore on the data of the initial problem), the right-hand side of the above inequality can be absorbed by the left-hand side of the energy estimate. The other linear terms can be treated in a very similar way.

• quadratic terms -We ε N ω g a (∇ R n , S n+1 ) : S n+1 .

Using the inequality

We

ε N ω g a (∇ R n , S n+1 ) : S n+1 ≤ We ε N ∇ R n L 2 (ωε) S n+1 2 L 4 (ωε) ≤ We ε N C S ∇ R n L 2 (ωε) ∇ S n+1 2 L 2 (ωε)
for ε sufficiently small (namely We ε N C S ∇ R n L 2 (ωε) < D), then the righthand side of the above inequality can be absorbed by the left-hand side of the energy estimate. From the above considerations, we deduce the following estimate

∇ R n+1 2 L 2 (ωε) + S n+1 2 H 1 (ωε) ≤ C F ε 2 L 2 (ωε) + C G ε 2 L 2 (ωε) .
By Remark 9, we obtain

(10) ∇ R n+1 2 L 2 (ωε) + S n+1 2
H 1 (ωε) ≤ Cε -2 .
Finally, in order to get the induction step on n, it is sufficient to guarantee that We ε N -2 C S C < D to get the uniform estimate on ( R n , S n ). Note also that this condition is satisfied if ε is small enough.

Step 2. We prove that ( R

n , S n ) n is a Cauchy sequence in H 1 (ω ε ). Introducing R n+1 := R n+1 -R n , S n+1 := S n+1 -S n , Q n+1 := Q n+1 -Q n ,
we get by subtraction in (R lin. )

ε N Re ( R n • ∇ R n+1 + R n • ∇ R n ) + L (A) ε ( R n+1 ) -(1 -r)∆ R n+1 + ∇ Q n+1 = div( S n+1 ), ε N We R n • ∇ S n+1 + R n • ∇ S n + g a (∇ R n , S n+1 ) + g a (∇ R n , S n ) + L (B) ε ( R n+1 ) + L (C) ε ( S n+1 ) + S n+1 -D∆ S n+1 = 2rD( R n+1 )
and div R n+1 = 0.

Performing an energy estimate consists, again, in multiplying the first equation by 2r R n+1 then integrate over ω ε , multiplying the second equation by S n+1 then integrate over ω ε and sum up the two contributions. We use the following estimates:

• The first contributions is easily controlled, as

ε N Re ωε ( R n • ∇ R n+1 ) • R n+1 = 0.
• The second contribution satisfies:

ε N Re ωε ( R n • ∇ R n )) • R n+1 ≤ ε N Re R n L 4 (ωε) ∇ R n L 2 (ωε) R n+1 L 4 (ωε) ≤ Cε N -1 ∇ R n L 2 (ωε) ∇ R n+1 L 2 (ωε) ≤ r(1 -r) ∇ R n+1 2 L 2 (ωε) + C 4r(1-r) ε 2N -2 ∇ R n 2 L 2 (ωε)
, where we have used the estimate established in Eq. ( 10) and then Sobolev imbedding and Young's inequality.

• The third contribution satisfies

ωε L (A) ε ( R n+1 ) • R n+1 ≤ ||| L (A) ε ||| ∇ R n+1 2 L 2 (ωε) ,
where ||| • ||| denotes the operator norm from L 2 (ω ε ) to itself. Recalling the expression of L ε ||| is arbitrarily small for sufficiently small data or ε.

• The other contributions can be treated in a straightforward way or by arguments similar to the previous ones. We thus deduce that, under smallness assumptions on the data and ε,

∇ R n+1 2 L 2 (ωε) + S n+1 2 H 1 (ωε) ≤ Cε 2N -2 ( ∇ R n 2 L 2 (ωε) + S n 2 H 1 (ωε) ). It means in particular that ( R n+1 -R n , S n+1 -S n ) tends to 0 as n goes to +∞. Consequently, ( R n , S n ) is a Cauchy sequence in H 1 (ω ε ). The sequence converges to some ( R ⋆ , S ⋆ ) in H 1 (ω ε )
which, by means of construction, satisfies the following estimate:

(11) ∇ R ⋆ 2 L 2 (ωε) + S ⋆ 2 H 1 (ωε) ≤ Cε -2 .
Step 3. Letting n tend to +∞ we come to the conclusion that ( R ⋆ , S ⋆ ) is the unique solution of the system satisfied by the lifted remainder. Therefore, the lifted remainder satisfies:

∇ R L 2 + S H 1 ≤ C ε -1
which concludes the proof.

Corollary 3. The asymptotic expansion is valid at any order.

Algorithm and numerical procedure

We show in this section that it is possible to effectively evaluate all the contributions of the ansatz. The only difficulty is to prove that the solution of each problem can be built using the only previous elementary solutions. For instance, let us prove that (u k , p k , σ k ), which is the solution of the problem pb (k) , only depends on (u j , p j , σ j ), j < k and on (U j , P j , Σ j ), j ≤ k. This is not so obvious since the problem pb (k) calls for the use of a parameter a k , which seems to be related to a forthcoming problem P B (k+2) through the relationship

a k = lim Y →+∞ T d F k+2 (X, Y ) dX.
However we know prove that the definition of a k is a consistent: Proposition 6. Coefficients a k (see the definition of pb (k) and PB (k+2) ) satisfying Eq. (6) can be written as

a k = lim Y →+∞ T d -Re u 0 • ∇ x U k + (1 -r)∆ x U k -Re k-1 i=1 u i • ∇ x U k-i -Re k i=1 U i • (∇ x U k-i + ∇u k-i ) .
Proof. The source contribution F k+2 can be written as

F A k+2 + F B k+2 with F A k+2 = -Re u 0 • ∇ x U k + (1 -r)∆ x U k + div x (Σ k ) -Re k-1 i=1 u i • (∇U k+1-i + ∇ x U k-i ) -Re k i=1 U i • (∇U k+1-i + ∇ x U k-i + ∇u k-i ) , F B k+2 = -Re u 0 • ∇U k+1 -Re u k • ∇U 1 + div(Σ k+1 ) + 2(1 -r)∇ x • ∇U k+1 -∇ x P k+1 .
The first contribution F A k+2 using the already defined elementary solutions ((u j , p j , σ j ), j < k and (U j , P j , Σ j ), j ≤ k), whereas the second ones F B k+2 using elementary solutions of problem pb (k) and P B (k+1) .

We now prove that lim

Y →+∞ T d F B k+2 = 0.
For instance, we treat the first contribution (the four other contributions are similarly treated)

u 0 • ∇U k+1 = d ℓ=1 u (ℓ) 0 ∂ X ℓ U k+1 -u (d+1) 0 ∂ Y U k+1 .
Taking the X-average, using the periodicity we obtain To conclude the proof, it suffices to note that some contributions of F A k+2 vanish too.

The problem pb (k) calls for the use of a parameter b k , which seem to be related to a forthcoming problem P B (k+1) through the relationship

b k = T d N k+1 (X) dX -D -1 Y <0
G k+1 (X, Y ) dX dY.

However we know prove that the definition of b k is a consistent: Proposition 7. Coefficients b k (see the definition of pb (k) and PB (k+1) ) satisfying Eq. (7) only depend on the elementary solutions (u j , p j , σ j ) j<k and (U j , P j , Σ j ) j≤k .

Proof. The source contribution G k+1 only depends on the already defined elementary solutions ((u j , p j , σ j ), j < k and (U j , P j , Σ j ), j ≤ k). Only one term in the boundary contribution N k+1 does not depend on these elementary solution: N B k+1 = (∇H • ∇ x )σ k-1 | γ - 0 . Nevertheless, its X-average is equals to 0. The resulting application of Propositions 6 and 7 leads to the following algorithm which allows us to build the approximation of the solution at any fixed order of precision:

Initialization:

Compute a 0 (compatibility of problem PB (2) at the top) Compute b 0 (compatibility of problem PB (1) at the bottom) Compute (u 0 , p 0 , σ 0 ) Compute (U 1 , P [START_REF] Olmsted | Johnson-Segalman model with a diffusion term in cylindrical Couette flow[END_REF], as suggested by the interpretation of the stress diffusion term as arising from the diffusion of polymeric dumbbells [START_REF] El-Kareh | Existence of solutions for all Deborah numbers for a non-Newtonian model modified to include diffusion[END_REF]. However some authors also considered Dirichlet boundary conditions [START_REF] Rossi | Slippage and migration in TaylorÐCouette flow of a model for dilute wormlike micellar solutions[END_REF] or mixed boundary conditions [START_REF] Adams | The interplay between boundary conditions and flow geometries in shear banding: hysteresis, band configurations, and surface transitions[END_REF]. We emphasize that the method that has been developped in this paper readily adapts to the consideration of Dirichlet boundary conditions: the definition of the elementary problems has to be adapted and, in particular, the behaviour at infinity of the boundary layer elastic tensor corrector is completely determined by an exponential decay towards a constant which can be counter-balanced by using a suitable elementary problem at next order. 7.2. Influence of the curvature of the channel. The analysis of the roughness effects has been led in a particular geometry: a channel. When considering spacevarying profiles of the boundary such as nozzles or more general converging-diverging profiles, additional coupling effects have to be taken into account. Although the methodology developped in this paper still applies, source terms have to be added in the elementary problems in order to include the scale effects at the macroscopic scale onto the boundary term which serves as a correction in the boundary layer problems. However, there is no additional difficulty from the mathematical point of view, although it tends to increase the complexity of the description of the elementary problems.

7.3.

The non-diffusive model. In many studies the standard Oldroyd model is considered without diffusion of the elastic stress tensor. The link between the standard model and the diffusive model has been investigated from the numerical point of view by the authors [START_REF] Chupin | Stationary Oldroyd model with diffusive stress: mathematical analysis of the model and vanishing diffusion process[END_REF] by considering the vanishing diffusion process in the diffusive model. In the context of the roughness issue, the adaptation of our analysis is questionable when considering the standard Oldroyd model only. Although the two models are very close (at least formally in the regime D → 0), the asymptotic expansion proposed in this paper does not apply to the case D = 0, even formally. This is due to the degeneracy of the boundary conditions: first, the loss of the boundary conditions in the standard model prevents us from developping the current strategy which is based upon the correction of the boundary terms at higher orders ; second, ignoring the specific treatment of the boundary conditions that was done in the diffusive model provides inconsistent elementary problems.

Figure 1 .

 1 Figure 1. a) Rough domain. b) Flattened domain: the rough boundary has been truncated by a smooth one. c) Boundary layer domain corresponding to a focus on the vicinity of the rough periodic boundary.

  Due to the behavior of the mean value T d U k+1 (see the Proposition 1), we have lim Y →+∞ T d u 0 • ∇U k+1 = 0.

  1 , Σ 1 ) Iterative process on k: FOR k = 1, ..., +∞, DO Compute a k (compatibility of problem PB (k+2) at the top) Compute b k (compatibility of problem PB (k+1) at the bottom) Compute (u k , p k , σ k ) Compute (U k+1 , P k+1 , Σ k+1 ) END 7. Concluding remarks 7.1. Boundary conditions. The diffusive Oldroyd model is generally associated to Neumann boundary conditions for the elastic stress tensor


Ning Wang
email: ning.wang@utc.fr

Marie-Hé Lè Ne Abel
email: marie-helene.abel@utc.fr

Jean-Paul Barthès
email: barthes@utc.fr

Elsa Negre
email: elsa.negre@dauphine.fr

Towards a Recommender System from Semantic Traces for Decision Aid

Keywords: Traces of Interaction, Recommender System, Semantic Modelling

Collaboration allows integrating intellectual resources and knowledge from all participants in order to achieve individual or collective goals. With the help of informational environments, we can better organize, realize and record collaboration. During interactions among users in such environments, each activity produces a set of traces. Such traces are recorded and classified, based on a model of traces and can be exploited to improve collaboration. In this article, we propose a semantic model of traces and analyze classified traces by means of TF-IDF. We exploit the results to offer users recommendations and decision aid.

INTRODUCTION

As an important form of sharing and exchanging information, collaboration is one of the sources of power for human society development and progress [START_REF] Grudin | Groupware and social dynamics: Eight challenges for developers[END_REF]. Thanks to information technology, nowadays collaboration can be organized and managed in an informational environment. In such an environment, users achieve their purposes by realizing different actions. We are interested in the results of actions as well as in the actions themselves. For example, not only the content of a document a user has created matters, but also knowing who shares and consults this document at what time, with what kind of frequency, etc. A set of actions, step by step, is defined as a trace [START_REF] Zarka | Trace replay with change propagation impact in client/server applications[END_REF]. After proper modelling and analysis, traces could in return help to indicate the strengths and weaknesses of an individual or of a group of users [START_REF] Tomaz | Bringing knowledge into recommendation systems[END_REF]. Thus, with the information exploited behind traces, we can improve collaboration, as mentioned by works focusing on the reuse of traces for different purposes such as decision aid [START_REF] Garcí A-Crespo | Sem-Fit: A semantic based expert system to provide recommendations in the tourism domain[END_REF], or recommendation [START_REF] Chang | A TV Program Recommender Framework[END_REF].

In this paper we propose a mechanism that models, records and analyzes users' traces and in return recommends and helps users making decisions which are personalized for either personal purpose or for the entire group. The following tasks are needed to achieve this objective: (i) propose a semantic structure to record the traces, (ii) evaluate the traces using TF-IDF and a semantic distance among the actions, (iii) give recommendations and provide some decision aid accordingly.

The remainder of this paper is organized as follows. We identify various limitations of the current studies on recommendation methods in Section 2. In Section 3 we propose a kind of recommender system for the need of exploiting the traces. In Section 4 we illustrate our method by giving a toy example. Section 5 provides conclusions and mentions directions for future works.

RELATED WORKS

Aiming at a better treatment and exploitation of traces, we need to analyze different types of traces as well as the structure of collaboration in a Collaborative Working Environment (CWE). The model of traces proposed by Li (2013) allows an elaborated analysis of interactions among users. It pays special attention to the exchanges of users in informational environment. Traces then can be exploited to feed a recommender system. Interests of recommender systems are justified by the need to manage the growing amount of information [START_REF] Adomavicius | Toward the next generation of recommender systems: A survey of the state-of-the-art and possible extensions[END_REF].

Recently, various articles were published aiming at exploiting the traces with the help of semantics. [START_REF] Chen | Adapting domain ontology for personalized knowledge search and recommendation[END_REF] present a mechanism for personalized knowledge search and recommendation adapting a suitable domain ontology according to the previous browsing and reading behavior of users. [START_REF] Vesin | Protus 2.0: Ontology-based semantic recommendation in programming tutoring system[END_REF] present a new approach for performing effective personalization based on semantic web technologies. Although more and more attention is focused on exploiting implicit information behind data (data mining), these recent studies did not work much on modelling users' actions. In fact a good model of organizing and storing users' actions can make future exploitation of traces more efficient.

OUR APPROACH

Figure 1 shows the structure of our proposed recommender system. First of all, with the model of actions we propose, users' actions are collected and modelled from an interactive platform. After being sifted by the filter of classification, we obtain classified traces, which allows a preliminary presentation back to the users. Alternatively, we apply an algorithm to calculate an index indicating the correlation between the classified traces of a certain user and a subject. These values can lead to useful information that are presented as personalized recommendation, either to a group defined as a set of users in the platform, or to an individual user.

Interface of Interactive Platform

User

Filter of classification Classified traces

User User

TF-IDF Ontology of application Action

Recommendations

Presentation of classified traces

Presentation of Recommendations

Model of actions

Figure 1: The structure of our proposed recommender system for the exploitation of semantic traces.

Modelling of Actions

We define the principal concepts as follows:

 Action: an interaction or an act performed by a user in a collaborative environment, e.g. sending a document to other users;  Classified trace: a set of actions performed by a user in the informational environment classified according to the model of traces (Li, 2013);  Set of traces: a set of classified traces.

According to our definition, an action is the most basic element forming a trace. As regarded as an important resource for our recommender system, we introduce the Resource Description Framework (RDF) to model actions [START_REF] Antoniou | A semantic web primer[END_REF]. RDF is used as a general method for conceptual description or modelling of information that is implemented in web resources. Figure 2 shows the basic structure in the RDFS graph of our model. An ellipse represents a class of resources and a rectangle represents an object property. For example, a person has the object property of "has_id_person" and the range of this property is the class called "id". In Figure 3, the class "creation" inherits all properties from "action". We apply the model of actions to a web-based collaborative platform E-MEMORAe2.0 [START_REF] Abel | Knowledge sharing via the E-MEMORAe2. 0 platform[END_REF]. Details are shown in Figure 3. This model of actions has two main advantages compared to a traditional form of history or log of users:

 Actions are presented in a labeled, directed multi-graph. In our model the actions are represented as resources in the RDF schema and they are no longer discrete but are connected by properties. This allows a better structure of storage and usage of actions. For example, a person "Ala" chats with "Ning". This action can be presented by a RDF instance showed in the lower part of Figure 3 where "Ala" and "Ning" are two instances of the class of resource "person". "Chat_1" is an instance of the class of resource "conversation" which is linked to the action "creation".  Normally different types of actions have different importance. For example creating a piece of Wiki is more important than consulting it. In our model the actions are all classified by three classes: creation, consultation and addition which enables us to treat different types of actions more efficiently.

Application of TF-IDF

In order to evaluate the importance of different traces, we apply TF-IDF as the method of evaluation. TF-IDF (Term Frequency-Inverse Document Frequency) [START_REF] Jones | A statistical interpretation of term specificity and its application in retrieval[END_REF]) is a statistical method for weighting usually used for the research of information from texts. This method measures the correlation of a term in presenting a document from a corpus. The weight of a term is proportionally higher when it appears more frequently in a document. It also varies with the frequency of the word in the corpus. The values are used for evaluating the reference of a document.

Figure 4: Relation between concepts of the method TF-IDF and the method adapted to our case.

Here we adapt this method to our case. Figure 4 illustrates the key-concepts of the method TF-IDF and those more relevant in our case. Typically TF-IDF focuses on the relation between words, documents and corpus. If a word appears more in a document and at the same time appears less in the other documents of the same corpus, it better represents this document. For our research, we are interested in evaluating the correlation between a trace of a given user and a certain subject. We propose to consider that if the actions of a user are more pertinent concerning a subject, the user has more knowledge about it. So we are able to recommend this user as an expert in this domain. In our case, we study the relation between actions, traces and the set of traces in a group of users working on the same subject.

We adapt the equation of TF-IDF:

𝑡𝑓 𝑖,𝑗 = 𝑛 𝑖,𝑗 ∑ 𝑛 𝑘,𝑗 𝑘 (1) 𝑖𝑑𝑓 𝑖 = log |𝑃| |{𝑝: 𝑡 𝑖 ∈ 𝑝}| (2)
Where:

 𝑛 𝑖,𝑗 : the number of actions concerning the subject i performed by the user j;  ∑ n k,j k : the number of actions concerning all k subjects performed by the user j;  |𝑃|: the number of users in a group;  |{p: t i ∈ p}|: the number of users in a group who have performed at least one action about the subject i. The index of TF-IDF, Cij indicating the competence of user j on subject i, which can be regarded as the relevance between a subject and a user, is defined as follows:

𝐶 𝑖,𝑗 = 𝑡𝑓 𝑖,𝑗 × 𝑖𝑑𝑓 𝑖

(3)

Application of Semantics

When we evaluate the competence of a user on a certain subject, we can also take into consideration the semantic relationship among subjects. As semantics indicates the real meaning, a smaller semantic distance between two concepts means a closer relation of them. For example if we need to find an expert on "Computer", even though a person hasn't left a weighted trace on "Computer" directly, he may have contributed to the subject "Tablet" which is quite close in a semantic view. So we can evaluate the weight of this user on the subject "Computer" by evaluating it on the subject "Tablet". We explain in detail in Section 4 by giving a toy example.

Classified Recommender System

We take the model of traces (Li, 2013) as the basis to carry out classified recommendations. According to this model, a trace is classified into 4 types: Private Trace, Collaborative Trace, Collective Trace and Personal Trace. A private trace is sent and received by the same user. A collaborative trace has one sender and at least two receivers. A collective trace has many senders and receivers while a personal trace is defined as having only one sender with no limit on its receivers. Our recommender system takes into consideration a semantic model of the system along with the traces of recorded interactions (e.g. who has shared a document concerning the subject S with whom, e.g., who usually interacts with the expert John?). It aims at realizing recommendations for a group (improve the collaboration, identify risks, opportunities of a set of users from a group), of an individual (how to improve its efficiency, the organization of a user among his tasks), for private purposes (how to improve the private organization of a user) and for collective purposes (how to improve the communication inside a group, etc.).

EXAMPLE

Figure 5 illustrates an example of interaction on different subjects of two groups of users with a histogram chart. Each line represents the collaborative trace of a user, for each subject. Now we evaluate the competence of the user "Ning" on the subject "WP". According to the histogram, "Ning" has realized 13 actions among which 1 action concerns "WP". In group 1, the number of users is 4 among which 3 have realized at least an action about "WP". According to our method of evaluation, we obtain:

𝐶 𝑊𝑃,𝑁𝑖𝑛𝑔 = 1 13 × log 4 3 = 0.645 (4) 𝐶 𝑊𝑃,𝐴𝑙𝑎 = 3 12 × log 3 2 = 2.491 (5)
Even coming from two different groups, we can still recommend that "Ala" is more competent than "Ning" on "WP" relying on 𝐶 WP,𝐴𝑙𝑎 > 𝐶 WP,Ning .

Similarly, we evaluate the competence of the two users on other subjects as shown in Table 1. Figure 6: Semantic relationship between subjects in which "Ning" and "Ala" are involved.

Figure 6 shows a part of the domain ontology in Information Technology. It shows that the subjects "Computer" and "Tablet" are subclasses of the subject "Hardware". Also, these subjects "iOS", "Android", "WP" and "Java" are subclasses of the concept "Software". To compare the capability on the subject "Java" between "Ning" and "Ala", as we are in lack of trace of "Ning" or "Ala" on "Java", we propose that such a subject be measured by the classes (i.e., "iOS", "Android" and "WP") having the same superclass. As they are closest in the view of semantics, evidently they have closest meaning.

We propose:

∑ 𝐶 sib S ,𝑈 1 > ∑ 𝐶 𝑠𝑖𝑏 S ,𝑈 2 => 𝐶 S,𝑈 1 > 𝐶 S,𝑈 2 (6)
Where:

 𝐶 S,𝑈 1 : the index of competence of user U1 about subject S, in case concerned traces is not given;  ∑ 𝐶 sib_S,𝑈 1 : the sum of index of competence about user U1 concerning subjects which are siblings of subject S; Here "Java" shares the same ancestor "Software" with "iOS", "Android", and "WP". We obtain: ∑ 𝐶 sib_Java,Ning = 𝐶 𝐴𝑛𝑑𝑟𝑜𝑖𝑑,𝑁𝑖𝑛𝑔 +𝐶 𝑖𝑂𝑆,𝑁𝑖𝑛𝑔 +𝐶 𝑊𝑃,𝑁𝑖𝑛𝑔 = 3.869 As ∑ 𝐶 sib_Java,Ala > ∑ 𝐶 sib_Java,Ning we can deduce that 𝐶 Java,Ala > 𝐶 Java,Ning so that we recommend "Ala" be more qualified than "Ning" on the subject "Java".

CONCLUSION AND FUTURE WORK

Traces are important records in collaborative environments. Fully exploitation of such information helps us organize and improve collaboration. In this article we propose recommendations based on evaluation of traces using TF-IDF. Moreover we demonstrate that we could solve the problem when there does not exist enough relevant traces with semantics. We illustrated our method on a toy example.

Future works include implementing our proposal of recommender system. In a collaborative environment, the date and time when an action is realized are also recorded. So, it is necessary to take into consideration the fact that a recent action has more weight than a previous action.

Figure 2 :

 2 Figure 2: Basic structure in RDFS graph presenting an action.

Figure 3 :

 3 Figure 3: The model of actions in the platform E-Memorae 2.0.

Figure 5 :

 5 Figure 5: An example of collaborative interactions of two groups.

Figure 7 :

 7 Figure 7: The index of Competence about sibling concepts of "Java" of "Ning".

Table 1 :

 1 The index of competence of "Ning" and "Ala" about the subjects involved.

		Ning	Ala
	WP	0.645	2.491
	Android	0	1.661
	iOS	3.224	0
	Tablet	1.290	0
	Computer	0.0463	0.0795

Information Technology Hardware Software Computer Tablet iOS Android WP JAVA is is is is is is is is