Task For each trial, two cues are presented. After the choice of a shape, reward is delivered proportional to its fixed reward probability. The cue symbols and probabilities remained the same during a session.

Experiments

Model In each simulation (out of 1 20), the model was tested first for 1 00 trials with inactive the main output of BG, GPi (D1 ), using the same set of stimuli. After-wards, it was tested for 1 20 more trials with active GPi (D2) this time, using the same set of stimuli as previously. FInally, the GPi was suppressed (D3) once more and the model was tested on the same stimuli for 1 20 trials.

Monkeys Two female macaque monkeys were tested on the task. 1 5 minutes before working session, a solution was injected bilaterally in GPi: either saline in saline condition (SC) or GABA agonist muscimol in muscimol condition (MC). On day 1 , the two monkeys were trained for 6 and 7 sessions respectively in MC; using the same set of stimuli for each session. On day 2, the same number of sessions were conducted but in SC, using the same set of stimuli as in day 1 .

Results

Model Upper figure shows that when the GPi output is suppressed (D1 ), the performance of the model is random with a 50% probability of choosing the best cue through session(chance level). However, after GPi inhibition has been removed, the model instantly reaches a very high performance level (D2). In the last test with suppressed GPi again (D3), the performances remain much higher than chance level.

Monkeys The lower figure shows that animals are unable to choose the best stimulus on D1 (in MC), however it shows a drastic change on behavior on D2 (in SC) when animals starts with near-optimal performance on the first 25 trials. The monkeys haven't tested to the third condition of the protocol (D3), but it is predicted their performanes to stay in a high level.

Model

The model is based on [1 ,2] and organized along three segregated cortico-basal-thalamic close loops (motor, associative and cognitive). A competition mechanism within each cortical group has been ad-ded [START_REF] Piron | The Globus Pallidus Pars Interna in Goal-Oriented and Routine Behaviors: Resolving a Long-Standing Paradox[END_REF], by using short range excitation and long range inhibitions, ensuring a unique cognitive and motor decision.

Learning occurs between the cognitive cortex and striatum through dopamine which modulates learning using reinforcement learning (RL; LTD & LTP). Hebbian learning (HL; LTP) at cortical level has also been added between cognitive and associative cortex. This learning is enforced once per trial, when a move has been made and independently of the actual reward.

Reinforcement learning

The model predicts!

The monkey confirms! Abstract Even though there was early interest in the neural basis of automatic behavior in the beginning of the previous century, it is still unclear how we obtain and express habits. The dominant view of the 20th century was that novel behaviors are generated by cortex, because they require attention and flexible thinking. Contrary, habits are automatic, so they are primarily mediated by subcortical structures, more precise basal ganglia (BG). Nowadays, there is arising bibliography that argues exactly the opposite position. So the essential question is "Who's the teacher and who's the pupil?". Though our computational model of cortical-basal-thalamic closed loop, we show how learning has been transferred from the basal ganglia to the cortex, simply as a consequence of the statistics of the choice. Our theory was also tested on monkeys, which confirmed our predicted results.

Who's the teacher? Who's the pupil?

Meropi Topalidou, Daisuke Kase, Thomas Boraud, Nicolas P. Rougier


Boris Haspot
email: haspot@ceremade.dauphine.fr

Navier-Stokes equation with axisymmetric initial data and zero Mach number limit

HAL is

Introduction

The motion of a slighty compressible barotropic fluid is described by the following system:

       ∂ t ρ ε + div(ρ ε u ε ) = 0, ∂ t (ρ ε u ε ) + div(ρ ε u ε ⊗ u ε ) -µ∆u ε -(λ + µ)∇divu ε + ∇P (ρ ε ) ε 2 = 0, (ρ ε , u ε ) -t = 0 = (ρ ε 0 , u ε 0 ) (1.1)
Here ρ = ρ (t, x) ∈ R and u = u (t, x) ∈ R N stand for the dimensionless density and velocity field, and the pressure P is a suitably smooth function of ρ . Unless otherwise specified, it will always be assumed that x belongs to the whole space R N (N ≥ 2). We denote by λ and µ the two Lamé coefficients of the fluid, which are constant and satisfy µ > 0 and ν = λ + 2µ > 0. Such a condition ensures ellipticity for the operator µ∆ + (λ + µ)∇div and is satisfied in the physical cases (where λ + 2µ N ≥ 0). The initial conditions (ρ 0 , u 0 ) are given. The parameter ε corresponds to the so called Mach number which is given by ε = LT -1 ξ -1 where L and T are the typical values of the length and time (before rescaling) and ξ stands for the sound speed. We refer to [START_REF] Hagstrom | All-time existence of classical solutions for slightly compressible flows[END_REF][START_REF] Hoff | Thezero-Mach limit of compressible flows[END_REF] or in the introduction of [START_REF] Lions | Mathematical Topics in Fluid Mechanics[END_REF] for more explanations on the derivation of the above model. In this paper our purpose is double, indeed in a first time we are interested in proving the global existence of strong solution for small initial data for suitable initial data such that 1 the rotational part of the velocity verifies the same assumption as the so called Fujita Kato theorem and in fact the more general theorem due to Cannone-Meyer-Planchon (see [START_REF] Cannone | Solutions autosimilaires des équations de Navier-Stokes[END_REF]) . In a second time we will study the limit of such solution when the Mach number ε goes to zero, and we will recover as a limit a solution of the incompressible Navier Stokes equations. When we study equations derived from the fluids mechanics, a first interesting question is to find the largest functional space X in local or global well-posedness may be proved and after to compare this space X with the energy space. It is then natural to understand in which sense a space is critical for the existence of strong solution. That it is why it is important to study the scaling of the equations and the notion of self similar solution. In view of the celebrated works by H. Fujita and T. Kato for the incompressible model (N SI) (see [START_REF] Fujita | On the Navier-Stokes initial value problem I[END_REF]), one can guess that the critical space for the velocity is Ḣ 1 2 . By critical, we mean that we can solve the system (N SI) in functional spaces with norm invariant by the changes of scales which leave (N SI) invariant. Indeed we can check that for a solution v we can associate via the following transformation: (v 0 (x), f (t, x)) → (lv 0 (lx), l 3 f (l 2 t, lx)), v(t, x) → lv(l 2 t, lx) a family of solution to (N SI) and that the norm in Ḣ N 2 -1 is invariant by the transformation v 0 (x) → lv 0 (lx).In the case of barotropic fluids, we can observe that the transformations: (ρ 0 (x), u 0 (x), f (t, x)) → (ρ 0 (lx), lu 0 (lx), l 3 f (l 2 t, lx)), (ρ(t, x), u(t, x)) → (ρ(l 2 t, lx), lu(l 2 t, , lx)).

(

have that property, provided that the pressure term has been changed accordingly.

Roughly speaking we expect that such spaces are optimal in term of regularity on the initial data in order to prove the well-posedness of the system (1.1).

It seems then natural to consider initial data (b 0 , u 0 ) in Ḣ N 2 × ( Ḣ N 2 -1 ) N . However Ḣ N

2

is not a subalgebra of L ∞ , in particular it does not allow to control the nonlinear term involving the density as the pressure since we can not use classical composition theorem. Furthermore if we wish to propagate any regularity on the density along the time, we have to observe that the density is governed by a transport equation. It requires then that the velocity should be Lipschitz if we want to apply classical theorem on the transport equation. These two restrictions led to consider slightly smaller space, the homogeneous

Besov space B N 2 2,1 × (B N 2 -1 2,1
) N (see the definition in section 2) which is also critical according (1.2). Indeed we can verify that the only critical Besov space

B N 2 2,r which is embedded in L ∞ is B N 2 2,1 .
Danchin in [START_REF] Danchin | Global existence in critical spaces for compressible Navier-Stokes equation[END_REF] shows for the first time a result of existence of global strong solution with small initial data in critical space for the scaling of the system. More precisely the initial data are chosen as follows (q 0 , u 0 ) ∈ (B

N 2 2,1 ∩ B N 2 -1 2,1 ) × B N 2 -1 2,1 .
The main difficulty consists in obtaining suitable estimates on the linearized system with convection terms. The crucial point is the proof of damping effect on the density which enables to control the pressure term. This last result has been generalized to the case of Besov space constructed on L p space by Charve and Danchin in [START_REF] Charve | A global existence result for the compressible Navier-Stokes equations in the critical L p framework[END_REF], Danchin in [START_REF] Danchin | A Lagrangian approach for the compressible Navier-Stokes equations[END_REF], Chen et al in [START_REF] Chen | Global well-posedness for the compressible Navier-Stokes equations with the highly oscillating initial velocity[END_REF] and the author in [START_REF] Haspot | Existence of global strong solutions in critical spaces for barotropic viscous fuids[END_REF] by using two different methods. We refer also to [START_REF] Haspot | Global existence of strong solution for viscous shallow water system with large initial data on the irrotational part[END_REF] for the existence of global strong solution for the shallow water system with a smallness condition which is supercritical. In [START_REF] Charve | Existence of strong solutions in a larger space for the shallow-water system[END_REF] we weaken the previous results in the case of the shallow water system by assuming that the initial velocity verifies only:

Pu 0 ∈ B N 2 -1 2,2 , Qu 0 ∈ B N 2 -1 2,1 and u 0 ∈ B -1 ∞,1 .
Here we can observe that the choice of u 0 in B -1 ∞,1 is absolutely crucial since it allows to control the Lipschitz norm of the velocity u. This last result has been extended by Danchin and He in [START_REF] Danchin | The incompressible limit in L p type critical spaces[END_REF] to the case of the system (1.1) with constant viscosity coefficients and with the rotational part of the velocity Pu 0 which belongs only to

B N 2 -1 2,r
with r suitably chosen and such that r > 2.

We can mention that all these previous results require a Lipschitz control on the velocity and in particular that u 0 belongs at least to B -1 ∞,1 . We would like to avoid this restriction and to consider only the so called Fujita Kato initial data which corresponds to Pu 0 ∈ Ḣ N 2 -1 . To do this, we are going to consider initial data with geometrical assumption in order to weaken the coupling in the mass equation between the density and the rotational part of the velocity Pu. More precisely we are going to consider axisymmetric initial data in dimension N = 2, 3. Let us give now more details on the form of the solution that we are going to consider. If we set ρ = 1 + εb , we are to study:

           ∂ t b + divu ε = -div(b u ), ∂ t u + u • ∇u - µ∆u + (λ + µ)∇divu 1 + εb + P (1 + εb ) 1 + εb ∇b ε = 0, (b , u ) t=0 = (b 0 , u 0 ). (1.3)
In dimension N = 2 we are interested in looking at solutions of the form:

ρ ε (t, x) = ρ ε (t, |x|), u ε (t, x) = x |x| u ε 1 (t, |x|) + x ⊥ |x| u ε 2 (t, |x|) = u 1,ε (t, x) + u 2,ε (t, x). (1.4)
We observe that the system (1.3) can be rewritten as follows:

                     ∂ t b + divu 1,e ε = -div(b u 1,e ), ∂ t u 1,ε + u 1,ε • ∇u 1,ε - µ∆u 1,ε + (λ + µ)∇divu 1,ε 1 + εb + P (1 + εb ) 1 + εb ∇b ε = u 2,ε • ∇u 2,ε , ∂ t u 2,ε + u 1,ε • ∇u 2,ε + u 2,ε • ∇u 1,ε - µ∆u 2,ε 1 + εb = 0 (b , u ) t=0 = (b 0 , u 0 ).
(1.5) We will also consider the case of axisymmetric solution when N = 3 which corresponds to rewrite (ρ, u) as follows:

u ε (t, x) = u r,ε (t, r, z)e r + u θ,ε (t, r, z)e θ + u z,ε (t, r, z)e z = u r,ε + u θ,ε + u z,ε and ρ(t, x) = ρ ε (t, r, z), (1.6)
with:

r = |x |, x = (x 1 , x 2 , 0), e r = x |x | ; e θ = x ⊥ |x |
and e z = (0, 0, 1).

As previously the system (1.3) can be rewritten as follows:

                                             ∂ t b + div(u r,ε + u z,ε ) ε = -div(b (u r,ε + u z,ε )) ∂ t u r,ε + (u r,ε + u z,ε ) • ∇u r,ε - µ∆u r,ε + (λ + µ)[∂ r divu r,ε + ∂ r divu z,ε ]e r 1 + εb + P (1 + εb ) 1 + εb ∂ r b e r ε + u θ,ε • ∇u θ,ε = 0 ∂ t u θ,ε + (u r,ε + u z,ε ) • ∇u θ,ε - µ∆u θ,ε 1 + εb + u θ,ε • ∇u r,ε = 0, ∂ t u z,ε + (u r,ε + u z,ε ) • ∇u z,ε - µ∆u z,ε + (λ + µ)[∂ z divu r,ε + ∂ z divu z,ε ]e z 1 + εb + P (1 + εb ) 1 + εb ∂ z b e r ε = 0. (1.7)
Setting u k,ε = u r,ε + u z,ε we can rewrite the previous system as follows:

                           ∂ t b + divu k,ε ε = -div(b u k,ε ) ∂ t u k,ε + u k,ε • ∇u k,ε - µ∆u k,ε + (λ + µ)∇divu k,ε 1 + εb + P (1 + εb ) 1 + εb ∂ r b e r ε + u θ,ε • ∇u θ,ε = 0. ∂ t u θ,ε + u k,ε • ∇u θ,ε - µ∆u θ,ε 1 + εb + u θ,ε • ∇u r,ε = 0. (1.8)
We can observe that in the system (1.3) and (1.7) the coupling between the density b ε and the velocity u ε is weaken since only the irrotationnal part of the velocity governes the behavior of the density in the mass equation (except in the 3D case where Pu z,ε plays also a role). This remark will be crucial in order to choose u 2 0 or u θ,0 in Ḣ N 2 -1 . It will be however mandatory to choose u 1 0 , u r,0 and u z,0 in

B N 2 -1 2,1
in order to ensure a Lipschitz control on the irrotational part which allows us to estimate the density via the mass equation. An another difficulty will consists in dealing with the coupling between the rotational and the irrotational part of the velocity which appears via the terms u 2,ε • ∇u 2,ε and u θ,ε • ∇u θ,ε . We will show some regularizing effects induced on these bilinear convection terms which enables us to propagate the regularity

B N 2 -1 2,1
on the irrotationnal part of the velocity.

Let us focus now on the zero Mach number limit when ε goes to 0. A large amount of literature has been devoted to this main feature.Roughly speaking, two different heuristics have been developed. The case of well-prepared data which corresponds to the assumption that ρ 0 = 1 + O(ε 2 ) and divu 0 = O(ε) has been investigated in [START_REF] Klainerman | Compressible and incompressible fluids[END_REF][START_REF] Kreiss | Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations[END_REF] and [START_REF] Hoff | Thezero-Mach limit of compressible flows[END_REF].

In this paper, we will consider the case of ill-prepared data, where it is only assumed that ρ 0 = 1 + εb ε 0 with (b 0 , u 0 , f ε ) uniformly bounded (in a convenient functional space), Pu 0 tending to some v 0 and Pf tending to some g when ε goes to 0. One expects that the limit v of u solves the the so called incompressible Navier-Stokes equations:

(N SI)      ∂ t v + v • ∇v -µ∆v + ∇Π = 0, divv = 0, v /t=0 = v 0 .
Let us recall now some results of global strong solution for the system (N SI) which are due to Fujita and Kato in [START_REF] Fujita | On the Navier-Stokes initial value problem I[END_REF] and Cannone, Meyer and Planchon in [START_REF] Cannone | Solutions autosimilaires des équations de Navier-Stokes[END_REF] (see also [START_REF] Danchin | The incompressible limit in L p type critical spaces[END_REF]).

Theorem 1.1 Let N = 2, 3 and v 0 ∈ B N p -1 p,r with divv 0 = 0, p < +∞ and r ∈ [1, +∞]. There exists ε 0 such that if: v 0 B N p -1 p,r ≤ ε 0 µ,
then (N SI) has a unique global solution u in the space:

L ∞ (R + , B N p -1 p,r ) ∩ L 1 (R + , B N p +1 p,r ).
Furthermore we have:

v L ∞ (R + ,B N p -1 p,r ) + µ v L 1 (R + ,B N p +1 p,r ) ≤ M v 0 B N p -1 p,r
, for M a constant depending only on N and p.

We can observe that under the condition (1.4) in dimension N = 2 and (1.6) in dimension N = 3 the previous system can be respectively rewritten as follows:

     ∂ t u 2 -µ∆u 2 = 0 u 2 • ∇u 2 = -∇Π divu 2 = 0, (1.9)
and:

               ∂ t u r + (u r + u z ) • ∇u r -µ∆u r + u θ • ∇u θ + ∂ r Π e r = 0 ∂ t u θ + (u r + u z ) • ∇u θ -µ∆u θ + u θ • ∇u r = 0, ∂ t u z + (u r + u z ) • ∇u z -µ∆u z + ∂ z Π e z = 0, div(u r + u z ) = 0.
(1.10)

The main difficulty in order to justify rigorously the convergence is that one has to face the propagation of acoustic waves with the speed ε -1 , a phenomenon which does not occur in the case of "well-prepared" data. Nevertheless, several remarkable results have been obtained recently. First of all, for initial data with minimal regularity assumptions, P-L Lions stated in [START_REF] Lions | Mathematical Topics in Fluid Mechanics[END_REF] the existence of global law considered is of type P (ρ) = aρ γ with certain restrictions on γ depending on the space dimension N . Since then, convergence results to the incompressible model have been proved by B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi. The case of periodic boundary conditions has been investigated in [START_REF] Lions | Incompressible limit for a viscous compressible fluid[END_REF], the case of bounded domains with Dirichlet conditions in [START_REF] Desjardins | Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions[END_REF] and the case of the whole space in [START_REF] Desjardins | Low Mach number limit of viscous compressible flows in the whole space[END_REF]. Some local weak convergence results are also available in a more general context (see [START_REF] Lions | Une approche locale de la limite incompressible[END_REF]). Roughly, the main difference between the whole space and the periodic case is that in the former case one can utilize the dispersion of sound waves to get strong convergence results whereas in the latter case, the sound waves will oscillate forever, leading only to weak convergence.

In the framework of strong periodic solutions, several recent works have to be mentioned. For smooth initial data and no external force, it has been stated in [START_REF] Hagstrom | All-time existence of classical solutions for slightly compressible flows[END_REF] that slightly compressible two-dimensional periodic incompressible Navier-Stokes equations and is unlikely to extend in higher dimension.

More recently Danchin was the first to justify rigorously the zero Mach limit in the framework of critical Besov space for the scaling of the equations (see [START_REF] Danchin | Zero Mach number limit for compressible flows with periodic boundary conditions[END_REF][START_REF] Danchin | Zero Mach number limit in critical spaces for compressible Navier-Stokes equations[END_REF]). More precisely he deals with initial data such that

(b ε 0 , u ε 0 ) is uniformly bounded in B N 2 -1, N 2 2,1,ε,∞ ×B N 2 -1
(see the definition 2.4 for the hybrid Besov spaces). Danchin and He (see [START_REF] Danchin | The incompressible limit in L p type critical spaces[END_REF]) have recently generalized this result to the case of Besov spaces constructed on L p Lebesgue spaces with a Lipschitz control on the velocity. To finish we would like to mention a very interesting and original result due to Danchin and Mucha [START_REF] Danchin | Compressible Navier-Stokes equations : large solutions and incompressible limit[END_REF] who propose a new approach to pass from the compressible Navier-Stokes equation to the incompressible Navier-Stokes equation by considering the limit of the solution when λ the second coefficient of viscosity goes to +∞. In this paper we would like to extend the previous result [START_REF] Danchin | The incompressible limit in L p type critical spaces[END_REF] for initial data of the type Cannone-Meyer-Planchon see [START_REF] Cannone | Solutions autosimilaires des équations de Navier-Stokes[END_REF] for axisymmetric initial data. From now for the sake of simplicity, we shall also assume that the data (b 0 , u 0 ) do not depend on ε and will be merely denoted by (b 0 , u 0 , f ) (therefore v 0 = Pu 0 ). Our first main results concern the existence of global strong solution for small initial data when the Mach number is fixed ε = 1. We refer to the section 2 for the definition of the hybrid Besov space.

Theorem 1.2 Let ε = 1. Let N = 2. Let P be a suitably smooth function of the density.

Assume that ρ 0 (x) = ρ 1 0 (|x|) and u 0 (x) = x |x| (u 1 ) 0 (|x|) + (u 2 ) 0 (|x|). Let b 0 ∈ B 0, 2 p 2,p,1 , u 2 0 ∈ B 2 p -1 p,r and u 1 0 ∈ B 0, 2 p -1 2,p,1 with 2 ≤ p < 4 with 1 ≤ r ≤ p p-2 then there exists ε 0 such that if: b 0 B N 2 -1, N p 2,p,1 + u 1 0 B N 2 -1, N p -1 2,p,1 + u 2 0 B N 2 -1, N p -1 2,p,2 ≤ ε 0 (1.11)
there exists a global solution (b, u 1 , u 2 ) of the system (1.5) with:

b ∈ L ∞ ( B 0, 2 p 2,p,1 ) ∩ L 1 ( B 2, 2 p 2,p,1 ), u 1 ∈ L ∞ ( B 0, 2 p -1 2,p,1 ) ∩ L 1 ( B 2, 2 p +1 2,p,1 ), u 2 ∈ L ∞ (B 2 p -1 p,2 ) ∩ L 1 (B 2 p +1
p,2 ).

(1.12)

If p = 2 the solution is unique.

Remark 1 To the best of our knowledge, it is the first result of global strong solution for the compressible Navier-Stokes equation with a velocity u which is not necessary Lipchitz.

In addition it allows to choose initial data only a the level of the energy concerning the rotational part.

Let us mention that in [START_REF]Global strong solution for the compressible Navier Stokes system with large axisymmetric initial data in dimension N = 2[END_REF] we prove the existence of global strong solution with large initial data for the system (1.1) in the case of viscosity coefficients of the type shallow water (µ(ρ) = µρ and λ(ρ) = 0).

Theorem 1.3 Let ε = 1 and N = 3. Let P be a suitably smooth function of the density. Assume that ρ 0 (x) = ρ 1 0 (|x|) and

u 0 = (u r ) 0 + (u θ ) 0 + (u z ) 0 . Let b 0 ∈ B 1 2 , 3 p 2,p,1 , Q(u r ) 0 , Q(u z ) 0 ∈ B 3 2 , 3 p -1 2,p,1 , P(u r ) 0 , P(u z ) 0 ∈ B 3 p -1 p,r , u k ∈ B -1 ∞,1 and (u θ ) 0 ∈ B 3 p -1 p,2 with 2 ≤ p ≤ 4 and 1 ≤ r ≤ p p-2 then there exists ε 0 such that if: b 0 B 1 2 , 3 p 2,p,1 + Q(u r ) 0 , Q(u z ) 0 ) B 1 2 , 3 p -1 2,p,1 + P(u r ) 0 , P(u z ) 0 ) B 3 p -1 p,r + (u θ ) 0 B 3 p -1 p,2 + u k B -1 ∞,1 ≤ ε 0 (1.13)
there exists a global solution (b, u r , u θ , u z ) of the system (1.7) with:

b ∈ L ∞ ( B 1 2 , 3 p 2,p,1 ) ∩ L 1 ( B N 2 +1, N p 2,p,1
),

u θ ∈ L ∞ (B 3 p -1 p,2 ) ∩ L 1 (B 3 p +1 p,2 ), Qu r , Qu z ∈ L ∞ ( B 1 2 , 3 p -1 2,p,1 ) ∩ L 1 ( B 5 2 , 3 p +1 2,p,1 ) Pu r , Pu z ∈ L ∞ ( B 1 2 , 3 p -1 2,p,r ) ∩ L 1 ( B 5 2 , 3 p +1 2,p,r ) u k ∈ L ∞ (B -1 ∞,1 ) ∩ L ∞ (B 1 ∞,1
).

(1.14)

If p ≤ 3 the solution is unique.
Remark 2 This result is a equivalent to the so-called Fujita-Kato theorem and Cannone-Meyer-Planchon theorem (see [START_REF] Fujita | On the Navier-Stokes initial value problem I[END_REF][START_REF] Cannone | Solutions autosimilaires des équations de Navier-Stokes[END_REF]) for the compressible Navier-Stokes system in the case of axisymmetric initial data.

Definition 1.1 We refer to (2.21) for the definition of low and high frequencies for Besov space. We denote by E p ε,ν the space of functions (b, u) such that :

• (b BF,εν , Qu BF,εν ) ∈ C(B N 2 -1 2,1 ) ∩ L 1 (B N 2 +1 2,1 ) × C(B N 2 -1 2,1 ) ∩ L 1 (B N 2 +1
2,1 ) ,

• (b HF,εν , Qu HF,εν ) ∈ C(B N p p,1 ) ∩ L 1 (B N p p,1 ) × C(B N p -1 p,1 ) ∩ L 1 (B N p +1 p,1 ) , • Pu ∈ C(B N p -1 p,2 ) ∩ L 1 (B N p +1 p,2 ). • u k ∈ C(B -1 ∞,1 ) ∩ L 1 (B 1 ∞,1 ) (only if N = 3
) . We endow that space with the following norm:

(b, u) E p ε,ν = b BF,εν C(B N 2 -1 2,1 )∩ L 1 (B N 2 +1 2,1
)

+ Qu BF,εν C(B N 2 -1 2,1 )∩ L 1 (B N 2 +1 2,1
)

ε b HF,εν C(B N p -1 p,1 ) + ε -1 b HF,εν L 1 (B N p +1 p,1
)

+ Qu HF,εν C(B N p -1 p,1 )∩ L 1 (B N p +1 p,1
)

+ Pu C(B N p -1 p,2 )∩ L 1 (B N p +1 p,2
)

+ u k C(B -1 ∞,1 )∩ L 1 (B 1 ∞,1 ) δ N,3 ,
with δ j,3 = 0 if j = 3 and δ j,3 = 1 if j = 3.

We can now consider the global convergence of the solutions of (1.1) to (N SI) in the critical functional setting described above.

Theorem 1.4 Assume now that the initial data (b ε 0 , u ε 0 ) verify the regularity conditions of theorem 1.2 and 1.3. Let ε = εν. There exists a constant η independent of ε and ν such that if1 :

• When N = 2: C εν 0 = (b ε 0 , (u 1 0 ) ε ) BF, ε B 0 2,1 + ε b ε 0 HF, ε B 2 p p,1 + (u 1 0 ) ε HF, ε B 2 p -1 p,1 + (u 2 0 ) ε B 2 p -1 p,2 ≤ η, (1.15)
• When N = 3

C εν 0 = (b ε 0 , (Qu r , Qu z ) ε 0 ) BF, ε B 1 2 2,1 + ε b ε 0 HF, ε B 3 p p,1 + (Qu r , Qu z ) ε 0 HF, ε B 3 p -1 p,1 + (Pu r , Pu z ) ε 0 B 3 p -1 p,2 + u ε k B -1 ∞,1 + (u θ ) ε 0 B 3 p -1 p,2 ≤ η, (1.16)
then the following results hold:

1. Existence: For all 0 < ε ≤ ε 0 , system (1.1) has a unique solution (b , u

) in E p ε,ν
such that for some constant M > 0:

(b , u

) E p ε,ν ≤ M C εν 0 .
Furthermore u 1 ε , u ε r , u ε z converge weakly to 0 and u 2 ε , u ε θ converges in the sense of distribution to v solution of (N SI).

Convergence:

• If N = 3 then choosing r = 2 we have:

µ Pu ε -v L 1 (B 4 p + 1 2 p,2 ) + Pu -v L ∞ (B 4 p -3 2 p,2 ) ≤ M ( Pu ε 0 -v 0 B 4 p -3 2 p,2 + C εν 0 ε 1 2 -1 p . ν 1 2 ( b L 2 (B 4 p -1 2 p,1 ) + Qu L 2 (B 4 p -1 2 p,1
)

) ≤ M C εν 0 ε 1 2 -1 p .
• If N = 2 then we have:

u 2 ε -v L 1 (B c+2 p -c 2 +1 p,2 ) + u 2 ε -v L ∞ (B c+2 p -c 2 -1 p,2 ) ≤ M ( (u 2 0 ) ε -v 0 B c+2 p -c 2 -1 p,2 + C εν 0 ε c( 1 2 -1 p ) ). ν 1 2 ( b L 2 (B c+2 p -c 2 p,1 ) + u 1 ε L 2 (B c+2 p -c 2 p,1 ) ≤ M C εν 0 ε c( 1 2 -1 p ) .
with c verifying 0 ≤ c < 1 2 and c < 8-2p p-2 .

Remark 3 It would be easy to generalize the previous result when N = 3 to the case

(Pv r ) 0 , (Pv z ) 0 ∈ B N p -1 p,r 1 with 1 ≤ r 1 ≤ p p-2 .
For simplicity we just take r 1 = 2.

It would be also possible to deal with different Lebesgue index on u r , u z and u θ .

Our paper is organized as follows. In the first section 2, we introduce the homogeneous hybrid Besov spaces (see also [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]) and recall the basis of Littlewood-Paley theory and the papraproduct. In section 3, we prove the theorem 1.2 and 1.3. In the last section, following the arguments of [START_REF] Danchin | The incompressible limit in L p type critical spaces[END_REF] we study the zero Mach limit of the solution when ε goes to 0 using Strichartz estimates in order to deal with the irrotationnal part which is governed roughly speaking by a wave equation.

2 Homogeneous and hybrid Besov spaces

Homogeneous Besov space and paraproduct

As usual, the Fourier transform of u with respect to the space variable will be denoted by F(u) or u. To build the Littlewood-Paley decomposition, we need to fix a smooth radial function χ supported in (for example) the ball B(0, 4 3 ), equal to 1 in a neighborhood of B(0, 3 4 ) and such that r → χ(r.e r ) is nonincreasing over R + . So that if we define ϕ(ξ) = χ(ξ/2)-χ(ξ), then ϕ is compactly supported in the annulus {ξ ∈ R d , 3 4 ≤ |ξ| ≤ 8 3 } and we have that,

∀ξ ∈ R d \ {0}, l∈Z ϕ(2 -l ξ) = 1.
(2.17)

Then we can define the dyadic blocks (∆ l ) l∈Z by ∆ l := ϕ(2 -l D) (that is ∆ l u = ϕ(2 -l ξ) u(ξ)) so that, formally, we have u = l ∆l u (2.18)

As (2.17) is satisfied for ξ = 0, the previous formal equality holds true for tempered distributions modulo polynomials. A way to avoid working modulo polynomials is to consider the set S h of tempered distributions u such that

u B s p,r := l 2 rls ∆ l u r L p 1 r if r < ∞ and u B s p,∞ := sup l 2 ls ∆ l u L p .
We then define the space B s p,r as the subset of distributions u ∈ S h such that u Ḃs p,r is finite.

Once more, we refer to [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF] (chapter 2) for properties of the inhomogeneous and homogeneous Besov spaces.

In this paper, we shall mainly work with functions or distributions depending on both the time variable t and the space variable x. We shall denote by C(I; X) the set of continuous functions on I with values in X. For p ∈ [1, ∞], the notation L p (I; X) stands for the set of measurable functions on I with values in X such that t → f (t) X belongs to L p (I).

The Littlewood-Paley decomposition enables us to work with spectrally localized (hence smooth) functions rather than with rough objects. We naturally obtain bounds for each dyadic block in spaces of type L ρ T L p . Going from those type of bounds to estimates in L ρ T Ḃs p,r requires to perform a summation in r (Z). When doing so however, we do not bound the L ρ T Ḃs p,r norm for the time integration has been performed before the r summation. This leads to the following notation:

Definition 2.3 For T > 0, s ∈ R and 1 ≤ r, σ ≤ ∞, we set u L σ T B s p,r := 2 js ∆q u L σ T L p r (Z) .
One can then define the space L σ T Ḃs p,r as the set of tempered distributions u over (0,

T ) × R d such that lim q→-∞ Ṡq u = 0 in L σ ([0, T ]; L ∞ (R d )) and u L σ T Ḃs p,r
< ∞. The letter T is omitted for functions defined over R + . All the properties of continuity for the product and composition which are true in Besov spaces remain true in the above spaces. The time exponent just behaves according to Hölder's inequality.

Let us now recall a few nonlinear estimates in Besov spaces. Formally, any product of two distributions u and v may be decomposed into

uv = T u v + T v u + R(u, v), where (2.19)
T u v := l Ṡl-1 u ∆l v, T v u := l Ṡl-1 v ∆l u and R(u, v) := l |l -l|≤1 ∆l u ∆l v.
The above operator T is called "paraproduct" whereas R is called "remainder". The decomposition (2.19) has been introduced by Bony in [START_REF] Bony | Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires[END_REF].

In this article we will frequently use the following estimates (we refer to [2] section 2.6, [START_REF] Danchin | Zero Mach number limit in critical spaces for compressible Navier-Stokes equations[END_REF], for general statements, more properties of continuity for the paraproduct and remainder operators, sometimes adapted to L σ T Ḃs p,r spaces): under the same assumptions there exists a constant C > 0 such that if 1/p 1 + 1/p 2 = 1/p, and 1/r 1 + 1/r 2 = 1/r:

Ṫu v B s 2,1 ≤ C u L ∞ v Ḃs 2,1 , Ṫu v B s+t p,r ≤ C u Ḃt p 1 ,r 1 v Ḃs p 2 ,r 2 (t < 0), Ṙ(u, v) B s 1 +s 2 -N 2 p,r ≤ C u Ḃs 1 p 1 ,r 1 v Ḃs 2 p 2 ,r 2 (s 1 + s 2 > 0).
(2.20)

Let us now turn to the composition estimates. We refer for example to [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF] (Theorem 2.59, corollary 2.63)):

Proposition 2.1 1. Let s > 0, u ∈ Ḃs 2,1 ∩ L ∞ and F ∈ W [s]+2,∞ loc (R d ) such that F (0) = 0. Then F (u) ∈ Ḃs 2,1
and there exists a function of one variable C 0 only depending on s, d and F such that

F (u) B s 2,1 ≤ C 0 ( u L ∞ ) u B s 2,1 . 2. If u and v ∈ Ḃ N 2 2,1 and if v -u ∈ Ḃs 2,1 for s ∈] -N 2 , N 2 ] and G ∈ W [s]+3,∞ loc (R d ), then G(v) -G(u)
G(v) -G(u) B s 2,1 ≤ C( u L ∞ , v L ∞ ) |G (0)| + u B N 2 2,1 + v B N 2 2,1 v -u Ḃs 2,1
.

Let us end this section by recalling the following estimates for the heat equation:

Proposition 2.2 Let s ∈ R, (p, r) ∈ [1, +∞] 2 and 1 ≤ ρ 2 ≤ ρ 1 ≤ +∞. Assume that u 0 ∈ Ḃs p,r and f ∈ L ρ 2 T ( Ḃs-2+2/ρ 2 p,r
). Let u be a solution of:

∂ t u -Au = f u /t=0 = u 0 ,
with A = µ∆ + (mu + λ)∇div. Then there exists C > 0 depending only on N, µ, ρ 1 and ρ 2 such that:

u L ρ 1 T (B s+2/ρ 1 p,r ) ≤ C u 0 B s p,r + f L ρ 2 T (B s-2+2/ρ 2 p,r
) .

If in addition r is finite then u belongs to C([0, T ], Ḃs p,r ).

Hybrid Besov spaces

As explained, in the compressible Navier-Stokes system, the density fluctuation has two distinct behaviours in some low and high frequencies, separated by a frequency threshold. This leads to the definition of the hybrid Besov spaces. Let us begin with the spaces that are introduced by R. Danchin in [START_REF] Danchin | Global existence in critical spaces for compressible Navier-Stokes equation[END_REF][START_REF] Danchin | Zero Mach number limit in critical spaces for compressible Navier-Stokes equations[END_REF] or [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]:

Definition 2.4 For α > 0, r ∈ [0, ∞] and s ∈ R we denote u B s 1 ,s 2 2,p,1,α,r def = l≤-ln α ln 2 2 rls 1 ∆l u L p + α 1-2 r l≥-ln α ln 2
2 ls 2 ∆l u r L p .

We will note in the sequel: Remark 1 As stated in [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF] we have the equivalence

z BF,α = l≤-ln α ln 2 ∆ l z and z BF,α = l>-ln α ln 2 ∆ l z. ( 2
1 2 u Ḃs-1 2,1 + α u Ḃs 2,1 ≤ u B s,∞ α ≤ u Ḃs-1 2,1 + α u Ḃs 2,1
.

Let us start now by recalling some estimates in Besov space for the following system:

     ∂ t h + u • ∇h + divu = F , ∂ t u -µ∆u -µ∇divu + a∇h = G , (h, u) /t=0 = (h 0 , u 0 ), (2.23)
with (F , G ) external forces. More precisely in [START_REF] Haspot | Existence of global strong solutions in critical spaces for barotropic viscous fuids[END_REF][START_REF] Danchin | Zero Mach number limit in critical spaces for compressible Navier-Stokes equations[END_REF][START_REF] Chen | Global well-posedness for the compressible Navier-Stokes equations with the highly oscillating initial velocity[END_REF] it has been proved the following proposition by some different methods. In particular in [START_REF] Haspot | Existence of global strong solutions in critical spaces for barotropic viscous fuids[END_REF], we introduce the notion of effective velocity which enables us to diagonalize the system (2.23).

Proposition 2.3 Let 2 ≤ p ≤ 4.Let (h 2 , u 2
) the solution of (2.23). There exists a constant C depending only on µ, N , s and s such that the following estimate holds:

(h, u)(t) L ∞ T ( B N 2 -1, N p 2,p,1 ×B N 2 -1, N p -1 2,p,1 ) + (h, u) L 1 T ( B N 2 +1, N p 2,p,1 × B N 2 +1, N p +1 2,p,1 ) ≤ Ce V (T ) (h 0 , u 0 ) B N 2 -1, N p 2,p,1 ×B N 2 -1, N p -1 2,p,1 + (F , G ) L 1 ( B N 2 -1, N p 2,p,1 ×B N 2 -1, N p -1 2,p ,1 )
. with:

V (T ) = t 0 ∇u(s) B N 2 , N p 2,p,1 ds.

Dispersive estimate

Following B. Desjardins and E. Grenier in [START_REF] Desjardins | Low Mach number limit of viscous compressible flows in the whole space[END_REF], we shall use some dispersive inequalities for the wave equation: the so-called Strichartz estimates (see [START_REF] Strichartz | Restriction of Fourier transform to quadratic surfaces and decay of solutions of wave equations[END_REF][START_REF] Ginibre | Generalized Strichartz inequalities for the wave equation[END_REF][START_REF] Keel | Endpoint Strichartz estimates[END_REF] and the references therein). Here is the statement that we need. The reader may refer to [START_REF] Danchin | Zero Mach number limit in critical spaces for compressible Navier-Stokes equations[END_REF] for a proof.

Proposition 2.4 Let (b, d) be a solution of the following system:

(W ε ) ∂ t b + ε -1 Λd = F, ∂ t d -ε -1 Λb = G.
Then for any s ∈ R and positive T (possibly infinite), the following estimate holds:

(b, d) L r T (B s+N ( 1 p -1 2 )+ 1 r p,1 ) ε 1 r (b 0 , d 0 ) B s 2,1 + (F, G) L 1 T (B s 2,1 ) with p ≥ 2, 2 r ≤ min(1, (N -1)( 1 2 -1 p )) and (r, p, N ) = (2, ∞, 3
). Now, we can expect proposition 2.4 combined with estimates for the heat equation to provide us with uniform estimates for (b ε , u ε ), thus uniform bounds for the right-hand side of (W ε ). According to proposition 2.4, this should give us some convergence result. No further arguments are needed to get global convergence in the small in critical spaces,or local convergence in the large for more regular data. In the next two sections, we shall develop these ideas and give complete statements of our convergence results.

By uniqueness we can observe that the solution (b n , u 1 n , u 2 n ) remains rotationnally invariant, and we have:

                 ∂ t b n + divu 1 n + u 1 n • ∇b n = -b n divu 1 n , ∂ t u 1 n + u 1 n • ∇u 1 n - µ∆u 1 n + (λ + µ)∇divu 1 n 1 + b n + P (1 + b n ) 1 + b n ∇b n = -u 2 n • ∇u 2 n , ∂ t u 2 n + u 1 n • ∇u 2 n + u 2 n • ∇u 1 n - µ∆u 2 n + (λ + µ)∇divu 2 n 1 + b n = 0, (b n , u 1 n , u 2 n ) t=0 = (S n b 0 , S n u 1 0 , S n u 2 0 ). (3.25)

Uniform bounds

In this part, we aim at getting uniform estimates on (b n , u 1 n , u 2 n ) in the following space F T with the norm • F T :

F T = L ∞ T ( B N 2 -1, N p 2,p,1 ) ∩ L 1 T ( B N 2 +1, N p 2,p,1 ) × L ∞ T ( B N 2 -1, N p -1 2,p,1 ) ∩ L 1 T ( B N 2 +1, N p +1 2,p,1 ) × L ∞ T (B N p -1 p,2 ) ∩ L 1 T (B N p +1 p,2 ) . (b n , u 1 n , u 2 n ) F T = b n L ∞ T ( B N 2 -1, N p 2,p,1 )∩ L 1 T ( B N 2 +1, N p 2,p,1 ) + u 1 n L ∞ T ( B N 2 -1, N p -1 2,p,1 )∩ L 1 T ( B N 2 +1, N p +1 2,p,1 ) + u 2 n L ∞ T (B N p -1 p,2 )∩ L 1 T (B N p +1 p,2
)

.

Let us start by estimating the incompressible part

u 2 n in L ∞ T (B N p -1 p,2 ) ∩ L 1 T (B N p +1
p,2 ). We observe that u 2 n verify the following equation:

∂ t u 2 n -µ∆u 2 n -(λ + µ)∇divu 2 n = -u 1 n • ∇u 2 n -u 2 n • ∇u 1 n -(µ∆u 2 n + (λ + µ)∇divu 2 n ) b n 1 + b n .
According to proposition 2.2, we have:

u 2 n L ∞ T (B N p -1 p,2 )∩ L 1 T (B N p +1 p,2 ) u 2 0 B N p -1 p,2 + u 1 n • ∇u 2 n L 1 T (B N p -1 p,2 ) + u 2 n • ∇u 1 n L 1 T (B N p -1 p,2 ) + ∆u 2 n b n 1 + b n L 1 T (B N p -1 p,2
) .

(3.26) From (2.20) and (2.1) we get:

u 1 n • ∇u 2 n L 1 T (B N p -1 p,2 ) (b n , u 1 n , u 2 n ) 2 F T u 2 n • ∇u 1 n L 1 T (B N p -1 p,2 ) (b n , u 1 n , u 2 n ) 2 F T ∆u 2 n b n 1 + b n L 1 T (B N p -1 p,2 ) b n L ∞ T (B N p p,1 ) u 2 n L 1 T (B N p +1 p,2
) .

(3.27)

Let us consider now the behavior of (b n , u 1 n ) which verify: the system (3.25):

     ∂ t b n + divu 1 n + u 1 n • ∇b n = F 1 n , ∂ t u 1 n -µ∆u 1 n -(λ + µ)∇divu 1 n + P (1)∇b n = F 2 n , (b n , u 1 n ) t=0 = (S n b 0 , S n u 1 0 ).
(3.28) with:

F 1 n = -b n divu 1 n , F 2 n = -u 2 n • ∇u 2 n -u 1 n • ∇u 1 n -(µ∆u 1 n + (λ + µ)∇divu 1 n ) b n 1 + b n + (P (1) - P (1 + b n ) 1 + b n )∇b n , (3.29
): b n L ∞ T ( B N 2 -1, N p 2,p,1 )∩ L 1 T ( B N 2 +1, N p 2,p,1 + u 1 n ) L ∞ T ( B N 2 -1, N p -1 2,p,1 )∩ L 1 T ( B N 2 +1, N p +1 2,p,1 ) ≤ Ce Vn(T ) b 0 B N 2 -1, N p 2,p,1 + u 1 0 B N 2 -1, N p -1 2,p,1 + F 1 n L 1 T ( B N 2 +1, N p 2,p,1 ) + F 2 n L 1 T ( B N 2 -1, N p -1 2,p,1
) , (3.30) with:

V n (T ) = C t 0 ∇u 1 n (s) B N 2 , N p 2,p,1 ds. (3.31)
Therefore, it is only a matter of proving appropriate estimates for F 1 n and F 2 n by using proposition 2.20. Let us start with F 1 n , we have according to the proposition 2.20:

F 1 n L 1 T ( B N 2 +1, N p 2,p,1 ) b n L ∞ ( B N 2 -1, N p 2,p,1 ) divu 1 n L 1 T ( B N 2 , N p 2,p,1 ) . (3.32)
Similarly concerning F 2 n , we have from proposition 2.20 and 2.1:

(µ∆u 1 n + (λ + µ)∇divu 1 n ) b n 1 + b n L 1 T ( B N 2 -1, N p -1 2,p,1 ) ≤ C( b n L ∞ ) b n L ∞ ( B N 2 -1, N p p,1 ) u 1 n L 1 T ( B N 2 +1, N p +1 2,p,1 ) (P (1) - P (1 + b n ) 1 + b n )∇b n L 1 T ( B N 2 -1, N p -1 2,p,1 ) b n L ∞ ( B N 2 -1, N p p,1 ) b n L 1 T ( B N 2 +1, N p 2,p,1 ) u 1 n • ∇u 1 n L 1 T ( B N 2 -1, N p -1 2,p,1 ) u 1 L 2 T ( B N 2 , N p 2,p,1 ) ∇u 1 n L 2 T ( B N 2 -1, N p -1 2,p,1
) . (3.33) As in [START_REF] Danchin | The incompressible limit in L p type critical spaces[END_REF] we set p * = 2p p-2 such that 1 p + 1 p * = 1 2 . Since 2 ≤ p < 4 we deduce that p ≤ p * , we can now sea with the convection term u 2 n • ∇u 2 n . We have then:

(u 2 n • ∇u 2 n ) i = T (u 2 n ) j ∂ j (u 2 n ) i + T ∂ j (u 2 n ) i (u 2 n ) j + ∂ j R((u 2 n ) j , (u 2 n ) i ). (3.34)
Next we have since 1 p + 1 p * = 1 2 and 1 2 + 1 2 = 1, using classical paraproduct law (see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF], Chap 2) and interpolation since B

T (u 2 n ) j ∂ j (u 2 n ) i L 1 (B N 2 -1 2,1 ) u 2 n L ∞ (B N p * -1 p,2
)

∇u 2 n L 1 (B N p p,2 )
. (3.35) then there exists a global strong solution (c, v) for the system (4.41). In addition we have:

(b ε , u ε ) E p ε,ν (c, v) E p 1,ν C εν 0 . (4.42)

The incompressible limit, weak convergence

Using the uniform estimates (4.42), we can show that the limit of (b ε , Qu ε ) when ε goes to 0 converges to 0 in the sense of the distribution. Similarly Pu ε converges to v in the sense of the distribution. The proof follows the same line than [START_REF] Danchin | The incompressible limit in L p type critical spaces[END_REF] (see also [START_REF] Lions | Une approche locale de la limite incompressible[END_REF]).

Convergence of b ε and Qu ε to zero

The case N = 3: The starting point is the dispersive inequality given in proposition 2.4 for (W ε ). Indeed denoting u ε k = u ε r + u ε z we have:

∂ t b ε + ε -1 divQu ε k = F ε , ∂ t Qu k + ε -1 ∇b ε = G ε , (4.43)
with

F ε = -div(b ε u ε ) and G ε = ν∆Qu ε k -Q(u ε k • ∇u ε k ) -Q( εb 1+εb Au ε k ) + K(εb ε )∇b ε ε -Q(u ε θ • ∇u ε θ ). Let us assume that ε = ν = 1 for the moment. Proposition 2.4 gives for all p 1 ∈ [2, +∞): (b, Qu) BF L 2p 1 p 1 -2 (B 2 p 1 -1 2 p 1 ,1 ) (b 0 , Qu 0 ) BF B 1 2 2,1 + (F, G) BF L 1 (B N 2 -1 2,1
) .

We have seen in the previous section that:

(F, G) BF L 1 (B N 2 -1 2,1 ) ≤ C 0 (b, Qu) BF L 1 (B N 2 +1 2,1 ) ≤ C 0 . ( 4

.44)

Using interpolation result with:

[L 1 (B 5 2 2,1 ), L 2p 1 p 1 -2 (B 2 p 1 -1 2 p 1 ,1 )] p 1 p 1 +2 = L 2 (B 4 p -1 2 p,1 ) with p = p 1 + 2 2 .
Similarly we have for α ≤ 0 small enough:

[L 1 (B 5 2 2,1 ), L 2p 1 p 1 -2 (B 2 p 1 -1 2 p 1 ,1 )] 2p 1 (1+α) (p 1 +2)(2+α) = L 2+α (B 4 p -1 2 -α 2+α p,1 ) with p = 2(2 + α)(p 1 + 2) 8 + 6α -αp 1 .
We deduce from (4.44) that:

ν 1 2 (b, Qu) BF L 2 (B 4 p -1 2 p,1 ) C 0 for all p ∈ [2, +∞) (b, Qu) BF L 2+α (B 4 p -1 2 -α 2+α p,1 ) C 0 for all p ∈ [2, +∞) with p = 2(2 + α)(p 1 + 2) 8 + 6α -αp 1 .
Coming back to the original variables in (4.40) we get:

ν 1 2 (b ε , Qu ε ) BF, ε L 2 (B 4 p -1 2 p,1 ) ε 1 2 -1 p C εν 0 (b, Qu) BF L 2+α (B 4 p -1 2 -α 2+α p,1 ) C εν 0 ε 1 2 -1 p .
We have by using (1.16) and the definition of the cut-off in (2.21):

(b ε , Qu ε ) L 2 (B 4 p -1 2 p,1 ) (b ε , Qu ε ) BF, ε L 2 (B 4 p -1 2 p,1 ) + (b ε , Qu ε ) HF, ε L 2 (B 4 p -1 2 p,1 ) ν -1/2 ε 1 2 -1 p C εν 0 + ε 1 2 -1 p (b ε , Qu ε ) HF, ε L 2 (B N p p,1 ) , ν -1/2 ε 1 2 -1 p C εν 0 .
Similarly we have: (b, Qu)

L 2+α (B 4 p -1 2 -α 2+α p,1 ) C εν 0 ε 1 2 -1 p .
The case N = 2:

As previously we have:

∂ t b ε + ε -1 divu 1 ε = F ε , ∂ t u 1 ε + ε -1 ∇b ε = G ε , (4.45)
with

F ε = -div(b ε u 1 ε ) and G ε = ν∆u 1 ε -(u 1 ε • ∇u 1 ε ) -( εb ε 1+εb Au 1 ε ) + K(εb ε )∇b ε ε -u 2 ε • ∇u 2 ε .
Taking ε = ν = 1 for the moment and using again proposition (2.4) to (1.3) we have:

(b, u 1 ) BF L r (B 2 p 1 -1+ 1 r p 1 ,1 ) (b 0 , u 1 0 ) BF B 1 2 2,1 + (F, G) BF L 1 (B N 2 -1 2,1 ) for 2 r ≤ 1 2 - 1 p 1 .
We take now r such that 1 r = c( 1 2 -1 p 1 ). We have seen in the previous section that:

(F, G) BF L 1 (B N 2 -1 2,1 ) ≤ C 0 (b, u 1 ) BF L 1 (B N 2 +1 2,1 ) ≤ C 0 . ( 4

.46)

Using interpolation as in the previous case, we deduce that for α < 0:

(b, u 1 ) BF L 2 (B c+2 p -c 2 p,1 ) C 0 with p = 4p 1 + (4 -2p 1 )c p 1 + 2 + (2 -p 1 )c (b, u 1 ) BF L 2+α (B c+2 p -c 2 -α 2+α p,1 ) C 0 with 1 p = p 1 + 2 + c(2 -p 1 ) 4p 1 + c(4 -2p 1 ) + α 2 -p 1 2(2 + α)(2p 1 -c(p 1 -2))
.

We recall that since c ∈ [0, 1 2 ] and p 1 ∈ [2, +∞] we can deal with p ∈ [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF][START_REF] Cannone | Solutions autosimilaires des équations de Navier-Stokes[END_REF]. Coming back to the original variables as in (4.40) it yields:

ν 1 2 (b ε , u 1,ε ) BF, ε L 2 (B c+2 p -c 2 p,1 ) ε c( 1 2 -1 p ) C εν 0 , (b, u 1,ε ) BF L 2+α (B c+2 p -c 2 -α 2+α p,1 ) ε c( 1 2 -1 p ) C εν 0 .
Proceeding as previously we get for α < 0 small enough:

(b ε , u 1 ε ) L 2 (B c+2 p -c 2 p,1 ) ν -1/2 ε c( 1 2 -1 p ) C εν 0 , (b, u 1,ε ) L 2+α (B c+2 p -c 2 -α 2+α p,1 ) ε c( 1 2 -1 p ) C εν 0 .

Convergence of the incompressible part

Let us give now the proof of the convergence of Pu ε to v. Setting2 ω ε = Pu ε -v we observe that in dimension N = 2 and N = 3, ω ε verifies:

∂ t ω ε -µ∆ω ε = -u 1,ε • ∇u 2,ε -u 2,ε • ∇u 1,ε - µεb ε 1 + εb ε ∆u 2,ε . (4.47) and:
∂ t ω ε -µ∆ω ε = -P(ω ε •∇v)-P(Pu ε •∇ω ε )-P(Qu ε •∇Pu ε )-P(u ε •∇Qu ε )-P( εb 1 + εb Au ε ).

(4.48)

The case N = 3:

Let us apply now the proposition 2.2 to the system (4.48), we get: . Furthermore using the cut-off (2.22), we have:

Y ε = ω ε L ∞ (B
εb ε L ∞ (B 4 p -1 2 p,1 ) ν -1 ε 1 2 -1 p b ε BF, ε L ∞ (B 1 2 2,1 ) + ε 1 2 -1 p εb ε HF, ε L ∞ (B 3 p p,1 ) ν -1 ε 1 2 -1 p C εν 0 .
Combining all the above estimates gives with α > 0 small:

Y ε ω ε 0 B 4 p -3 2 p,2 + ν -1 ε 1 2 -1 p (1 + ν -1 C εν 0 )(C εν 0 ) 2 + µ -1 ( v L ∞ (B 3 p -1 p,2 ) + µ v L 1 (B 3 p +1 p,2 )
)Y ε .

Using the theorem 1.1 and the smallness assumption, we deduce that:

Y ε ω ε 0 B 4 p -3 2 p,2 + ε 1 2 -1 p C εν 0 .
which concludes the proof of the case N = 3.

The case N = 2:

As previously applying proposition 2.2 to the system (4.47), we obtain:

Y ε = ω ε L ∞ (B c+2 p -c 2 -1 p,2 ) + ω ε L 1 (B c+2 p -c 2 +1 p,2 ) ω ε 0 B c+2 p -c 2 -1 p,2 + G ε L 1 (B 4 p -c 2 -1 p,2
) , with G ε = -u 1,ε • ∇u 2,ε -u 2,ε • ∇u 1,εµεb ε 1+εb ε ∆u 2,ε . According to (2.20) and the fact that .

Definition 2 . 2

 22 Ṡl u L ∞ = 0, where Ṡl stands for the low frequency cut-off defined by Ṡl := χ(2 -l D). If u ∈ S h , (2.18) is true and we can write that Ṡl u = q≤l-1 ∆q u. We can now define the homogeneous Besov spaces used in this article: For s ∈ R and 1 ≤ p, r ≤ ∞, we set

  .21) and: z BF,α B s p,r = z BF,α B s p,r and z HF,α B s p,r = z BF,α B s p,r . (2.22)

,L 1

 1 with G ε = -P(ω ε • ∇v) -P(Pu ε • ∇ω ε ) -P(Qu ε • ∇Pu ε ) -P(u ε • ∇Qu ε ) -P( εb 1+εb Au ε ). Next it remains essentially to estimate G ε

  belongs to Ḃs 2,1 and there exists a function of two variables C only depending on s, d and G such that

  ) Using proposition 2.3 and 2.2, we have the following estimate on (b n , u 1 n , u 2 n

We refer to the next section and in particular (2.21),(2.22) for the notations.

It implies that ω ε = u 2,ε -u 2 when N = 2 and ω ε = Pu k,ε + u θ,ε -u k -u θ when N =

3, we refer to (1.9) and (1.10) for the definition of u 2 , u k and u θ .

Acknowledgement:

The author has been partially funded by the ANR project INFAMIE ANR-15-CE40-0011.

3 Proof of the theorem 1.2 and 1.3

Let us start with the proof of the theorem 1.2, the proof of theorem 1.3 will follow the same lines. We fix now the cute off frequency in (2.4) which is equal now to l 0 .

Proof of the existence

Our goal now consists in proving the existence of global strong solution for the system (1.5) with ε = 1 under the smallness condition (1.11). Let us briefly explain the main arguments of the proof.

1. We smooth out the data and get a sequence of local solutions (b n , u 1 n , u 2 n ) n∈N on [0, T n ] to (1.5) by using the result of [START_REF] Haspot | Global existence of strong solution for viscous shallow water system with large initial data on the irrotational part[END_REF].

We prove uniform estimates on (b

,ν by using the proposition 2.3 and by studying the behavior of u 1 n in low and high frequencies. We deduce then that T n = +∞ .

3. We use compactness arguments to prove that the sequence (b n , u 1 n , u 2 n ) converges, up to extraction, to a solution of (1.5).

Construction of approximate solutions

We smooth out the data as follows:

We observe that (u 1 0 ) n and (u 2 0 ) n are alway invariant by rotation since the Fourier transform conserves the property of radial functions and since we work with ϕ a radial function in section 2.

Note that we have:

, and similar properties for (u 1 0 ) n , (u 2 0 ) n , a fact which will be used repeatedly during the next steps. Now, according [START_REF] Haspot | Global existence of strong solution for viscous shallow water system with large initial data on the irrotational part[END_REF], one can solve (1.5) with the smooth data ((b 0 ) n , (u 1 0 ) n , (u 2 0 ) n ). We get a solution (b n , u 1 n , u 2 n ) on a non trivial time interval [0, T n ] which verifies by uniqueness the condition (1.4) such that:

),

p,2 ).

(3.24)

Similarly we have:

) .

(3.36)

For the remainder term we have:

.

(3.37)

We proceed similarly in high frequencies, and using (3.35), (3.36) and (3.37) we get:

) .

(3.38)

Finally combining (3.32), (3.33) and (3.38) we obtain that:

(3.39) By a basic bootstrap argument we deduce that the sequence (b n , u

we deduce by using classical continuation criterion (see [START_REF] Danchin | Well-Posedness in critical spaces for barotropic viscous fluids with truly not constant density[END_REF][START_REF] Haspot | Global existence of strong solution for viscous shallow water system with large initial data on the irrotational part[END_REF]) that for any n, T n = +∞. It is now classical to prove that the sequence (b n , u 1 n , u 2 n ) n∈N converges in the sense of the distributions to a solution (b, u 1 , u 2 ) of (1.5) (see for more details [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]). The uniqueness of the solution is also classical (see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]).

The proof of the theorem 1.3 follows exactly the same lines.

4 Low Mach limit, Proof of the theorem 1.4

We are now going to prove the theorem 1.4 by following essentially the arguments of the proof of Danchin and He in [START_REF] Danchin | The incompressible limit in L p type critical spaces[END_REF].

Global uniform estimates in ε

Let us make the following change of functions c(t, x) = εb ε (ε 2 t, εx) and v(t, x) = εu ε (ε 2 t, εx).

(4.40)

Then (c, v) must verify:

.41)

Using the theorem 1.2, 1.3 and by scaling invariance, we know that there exists η > 0 such that if:

Similarly we have:

. Furthermore using the cut-off (2.22), we have:

We conclude by using the same arguements than for the case N = 3.


Pierre-Emmanuel Arduin
email: pierre-emmanuel.arduin@lamsade.dauphine.fr

Elsa Negre
email: elsa.negre@lamsade.dauphine.fr

Camille Rosenthal

Camille Rosenthal-Sabroux
email: camille.rosenthal-sabroux@lamsade.dauphine.fr

Towards a new

Keywords: tacit knowledge, decision process, cost-benefit analysis, multicriteria analysis

come

Introduction

We know that tacit knowledge plays an important role in decision making that is why, in this paper, we will discuss a new way to take into account tacit knowledge for decision-making process. Our approach is not yet a formal method but is based on a practical case study in the field of second hand vehicle purchase. To the best of our knowledge, in the literature, no research addresses this kind of approach. This paper proposes ideas and research issues on how to tackle tacit knowledge for decision process with cost-benefit and multicriteria analysis based on a real-world case.

In this paper we present the decision making process, then we emphasize tacit knowledge in the decision making process. Our work is based on three postulates, which are (i) knowledge is not an object, (ii) within a company there exists two main knowledge categories, and (iii) knowledge is linked to action. We also present the industrial context of our study: purchase and sale of second-hand vehicles. Actually in the company they have already defined a decision process to sell a car but tacit knowledge is not taken into account.

We propose to combine multicriteria analysis ( [START_REF] Roy | Multicriteria Methodology for Decision Aiding[END_REF]) and cost-benefit analysis ( [START_REF] Nas | Cost-benefit analysis: Theory and application[END_REF]). Because tacit knowledge cannot always be explicited, we use the multicriteria analysis to fittingly explicit tacit knowledge which can be explicited. In order to respect the financial policy of the company, we use the cost-benefit analysis. So it is possible to make the best decision regarding the purchase price according to the different available criteria. Thus we help the company that purchases/sales second hand vehicles, to realize the best deal. This paper is organized as follows: Section 2 introduces tacit and explicit knowledge and decision process. Section 3 presents the context of our real-world case while, in section 4, we describe the existing problem in such a case. Section 5 reports our approach combining multicriteria and cost-benefit analysis to take into account tacit knowledge in decision process. Finally, Section 6 concludes this paper and provides future work.

2 Tacit and explicit knowledge and decision process

The Company's Knowledge

The company's knowledge consists of tangible elements (databases, procedures, drawings, models, algorithms, documents used for analyzing and synthesizing data) and intangible elements (people's abilities, professional knack, "trade secrets", "routines" -unwritten rules of individual and collective behavior patterns [NW82] -, knowledge of the company's history and decision-making contexts, knowledge of the company environment (clients, competitors, technologies, influential socio-economic factors)) [START_REF] Michel Grundstein | Reinforcing decision aid by capitalizing on company's knowledge: Future prospects[END_REF]. They characterize a company capability to design, produce, sell, support its products and services. They are representative of the company's experience and culture [START_REF] Davenport | Working Knowledge: How Organizations Manage What They Know[END_REF]. They constitute and produce the added-value of its organizational and production business processes. Tangible elements are "explicit knowledge". Heterogeneous, incomplete or redundant, they are often marked by the circumstances under which knowledge was created. They do not express the unwritten rules of those who formalized knowledge, the "unspoken words". They are stored and disseminated in archives, cabinets, and databases.

Intangible elements are "tacit knowledge". Acquired through practice, they are adaptable to the situations. Explicitable or non-explicitable, they are often transmitted by implicit collective apprenticeship or by a master-apprentice relationship. They are located in people's minds. Here we are referring to the knowledge classification of Michael Polanyi [START_REF] Polanyi | Sense-giving and sense-reading[END_REF]. He classifies the human knowledge into two categories: tacit knowledge and explicit knowledge. "Tacit knowledge is personal, context-specific, and therefore hard to formalize and communicate. Explicit or 'codified' knowledge, on the other hand, refers to knowledge that is transmittable in formal, systematic language" [START_REF] Polanyi | Sense-giving and sense-reading[END_REF]. Our point of view can be found in the work of Ikujiro Nonaka and Hirotaka Takeuchi [START_REF] Nonaka | The knowledge-Creating Company[END_REF], two Japanese authors who, with reference to [START_REF] Polanyi | Sense-giving and sense-reading[END_REF] consider that "tacit knowledge and explicit knowledge are not totally separated but mutually complementary entities." For [START_REF] Nonaka | The knowledge-Creating Company[END_REF], explicit knowledge can be easily expressed in written documents but is less likely to result in major decisions than tacit knowledge, which is to say that the decision process stems from knowledge acquired through experience, albeit difficult to express in words. These observations concerning knowledge in the company context highlight the importance of tacit knowledge. They point out the interest in taking into account tacit knowledge in decision process. In the following paragraphs, we will first present the general principles that underlie decision support research, and especially the formalized decision-making process by Herbert Simon [START_REF] Alexander | The New Science of Management Decision[END_REF].

The Decision-making Process

A process is a combination of phenomena, conceived to be active and organized in time. As such, a decision-making process comprises everything that takes place in reality, like actions, activities and phenomena that lead to the choice of the final decision. According to [START_REF] Alexander | The New Science of Management Decision[END_REF] a decision-making process passes through four different phases: the "Investigation" (Intelligence) phase, the "Design" phase, the "Choice" phase, and the "Review" phase. With respect to the process defined in [START_REF] Alexander | The New Science of Management Decision[END_REF], an additional important aspect, which underlies all phases, is the communication and interaction dimension. The emphasis on communication and interaction means fostering a minimum on tacit knowledge. In this respect, the "Investigation" (Intelligence) phase, being the first phase in Simon's decision-making process [START_REF] Alexander | The New Science of Management Decision[END_REF], is primordial. The relevance of the entire process depends on this phase. It leads to the tacit knowledge.

Tacit Knowledge and the decision process

When [START_REF] Polanyi | Sense-giving and sense-reading[END_REF] introduces the concepts of sense-giving and sense-reading, we simply observe that we continuously appropriate information which is not ours. He defined them as follows: "Both Figure 1: Tacit knowledge transfer the way we endow our own utterance with meaning and our attribution of meaning to the utterances of others are acts of tacit knowing. They represent sense-giving and sense-reading within the structure of tacit knowing" [START_REF] Polanyi | Sense-giving and sense-reading[END_REF]. As the authors of this paper, we have got tacit knowledge that we have structured into information during a process of sense-giving. As the readers of this paper, you have interpreted this information perceiving forms and colours, integrated words, data, during a process of sense-reading possibly creating new tacit knowledge for you (see Figure 1).

When a person P1 structures its tacit knowledge and externalizes it, he creates information. A person P2 perceiving some data from this information and internalizing it, possibly creates new tacit knowledge. Thus knowledge is the result of the interpretation by someone of information ( [START_REF] Arduin | From knowledge sharing to collaborative decision making[END_REF]). This interpretation is done through an interpretative framework that filters data contained in the information and the use of previous tacit knowledge as presented by [START_REF] Tsuchiya | Improving knowledge creation ability through organizational learning[END_REF] and by [START_REF] Grundstein | New Research on Knowledge Management Models and Methods, chapter Three Postulates That Change Knowledge Management Paradigm[END_REF].

Our work is based on three postulates:

Postulate 1: Knowledge is not an object1 Generally, knowledge cannot be treated as an object. It is the result of the encounter of data and a subject and it is processed within the interpretative scheme of the subject's memory. This postulate is based on the theories developed by Tsuchiya, who deals with the construction of tacit individual knowledge ( [START_REF] Tsuchiya | Improving knowledge creation ability through organizational learning[END_REF]. According to his research, the tacit knowledge which resides within our brain is the result of the meaning we attribute -through our interpretative schemes -to the data that we perceive as part of the information that we receive. This individual knowledge is tacit and it can or cannot be expressed.

Postulate 2. Within a company, there exist two main knowledge categories Within a company, knowledge consists of, on the one hand, explicit knowledge comprising all tangible elements and, on the other hand, tacit knowledge, which comprises intangible knowledge. Hence, knowledge within a business organization comprises two main categories: formalized and explicit knowledge, which can be called "business know-how", and the individual and collective tacit knowledge, which can be called the "business skills", and which can either be made explicit or not (cf. Table 1), which can be explicitable or not.

Postulate 3: Knowledge is linked to action From a business perspective, knowledge is created through action. Knowledge is essential for the business's functioning and is finalized through its actions. Hence, one has to be interested in the activities of the actors -decision-makers, engaged in the process contained in the company's mission. This vantage point is included in the use of the concept of knowledge, which cannot be detached from the individual placed within the company, his/her actions, decisions and relations with the surrounding systems (people and artifacts).

3 Context: Purchase/sale of second-hand vehicles

The idea of buying a second-hand vehicle

The company we are interested here, offers a purchase service of second-hand vehicles without conditions. Indeed, to sell a vehicle to a distributor, this vehicle must be in good condition and not too old. More generally, in France, a distributor buys your car only if you buy a new one in his company. He can afford to offer a higher purchase price because he will make a significant profit margin with the car you will buy. The promise of this company is to buy all cars, regardless of their condition and without obligation to buy an other vehicle.

Sell a car, how does it work?

When a private individual, Elsa, wants to sell her vehicle, she asks a prior valuation of the second-hand vehicle on the company website. She gives information about the main characteristics of the vehicle such as brand, model and age. Following this query, she obtains a valuation of the purchase price of the vehicle. The system does not take into account the wear on the vehicle and it believes that the vehicle is in good condition. When querying to obtain a first valuation on the website, Elsa leaves her address and telephone number, she will be contacted by the company to have an appointment in an agency of the company. When Elsa goes to the agency, she is received by a vehicle expert, John. He examines the vehicle and values the costs to recondition the vehicle. This step is very important because it controls the working state of the vehicle, John must do a road test and identify anomalies. The data are inputed and stored through a computer platform in order to ask the quotation experts the vehicle valuation. The value that John will get from the purchasing department through this platform will be the best proposal and John will not have scope on this proposal. Figure 2 displays this process. Figure 3: Difference between the real and the predicted profit margin [dlB12] 4 Problem: Tacit knowledge is not taken into account

The vehicle valuation method is not discussed here, but for now, it is unpredictable because this valuation depends on each quotation expert. To estimate the market, these quotation experts go on online vehicle ad websites and evaluate market prices. They valuate the vehicle so that it is well placed among similar ads. This approach sometimes leads to the purchase price offered to the private individual is not consistent with the real resale price.

The profitability of the purchase/sale business is based on one principle: buy second-hand cars and sell them more expensive, with a profit margin nearly constant. The issue of valuating vehicles is to know how much the vehicle may be sold and allows therefore to calculate a purchase price, removing the wanted profit margin. Nowadays, quotation expert valuations do not always correspond to the reality of the market. Sometimes the company does not break even, because the purchase price is too high compare to the final resale price or profit margins are too large while the company could offer a better price. Figure 3 shows that the wanted profit margin is achieved in less than half of the vehicles. These errors, whether against or in favor of the private individual make the activity uncertain and decrease the quality of the service offered by the company. In addition to make the business management difficult to predict, it changes the behavior of the quotation experts. To protect themselves from a possible market decline, the quotation experts propose purchase prices lower than they should. Private individuals are disadvantaged and it creates dissatisfaction and discontent among the private individuals who come to sale their vehicle.

Taking into account some tacit knowledge would improve the vehicle price valuation:

• The context of a particular sale. For example, financial or family problems can cause an urgent need to sell the vehicle regardless of the price or otherwise can cause an attempt to negotiate the price as high as possible,

• The repair costs. For example, a simple scratch can cause the replacement of an important part of the vehicle and only an expert can know that,

• The location of the vehicle. For example, the experts know that some vehicles will be sold differently depending on their geographical location. Figure 4 shows that the same vehicle is not sold at the same price in the north or the south of France, the same difference exists between countries, as displayed Figures 5 and6, • The market itself. For example, if a vehicle is very popular, the company will keep it in stock less time, so it will be cheaper. However, if a vehicle is not popular, it will spend a longer time between the purchase of the vehicle and its resale, so there is an additional storage cost for the company.

Our proposition

The context presentation shows that John, the vehicle expert who receives the private individual seller Elsa, has no leeway on the proposed purchase price. However, it would be interesting to give him a minimum of opportunities to try to reconcile the purchase price and the sale price wanted by Elsa. Indeed, sometimes, because of a few euros, a sale may or may not occur. However, John must also stay close to the expectations of company, namely, "business is business" and because the company will not accept losing money.

Our proposal is to develop a tool for risk analysis in terms of decision making. In such a way that this tool would allow John to slightly higher modify the proposed purchase price while ensuring that the company will still be able to make his profit margin and especially not to loose money. It is therefore to take into account the various available indicators (relational context, emergency of the sale for the private individual, ...) to cross these indicators with the vehicle-related criteria (the market, repairs, ...) and test the impact on the financial expectations of the company. More specifically, it is to combine the cost-benefit analysis ( [START_REF] Nas | Cost-benefit analysis: Theory and application[END_REF]) to respect the financial policy of the company and the multicriteria analysis ( [START_REF] Roy | Multicriteria Methodology for Decision Aiding[END_REF]), so that John makes the best decision regarding the purchase price according to the different available criteria.

Cost-Benefit Analysis

Cost-benefit analysis (see [START_REF] Nas | Cost-benefit analysis: Theory and application[END_REF] for more details) is a systematic process for calculating and comparing benefits and costs of a project. The main purpose is to provide a basis for comparing projects, i.e. comparing the total expected cost of each option against the total expected benefits. Note that benefits and costs are expressed in monetary terms. It helps predict whether the benefits outweigh its costs, and by how much relative to other alternatives.

[Arg10] breaks down the cost-benefit analysis into 10 steps:

1. Determine if the cost-benefit analysis worth doing 2. Identify objectives and policy alternatives 3. Determine who has standing 4. Identify the costs and benefits of each alternative 5. Sort into measurable and non-measurable costs and benefits 6. Measure costs and benefits that can be measured in money terms 7. Conduct sensitivity analysis 8. Compare costs-benefits across alternatives 9. Adjust for non-measurable costs and benefits

Make a decision

The cost-benefit analysis recognizes that each choice has a cost and makes explicit hidden costs and benefits. Unfortunately, the valuation of life is difficult or impossible and this kind of approach is ignorant of the political context of decision-making.

Multicriteria analysis

Multicriteria analysis (see [START_REF] Roy | Multicriteria Methodology for Decision Aiding[END_REF] for more details) provides a framework to enable decisionmakers to overcome difficulties in handling large amounts of complex information and establishes preferences between options, i.e. a set of identified objectives, and for which measurable criteria have been established to assess the extent to which these objectives have been achieved. In fact, there does not exist a unique optimal solution for such problems and it is necessary to use decision maker's preferences to differentiate between solutions. That is why "solving" such a problem can be interpreted in different ways. The two main purposes are: 1) the "best" alternative from a set of available alternatives, 2) a small set of good alternatives.

[Arg10] breaks down the multicriteria analysis into 7 steps:

1. Establish the decision context 2. Identify the value/performance criteria 3. Describe/rate the performance of each option against the criteria 4. Assign weights across criteria 5. Combine the scores and weights 6. Examine the results and review

Conduct sensitivity analysis

The multicriteria analysis provides an audit trail, especially in situations where decision-making is required to follow rules and to be justified in explicit terms and is based on weightings. Unfortunately, weightings are hard to obtain and sometimes, there is a lack of methodological rigor.

Combining Cost-benefit and multicriteria analysis

Our proposition consists in combining cost-benefit analysis and multicriteria analysis ([KBV + 05], [START_REF] Dobes | Multi-Criteria Analysis: "Good Enough" for Government Work[END_REF], [START_REF] Paolo | Comparing cost benefit and multi-criteria analysis: the evaluation of neighbourhoods' sustainable mobility[END_REF]). Even though, [START_REF] Dobes | Multi-Criteria Analysis: "Good Enough" for Government Work[END_REF] seems to interpret that the multicriteria approach goal is worst than the benefit-cost approach goal, cost-benefit analysis and multicriteria analysis share some similarities:

• Both are frameworks for assessing options facing decision-makers,

• Both try to construct a common metric for comparing options

• Both are sensitive to assumptions, but the assumptions are different.

We argue to benefit of the advantages of each kind of analysis. The cost-benefit analysis is pertinent and easy-to-use for measurable costs and takes into account the expectations of the company ("business is business"). The multicriteria analysis is pertinent to make decision and allows to bring into play some "non-measurable" costs. Thus, our proposition consists in using the cost-benefit analysis for the measurable costs (quantitative ones) and using the multicriteria analysis for the "non-measurable" costs (qualitative ones), i.e. the tacit knowledge that can affect the purchase price.

Using cost-benefit analysis will not be discussed in this paper which will focus now on the strengths of multicriteria analysis in order to consider non measurable costs and benefits, i.e. the tacit knowledge embodied in John's mind. Pseudo-criterion such as context, repair costs, location and market will be confronted in order to decide if and how much the final price could be increased by John, the vehicle expert. The possible actions are then: no increase, increase at most 5% of the vehicle estimated price, increase at most 10% of the vehicle estimated price, increase at most 15% of the vehicle estimated price, increase at most 20% of the vehicle estimated price, and increase at most 25% of the vehicle estimated price. These levels have been defined by us according to the circumstances of a vehicle sale and the influence we consider that tacit knowledge can have on a vehicle final price. They could of course be refined with John, the vehicle expert, in future works.

The context of the sale could have a strong influence on the final price and that is the reason why it has to be considered by our proposition. More the decision is supported by the context, more the rating will be hight, as shown in Table 2. Repair costs can be difficult to evaluate notably when the car cannot be studied in detail. Our proposition gives to John, the vehicle expert, the opportunity to say rapidly if and how much according to him the repair costs influence the vehicle final price, although the car is not studied in detail. John can be able to say, for example, if a simple scratch will cause the replacement of an important part of the vehicle body. More the decision is supported by the repair costs, more the rating will be hight, as shown in Table 2. The same reasoning is followed to rate the influence of location and market on the vehicle final price.

So, regarding on the pseudo-criteria John has to rate the different possible actions (no increase, increase at most X%, etc.). We have also to determine with him the value of the preference and the indifference thresholds for each pseudo-criterion. We can then compute the partial concordance indexes c j (a, b) for every couple of actions (a, b) for each pseudo-criterion j. The partial concordance indexes are defined as follows [START_REF] Roy | Multicriteria Methodology for Decision Aiding[END_REF]:

• c j (a, b) = 1 if g j (a) ≥ g j (b) -q j (a is preferred to b) Thereby we can construct a payoff matrix and study the relations between actions and pseudocriteria. It is interesting for us to see how John, the vehicle expert, will construct the pseudo-criteria according to its tacit knowledge. Within this process he will try to elicit it although he is not using it as if he was methodically estimating a car for example. In so far as new non-measurable costs and benefits can be identified, our proposition may have to grow estimation after estimation.

In the case of Elsa's vehicle example, for the multicriteria part, there is no emergency for her to sell her car, that is the reason why the context of the sale could influence the final price to grow, in a reasonable measure (see Table 3, indicated values come from Table 2). For example, the last line of Table 3 ("increase at most 25% ") means that, because of the current context, repair costs, location and market (1), it is out of the question to propose an increase of at most 25%. Repair costs have been identified as being minor by the expert, that is the reason why they could influence the final price to grow. Furthermore and according to the vehicle-expert conclusions, this sale is located in an area where Elsa's vehicle is very demanded, instead of having no popularity in the market. It is the case for some vintage vehicles, which are demanded but not popular. It induces

Figure 2 :

 2 Figure 2: Sell a car, How does it work?

Figure 4 :Figure 5 :

 45 Figure 4: Two similar vehicles sold in France (North and South) (www.lacentrale.com)

Figure 6 :

 6 Figure 6: Similar vehicle sold in Spain (www.ooyyo.es)

  Figure 7: Partial concordance index behavior for a pseudo-criterion j

•

  c j (a, b) = 0 if g j (a) ≤ g j (b) -p j (b is preferred to a) • c j (a, b) = p j -(gj (b)-gj (a)) pj -qjelse (indifference, see figure7)

Table 1 :

 1 Company Knowledge[START_REF] Michel Grundstein | Reinforcing decision aid by capitalizing on company's knowledge: Future prospects[END_REF]

	KNOW-HOW		SKILLS
	(explicit knowledge)	(tacit knowledge embodied by individuals)
	Collective knowledge	Collective knowledge	Personal knowledge
			People's abilities,
			Professional knack,
	Knowledge that is	Knowledge that is	Knowledge of company
	formalized within	incorporated within models	history and decisional contexts,
	documents and/or codified	and regular and predictable	Knowledge of the environment
	into data bases	behaviors	(customers, competitors,
			technologies) and
			socio-economical factors
		Defensive routines	Specific knowledge
		Knowledge that are obstacles	belonging to each individual
	Information source of	to change	Knowledge that is volatile
	knowledge for someone	Constructive routines	intangible resource, which depends
		Knowledge that favors innovation on the continuity of the presence of
		and change	employees in the company

We consider, according to the distionary, that an object is anything visible or tangoible; a material product or substance.

 2)

the behaviors of the pseudo-criteria location and market in the Table 3. If Elsa was hurried to sell her vehicle, the behavior of the context pseudo-criterion would have been different (for example, if she was hurried, she should accept a cheaper price). If a scratch on her vehicle could lead to machine the body of the vehicle, the behavior of the repair costs pseudo-criterion would also have been different (for example, the more repairs there are, the less the purchase price increases). So the multicriteria part of our approach considers all of these non-measurable elements to estimate the value of a vehicle and thus highlights tacit knowledge. The aim is to obtain a value closer as possible as the seller could wish it, and a value higher as possible for the company. The confrontation between the different issues (no increase, increase at most X%, etc.) and the computation of the partial concordance indexes c j (a, b) will not be presented in this paper due to confidentiality aspects of the preference and indifference thresholds for each pseudo-criterion. Finaly, in our example, John can propose an increase of at most 5% to Elsa for her car thanks to the materialization of his tacit knowledge about the context (4), the repair costs (4), the location (3) and the market (3).

A real strength of our proposition is that not only it considers all the measurable costs and benefits of a vehicle sale, but also its non-measurable costs and benefits, which are in the head of John, the vehicle expert, and that he is not always aware. Our proposition facilitates the use of tacit knowledge by John, the vehicle expert, because it encourages him to consider it, to take it into account to propose a vehicle price as closer as the seller could wish it according to the real state of its vehicle and not only following an evaluation proposed by the purchasing department on which he had no scope in the past. The increased price should ensure that the transaction will be done, that it will satisfy the seller, and most of all that it will be profitable for the company. Cost-benefit analysis gives to John, the vehicle expert, an estimation regarding on measurable data, and multicriteria analysis allows him to know, regarding on the considered tacit knowledge, if and how much he can increase this estimation in order to ensure the transaction will be more profitable as possible for the company. We are aware that our approach does not take into account all tacit knowledge but only a part, the explicitable part of tacit knowledge.

Conclusion and future work

In this paper we emphasise the strengths of tacit knowledge in the decision making process in addition to decision support systems. Thus, we propose to combine multicriteria analysis and cost-benefit analysis. Because tacit knowledge cannot always be explicited, we use the multicriteria analysis to fittingly explicit tacit knowledge which can be explicited. In order to respect the financial policy of the company, we use the cost-benefit analysis. So it is possible to make the best decision regarding the purchase price according to the different available criteria.

In the future, our works will focus on identifying and formalizing the explicitable part of tacit knowledge. This work is a first step for a more formalized method. The materialization of tacit knowledge opens the way towards some new research perspectives, specially for user-centered design of information and knowledge systems.