Boris Haspot
email: haspot@ceremade.dauphine.fr

Strong solution for Korteweg system in bmo -1 (R N ) with initial density in L ∞

In this paper we investigate the question of the existence of strong solution in finite time for the Korteweg system for small initial data provided that the initial momentum ρ 0 u 0 belongs to bmo -1 T (R N ) for T > 0 and the initial density ρ 0 is in L ∞ (R N ) with N ≥ 1 and far away from the vacuum. This result extends the so called Koch-Tataru theorem for the Korteweg system. It is also interesting to observe that any initial shock on the density is instantaneously regularized inasmuch as the density becomes Lipschitz for any ρ(t, •) with t > 0. We also prove the existence of global strong solution for initial data (ρ 0 -1,

This result allows in particular to extend the notion of Oseen solution (corresponding to particular solution of the incompressible Navier Stokes system in dimension N = 2) to the Korteweg system provided that the vorticity of the momentum ρ 0 u 0 is a Dirac mass αδ 0 with α sufficiently small. IHowever unlike the Navier Stokes equations the property of self similarity is not conserved for the Korteweg system since there is no invariance by scaling because the term of pressure.

Introduction

We are concerned with compressible fluids endowed with internal capillarity. The model we consider originates from the XIXth century work by J. F. Van der Waals and D. J. Korteweg [START_REF] Van Der Waals | Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung[END_REF][START_REF] Korteweg | Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires par des variations de densité[END_REF] and was actually derived in its modern form in the 1980s using the second gradient theory (see [START_REF] Dunn | On the thermomechanics of interstitial working[END_REF][START_REF] Jamet | The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change[END_REF][START_REF] Truedell | The nonlinear field theories of mechanics[END_REF]). Korteweg-type models are based on an extended version of nonequilibrium thermodynamics, which assumes that the energy of the fluid not only depends on standard variables but also on the gradient of the density. We are now going to consider the so-called local Korteweg system which is a compressible capillary fluid model, it can be derived from a Cahn-Hilliard like free energy (see the pioneering work by J.-E. Dunn and J. Serrin in [START_REF] Dunn | On the thermomechanics of interstitial working[END_REF] and also [START_REF] Anderson | Diffuse-interface methods in fluid mech[END_REF][START_REF] Cahn | Free energy of a nonuniform system, I. Interfacial free energy[END_REF][START_REF] Gurtin | Two-phases binary fluids and immiscible fluids described by an order parameter[END_REF]). The conservation of mass and of momentum write:

     ∂ ∂t ρ + div(ρu) = 0, ∂ ∂t (ρu) + div(ρu ⊗ u) -div(2µD(u)) + ∇P(ρ) = divK.
(1.1)

Here u = u(t, x) ∈ R N stands for the velocity field with N ≥ 1, ρ = ρ(t, x) ∈ R + is the density, D(u) = 1 2 (∇u + t ∇u) is the strain tensor and P (ρ) is the pressure (we will only consider regular pressure law). We denote by µ the viscosity coefficients of the fluid. We supplement the problem with initial condition (ρ 0 , u 0 ). The Korteweg tensor reads as:

divK = 2κ 2 ρ∇( ∆ √ ρ √ ρ ), (1.2)
where κ(ρ) = κ 2 ρ is the coefficient of capillarity with κ 2 ∈ R + . The term divK allows to describe the variation of density at the interfaces between two phases, generally a mixture liquid-vapor. In our case it is also called quantum pressure.

We briefly mention that the existence of global strong solutions for the system (1.1) with small initial data for N ≥ 2 is known since the works by Hattori and Li [START_REF] Hattori | The existence of global solutions to a fluid dynamic model for materials for Korteweg type[END_REF] in the case of constant capillary coefficient κ(ρ). Danchin and Desjardins in [START_REF] Danchin | Existence of solutions for compressible fluid models of Korteweg type[END_REF] improved this result by working with initial data (ρ 0 -1, ρ 0 u 0 ) belonging to the homogeneous Besov spaces 1

B N 2 2,1 × B N 2 -1 2,1
(it is important to point out that B N 2 2,1 is embedded in L ∞ which allows to control the L ∞ norm of the density, it is even better since it implies that ρ 0 is necessary a continuous function). In [START_REF] Haspot | Existence of global strong solution for Korteweg system with large infinite energy initial data[END_REF], it is proved the existence of global strong solution with small initial data provided that (ρ 0 -1,

ρ 0 u 0 ) is in (B N 2 2,2 ∩ L ∞ ) × B N 2 -1 2,2
. This result extends [START_REF] Danchin | Existence of solutions for compressible fluid models of Korteweg type[END_REF] but does not allow to deal with general shocks on the initial density.

We wish now rewrite the system (1.1) using the formulation introduced in [START_REF] Antonelli | Global existence of finite energy weak solutions of quantum Navier-Stokes equations[END_REF] (see also the existence of global energy weak solution), we consider the following effective velocity with c > 0:

v c = u + c∇ ln ρ, (1.3)
which enables us to rewrite the system (1.1) as follows when κ 2 ≤ µ 2 2 using the fact that div(ρ∇∇ ln ρ) = 2ρ∇(

∆ √ ρ √ ρ ):    ∂ t ρ -c∆ρ + div(ρv c ) = 0 ρ∂ t v + ρu • ∇v -µdiv(ρ∇v) -(µ -c)div(ρ t ∇v) + ∇P (ρ) -2κ 2 1 ρ∇( ∆ √ ρ √ ρ ) = 0.
(1.4) with κ 2 1 = κ 2 -2µc + c 2 . We specify now the value of c and we want to deal with a c such that κ 2 1 = 0, we take now:

c 1 = µ -µ 2 -κ 2 and c 2 = µ + µ 2 -κ 2 .
1 In the sequel we will only consider homogeneous Besov space that we will note B s p,r even if they are generally written Ḃs p,r with s ∈ R and (p, r) ∈ [1, +∞] 2 . We refer to [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF] for the definition of the homogeneous Besov spaces and for the notion of product in the Besov spaces related to the so called paraproduct. We will use the same notation as in [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]. 2 We will only consider this case in the sequel.

2

Setting now v 1 = u + c 1 ∇ ln ρ and v 2 = u + c 2 ∇ ln ρ , we have:

                       ∂ t ρ -c 1 ∆ρ + div(ρv 1 ) = 0 ∂ t (ρv 1 ) + 1 2 div(ρv 1 ⊗ v 2 ) + 1 2 div(ρv 2 ⊗ v 1 ) -µ∆(ρv 1 )
-µ 2 -κ 2 ∇div(ρv 1 ) + ∇P (ρ) = 0,

∂ t (ρv 2 ) + 1 2 div(ρv 1 ⊗ v 2 ) + 1 2 div(ρv 2 ⊗ v 1 ) -µ∆(ρv 2 )
+ µ 2 -κ 2 ∇div(ρv 2 ) + ∇P (ρ) = 0.

(1.5)

In this paper our main goal consists in proving the existence of global strong solution for the system (1.1) with minimal regularity assumption on the initial data. More precisely since this system models a mixture liquid vapor with different density, we wish to show the existence of strong solution for initial density ρ 0 belonging only to the set L ∞ (R N ) (and in addition for initial density far away from the vacuum). In particular it implies that our functional setting will include the case of initial density admitting shocks what is also fundamental both in the physical theory of non-equilibrium thermodynamics as well as in the mathematical study of inviscid models for compressible flow. In addition we wish also to deal with momentum ρ 0 u 0 exhibiting specific structure, typically we have in mind the case of initial vorticity belonging to the set of finite measure in dimension N = 2 (we will recall later that there is such a theory for Navier-Stokes equations and even explicit such solutions, the so called Oseen tourbillon). In order to obtain such results, it seems necessary to work in space with minimal regularity assumptions. Typically the space B N p -1 p,r with third index r = +∞ is a good candidate for the initial momentum ρ 0 u 0 . To do this, let us now recall the notion of scaling for the Korteweg's system (1.1). Such an approach is now classical for incompressible Navier-Stokes equation and yields local well-posedness (or global well-posedness for small initial data) in spaces with minimal regularity. In our situation we can easily check that, if (ρ, u) solves (1.1), then (ρ λ , u λ ) solves also this system:

ρ λ (t, x) = ρ(λ 2 t, λx) , u λ (t, x) = λu(λ 2 t, λx) (1.6)
provided the pressure laws P have been changed to λ 2 P .

Remark 1 It is very important to point out that since there is only a scaling invariance up to the pressure term, we can not hope to show the existence of global strong self similar solution for the Korteweg system. Indeed in comparison with the Navier-Stokes system, we know that it exists global in time self similar solution provided that the initial velocity u 0 is homogeneous of degree -1 (such initial data exists for example in the set

B N p -1 p,∞ with 1 ≤ p < +∞).
In particular in the sequel we will able to deal with initial data (ρ 0 , ρ 0 u 0 ) such that ρ 0 is homogeneous of degree 0 and ρ 0 u 0 of degree -1, however the associated strong solution will be a priori not self similar.

The previous transformation (1.6) suggests us however to choose initial data (ρ 0 , u 0 ) in spaces whose norm is invariant for all λ > 0 by the transformation (ρ 0 , u 0 ) -→

(ρ 0 (λ•), λu 0 (λ•)). A natural candidate is for example the space L ∞ (R N ) × Ḣ N 2 -1 .
As we mentioned previously this invariance was also used for the first time by Kato [START_REF] Kato | The Navier-Stokes equation for an incompressible fluid in R 2 with a measure as the initial vorticity[END_REF] to prove that the Navier-Stokes equation is locally well-posed for arbitrary data in L N (R N ) and globally well-posed for small initial data. Kato's result was extended to larger scale invariant function spaces (one interest of dealing with larger function spaces is that they may contain initial data which are homogeneous of degree -1 and therefore give rise to self-similar solutions), in particular in [START_REF] Cannone | Solutions auto-similaires des équations de Navier-Stokes. Séminaire sur les équations aux dérivées partielles[END_REF] Cannone, Meyer and Planchon proved the existence of global strong solution with small initial data in B N p -1 p,∞ with p < +∞. A similar analysis was carried out for the vorticity equation in Morrey spaces by Giga and Miyakawa [START_REF] Giga | Two-dimensional Navier-Stokes flow with measures as initial vorticity[END_REF]. Finally this approach has been generalized by Koch and Tataru in [START_REF] Koch | Well-posedness for the Navier-Stokes equations[END_REF] when the initial data is small in BM O -1 (R N ). As explained these results allow to obtain the existence of global self similar solution for small initial data when N ≥ 2 for the incompressible Navier-Stokes equations. In dimension N = 2, it allows in particular to prove the existence and the uniqueness of the solution for initial data u 0 verifying curlu 0 = αδ 0 or u 0 = α x ⊥ |x| 2 with α small enough. These solutions are the so-called Lamb-Oseen solutions which are self-similar. However we have even an explicit formula for these solutions even when |α| is large, the Lamb-Oseen vortex are given by:

curlu α (t, x) = α t G( x √ t ), u α (t, x) = α √ t v G ( x √ t ), x ∈ R 2 , t > 0, (1.7)
where:

G(ξ) = 1 4π e -|ξ| 2 4 , v G (ξ) = 1 2π ξ ⊥ |ξ| 2 (1 -e -|ξ| 2 4 ), ξ ∈ R 2 ,
with ξ ⊥ = (-ξ 2 , ξ 1 ). In passing, we mention that the existence of global solution with initial vorticity in the set M(R 2 ) of all finite measures on R 2 was first proved by Cottet [START_REF] Cottet | Équations de Navier-Stokes dans le plan avec tourbillon initial mesure[END_REF] and independently by Giga, Miyakawa and Osada [START_REF] Giga | Two-dimensional Navier-Stokes flow with measures as initial vorticity[END_REF]. In [START_REF] Giga | Two-dimensional Navier-Stokes flow with measures as initial vorticity[END_REF], the authors proved also the uniqueness when the atomic part of the initial vorticity is sufficiently small. The uniqueness for any curlu 0 ∈ M(R 2 ) is proved in [START_REF] Gallagher | Uniqueness for the two-dimensional Navier-Stokes equation with a measure as initial vorticity[END_REF], it allows in particular to obtain the existence and the uniqueness of global self similar solution for large initial data when N = 2.

In this paper we are interested in adapting the technics of Koch-Tataru to the case of the Korteweg system (1.5). More precisely, we wish to prove the existence of strong solution in finite time for (1.1) provided that ρ 0 v 1 (0, •) and ρ 0 v 2 (0, •) are sufficiently small in norm

• bmo -1 T (R N )
for T > 0 and when the initial density is far away from the vacuum and is bounded in L ∞ norm . We recall that bmo -1 (R N ) is the set of temperated distribution u 0 for which for all T > 0 we have:

sup x∈R N ,0<t<T t -N 2 t 0 B(x, √ t) |e s∆ u 0 (y)| 2 dyds 1 2 < +∞.
We define the norm of • bmo -1 T (R N ) by:

u 0 bmo -1 (R N ) = sup x∈R N ,0<t<T t -N 2 t 0 B(x, √ t) |e s∆ u 0 (y)| 2 dyds 1 2
, with e t∆ u 0 be the solution to the heat equation with initial data u 0 :

e t∆ u 0 = u 0 * φ √ 4t with φ(x) = π -N 2 e -|x| 2 and φ t (x) = t -N φ( x t
).

In the sequel we will denote by E T the space of temperated distribution associated to the following norm:

u E T = sup 0<t≤T √ t u(t, •) L ∞ (R N ) + sup x∈R N ,0<t≤T t -N 2 t 0 B(x, √ t) |u| 2 (s, y)dtdy 1 2 , (1.8)
More precisely we will show ρv 1 and ρv 2 belong to E T for T > 0 the time of existence and that (ρ,

1 ρ ) is in L ∞ T (L ∞ (R N ))
. We recall that ρv 1 = ρu + c 1 ∇ρ and ρv 2 = ρu + c 2 ∇ρ, it implies in particular that ∇ρ =

1 2 √ µ 2 -κ 2 (ρv 2 -ρv 1 )
and we deduce from the definition of E T that it exists C > 0 independent on t such that for all 0 < t ≤ T we have:

∇ρ(t, •) L ∞ (R N ) ≤ C √ t .
(1.9)

Combining (1.9) and the fact that ρ remains in L ∞ T (L ∞ (R N )) it implies that instantaneously the density ρ is regularized and becomes necessary Lipschitz even if the initial density ρ 0 admits shocks. The second point is that with our choice on the initial data, it implies that ρ 0 u 0 is in bmo -1 T . In other way we can work with Dirac mass αδ 0 for the initial velocity associated to the momentum ρ 0 u 0 in dimension N = 2 provided that α is sufficiently small, it consists to prescribe the initial momentum as follows ρ 0 u 0 (x) = α x ⊥ |x| 2 with x ∈ R 2 . It means that we can extend the notion of Lam-Oseen tourbillon to the case of the Korteweg system at least when α is sufficiently small. It is obvious that in this compressible framework the divergence of such Lamb-Oseen tourbillon does not remain null all along the time, this is due to the coupling between vorticity and divergence of the velocity. Similarly we can combine vorticity vortex and divergnece vortex if we take the following initial momentum ρ 0 u 0 (x) = α x ⊥ |x| 2 + α 1

x ⊥ |x| 2 with x ∈ R 2 and |α|, |α 1 | sufficiently small.

Mathematical results

In this section we state our main result.

Theorem 1.1 Let 0 < κ 2 ≤ µ 2 and N ≥ 1. Let ρ 0 ∈ L ∞ (R N ) with ρ 0 ≥ c > 0 and ρ 0 v 1 (0, •), ρ 0 v 2 (0, •) ∈ bmo -1 (R N ).
Then there exists T > 0 sufficiently small such that there exists a unique solution (ρ, ρu) of the system (1.1) on [0, T ] provided that for ε 1 > 0 we have :

ρ 0 v 1 (0, •) bmo -1 T + ρ 0 v 2 (0, •) bmo -1 T ≤ ε 1 , (1.10)
In addition it exists C > 0 such that:

ρv 1 E T + ρv 2 E T + ρ L ∞ T (L ∞ ) + 1 ρ L ∞ T (L ∞ ) ≤ C. (1.11)
Remark 2 It is important to observe that the condition (1.10) implies that ∇ρ 0 is small in bmo -1 (R N ) with the norm

• bmo -1 T when 0 < κ 2 < µ 2 .
It is also interesting to observe that there is no smallness assumption on ρ 0 when κ 2 = µ 2 since in this case vρv 1 (0, •) = ρv 2 (0, •). We deduce also on this case that ρu(0, •) is not necessary small in bmo -1 but the smallness carries on the momentum of the effective velocity ρv 1 (0, •) which describes the coupling between the velocity u 0 and the density ρ 0 .

Corollary 1 Let 0 < κ 2 ≤ µ 2 and N ≥ 1. Let ρ 0 ∈ L ∞ (R N ) with ρ 0 ≥ c > 0 , assume that ∇ρ 0 and ρ 0 u 0 are in D(R N ) bmo -1
then there exists T > 0 such that there exists a unique solution (ρ, ρu) of the system (1.1). We have in addition the estimate (1.11).

We prove now a result of global strong solution with small initial data.

Theorem 1.2 Let 0 < κ 2 ≤ µ 2 , ρ > 0, N ≥ 2 and P (ρ) > 0. We assume that (ρ 0 -ρ, u 0 ) ∈ (B N 2 -1 2,∞ ∩ B N 2 2,∞ ∩ L ∞ ) × (B N 2 -1 2,∞ ) N . There exists ε 0 such that if: ρ 0 -ρ B N 2 -1 2,∞ ∩B N 2 2,∞ ∩L ∞ + u 0 B N 2 -1 2,∞ ≤ ε 0 , (1.12)
then there exists a global strong solution (ρ, u) of the system (1.1). In addition it exists C > 0 such that for s 1 ∈ ( 3 4 , 1):

           (ρv 1 , ρv 2 ) L ∞ (R + ,B N 2 -1 2,∞ )∩ L 1 (R + ,B N 2 +1 2,∞ ) + ρ -ρ L ∞ (R + ,B N 2 -1 2,∞ ∩B N 2 2,∞ )∩ L 1 (R + ,B N 2 +1 2,∞ ∩B N 2 +2 2,∞ ) ≤ C, ρ -ρ L ∞ (R + ,L ∞ (R N )) ≤ Cε 0 sup t∈R + t s 1 2 (ρ(t, •) -ρ, ρv 1 (t, •), ρv 2 (t, •) B N 2 -1+s 1 2,∞ ≤ C.
Remark 3 As previously, this theorem allows to prove the existence of global strong Oseen solution provided that curl(ρ 0 u 0 ) = αδ 0 with |α| small enough in dimension N = 2. When κ 2 < µ 2 it would be possible to extend this result by working with Besov spaces constructed on L p Lebesgue spaces in high frequencies as in [START_REF] Haspot | Existence of global strong solutions in critical spaces for barotropic viscous fluids[END_REF] for compressible Navier-Stokes equations.

2 Proof of the Theorem 1.1 and the Corollary 1

We are going now to prove the Theorem 1.1 in the case 0 < κ 2 < µ 2 . The case

κ 2 = µ 2 is similar except that v = v 1 = v 2
and we apply the estimates in bmo -1 T on ρv.

From (1.5), we observe that we have for t > 0, Q = ∇(∆) -1 div and P = Id -P:

                                   ρ(t, •) = e c 1 t∆ ρ 0 - t 0 e c 1 (t-s)∆ div(ρv 1 )ds, Pρv 1 (t, •) = e µt∆ P(ρv 1 (0, •)) - 1 2 t 0 e µ(t-s)∆ P(div(ρv 1 ⊗ v 2 ) + div(ρv 2 ⊗ v 1 ))ds, Pρv 2 (t, •) = e µt∆ P(ρv 2 (0, •)) - 1 2 t 0 e µ(t-s)∆ P(div(ρv 1 ⊗ v 2 ) + div(ρv 2 ⊗ v 1 ))ds, Qρv 1 (t, •) = e c 2 t∆ Q(ρv 1 (0, •)) - t 0 e c 2 (t-s)∆ Q(F (ρ, v 1 , v 2 )(s))ds, Qρv 2 (t, •) = e c 1 t∆ Q(ρv 2 (0, •)) - t 0 e c 1 (t-s)∆ Q(F (ρ, v 1 , v 2 )(s))ds, with: F (ρ, v 1 , v 2 )(s) = 1 2 (div(ρv 1 ⊗ v 2 ) + div(ρv 2 ⊗ v 1 ))(s, •) + ∇P (ρ(s, •)).
We are going now to work in the following space:

(ρ, v 1 , v 2 ) X T = ρv 1 E T + ρv 2 E T + (ρ, 1 ρ ) L ∞ (R N ) ,
for T small enough.We observe in particular that for any ρ ∈ L ∞ T (L ∞ (R N )) we have:

ρm E T ≤ ρ L ∞ T (L ∞ (R N )) m E T . (2.13)
We shall use in the sequel a contracting mapping argument for the function ψ defined as follows:

ψ(ρ, ρv 1 , ρv 2 ) =   e c 1 t∆ ρ 0 e µt∆ P(ρv 1 (0, •)) + e c 2 t∆ Q(ρv 1 (0, •)) e µt∆ P(ρv 2 (0, •)) + e c 1 t∆ Q(ρv 2 (0, •))   - t 0   e c 1 (t-s)∆ div(ρv 1 ) 1 2 e µ(t-s)∆ P(div(ρv 1 ⊗ v 2 ) + div(ρv 2 ⊗ v 1 )) + e c 2 (t-s)∆ Q(F (ρ, v 1 , v 2 )) 1 2 e µ(t-s)∆ P(div(ρv 1 ⊗ v 2 ) + div(ρv 2 ⊗ v 1 )) + e c 1 (t-s)∆ Q(F (ρ, v 1 , v 2 ))   ds. (2.14) More precisely we want to prove that ψ is a contractive map from E R,M,T to E R,M,T with R, M > 0: E R,M,α,T = {(ρ, ρv 1 , ρv 2 ) ∈ L ∞ T (L ∞ (R N )) × E T × E T / ρ ≥ 1 M on [0, T ] × R N , ρv 1 E T + ρv 2 E T ≤ 2R, ρ L ∞ T (L ∞ (R N ) ≤ 2 ρ 0 L ∞ } (2.15)
E R,M,T is endowed with the following norm:

(ρ, ρv 1 , ρv 2 ) E R,M,α,T = (ρv 1 , ρv 2 ) E T + 1 β ρ L ∞ T (L ∞ ) , (2.16)
with β > 0 sufficiently large that we will defined later and E R,M,T is a Banach space. Let c > 0, we know that it exists C > 0 (depending on c) such that for all 0 < t ≤ T we have (see [START_REF] Lemarié | Recent developments in the Navier-Stokes problem[END_REF] p163):

t 0 e c(t-s)∆ Pdiv(v ⊗ w)ds E T ≤ C v E T w E T t 0 e c(t-s)∆ Qdiv(v ⊗ w)ds E T ≤ C v E T w E T .
(2.17)

Similarly we have (see [START_REF] Koch | Well-posedness for the Navier-Stokes equations[END_REF]) for C, C 1 > 0:

t 0 e c(t-s)∆ ∇P (ρ)(s, •)ds E T ≤ C sup 0<t≤T t P (ρ)(t, •) L ∞ + sup 0<t≤T,x∈R N 1 t N 2 t 0 B(x, √ t) |P (ρ)(s, y)|dsdy ≤ C 1 T P (ρ) L ∞ T (L ∞ (R N )) .
(2.18)

We deduce from (2.17), (2.13) and (2.18) that it exists C > 0 such that for any T > 0 we have:

(ψ 2 (ρ, ρv 1 , ρv 2 ), ψ 3 (ρ, ρv 1 , ρv 2 )) E 2 T ≤ C ρ 0 v 1 (0, •) bmo -1 T + ρ 0 v 2 (0, •) bmo -1 T + 1 ρ L ∞ T (L ∞ ) ρv 1 E T ρv 2 E T + T P (ρ) L ∞ T (L ∞ (R N )) .
(

2.19)

Let us estimate now the L ∞ norm of (ψ 1 (ρ, ρv 1 , ρv 2 ),

1 ψ 1 (ρ,ρv 1 ,ρv 2 )
), we have then for any t ∈ [0, T ] and using the maximum principle:

       ψ 1 (ρ, ρv 1 , ρv 2 )(t, •) L ∞ ≤ ρ 0 L ∞ + t 0 e c 1 (t-s)∆ div(ρv 1 )ds L ∞ ψ 1 (ρ, ρv 1 , ρv 2 )(t, •) ≥ min x∈R N ρ 0 (x) - t 0 e c 1 (t-s)∆ div(ρv 1 )ds L ∞ .
(2.20)

We have then using integration by parts and for 0 < t ≤ T it exists C, C 1 , C 2 , C 3 > 0 such that:

| t 0 e c 1 (t-s)∆ div(ρv)ds| ≤ | t 0 R N 1 (4c 1 π(t -s)) N 2 e - |x-y| 2 4c 1 (t-s) div(ρv 1 )(s, y)dyds| ≤ C t 0 R N 1 (4πc 1 (t -s)) N 2 + 1 2 |x -y| 4c 1 (t -s) e - |x-y| 2 4c 1 (t-s) |ρv 1 (s, y)|dyds| ≤ C 1 t 0 1 √ t -s 1 (4π(t -s)) N 2 | • | 4(t -s) e - |•| 2 4(t-s) L 1 ρv 1 (s, •) L ∞ ds ≤ C 2 ρv 1 E T t 0 1 √ t -s 1 √ s ds ≤ C 3 ρv 1 E T .
(2.21)

We deduce now from (2.20) and (2.21) that for C > 0 large enough we have for t ∈ [0, T ]:

ψ 1 (ρ, ρv 1 , ρv 2 )(t, •) L ∞ ≤ ρ 0 L ∞ + C ρv 1 E T ψ 1 (ρ, ρv 1 , ρv 2 )(t, •) ≥ min x∈R N ρ 0 (x) -C ρv 1 E T . (2.22)
Let us prove now that for M , T , R well chosen, we have:

ψ(E R,M,T ) ⊂ E R,M,T . (2.23) Taking 1 M = c 2 and R such that 2CR ≤ min( c 2 , ρ 0 L ∞ (R N ) ) (with C > 0 defined in (2.22
)) and where min x∈R N ρ 0 (x) ≥ c > 0, we deduce from (2.22) that for any t ∈ [0, T ] and for any (ρ, ρv 1 , ρv 2 ) ∈ E R,M,α,T :

   ψ 1 (ρ, ρv 1 , ρv 2 )(t, •) L ∞ ≤ 2 ρ 0 L ∞ ψ 1 (ρ, ρv 1 , ρv 2 )(t, •) ≥ c 2 = 1 M . (2.24)
Now we deduce from (2.19) that we have for any (ρ, ρv 1 , ρv 2 ) ∈ E R,M,T and C 1 > 0 large enough:

(ψ 2 (ρ, ρv 1 , ρv 2 ), ψ 3 (ρ, ρv 1 , ρv 2 )) E 2 T ≤ C 1 ρ 0 v 1 (0, •) bmo -1 T + ρ 0 v 2 (0, •) bmo -1 T + 4M R 2 + T M 1 .
(2.25)

We set now R = 4C 1 ρ 0 v 1 (0, •) bmo -1 T + ρ 0 v 2 (0, •) bmo -1 T ) and M 1 = sup x∈[0,2 ρ 0 L ∞ ] P (x). Now choosing R such that 4C 1 M R 2 ≤ R 2 and T such that C 1 M T ≤ R 4 we deduce from (2.25) that we have for any (ρ, ρv 1 , ρv 2 ) ∈ E R,M,T : (ψ 2 (ρ, ρv 1 , ρv 2 ), ψ 3 (ρ, ρv 1 , ρv 2 )) E 2 T ≤ 2R.
(2.26)

Fr In conclusion we have chosen R and T such that:

R ≤ min( c 2 , ρ 0 L ∞ (R N ) ) 2C , R ≤ c 2 32C 1 and T ≤ R 4C 1 M , (2.27) with R = 4C 1 ( ρ 0 v 1 (0, •) bmo -1 T + ρ 0 v 2 (0, •) bmo -1 T
). We deduce (2.23) from (2.24) and

(2.26). It suffices then in practice to take

T ≤ R 4C 1 M with R = min( c 4C , c 2 32C 1 ).
Let us prove now that ψ is a contractive map. More precisely let us estimate ψ(ρ, ρv 1 , ρv 2 )-

ψ(ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) E R,M,T with ((ρ, ρv 1 , ρv 2 ), (ρ 1 , ρ 1 w 1 , ρ 1 w 2 )) ∈ E R,M,T .
We have then:

ψ(ρ, ρv 1 , ρv 2 ) -ψ(ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) = t 0   e c 1 (t-s)∆ div(ρ 1 w 1 -ρv 1 ) e µ(t-s)∆ P(F 1 (ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) -F 1 (ρ, v 1 , v 2 )) + e c 2 (t-s)∆ Q(F (ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) -F (ρ, v 1 , v 2 )) e µ(t-s)∆ P(F 1 (ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) -F 1 (ρ, v 1 , v 2 )) + e c 1 (t-s)∆ Q(F (ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) -F (ρ, v 1 , v 2 ))   ds.
(2.28) with:

F 1 (ρ, v 1 , v 2 ) = 1 2 (div(ρv 1 ⊗ v 2 ) + div(ρv 2 ⊗ v 1 )).
Now we have:

div(ρv 1 ⊗ v 2 ) -div(ρ 1 w 1 ⊗ w 2 ) = div( 1 ρ (ρv 1 -ρ 1 w 1 ) ⊗ ρv 2 ) + div(ρ 1 w 1 ⊗ (ρv 2 -ρ 1 w 2 ) 1 ρ ) + div(ρ 1 w 1 ⊗ ρ 1 w 2 (ρ 1 -ρ) ρρ 1 ) (2.29)
From (2.17) and (2.37), we deduce that for C, C 1 > 0 large enough and for any (ρ

1 , ρ 1 w 1 , ρ 1 w 2 ), (ρ, v 1 , v 2 )) in E R,M,T we have for C, C 1 > 0 large enough: t 0 e µ(t-s)∆ P(F 1 (ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) -F 1 (ρ, v 1 , v 2 ))ds E T ≤ C 1 ρ L ∞ T (L ∞ ) ρv 1 -ρ 1 w 1 E T ρv 2 E T + ρv 2 -ρ 1 w 2 E T ρ 1 w 1 E T + ρ 1 w 1 E T ρ 1 w 2 E T 1 ρ 1 L ∞ T (L ∞ ) ρ -ρ 1 L ∞ T (L ∞ ) + ρv 2 -ρ 1 w 2 E T ρv 1 E T + ρv 1 -ρ 1 w 1 E T ρ 1 w 2 E T ≤ C 1 M (R + M βR 2 ) (ρ, ρv 1 , ρv 2 ) -(ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) E R,M,T (2.30)
We proceed similarly for the part

t 0 e c i (t-s)∆ Q(F 1 (ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) -F 1 (ρ, v 1 , v 2 ))ds E T with i = 1, 2.
Similarly from (2.18), we have for c = c 1 , c 2 and C > 0 large enough:

t 0 e c 1 (t-s)∆ ∇(P (ρ) -P (ρ 1 ))ds E T ≤ CT P (ρ) -P (ρ 1 ) L ∞ T (L ∞ (R N )) ≤ CM 2 T β (ρ, ρv 1 , ρv 2 ) -(ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) E R,M,T , (2.31) with M 2 = sup x∈[0,2 ρ 0 L ∞ (R N ) ] |P (x)|.
From (2.28), (2.37), (2.30) and (2.31) we deduce that it exists C > 0 sufficiently large such that:

ψ 2 (ρ, ρv 1 , ρv 2 ) -ψ 2 (ρ 1 , ρ 1 w 1 , ρ 1 w 2 ), ψ 3 (ρ, ρv 1 , ρv 2 ) -ψ 3 (ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) E T ≤ C[M (R + M βR 2 ) + M 2 βT ] (ρ, ρv 1 , ρv 2 ) -(ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) E R,M,T (2.32)
From (2.21), it yields that for C > 0 large enough:

1 β ψ 1 (ρ, ρv 1 , ρv 2 ) -ψ 1 (ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) L ∞ T (L ∞ (R N )) ≤ C β (ρv 1 -ρ 1 w 1 ) E T (2.33)
Combining (2.32) and (2.33), we deduce that we have for C > 0 large enough:

ψ(ρ, ρv 1 , ρv 2 ) -ψ(ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) E T ≤ C M (R + M βR 2 ) + M 2 βT + 1 β (ρ, ρv 1 , ρv 2 ) -(ρ 1 , ρ 1 w 1 , ρ 1 w 2 ) E R,M,T (2.34)
It suffices now to choose β, T and R such that:

C M (R + M βR 2 ) + M 2 βT + 1 β < 1 (2.35)
It proves in particular that the map ψ is contractive and it concludes the proof of the theorem 1.2.

Let us prove now the corollary 1, if

ρ 0 v 1 (0, •) ∈ (D(R N )) N (bmo -1 (R N )) N
then we can observe that: lim

T →0 e t∆ ρ 0 v 1 (0, •) E T = 0. (2.36)
Indeed assume that w 0 ∈ (D(R N )) N then we have using the maximum principle for the heat equation:

e t∆ w 2 E T ≤ w L ∞ sup 0<t<T,x 0 ∈R N 1 t N 2 t 0 B(x 0 , √ t) dxds ≤ T w L ∞ .
By density we can conclude. In particular it implies that for T > 0 small enough we have:

e t∆ ρ 0 v 1 (0, •) E T + e t∆ ρ 0 v 2 (0, •) E T < ε 1 .
(2.37)

Using the proof of the Theorem 1.1), we conclude that there exists a strong solution (ρ, ρu) on [0, T ]. Indeed it we follow the previous proof, it suffices to fix R and T sufficiently small and verifying the previous estimate. By density it exist w 1 ∈ (D(R N )) N such that:

e t∆ (w 1 -ρ 0 v 1 (0, •)) E T ≤ w 1 -ρ 0 v 1 (0, •) bmo -1 T ≤ R 2 .
Now we choose T 1 ≤ T sufficiently small such that:

e t∆ w 1 E T ≤ R 2
We have now:

e t∆ ρ 0 v 1 (0, •)) E T 1 ≤ R,
and R, T 1 verifies the estimates of the proof of the Theorem 1.1.

3 Proof of the Theorem 1.2

In order to prove the theorem 1.2, we are going to start by studying the linear system associated to (1.5):

∂ t q -c∆q + divm = F ∂ t m -µ∆m -α∇divm + β∇q = G, (3.38)
with c > 0, µ > 0, µ + α > 0 and β > 0. In the sequel we will note W c,µ,α,β the semigroup associated to the previous system and we have in particular from the Duhamel formula:

q m (t, •) = W c,µ,α,β (t) q 0 m 0 + t 0 W c,µ,α,β (t -s) F G (s) ds. (3.39)
This system has been studied by Bahouri et al (see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]) in the framework of the Besov space B s 2,1 when c = 0. We are going now to extend this study to the case of general Besov space of the form B s 2,r . Setting:

d = Λ -1 divm and Ω = Λ -1 curlm,
with by definition Λ s f (ξ) = |ξ| s f (ξ) when α ∈ R and for f a temperated distribution, we will study the following system:

     ∂ t q -c∆q + Λd = F ∂ t d -ν∆d -βΛq = Λ -1 divG ∂ t Ω -µ∆Ω = Λ -1 curlG.
(3.40) with ν = (µ + α). We refer to [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF] for the definition of the Chemin-Lerner spaces L ρ T (B s p,r ) with (ρ, p, r) ∈ [1, +∞] 3 , T > 0 and s ∈ R. Proposition 3.1 Let T ∈ (0, +∞]. We assume that (q 0 , u 0 ) belongs to

B s-1,s 2,∞ ×(B s-1 2,∞ ) N with the source terms (F, G) in L 1 T (B s-1 2,∞ ∩ B s 2,∞ ) × ( L 1 T (B s 2,∞ )) N . Let (q, u) ∈ L ∞ T (B s-1 2,∞ × B s 2,∞ ) ∩ L 1 T (B s+1 2,∞ × B s+2 2,∞ ) × L ∞ T (B s-1 2,∞ ) ∩ L 1 T (B s+1 2,∞ )
N be a solution of the system (3.38), then there exists a universal constant C such that for any T > 0 we have:

(q, ∇q, u) L ∞ T (B s-1 2,∞ )∩ L 1 T (B s+1 2,∞ ) ≤ C( (q 0 , ∇q 0 , u 0 ) B s-1 2,∞ + (F, ∇F, G) L 1 T (B s-1 2,∞ ) ). ( 3

.41)

Proof: Let (q, m) be a solution of (3.38), we are going to separate the case of the low and high frequencies, which have a different behavior concerning the control of the derivative index for the Besov spaces. Our goal consists now in studying the system (3.40) and in particular to estimate (q, d) and Ω. We observe that Ω verifies simply an heat equation and classical estimates on the heat equation in Besov spaces give (see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]):

Ω L ∞ T (B s-1 2,∞ )∩ L 1 T (B s+1 2,∞ ) ≤ C( Ω 0 B s-1 2,∞ + G L 1 T (B s-1 2,∞ ) ). (3.42)
Let us study now the unknowns (q, d).

Case of low frequencies

We assume here that l ∈ Z with l ≤ l 0 (we will determine later l 0 ∈ Z). Applying operator ∆ l (see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF] for the definition of ∆ l ) to the system (3.40) and denoting g l = ∆ l g, we obtain the following system:

       d dt q l + cΛ 2 q l + Λd l = F l , d dt d l + νΛ 2 d l -βΛq l = G l . (3.43)
We set:

f 2 l = β q l 2 L 2 + d l 2 L 2 . (3.44)
Taking the L 2 scalar product of the first equation of (3.43) with q l and of the second equation with d l , we get the following two identities:

       1 2 d dt q l 2 L 2 + (Λd l , q l ) + c Λq l 2 L 2 = (F l , q l ), 1 2
d dt d l 2 L 2 + ν Λd l 2 L 2 -β(Λq l , d l ) = (G l , d l ).
(3.45)

We deduce that:

1 2 d dt f 2 l + (βc Λq l L 2 + ν Λd l 2 L 2 ) ≤ G l L 2 d l L 2 + β F l L 2 q l L 2 . (3.46)
From the definition of f 2 l we deduce that it exists α 1 , C > 0 independent on l 0 such that:

1 2 d dt f 2 l + α 1 2 2l f 2 l ≤ Cf l ( G l L 2 + F l L 2 ). (3.47)
We have in particular for l ≤ l 0 :

1 2

d dt f l + α 1 2 2l f l ≤ C( G l L 2 + F l L 2 ). (3.48)

Case of high frequencies

We consider now the case where l ≥ l 0 + 1 and we define now f l as follows:

f 2 l = Λq l 2 L 2 + A d l 2 L 2 - 2 c + ν (Λq l , d l ),
with A > 0 to be determinated. We apply the operator Λ∆ l to the first equation of (3.40), multiply by Λq l and integrate over R N , so we obtain:

1 2 d dt Λq l 2 L 2 + c Λ 2 q l 2 L 2 + (Λ 2 d l , Λq l ) = (ΛF l , Λq l ). (3.49)
Moreover we have in a similar way:

     1 2 d dt d l 2 L 2 + ν Λd l 2 L 2 -β(Λq l , d l ) = (G l , d l ). d dt (Λq l , d l ) + Λd l 2 L 2 -β Λq l 2 L 2 + (c + ν)(Λ 2 d l , Λq l ) = (ΛF l , d l ) + (G l , Λq l ).
(3.50) By linear combination of (3.49)-(3.50) we have:

1 2 d dt f 2 l + c Λ 2 q l 2 L 2 + Aν - 1 c + ν Λd l 2 L 2 + β c + ν Λq l 2 L 2 -Aβ(Λq l , d l ) = - 1 c + ν (ΛF l , d l ) - 1 c + ν (G l , Λq l ) + (ΛF l , Λq l ) + A(G l , d l ).
(3.51)

We have now in using Young inequalities for all a > 0:

|(d l , Λq l )| ≤ a 2 Λq l 2 L 2 + 1 2a d l 2 L 2 , 1 2
d dt f 2 l + c Λ 2 q l 2 L 2 + Aν - 1 c + ν Λd l 2 L 2 + ( β c + ν - Aaβ 2 ) Λq l 2 L 2 - Aβ 2a d l 2 L 2 ≤ - 1 c + ν (ΛF l , d l ) - 1 c + ν (G l , Λq l ) + (ΛF l , Λq l ) + A(G l , d l ). ( 3

.52)

We have now since l ≥ l 0 + 1 for c 1 > 0:

1 2

d dt f 2 l + c Λ 2 q l 2 L 2 + Aν - 1 c + ν -2 -2l 0 Ac 1 β 2a Λd l 2 L 2 + ( β c + ν - Aaβ 2 ) Λq l 2 L 2 ≤ - 1 c + ν (ΛF l , d l ) - 1 c + ν (G l , Λq l ) + (ΛF l , Λq l ) + A(G l , d l ).
(3.53) We choose now a, A and l 0 as follows:

Aν = 2M c + ν , β c + ν - Aaβ 2 ≥ 0, Aν - 1 c + ν -2 -2l 0 Aβ 2a ≥ 1 2(c + ν) . (3.54)
with M > 1 sufficiently large to determine later. Now from the definition of f 2 l and using Young inequality, we have:

1 2 Λq l 2 L 2 + ( 2M ν(c + ν) - 2 (c + ν) 2 ) d l 2 L 2 ≤ f 2 l ≤ 3 2 Λq l 2 L 2 + ( 2M ν(c + ν) + 2 (c + ν) 2 ) d l 2 L 2 (3.55)
We choose now M sufficiently large such that 2M ν(c+ν) -2 (c+ν) 2 ≥ 1 2 . Using (3.53) and (3.55) we deduce that there exists constants α 2 > 0 and C 1 > 0 such that for l ≥ l 0 + 1 we have:

1 2

d dt f 2 l + α 2 2 2l f 2 l ≤ C 1 f l (2 l F l L 2 + G l L 2 ). (3.56) It yields that: 1 2 d dt f l + α 2 2 2l f l ≤ C 1 (2 l F l L 2 + G l L 2 ). (3.57)

Final estimates

Integrating over [0, t] the estimates (3.48) and (3.57) and multiplying by 2 l(s-1) , we have for C 2 > 0 large enough and any l ∈ Z:

1 2 2 l(s-1) f l (t) + min(α 1 , α 2 )2 2ls t 0 f l (s)ds ≤ 1 2 f l (0) + C 2 ( G l L 1 t (B s-1 2,∞ ) + F l L 1 t (B s-1 2,∞ ∩B s-1 2,∞ ) ). (3.58)
From the definition of f l we deduce the estimate (3.41) using in particular (3.55). It concludes the proof of the proposition.

Let us study again the system (3.38) and applying div to the momentum equation, we have:

∂ t q -c∆q + divm = 0 ∂ t divm -ν∆divm + β∆q = 0, (3.59)
We assume now that µ, c, ν, β > 0 and ν = c. The only case where ν = c will be in the sequel the case κ 2 = µ 2 . We will study this case later. When we apply the Fourier transform F, we have then:

F q m (t, ξ) = e -tA(ξ) F q 0 m 0 (ξ) (3.60)
with:

A(ξ) = c|ξ| 2 1 -β|ξ| 2 ν|ξ| 2
The characteristic polynom is:

P A(ξ) (λ) = λ 2 -λ|ξ| 2 (c + ν) + cν|ξ| 4 + β|ξ| 2 .
The eigenvalues are:

• if |ξ| 2 ≥ 4β (ν-c) 2 : λ 1h (ξ) = 1 2 |ξ| 2 (ν + c)| + |ν -c| 1 - 4β (ν -c) 2 |ξ| 2 , λ 2h (ξ) = 1 2 |ξ| 2 (ν + c)| -|ν -c| 1 - 4β (ν -c) 2 |ξ| 2 . • if |ξ| 2 < 4β (ν-c) 2 : λ 1l (ξ) = 1 2 |ξ| 2 (ν + c)| + i|ν -c| 4β (ν -c) 2 |ξ| 2 -1 , λ 2l (ξ) = 1 2 |ξ| 2 (ν + c)| -i|ν -c| 4β (ν -c) 2 |ξ| 2 -1 .

High Frequencies

When |ξ| 2 ≥ 4β (ν-c) 2 we have:

F q divm (t, ξ) = P (ξ)
e -tλ 1h (ξ) 0 0 e -tλ 2h (ξ) (P (ξ)) -1 F q 0 divm 0 (ξ). (3.61) with:

P (ξ) = -2 |ξ| 2 c-ν-|c-ν|α(ξ) -2 |ξ| 2 c-ν+|c-ν|α(ξ) 1 1 (P (ξ)) -1 = - β |c -ν|α(ξ)   1 2 |ξ| 2 c-ν+|c-ν|α(ξ) -1 -2 |ξ| 2 c-ν-|c-ν|α(ξ)   We have denoted α(ξ) by α(ξ) = 1 - 4β (ν-c) 2 |ξ| 2 . Finally we get: F q m (t, ξ) = - β |c -ν|α(ξ)    -2e -tλ 1h (ξ) |ξ| 2 c-ν-|c-ν|α(ξ) + 2e -tλ 2h (ξ) |ξ| 2 c-ν+|c-ν|α(ξ) 1 β|ξ| 2 (e -tλ 2h (ξ) -e -tλ 1h (ξ) ) e -tλ 1h (ξ) -e -tλ 2h (ξ) 2e -tλ 1h (ξ) |ξ| 2 c-ν+|c-ν|α(ξ) - 2e -tλ 2h (ξ) |ξ| 2 c-ν-|c-ν|α(ξ)    × F q 0 divm 0 (ξ).
(3.62)

It yields then:

Fq(t, ξ) = (e -tλ 1h (ξ) -e -tλ 2h (ξ) ) sgn(c -ν) 2α(ξ) Fq 0 (ξ) + 1 |ξ| 2 |c -ν|α(ξ) Fdivm 0 (ξ) + 1 2
(e -tλ 2h (ξ) + e -tλ 1h (ξ) )Fq 0 (ξ)

Fdivm(t, ξ) = (e -tλ 2h (ξ) -e -tλ 1h (ξ) ) β |c -ν|α(ξ) Fq 0 (ξ) + sgn(c -ν) 2α(ξ) Fdivm 0 (ξ) + 1 2
(e -tλ 2h (ξ) + e -tλ 1h (ξ) )Fdivm 0 (ξ).

(3.63)

Low Frequencies

When |ξ| 2 < 4β (ν-c) 2 we have:

F q divm (t, ξ) = P 1 (ξ) e -tλ 1l (ξ) 0 0 e -tλ 2l (ξ) (P 1 (ξ)) -1 F q 0 divm 0 (ξ). (3.64)
In addition if we denote α 1 (ξ) = 4β (ν-c) 2 |ξ| 2 -1 , we have:

P 1 (ξ) = -2 |ξ| 2 c-ν-i|c-ν|α 1 (ξ) -2 |ξ| 2 c-ν+i|c-ν|α 1 (ξ) 1 1 (P 1 (ξ)) -1 = βi |c -ν|α 1 (ξ)   1 2 |ξ| 2 c-ν+i|c-ν|α 1 (ξ) -1 -2 |ξ| 2 c-ν-i|c-ν|α 1 (ξ)
  Finally we have:

F q m (t, ξ) = βi |c -ν|α 1 (ξ)    -2e -tλ 1l (ξ) |ξ| 2 c-ν-i|c-ν|α 1 (ξ) + 2e -tλ 2l (ξ) |ξ| 2 c-ν+i|c-ν|α 1 (ξ) 1 β|ξ| 2 (e -tλ 2l (ξ) -e -tλ 1l (ξ) ) e -tλ 1l (ξ) -e -tλ 2l (ξ) 2e -tλ 1l (ξ) |ξ| 2 c-ν+i|c-ν|α 1 (ξ) - 2e -tλ 2l (ξ) |ξ| 2 c-ν-i|c-ν|α 1 (ξ)    × F q 0 divm 0 (ξ).
(3.65) We obtain finally:

Fq(t, ξ) = i(e -tλ 2l (ξ) -e -tλ 1l (ξ) ) sgn(c -ν) 2α 1 (ξ) Fq 0 (ξ) + 1 |ξ| 2 |c -ν|α 1 (ξ) Fdivm 0 (ξ) + 1 2 (e -tλ 2l (ξ) + e -tλ 1l (ξ) )Fq 0 (ξ) Fdivm(t, ξ) = i(e -tλ 1l (ξ) -e -tλ 2l (ξ) ) β |c -ν|α 1 (ξ) Fq 0 (ξ) + sgn(c -ν) 2α 1l (ξ) Fdivm 0 (ξ) + 1 2
(e -tλ 2l (ξ) + e -tλ 1l (ξ) )Fdivm 0 (ξ).

(3.66)

Proposition 3.2 Let φ be a smooth function supported in the shell C(0, R 1 , R 2 ) with 0 < R 1 < R 2 .
There exist two positive constants κ and C depending only on φ and such that for all t ≥ 0 and λ > 0, we have:

φ(λ -1 D)W (t) q 0 m 0 L 2 ≤ Ce -κtλ 2 φ(λ -1 D) q 0 m 0 L 2 (3.67)
Proof: From (3.63), we deduce that for |ξ| 2 ≥ 4β (ν-c) 2 (with R k the Riez transform such that FR k g(ξ) = iξ k |ξ| Fg(ξ) for g a temperated distribution):

Fφ(λ -1 D)W (t) q 0 m 0 1 (t, ξ) = φ( |ξ| λ ) (e -tλ 1h (ξ) -e -tλ 2h (ξ) ) sgn(c -ν) 2α(ξ) Fq 0 (ξ) + 1 |ξ||c -ν|α(ξ) N j=1 FR j (m 0 ) j (ξ) + 1 2 (e -tλ 2h (ξ) + e -tλ 1h (ξ) )Fq 0 (ξ) Fφ(λ -1 D)W (t) q 0 m 0 1+i (t, ξ) = φ( ξ λ ) (e -tλ 2h (ξ) -e -tλ 1h (ξ) ) - β |ξ||c -ν|α(ξ) FR i q 0 (ξ) + sgn(c -ν) 2α(ξ) F(Qm 0 ) i (ξ) + 1 2 (e -tλ 2h (ξ) + e -tλ 1h (ξ) )F(Qm 0 ) i (ξ) + e -µt|ξ| 2 F(Pm 0 ) i (ξ) , (3.68)
with i ∈ [1, N ].
We have used the fact in the previous formula that Pm verifies simply an heat equation. Similarly from (3.66), we have for |ξ| 2 < 4β (ν-c) 2 :

Fφ(λ -1 D)W (t) q 0 m 0 1 (t, ξ) = φ( |ξ| λ ) i(e -tλ 2l (ξ) -e -tλ 1l (ξ) ) sgn(c -ν) 2α 1 (ξ) Fq 0 (ξ) + 1 |ξ||c -ν|α 1 (ξ) N j=1 FR j (m 0 ) j (ξ) + 1 2
(e -tλ 2l (ξ) + e -tλ 1l (ξ) )Fq 0 (ξ)

Fφ(λ -1 D)W (t) q 0 m 0 1+i (t, ξ) = φ( ξ λ ) i(e -tλ 1l (ξ) -e -tλ 2l (ξ) ) -β |c -ν||ξ|α 1 (ξ) FR i q 0 (ξ) + sgn(c -ν) 2α 1 (ξ) F(Qm 0 ) i (ξ) + 1 2 (e -tλ 2l (ξ) + e -tλ 1l (ξ) ) F(Qm 0 ) i (ξ) + e -µt|ξ| 2 F(Pm 0 ) i (ξ) .
(3.69) Applying Plancherel Theorem, we observe easily that it exists ε > 0 small enough and

C ε , κ ε > 0 such that: 1 R N /C(0, 4β |ν-c| 2 -ε, 4β |ν-c| 2 +ε) Fφ(λ -1 D)W (t) q 0 m 0 L 2 ≤ C ε e -κελ 2 t φ(λ -1 D) q 0 m 0 L 2 (3.70)
Indeed we use the fact that when ξ ∈ R N /C(0,

4β |ν-c| 2 -ε, 4β |ν-c| 2 + ε) we have: α 1 (ξ) ≥ |ν -c| ε 4β(1 -ε(ν-c) 2 4β ) , α(ξ) ≥ |ν -c| ε 4β(1 + ε(ν-c) 2 4β ) , α 1 (ξ)|ξ| ≥ √ ε.
The only difficulty is the behavior of the solution in the region C(0,

4β |ν-c| 2 -ε, 4β |ν-c| 2 + ε), in particular when C(0, R 1 λ, R 2 λ) ∩ C(0, 4β |ν-c| 2 -ε, 4β |ν-c| 2 + ε) = ∅. We have in par- ticular when |ξ| 2 ∈ ( 4β |ν-c| 2 , 4β |ν-c| 2 + ε) that α(ξ) ∈ (0, √ ε): e -tλ 1 (ξ) -e -tλ 2 (ξ) α(ξ) = e -t|ξ| 2 (ν+c) 2 e -t|ξ| 2 |ν-c|α(ξ) 2 -e t|ξ| 2 |ν-c|α(ξ) 2 α(ξ)
When tα(ξ) ≤ 1, it exists C > 0 sufficiently large such that for any x ∈ [0, |ν -c|( 4β |ν-c| 2 + ε)] we have: e x -e -x ≤ Cx.

It implies that we have when tα(ξ) ≤ 1, it exists C 1 , C 2 > 0 large enough such that:

e -tλ 1 (ξ) -e -tλ 2 (ξ) α(ξ) ≤ C 1 t|ξ| 2 e -t|ξ| 2 (ν+c) 2 ≤ C 2 e -t|ξ| 2 (ν+c) 4 . (3.71)
When tα(ξ) ≥ 1 (and 1 α(ξ) ≤ t), we deduce that for C 3 > 0 large enough and ε ≤ 1 2 : 3.72), we deduce that it exists ε > 0 small enough and

e -tλ 1 (ξ) -e -tλ 2 (ξ) α(ξ) ≤ |ν -c| 2 4β t|ξ| 2 e -t|ξ| 2 (ν+c) 2 e -t|ξ| 2 |ν-c|α(ξ) 2 -e t|ξ| 2 |ν-c|α(ξ) 2 ≤ |ν -c| 2 4β t|ξ| 2 e -t|ξ| 2 (ν+c) 2 + |ν -c| 2 4β t|ξ| 2 e -t|ξ| 2 (ν+c-√ ε|ν-c|) 2 ≤ C 3 e -
C ε , κ ε > 0 such that: 1 R N /C(0, 2 √ β |ν-c| , 4β |ν-c| 2 +ε) Fφ(λ -1 D)W (t) q 0 m 0 L 2 ≤ C ε e -κελ 2 t φ(λ -1 D) q 0 m 0 L 2 (3.73) Let us deal now with the case |ξ| 2 ∈ ( 4β |ν-c| 2 , 4β |ν-c| 2 -ε) whoch corresponds to α 1 (ξ) ∈ (0, √ ε),
we have then:

e -tλ 1 (ξ) -e -tλ 2 (ξ) α 1 (ξ) = e -t|ξ| 2 (ν+c) 2 e -it|ξ| 2 |ν-c|α 1 (ξ) 2 -e it|ξ| 2 |ν-c|α(ξ) 2 α 1 (ξ) = 2e -t|ξ| 2 (ν+c) 2 | sin t|ξ| 2 |ν-c|α 1 (ξ) 2 | |α 1 (ξ)| . When tα 1 (ξ) ≤ 1, it exists C 1 , C 2 > 0 large enough such that: e -tλ 1 (ξ) -e -tλ 2 (ξ) α 1 (ξ) ≤ C 1 t|ξ| 2 e -t|ξ| 2 (ν+c) 2 ≤ C 2 e -t|ξ| 2 (ν+c) 4 . (3.74)
When tα 1 (ξ) ≥ 1, we deduce that for C 3 > 0 large enough: 3.75) we obtain finally from the Plancherel Theorem that there exist C, κ > 0 such that for any t > 0 we have:

e -tλ 1 (ξ) -e -tλ 2 (ξ) α 1 (ξ) ≤ 2 4β |ν-c| 2 -ε t|ξ| 2 e -t|ξ| 2 (ν+c) 2 ≤ C 3 e -
φ(λ -1 D)W (t) q 0 m 0 L 2 ≤ Ce -κλ 2 t φ(λ -1 D) q 0 m 0 L 2 (3.76)
It concludes the proof of the proposition 3.2. We are going now to prove time decay estimates in Besov spaces for the semi group W (t).

Proposition 3.3 Let s ∈ R, r ∈ [1, +∞] and s 1 > s, (q 0 , m 0 ) ∈ (B s 2,∞ ) N +1 then it exists C s 1 > 0 such that for all t > 0 we have:

W (t) q 0 m 0 B s 1 2,r ≤ C s 1 t s 1 -s 2 q 0 m 0 B s 2,r . (3.77)
Proof: From proposition 3.2, we have for κ, C > 0 and any l ∈ Z:

∆ l W (t) q 0 m 0 L 2 ≤ Ce -κ2 2l t ∆ l q 0 m 0 L 2 . (3.78)
We deduce that for any l ∈ Z, we have:

2 ls 1 ∆ l W (t) q 0 m 0 L 2 ≤ C t s 1 -s 2 (2 2l t) s 1 -s 2 e -κ2 2l t 2 ls ∆ l q 0 m 0 L 2 . (3.79)
We use now the fact that x s 1 -s 2 e -κx is bounded in L ∞ (R + ) to deduce that it exists C s 1 > 0 such that:

2 ls 1 ∆ l W (t) q 0 m 0 L 2 l r ≤ C s 1 t s 1 -s 2 2 ls ∆ l q 0 m 0 l r . (3.80)
It concludes the proof of the proposition 3.3.

Proof of the Theorem 1.2

We shall use a contracting mapping argument to prove the Theorem 1.2 ( for the moment we only consider the case 0 < κ 2 < µ 2 ) and we consider the following map ψ 1 defined as follows with q = ρ -1:

ψ 1 (q, ρv 1 , ρv 2 ) = W c 1 ,µ, √ µ 2 -κ 2 ,P (1) (t) q 0 ρv 1 (0, •) + t 0 W c 1 ,µ, √ µ 2 -κ 2 ,P (1) (t -s) 0 F (ρ, v 1 , v 2 ) (s) ds. ψ 2 (q, ρv 1 , ρv 2 ) = W c 2 ,µ,- √ µ 2 -κ 2 ,P (1) (t) q 0 ρv 2 (0, •) + t 0 W c 2 ,µ,- √ µ 2 -κ 2 ,P (1) (t -s) 0 F (ρ, v 1 , v 2 )
(s) ds.

(3.81) with

F (ρ, v 1 , v 2 ) = - 1 2 (div(ρv 1 ⊗ v 2 ) + div(ρv 2 ⊗ v 1 )
) -(P (ρ) -P (1))∇ρ.

We define finally ψ 3 as follows:

ψ 3 (q, ρv 1 , ρv 2 ) = ψ 1 (q, ρv 1 , ρv 2 ) (ψ 2 ) 2 (q, ρv 1 , ρv 2 ) . (3.82)
Let us prove now that ψ 3 is a map from X N 2 in itself with s 1 ∈ ( 3 4 , 1):

E N 2 = L ∞ (R + , B N 2 -1 2,∞ ∩ B N 2 2,∞ ) ∩ L 1 (R + , B N 2 +1 2,∞ ∩ B N 2 +2 2,∞ ) × L ∞ (R + , B N 2 -1 2,∞ ) ∩ L 1 (R + , B N 2 +1 2,∞ 2N (q, ρv 1 , ρv 2 ) W N 2 = sup t∈R + t s 1 2 (q(t, •), ρv 1 (t, •), ρv 2 (t, •) B N 2 -1+s 1 2,∞ (q, ρv 1 , ρv 2 ) X N 2 = (q, ρv 1 , ρv 2 ) E N 2 ∩W N 2 + q L ∞ (R + ,L ∞ (R N )) .
The space E N 2 in which we work is more complicated as in [START_REF] Danchin | Existence of solutions for compressible fluid models of Korteweg type[END_REF], indeed we need decay estimate in time in Besov space on the solution in order to control the L ∞ norm of q. In [START_REF] Danchin | Existence of solutions for compressible fluid models of Korteweg type[END_REF], the control of the L ∞ norm of q is a direct consequence of Besov embedding

B N 2
2,1 → L ∞ since the third index of the Besov spaces are 1. From the proposition (3.1), we deduce that for C > 0 large enough:

ψ 3 (q, ρv 1 , ρv 2 ) E N 2 ≤ C q 0 B N 2 -1 2,∞ ×B N 2 2,∞ + ρv 1 (0, •) B N 2 -1 2,∞ + ρv 2 (0, •) B N 2 -1 2,∞ + F (ρ, v 1 , v 2 ) L 1 (R + ,B N 2 -1 2,∞ )
.

(3.83)

Next using classical paraproduct law and composition theorems (see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]), we get for C 1 > 0 large enough and a continuous function C:

div(ρv 1 ⊗ v 2 ) L 1 (R + ,B N 2 -1 2,∞ ) ≤ C 1 ρv 1 E N 2 ρv 2 E N 2 (1 + 1 ρ -1 L ∞ (R + ,L ∞ (R N ))∩ L ∞ (R + ,B N 2 2,∞ ))
),

(P (ρ) -P (1))∇q

L 1 (R + ,B N 2 -1 2,∞ ) ≤ C 1 ∇q L 4 3 (R + ,B N 2 -1 2
2,∞ )

P (1 + q) -P (1)

L 4 (R + ,B N 2 -1 2 2,∞ ) ≤ C 2 ∇q L 4 3 (R + ,B N 2 -1 2 2,∞ ) q L 4 (R + ,B N 2 -1 2 2,∞ ) C( q L ∞ (R + ,L ∞ (R N )) ).
(3.84) Combining (3.83), (3.84), interpolation and composition theorems, we obtain for C 1 > 0 large enough and a continuous function C:

ψ 3 (q, ρv 1 , ρv 2 ) E N 2 ≤ C 1 q 0 B N 2 -1 2,∞ ×B N 2 2,∞ + ρv 1 (0, •) B N 2 -1 2,∞ + ρv 2 (0, •) B N 2 -1 2,∞ (q, ρv 1 , ρv 2 ) 2 X N 2 (1 + C( (ρ, 1 ρ ) L ∞ (R + ,L ∞ (R N )) (1 + (q, ρv 1 , ρv 2 ) E N 2
) .

(3.85)

We get finally:

ψ 3 (q, ρv 1 , ρv 2 )(t, •) B N 2 -1+s 1 2,∞ ≤ C 1 t s 1 2 (q 0 , ρv 1 (0, •), ρv 2 (0, •)) B N 2 -1 2,∞ + (q, ρv 1 , ρv 2 ) 2 W N 2 t 0 C 1 (t -s) 1-s 1 2 1 s s 1 C( ( 1 ρ (s, •), ρ(s, •)) L ∞ (R N ) )(1 + q(s, •) B N 2 2,∞ ∩L ∞ )ds ≤ C 1 t s 1 2 (q 0 , ρv 1 (0, •), ρv 2 (0, •)) B N 2 -1 2,∞ + (q, ρv 1 , ρv 2 ) 2 W N 2 C( (ρ, 1 ρ ) L ∞ (R + ,L ∞ ) )(1 + (q, ρv 1 , ρv 2 ) X N 2 ) t 0 1 (t -s) 1-s 1 2 1 s s 1 ds (3.90) We have now since s 1 ∈ ( 3 4 , 1) t 0 1 (t -s) 1-s 1 2 1 s s 1 ds = 1 t s 1 2 1 0 1 (1 -u) 1-s 1 2 1 u s 1 du.
It comes that for C > 0 large enough we have:

ψ 3 (q, ρv 1 , ρv 2 ) W N 2 ≤ C (q 0 , ρv 1 (0, •), ρv 2 (0, •)) B N 2 -1 2,∞ + (q, ρv 1 , ρv 2 ) 2 W N 2 C( (ρ, 1 ρ ) L ∞ (R + ,L ∞ ) )(1 + (q, ρv 1 , ρv 2 ) X N 2
) .

(3.91)

It remains now to estimate the L ∞ (R + , L ∞ (R N )) norm on [ψ 3 (q, ρv 1 , ρv 2 )] 1 . From (3.81)
and using the fact that B

N 2 2,1 (R N ) is embedded in L ∞ (R N ), we have for C > 0 large enough: [ψ 3 (q, ρv 1 , ρv 2 )] 1 (t, •) L ∞ (R + ,L ∞ (R N )) ≤ [W c 1 ,µ, √ µ 2 -κ 2 ,P (1) (t) q 0 ρv 1 (0, •) ] 1 L ∞ (R N ) + t 0 [W c 1 ,µ, √ µ 2 -κ 2 ,P (1) (t -s) 0 F (ρ, v 1 , v 2 ) ] 1 (s) B N 2 2,1 ds.
(3.92)

W c 1 ,µ, √ µ 2 -κ 2 ,P (1)
q 0 ρv 1 (0, •)

is the solution of the system (3.38), we deduce from the first equation of (3.38) and using the maximum principle and the proposition 3.3 that for t > 0 and C, C 1 , C 2 , C 3 > 0 large enough:

[W c 1 ,µ, √ µ 2 -κ 2 ,P (1)
q 0 ρv 1 (0, •) ] 1 (t, •) L ∞ (R N ) ≤ q 0 L ∞ (R N ) + t 0 e c 1 (t-s)∆ div[W c 1 ,µ, √ µ 2 -κ 2 ,P (1) (s) q 0 ρv 1 (0, •) ] 2 B N 2 2,1 ds ≤ q 0 L ∞ (R N ) + t 0 C (t -s) 1-s 1 4 [W c 1 ,µ, √ µ 2 -κ 2 ,P (1) (s) q 0 ρv 1 (0, •) ] 2 B N 2 -1+ s 1 2 2,1
ds Combining (3.92), (3.93) and (3.98), we have for C 1 > 0 sufficiently large and C a continuous function we have for any t > 0:

[ψ 3 (q, ρv 1 , ρv 2 )] 1 (t, •) L ∞ (R + ,L ∞ (R N )) ≤ q 0 L ∞ (R N ) + C 1 q 0 ρv 1 (0, •) B N 2 -1 2,∞ + 1 √ t C( ( 1 ρ , ρ)(s, •) L ∞ (R N ) )(1 + (q, ρv 1 , ρv 2 ) X N 2 ) (q, ρv 1 , ρv 2 ) W N 2 (q, ρv 1 , ρv 2 ) E N 2
(3.99) This estimate proves that we need to distinguish the short time and the long time, indeed (3.99) is interesting only for the long time. We are going now to apply a fixed point theorem and we define the map ψ 4 as follows for any t > 0:

ψ 4 (q, ρv 1 , ρv 2 )(t, •) = ψ(q, ρv 1 , ρv 2 )(t, •) if t ∈ (0, T ], ψ 4 (q, ρv 1 , ρv 2 )(t, •) = ψ 3 (q, ρv 1 , ρv 2 )(t, •) if t > T,
with ψ defined as in the previous section:

ψ(q, ρv 1 , ρv 2 ) =   e c 1 t∆ q 0 e µt∆ P(ρv 1 (0, •)) + e c 2 t∆ Q(ρv 1 (0, •)) e µt∆ P(ρv 2 (0, •)) + e c 1 t∆ Q(ρv 2 (0, •))   - t 0   e c 1 (t-s)∆ div(ρv 1 ) 1 2 e µ(t-s)∆ P(div(ρv 1 ⊗ v 2 ) + div(ρv 2 ⊗ v 1 )) + e c 2 (t-s)∆ Q(F (ρ, v 1 , v 2 )) 1 2 e µ(t-s)∆ P(div(ρv 1 ⊗ v 2 ) + div(ρv 2 ⊗ v 1 )) + e c 1 (t-s)∆ Q(F (ρ, v 1 , v 2 ))   ds.
and T > 0 to determine later. We are going now to prove that ψ is a map from X N 2 ,R ∩ E R 1 ,M,T in itself with s 1 ∈ ( 3 4 , 1) and 0 < R, R 1 , T < 1 2 sufficiently small that we will define later. It is important to mention that we can work in small time on (0, T ) in

E R 1 ,M,T since (ρv 1 (0, •), ρv 2 (0, •)) are in BM O -1 (R N ), indeed we know that B N 2 -1 2,∞ is embedded in BM O -1 (R N ). We define X N 2 ,R as follows: X N 2 ,R = {(q, ρv 1 , ρv 2 ) ∈ X N 2 , (q, ρv 1 , ρv 2 ) X N 2 ≤ R}
In the sequel we will note X + N 2 ,T and X -

(q, ρv 1 , ρv 2 ) ∈ X N 2 ,R ∩ E R 1 ,M,T : ψ 4 (q, ρv 1 , ρv 2 ) X + N 2 ≤ ψ 3 (q, ρv 1 , ρv 2 ) X + N 2
≤ C 1 (q 0 , ρv 1 (0, •), ρv 2 (0, •))

B N 2 -1 2,∞ + q 0 B N 2 2,∞ ∩L ∞ + 1 √ T (q, ρv 1 , ρv 2 ) 2 X N 2 (1 + C( ρ, 1 ρ L ∞ (R + ,L ∞ ) ))(1 + (q, ρv 1 , ρv 2 ) X N 2 )
≤ C 1 (q 0 , ρv 1 (0, •), ρv 2 (0, •))

B N 2 -1 2,∞ + q 0 B N 2 2,∞ ∩L ∞ + R 2 √ T (1 + sup y∈[0,max(R+1, 1 1-R )]
C(y))(1 + R)

(3.100) We take now R such that: R = 2C 1 (q 0 , ρv 1 (0, •), ρv 2 (0, •))

B N 2 -1 2,∞ + q 0 B N 2 2,∞ ∩L ∞ .
We assume now that 0 < R < 1 2 and then the initial data are sufficiently small such that: ∩X T . It suffices to apply exactly the same estimates than in the previous section for the norm X T . For the norm X - we have to repeat the same estimates in Besov spaces on a finite time interval (0, T ) and to choose T, R 1 > 0 sufficiently small as previously. We just say few words on the case of the norm W - [ψ 4 (q, ρv 1 , ρv 2 )(t, •)] 1

B N 2 -1+s 1 2,∞ ≤ T s 1 2 [ψ 4 (q, ρv 1 , ρv 2 )(t, •)] 1 L ∞ ([0,T ],B N 2 -1 2,∞ ∩B N 2 2,∞ ) , ≤ T s 1 2 ψ 4 E - N 2
.

The norm E - is classical to estimate (we refer to [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF] or to the previous estimates). We deduce now that for C > 0 large enough: t 0 e β i (t-s)∆ div(ρv

1 ⊗ v 2 + v 2 ⊗ v 1 )(s) B N 2 -1+s 1 2,∞ ds ≤ C 1 t 0 1 (t -s) 1-s 1 2 ρv 1 ⊗ v 2 (s) B N 2 -2+2s 1 2,∞ ≤ t 0 1 (t -s) 1-s 1 2 1 s s 1 (q, ρv 1 , ρv 2 ) 2 W - N 2 C( 1 ρ (s, •) L ∞ )(1 + (q, ρv 1 , ρv 2 ) E - N 2
)ds

≤ C 2 t s 1 2 (q, ρv 1 , ρv 2 ) 2 W - N 2 C( 1 ρ (s, •) L ∞ )(1 + (q, ρv 1 , ρv 2 ) E - N 2
).

(3.102) Proceeding as previously, we have also for C 1 , C 2 > 0 large enough, C a continuous function and t ∈ [0, T ]:: t 0 e β i (t-s)∆ ∇P (ρ)(s) P (1 + q) -P (1)

B N 2 -1+s 1 2,∞ ds ≤ C 2 t 0 1 (t -s)
B N 2 2,∞ ds ≤ C 2 T t s 1 2 C( ρ(s, •) L ∞ ) (q, ρv 1 , ρv 2 ) E - N 2
.

(3.103)

We obtain then the stability for the norm X - In a similar way, we can prove that ψ 4 is contractive. More precisely taking (q, ρv 1 , ρv 2 ), (q 1 , ρ 1 w 1 , ρ 1 w 2 ) in X N 2 ,R ∩ E R 1 ,M,T we have for t > T :

ψ 1 (q, ρv 1 , ρv 2 ) -ψ 1 (q 1 , ρ 1 w 1 , ρ 1 w 2 ) = t 0 W c 1 ,µ, √ µ 2 -κ 2 ,P (1) (t -s) 0 F (ρ, v 1 , v 2 ) -F (ρ 1 , w 1 , w 2 ) (s) ds.

ψ 2 (q, ρv 1 , ρv 2 ) -ψ 2 (q 1 , ρ 1 w 1 , ρ 1 w 2 )

= t 0 W c 1 ,µ,- √ µ 2 -κ 2 ,P (1) (t -s)   0 F (ρ, v 1 , v 2 ) -F (ρ 1 , w 1 , w 2 )   (s) ds.
(3.104) As previously we show that for C 1 > 0 large enough we have:

ψ 4 (q, ρv 1 , ρv 2 ) -ψ 4 (q 1 , ρ 1 w 1 , ρ 1 w 2 ) X + N 2 ≤ C 1 R(1 + R) (q, ρv 1 , ρv 2 ) -(q 1 , ρ 1 w 1 , ρ 1 w 2 ) X N 2 .
(3.105) Taking again R sufficiently small we deduce that the map ψ 4 is contractive (it suffices again to repeat the same process on [0, T ] with T > 0 sufficiently small). It concludes the proof of the Theorem 1.2 in the case 0 < κ 2 < µ 2 . The proof in the case κ 2 = µ 2 is similar except that v 1 = v 2 , in addition when we study the system (3.59) there is no distinction between high and low frequencies. The eigenvalues associated to the matrix A(ξ) are always complex and the system has the same behavior than a equation mixing heat and Schrödinger equation.

  68), (3.70), (3.71) and (

From ( 3 .

 3 66), (3.70), (3.73), (3.74) and (

N 2 , 2 ≤ • X - N 2 + • X + N 2 .Combining ( 3 .

 22223 T to define the subsets of X N 2 where the norms are respectively only considered on the time interval [T, +∞[ and [0, T ]. We have obviously • X N 85), (3.91) and (3.99) we have for C a continuous function and C 1 > 0 large enough and for all

R 2 ( 1

 21 way we can estimate ψ 4 X - N 2

N 2

 2

N 2 . 4 ,

 24 First we have by interpolation since s 1 ∈ ( 3

N 2

 2

ψ 4 ewith β 1 ∈

 41 (q, ρv 1 , ρv 2 )(t, •) β i (t-s)∆ [div(ρv 1 ⊗ v 2 + v 2 ⊗ v 1 )(s) + ∇P (ρ)(s)] (c 1 , c 2 , µ).We have now for C 1 , C 2 > 0 large enough, C a continuous function and t ∈ [0, T ]:

N 2 using ( 3 .

 23 102) and (3.103).

Acknowledgements

The author has been partially funded by the ANR project INFAMIE ANR-15-CE40-0011. This work was realized during the secondment of the author in the ANGE Inria team.

It remains now to estimate the ψ 3 (q, ρv 1 , ρv 2 )

norm, using the definition of ψ 1 and ψ 2 (see (3.81)) we have for t > 0:

ds.

ψ 2 (q, ρv 1 , ρv 2 )(t, •)

q 0 ρv 2 (0, •)

ds.

(3.86) From proposition 3.3, it yields that for C > 0 large enough we have:

We have now using classical paraproduct laws for s > 0 and C a continuous function: (P (ρ) -P (1))∇q(s, •)

(3.89) From (3.87), (3.88) and (3.89), we get for C 1 > 0 large enough and C a continuous function:

And it yields:

The third inequality corresponds to an interpolation inequality in Besov spaces. In a similar way, we have for C, C 1 , C 2 , C 3 > 0 large enough and using proposition 3.3 and interpolation in Besov spaces (see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]):

) and F 3 (ρ) = (P (ρ) -P (1))∇ρ. Using classical paraproduct law, composition Theorem and interpolation (see [START_REF] Bahouri | Fourier analysis and nonlinear partial differential equations[END_REF]), we have now for s > 0, a function C continuous and C 1 > 0 large enough:

Similarly we have using composition, interpolation theorems and paraproduct laws (we can since s 1 > 3 4 ), it exists a continuous function C and C 1 > 0 large enough such that:

(P (1 + q) -P (1))∇q(s, •)

,

(q, ρv 1 , ρv 2 )

(q, ρv 1 , ρv 2 )

W N 2 (q, ρv 1 , ρv 2 )

(3.97) Plugging (3.97) in (3.94), we obtain for a continuous fonction C since s 1 > 3 4 :

(3.98)


Jean Simon

Laura Henriette

Analysis of the Activity of Preservice Teachers and Trainers on an ePortfolio Platform

Keywords: teachers training, traces analysis, ePortfolio

The Reunion Island teachers training school has decided to set up an ePortfolio platform to validate the "C2i2e" certificate for trainees. This certificate confirms their competencies in technology enhanced learning and digital education. Trainers and trainees of the school have been using this platform for the past three years.

Here, we study their activities to understand the real use of the platform that has been done. To analyze their behavior, we look at the traces they left on the platform. These traces show that, while all trainees go on the platform to get the certificate very few of them do it for their own goals and when they do it, it is usually for sharing documents. For their part, only a few trainers of the school work with the platform. When they do it, it is to make documents available to their trainees or to collect their students' work.

Introduction

The Reunion Island teachers training school (Ecole Supérieure du Professorat et de l'Education de la Réunion, "ESPE") trains preservice teachers of elementary and secondary school. This training alternates according to two periods throughout the year, a period in the ESPE when trainees follow the courses and a period in schools when they are in charge of a class. In their cursus, trainees have to validate a certificate (Certificat Informatique et Internet Niveau 2 Enseignant : C2i2e) that confirms their competencies in Technology Enhanced Learning (TEL) and in Digital Education. To do this, they have to perform a set of activities and to deposit the results on an ePortfolio platform. Trainers evaluate these works and propose whether or not to award the certificate to the trainee.

We have chosen Mahara as our ePortolio platform. This platform is a complete system of e-portfolios, curriculum vitae and social networks that allows connecting users and creating online communities. This platform is very intuitive and very easy to practice. It takes less than two hours to train trainees and trainers to its use. The ESPE has set up this platform for the TEL courses but the platform is also at disposal to all stakeholders of the school to meet their own needs. Thus, some volunteer trainers and all trainees have followed this training. The goal of this study is to see if this platform has been used only for the TEL courses as preconized by the institution or if actors have used it freely for other goals. In the latter case, we try to understand how they use it and what these goals are.

When they use the platform, actors can create groups. In these groups, they can share views of portfolios, but also files in folders or launch forums. A view contains blocks that can contains text or artefacts (files, pictures, movies, and so on). All these objects and all the activity on these objects leave traces on the platform. Thus, to understand this activity we analyze these traces. In a first step, we explain the methodology we employed. In a second step, we present the results we got. Finally, in a third step, we propose an interpretation of these results.

Methodology

Users have left more than 800000 traces on the platform. To study these traces, we preprocess them according to Activity Theory [START_REF] Engeström | Learning by expanding[END_REF]. This theory provides a theoretical framework that makes it possible to identify easily the various aspects of human activity. It is often used to analyze computer supported collaborative work. In the Activity Theory, to achieve his goal, the subject uses tools and relies on a community. This community follows rules and some division of labor. Thus, we linked all the traces we find on the platform to the user who has left them and to the group in which he was. The traces we got are the traces of "objects": users, views, forums, blocks, artefacts… They are also the traces of actions on these objects: creation, modification…

In a first time, we will distinguish two categories of users: trainees and trainers. Then, among trainees, we will distinguish between preservice elementary teachers and preservice secondary teachers. We will also consider two types of groups: with or without a trainer. We want to see if a trainee has the same behavior in a group with a teacher than in a group without teacher. For all these categories of users, we have computed the average number and the coefficient of variation of a whole series of indicators: average number of sessions they have opened, average number of views they have created, average number of blocks and artefacts they have deposited in their views and in their folders…

Results

There are 2111 users on the platform: 2081 trainees and 30 trainers. Trainees can stay on the platform during three years, maximum duration of their training, whereas there is no limit of duration for trainers. All the trainees of the school have to use the platform to validate the C2i2e certificate but only trainers in charge of this certificate have to use it. There are only 13 trainers in charge of the certificate, thus we could consider than the 17 others use it for their own needs: it is not really the case. In the following, we present the users activity according to different indicators.

Number of opened sessions

Trainers go on platform 6 times more than trainees do. The average number of sessions for a trainer is about 120 and, for a trainee, only 20. However we have to be careful because the coefficient of variation for trainers is very high (189%). It means that some trainers go on platform very often and others very rarely. Indeed, when we look at data, we can see than one trainer have opened a session on the platform 975 times whereas 14 others less than 10 times.

Among trainees, there is also a difference but much smaller. On average, a preservice elementary teacher has opened 23 sessions and a preservice secondary only 14. This difference indicates probably a different behavior according to the category of trainees. Inside each subcategory of trainees, the coefficients of variation, that are around 90%, show that their behaviors are more similar to each other than the ones of trainers. Only 10 preservice elementary teachers among 1330 went on the platform more than 100 times and 18 secondary, among 751, more than 50 times. This similar behavior is due to the C2i2e that preconizes a regular use of the platform. However, the little average number of sessions (20) seems to indicate that most of the trainees use it only for this purpose.

Number of views, blocks and artefacts

The number of views blocks and artefacts gives an idea of the production of the user on the platform. For example, a trainee has to create at least one view with 10 blocks and eventually 10 artefacts to validate his C2i2e.

Once again, there is a big difference between trainers and trainees. Trainers have created five more times views than trainees (11 vs 2) and two more times blocks (49 vs 21) and artefacts (57 vs 22). Once again, the high coefficient of variation for trainers (143%) indicates that from one trainer to another there could be very big differences. It is the case; if we look at data, one teacher have created 72 views with 134 blocks and 159 artefacts and another one, respectively 73 views, 331blocks and 255 artefacts whereas, on the opposite, another trainer has only created 2 views with 9 blocks and 15 artefacts.

What is very surprising is the homogeneity of the results of the trainees. Whatever the degree, elementary or secondary, the average of views, blocks and artefacts are the same. They have created on average 2 views with 21 blocks and 22 artefacts. Moreover, the coefficients of variations are low, around 30% for views and blocks and 50% for artefacts. Therefore, the behavior of one subcategory of trainees versus the other is very similar, but also inside each subcategory, from one trainee to the other, for most of them. Other information that we got, is that on average each trainee has created one more view than the C2i2e view. This additional view shows that the ePortfolio is not used only for the C2i2e.

Groups: creation, membership

As we have stated before, in the Activity Theory, we consider that the subject will rely on a group to perform his goal. Therefore studying the creations of groups or the memberships is necessary to understand the activity on the platform. For example, to validate their certificate, trainees will have to participate to one C2i2e group and to deposit in this group a view of their ePorfolio. Thus, each trainee has to participate at least to one group. If he participates to more than one group, it means that the platform is used for other goals than the C2i2e. As we will see, it is the case.

Concerning creation of groups, trainers create 70 more times groups than trainees. The average number of creations of groups by a trainer is 12 whereas it is only of 0.17 for trainees. It is easy to understand why. If they want to work with their trainees, trainers have to create groups otherwise, they will not be able to see the productions of their students. On the other side, trainees have just to deposit their work in the group created by the trainer. They do not need to create group. Thus, the average number of creations of groups by trainee (0.17) means that some trainees have freely created groups for their own needs and not for their trainer. However, 0.17 creation is very few. It means that not all trainees have created groups.

If we look at subcategories of trainees, elementary vs secondary, there is not a big difference between them. The average number of creations for preservice elementary teacher is 0.18 and 0.16 for secondary.

Whatever the category of users, the very high coefficient of variation is to point out: 211% for trainers and 258% for trainees. When we look at data, we see that only 11 trainers among 30 have created more than 5 groups and 5 of them have created more than 20 groups. More interesting is that 3 of these "active" trainers are not "digital" trainers because they do not have in charge the C2i2e. Thus, we can consider that these trainers use the ePortfolio platform for their own needs. Only, 311 trainees among 2018 have created at least one group, 36 of them have created more than one group and 5 more than 2 groups. For them also, we can consider they have used the ePortfolio for their own needs.

Group membership is another useful indicator. The average number of memberships of a trainer is 7 times more than the one of a trainee (14 vs 2) and the coefficient of variation is much higher (209% vs 91%). If we look at the data for trainers, we see that the number of memberships in a group is more or less the same than the number of creations what is not surprising. Therefore, we find the same differences between trainers according to the membership indicator than according to the creation one. For trainees, the results are more interesting. First, there is a difference between elementary (2.71 memberships) and secondary (1.42 memberships). Therefore, on average, one preservice elementary teacher belongs to more groups. To understand this difference we have looked at the membership in groups without a trainer. A preservice elementary teacher belongs to 0.35 groups without a trainer whereas a secondary belongs to 0.25. Thus, they belong more or less to the same number of groups without a trainer. The difference in group memberships (2.71 vs 1.42) comes necessarily from the groups with a trainer. It means that trainers for elementary training use the platform more than the ones for secondary training.

Trainees' membership in groups without a trainer is an indicator that they have used freely the platform for their own needs. However, the high coefficient of variation shows that not all the trainees belong to such groups. Actually, they are only one third out of them to do it; near seven hundreds trainees out of 2081.

The analysis of the group creations and the group memberships shows that trainers and trainees use the ePortfolio platform for their own needs. However, it is clear that this is only a small part of the trainers and a third part of the trainees. Among trainers, those working in the elementary courses use more the platform than those working in the secondary courses.

Forums and posts

Forums and posts are good indicators of the type of work on the platform. Here, these tools are very little used. It means that the work on the platform should be more cooperative than collaborative in the sense of Dillenbourg [START_REF] Dillenbourg | The evolution of research on collaborative learning[END_REF]. The average number of forum creations is 0.19 and, for once, trainees create, on average, more forums (0.19) than trainers (0.13). However, trainers deposit more posts (2.13 on average per trainer) than trainees do (0.58). Here also, the coefficients of variations are very high: 347% for creations and 282% for deposited posts. The very little number of creations or deposited posts and these very high coefficients of variation indicate that only a few actors use the forums. If we look at data, we see that only 2 trainers and 211 trainees have created forums. 14 trainers and 414 trainees have deposited posts. Preservice elementary teachers create ten times more forums (0.28 creation per user; 191 creators) than secondary (0.03 posts; 20 creators) and deposit posts 6 times more (0.84 post ;363 creators) than secondary (0.13 posts; 51 creators). As we can see, the use of the forum by elementary teachers is much more important than the one of secondary school teachers. Anyway, these figures are very low. The fact that the training is a hybrid formation could explain this; most part of the negotiation and discussion take place during face-to-face courses.

Interpretation of results

If we summarize the previous results, we see there is only 16 trainers who really used the platform. The use by the other 14 trainers is too insignificant to be taken into account. It means that among all the trainers of the ESPE, around 50, only a third of them use the platform. Among the 16 "active" trainers, 13 of them have to use it to train the students in TEL courses and to validate their certificate C2i2e. Most of the time, these "digital" trainers have also used the platform for other goals than the C2i2e. If we consider the "non digital" trainers, only 3 of them have used the platform for their own needs. So, we can consider that most of trainers who do not teach TEL, tend not to use it themselves.

We wanted to understand for which purpose trainers use the platform when it was not to validate the C2i2e. Thus, we have looked at the name of their groups, the title of the views in these groups and who has deposited the views. There are two major kinds of use. The first consist for the trainer to make available to trainees documents they use in courses in face-to-face, or documents that trainees can exploit in their own class during internship at schools. In this case, the platform replaces the photocopier: the ePorfolio platform is rather a sharing platform. This is the way the 3 "non digital" trainers use the platform. They are the creators of most of the views and artefacts in the groups they have created and shared with their trainees.

The second type of use is done essentially by a few "digital" trainers who must provide other courses than the TEL courses. They ask their trainees to return their work to them through the platform. In this case, we can consider that the platform is actually used as an ePortfolio platform. In this case, trainees deposit most of the views and artefacts in the groups shared with their trainers.

Concerning the trainees, most of the time, they go on the platform to validate their certificate C2i2e or because one of the previous trainers asked them to do it. However, we have seen that there are only a few trainees who have worked on the platform among themselves in groups where there is no trainer. To understand which usage of the platform they have, we have looked at the name of the groups and the title of the views. It appears that most of the time the platform is used as a sharing platform where they deposit documents useful to teach when they are in internship: the words "sequences", "lesson" or the name of the lessons appear in many titles.

We have seen that there was a difference between the subcategories of trainees: preservice elementary teachers use the platform more than preservice secondary teachers do. We have seen also that this difference comes from the trainers. In fact, the three "non-digital" trainers work only with preservice elementary teachers and the difference comes from them. It means that "non-digital" trainers of preservice secondary teachers do not use the platform for their own needs.

Conclusion

We studied the traces left on an ePorfolio platform by trainers and trainees of the Reunion Island teachers training school. The analysis shows that if all trainees use the platform to validate their certificate C2i2e, few trainees use it for their own needs and when they do it, it is to share documents for their traineeship. A third part of the trainers, only, uses it. When they do not use it to award the C2i2e, they do it either to make documents available to trainees or to retrieve their students' work.


Anny Becerra-Romero

Miguel Díaz-Rodríguez

Octavio Andrés González-Estrada

Numerical Methods is a core subject for most graduate and undergraduate engineering curricula. The course topics are transversal to most of the applied science courses. That is, any specific topic can include a problem requiring a numerical solution. Thus, developing an improved educational strategy for this subject is of paramount importance. Virtual learning environments (VLE) offer a tool for improving students learning outcomes. In this paper, we describe how we build a VLE for the subject numerical methods, based on Moodle. We use GeoGebra software and R programming language to develop virtual-learningobjects (VLO). To engage students, we propose a problem-solving activity based on movie action sequences. Each topic starts out with a movie clip that using physics principles, one can get to a equation requiring a numerical solution. For example, we explain the methods for solving ordinary differential equation by modeling a parachute sequence from a James Bond movie. The activity allows students to discuss whether the movie sequence is possible or not. Finally, we present preliminary feedback obtained on a graduate course. Also, we discuss the answers from a instructional design questionnaire applied to the course.

Development of a virtual learning environment for the subject Numerical Methods under Moodle 1. Introduction Nowadays, it is almost taken for granted that any subject within curricula is taught along with virtual learning environments (VLEs). Even if the course follows traditional lesson type where teachers deliver the content through a master class, the VLE helps as supporting material or at least it helps for managing the course content and marks. Moreover, VLEs are useful when teaching Science and Technology, in general because they permit the implementation of many learning objects such as: youtube videos, podcast, presentation slides, programming codes, and links to other websites or animations showing in a dynamic way simulations and concepts very difficult to understand by the students [1]. A VLE or also known as learning management system (LMS) is the environment where students and professors participate, and the e-learning takes place [START_REF] Weller | Virtual Learning Environments[END_REF]. The implementation of LMS has been proved to be a successful for Knowledge Management within organizations [START_REF] Lache | [END_REF].

On the other hand, in most of the engineering degree curricula, Numerical Methods is traditionally taught following comprehensive textbooks such as [START_REF] Chapra | Numerical Methods For Engineers[END_REF]. The subject presents fundamental knowledge for most engineering programs at undergraduate and graduate level; thus, any strategy to enhance its understanding is valuable. In this sense, Ref. [START_REF] Handayanto | [END_REF] reported that teaching numerical methods using VLE compared to traditional approaches presents a significant improvement in learning outcomes. Ref. [6] developed a meeting point in Moodle, for students of Numerical Methods for Computer Science Engineering at the University of Malaga. Results have also shown that VLE enhances students outcomes.

At Universidad de los Andes, Venezuela (ULA), the Coordinación General de Estudios Interactivos a Distancia (CEIDIS) (http://moodle.ula.ve/) have developed a Moodle platform for online courses such that it can serve as an alternative or complementary way of teaching, covering levels of undergraduate and graduate. The motivation of this paper is to showcase the VLE developed for teaching Numerical Methods which has been put forward following instructional design concepts described by CEIDIS as well as the authors' own contributions. One of the novel aspects of the course is to capture students attention by introducing a problem-solving activity based on movie action sequences. Each topic presents a movie clip that can be stated as an engineering problem requiring a numerical solution, e.g., the motorbike jump sequence from Point Break movie. The jump can be described using a nonlinear equation admitting a numerical solution. Also, for enhancing the teaching experience, several virtual learning objects (VLO) have been selected from online resources or developed by the authors.

The paper is organized as follows, Section 2 introduces Numerical Methods course content. Section 3 describes the basic features of the Numerical Methods VLE. Section 4 shows the VLOs selected or developed for each course topic. Section 5 discusses the preliminary assessment of the course. Finally, conclusions and further works are presented.

Subject description

The subject Numerical Methods for Engineers provides a basic introduction to the methods available for the numerical solution of engineering problems. In order to implement the numerical solution, students have to develop programming skills. Software such as: Python, R, Matlab, C++, FORTRAN, can be used in laboratory sessions to learn how to program while implemented the specific numerical method being studied. The students take the course after the block of basic science knowledge: mathematics, calculus, physics, and before the block of subjects in which specific applied engineering knowledge are taught. Then, in subsequent courses students are able to implement numerical methods. For instance, in the Mechanical Engineering curricula at the Universidad de los Andes, Venezuela, the students can take the course in the fifth term (to obtain the degree the students have to course ten terms where each term last a semester). The topics included in the course are: solution of nonlinear equations and linear equations, data fitting through linear regression, interpolation, numerical integration, numerical differentiation, and solution of ordinary differential equations. The course is not intended to cover all numerical methods but rather to provide a basic skill set useful throughout their education.

As an example, one of the courses following numerical methods, we have the mechanism and machine theory course, specifically the position problem of a mechanism. When developing the equations related to the position problem, students often arrive at a set of nonlinear equations [7]. In a robotic course, the solution of the kinematic problem is related to solving a set of linear equations [START_REF] Quintero-Riaza H F | Manipuladores paralelos: Síntesis, análisis y aplicaciones[END_REF]. Further examples can be presented to highlight the importance of numerical methods which to our opinion is a transversal axis within engineering curricula.

At the end of the course, students are expected to understand and apply the methods for:

• Assessing the error or approximation of a numerical solution.

• Finding the numerical solution of a nonlinear equation.

• Finding the numerical solution of a linear equation.

• Finding the parameters of a linear model through linear regression.

• Computing the numerical integration of an equation.

• Computing the numerical derivative of an equation.

• Solving ordinary differential equations.

The course content is presented through six thematic blocks. The first block addresses the introduction putting emphasis on the importance of numerical methods, describing many examples where numerical methods can be applied. The second block describes methods for the solution of nonlinear equations, while the third block describes the solution of linear equations. The fourth block deals with the data fitting through linear regression, and how to model experimental data using linear models. the fifth block describes how to compute the numerical solution for differentiation or integration of equations. Finally, the last block refers to the solution of ordinary differential equations.

The virtual learning environment

We developed the structure of the VLE following the set of courses provided by CEIDIS, and the best practices on how to organize a Moodle course [START_REF] Nash | Moodle course design best practices[END_REF]. First, after studying the course's content, we decided to take the Moodle Topic format. We split the course content in 10 topics. Each topic covering a specific numerical method, and they will be visible to the student after finishing the corresponding assignment of the previous topic. The topic starts out by setting the goals, answering what are the competencies students will expect to learn. Then, it follows the presentation of the learning material. After that, the topic includes an activity section where the learning objects are presented. The topic ends by assessing the learned skill through an evaluation activity. The structure of the topic is similar to the CEIDIS basic design course structure, 1 shows an example of each topic structure, the figure specifically presents the content of topic two which is related to the errors and approximations underlying in a numerical solution. The topic follows the Introduction and precedes the topic on solving nonlinear equations.

• Basic Competencies • Content • Activity • Evaluation Figure
The topics in the VLE are organized as follows:

• Introduction (Block 1)

• Errors and approximation (Block 1)

• Bisection and fixed-point iteration methods for solving of nonlinear equations (Block 2)

• Newton-Raphson and Secant method for solving of nonlinear equations (Block 2)

• Simple Gauss elimination method for solving linear equations (Block 3)

• Gauss-Jordan and Gauss-Seidel method for solving linear equations (Block 3)

• Data fitting (Block 4)

• Numerical Interpolation (Block 5)

• Numerical differentiation (Block 5)

• Numerical solution of Ordinary Differential Equations (Block 6) First, each topic starts by describing the competencies. Then, the learning material is introduced after showing a movie clip related to a problem requiring a numerical solution. For instance, the teacher can start topic 3 by playing the video of the high-altitude low-opening (HALO) parachute jump sequence from the James Bond movies Tomorrow Never Dies. The HALO jump can be modeled through a differential equation that in the evaluation section will be solved by a numerical method. Due to the paper length limitation, we refer to see Ref. [START_REF] Díaz-Rodríguez | Ecuaciones diferenciales aplicadas al salto HALO de bond (tomorrow never dies) Available from[END_REF] to understand how mathematical equation for the movie sequence is developed. The topic include a forum section where students discuss whether the movie sequence is feasible or not, which engages his/her attention in understanding the problem and its solution. After presenting the motivation activity, each topic includes readings taken from the book Numerical Methods for Engineers [START_REF] Chapra | Numerical Methods For Engineers[END_REF]. As can be seen in Figure 1, the first topic includes three specific readings. Then, the topic presents activities so that the student can put in practices the knowledge presented in the content section. In the section of activity presents the learning objects to consolidate the learning outcomes. Then, the evaluation section include a forum where student can discuss the results found in the evaluation activity. The discussion section is open to any student so they can interact in a chat format.

Due to the fact that each block presents not only similarities but also differences, we develop for each block different learning objects which are presented in the following section.

Virtual learning objects

The similarity in all blocks is given by the fact the student must implement the corresponding method learned in the topic. Thus, a learning objective transversal to all the topic should be an activity based on developing programming skills. In our case, we take advantages of the R programming language; however, any high-level programming language can be selected for such activities.

Learning objects based on R programming language

We developed the programming objects in R based on the fact that is a free software available under the GNU General Public License, also it has high statistical and graphics tools that we want to implement in the Mechanical Engineering curriculum. When working at home, the students can combine R package with the graphical front-end RStudio which is one of the most notably available free and open-source Integrated Development Environment (IDE). How to download and install R and RStudio is presented in topic one. Before sending the evaluation assignments, we demand the student to first test the code on SageMathCell [START_REF]CoCalc Sagemathcell project[END_REF] which is a web interface software system where you can run code in python, sage, R, and many other programming languages. In this way, we are able to evaluate the students by asking them to modify or implement a set of code in R-language. Then, they can copy and paste the code on the Moodle activity task tool. Thus, we avoid the issue of uploading or sending any file.

To exemplify the use of R as a learning object, the code for the topic 3 is presented below. The above example presents the code for solving a nonlinear equations through the Bisection method. The nonlinear equation is taken from the engineering problem presented in Ref. [START_REF] Ramirez-Matheus | [END_REF]. In the activity, students can copy and paste the code in his/her one IDE. Then, they can run it to interact with the results by modifying parameters of the numerical method, for instance changing the tolerance threshold parameters (tol). Finally, students can reuse the code when performing the evaluation task. For topic three, the task is based on solving the equation for projectile motion applied to the bike jump sequence presented in the movie Point Break [START_REF] Díaz-Rodríguez | Aprendiendo métodos numéricos por medio de películas Available from[END_REF].

Learning object based on GeoGebra

We use GeoGebra software for developing some of the learning objects. Objects created using GeoGebra have the potential that students can interact with the parameters of the numerical method being taught in a free and dynamic way, which helps to visualize the different behaviors through graphs. Students can estimate by their own guesses, ideas and conclusions, achieving a more lasting and significant learning through manipulation, exploration and experimentation [START_REF] Allan | [END_REF]. The core of the method for blocks two and five of the course can be better explained using a graphical solution. Those blocks are suitable to develop a solution in GeoGebra making visual the interpretation of the method. As an example, Figure 2 shows the solution of a nonlinear equation through the fixed-point iteration method.

The use of GeoGebra presents the advantage that we can reuse lots of resources and objects already on the web. Thus, we did not have to develop new objects, remember that reuse of object is one of the advantages when developing web-based courses. Instead, we search in the Classroom Resources page of GeoGebra [START_REF]GeoGebra Classroom resources[END_REF], which also provide the capability of sharing the object such that we can embed the object within the specific topic in the Moodle VLE.

The list of objects, topic and link to interact with the object is listed below,

• Topic 3, Bisection method [15] • Topic 4, Fixed-point iteration method [START_REF]GeoGebra Classroom resources[END_REF] • Topic 4, Newton-Raphson method [START_REF] Sreekumar K Newton | Rapshon method[END_REF] • Topic 8, Numerical integration [START_REF] Sanchis | [END_REF] Figure 2. GeoGebra Learning Objects for topics three [15] and four [START_REF] Bullock | Fixed point iteration: Rearrangement method x=g[END_REF] 4.

Additional learning objects

The web offers lots of material we can reuse for different purposes, e.g., block one deals with the fact of how the significant figures are relevant to the numerical solution and the error.

After conducting a web search, we found out an application to play and learn about significant figures [20]. The app randomly presents a number and the user has to answer how many significant figures have the displayed number. In this way, the students self-evaluate his/her ability to identified significant figures. The activity section of topic one includes a link to the app, see Fig. 1, the app can serve as a learning object.

Preliminary assessment of the course

The course was taught to graduate students following an hybrid approach. That is, both online and face-to-face components were implemented. The course normally is based on 4 hours faceto-face week lectures. In the hybrid approach, students interact only 2 hours. Seven students, which scored above 80 scale from 0 to 100 points, participated. Each assignment was performed in due time. The activities showed positive feedback, and lots of discussions were presented in the forum section. Participants gave a mean score of 4.4, scale from 1 to 5, when asked for the activity regarding the movie sequence. In some movie clips, the students agreed clips were feasible, based on their own numbers, while others found out the clip was unfeasible. However, one of the students claimed that at the end it is a movie, so action sequences even if they are not feasible as long as they are good enough to entertain they are worth.

In addition, the VLE was evaluated by an expert under the subject on analysis of an instructional design of the Master Degree in Educación Informática y Diseño Instructional, at the Universidad de los Andes. Overall, the expert provided positive feedback. With respect to the cognitive aspects, the VLE was developed following a constructivist approach. The comprehension of the learning material is based on cognitive operations such as logic operation and reasoning. The VLE promotes active participation through forums using written language, the resolution of the assigned activities, review of the material in a visual way or through podcasts, reading skills, the searching of information for choice concepts, similar applications, observation and discussion of results, and also the VLE presents results graphically in on-line simulator.

With respect to the behavioral aspects, the VLE presents a sequential order to each assigned activity and questions so that the students achieve the competencies that will enable them to provide solutions to the problems presented. Students have to repeat the step shown through tutorials, e.g., to install the R program used during the programming assignments. The VLE promotes the interaction with their peers through the chat platform. To enhance the student mental processes, the VLE presents the instructions through readings in which the teacher developed the particular activity. Thus, students can take notes, view and listen to the tutorial videos on their computer and install the program to use, participate in written in forums either by giving their perception or asking questions, clicking on the documents to download. Students have to complete written activities, practical exercises in the R-program and write the conclusions based on the results.

With respect to the constructive aspect, student are expected to ask questions on the forum section, comment on what they have learned relying on previous knowledge, provide solutions to practical exercises, build numerical calculation using RStudio, develop numerical solutions to engineering problems, interpret the results obtained from a numerical solution, and recognize that numerical methods represent approximate solutions.

Finally, the expert recommended that the VLE has to extend the constructive approach from individual learning as it is right now for an approach based on collaborative learning. For instance, the VLE can include a project where each student can write a part of code that combined together they can solve the task at hand. In addition, the actual design of the VLE only allow the construction of knowledge within the VLE, they recommend building knowledge outside the VLE.

Conclusions

In this paper, we have showcased a VLE developed in Moodle for teaching Numerical Methods. The course structure, and the learning objects designed for the course have been also presented. In addition, we have discussed the preliminary assessment of the course based on its implementation in a graduate course, and also from a questionnaire applied for an expert on instructional design. The developed VLE represents a tool for e-learning, blended learning or flipping classroom strategies. Currently, the course is being evaluated on undergraduate courses, future works will provide the assessment from students feedback.

Figure 1 .

 1 Figure 1. Screen-shot of topic two: Errors and Approximations (text in Spanish).

  -b^2+c^2+d^2)/(2*a*c); theta_2=45*pi/180 f=function(x)K1*cos(x)-K2*cos(theta_2) +K3-cos(theta_2-x); curve(f,0,2*pi,n=2001) abline(h=0,v=0,col="gray60") Resultados') print('*****************************') print('theta_4_Iter_Error') print('*****************************') c((a+b)/2*180/pi,iter,(b-a))