Roberta Bianchini
email: roberta.bianchini@ens-lyon.fr

Roberta Bianchini Strong Convergence

STRONG CONVERGENCE OF A VECTOR-BGK MODEL TO THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS VIA THE RELATIVE ENTROPY METHOD

Keywords: Vector-BGK models, incompressible Navier-Stokes equations, dissipative entropy, relative entropy, entropy inequality, diffusive relaxation

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Introduction.

In this paper we deal with the incompressible Navier-Stokes equations in two space dimensions, (1.1) ∂ t u N S + ∇ • (u N S ⊗ u N S ) + ∇P N S = ν∆u N S , ∇ • u N S = 0, with (t, x) ∈ [0, +∞) × T 2 , and initial data (1.2) u N S (0, x) = u 0 (x), ∇ • u 0 = 0.

In (1.1), u N S and ∇P N S are respectively the velocity field and the gradient of the pressure term, and ν > 0 is the viscosity coefficient. Here we consider a vector-BGK model for the incompressible Navier-Stokes equations, i.e. a dicrete velocities BGK system endowed with a vectorial structure, whose general formulation has been introduced in [START_REF] Carfora | A discrete kinetic approximation for the incompressible Navier-Stokes equations[END_REF], while further developments were presented in [START_REF] Bouchut | Second-order entropy satisfying BGK-FVS schemes for incompressible Navier-Stokes equations[END_REF] from the numerical side and in [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF] from the analytical point of view. Precisely, we study the following five velocities (15 equations) vector-BGK approximation to the incompressible Navier-Stokes equations, (1.3)

               ∂ t f ε 1 + λ ε ∂ x f ε 1 = 1 τ ε 2 (M 1 (w ε ) -f ε 1 ), ∂ t f ε 2 + λ ε ∂ y f ε 2 = 1 τ ε 2 (M 2 (w ε ) -f ε 2 ), ∂ t f ε 3 -λ ε ∂ x f ε 3 = 1 τ ε 2 (M 3 (w ε ) -f ε 3 ), ∂ t f ε 4 -λ ε ∂ y f ε 4 = 1 τ ε 2 (M 4 (w ε ) -f ε 4 ), ∂ t f ε 5 = 1 τ ε 2 (M 5 (w ε ) -f ε 5 ),
where (1.4)

w ε = (ρ ε , ερ ε u ε 1 , ερ ε u ε 2 ) = (ρ ε , ερ ε u ε ) = (ρ ε , q ε ) = 5 i=1 f ε i .
Its main properties are as follows:

• f ε i , M i (w ε ), i = 1, • • • 5, are vector-valued functions taking values in R 3 ; • ρ ε (t, x) on R + × T 2 is the approximating density, taking values in R + ; • u ε (t, x) = (u ε 1 (t, x), u ε 2 (t, x)) on R + × T 2 is the approximating vector field, taking values in R 2 . Precise compatibility conditions to be satisfied by the constant parameters of the model and the Maxwellian functions, together with their explicit expressions, will be provided in details in Section 2. BGK models were introduced by Bhatnagar, Gross and Krook as a modified version of the Boltzmann equation, characterized by the relaxation of the collision operator. Since they present most of the basic properties of hydrodynamics, they are considered interesting models even though they do not contain all of the relevant features of the Boltzmann equation. Essentially, vector-BGK models are inspired by the hydrodynamic limits of the Boltzmann equation [START_REF] Bardos | Fluid dynamic limits of hyperbolic equations. I. Formal derivations[END_REF][START_REF] Bardos | Fluid dynamic limits of kinetic equations -II Convergence proofs for the Boltzmann-equation[END_REF][START_REF] Cercignani | The Mathematical Theory of Dilute Gases[END_REF][START_REF] Demasi | Incompressible Navier-Stokes and Euler Limits of the Boltzmann equation[END_REF][START_REF] Golse | The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels[END_REF], but later they have been generalized as approximating equations for different kinds of systems. In this regard, one of the main directions has been the approximation of hyperbolic systems with discrete velocities BGK models, as in [START_REF] Brenier | Averaged multivalued solutions for scalar conservation laws[END_REF][START_REF] Jin | The relaxation schemes for system of conservation laws in arbitrary space dimensions[END_REF][START_REF] Natalini | A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF][START_REF] Bouchut | Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF][START_REF] Perthame | Kinetic Formulation of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications 21[END_REF]. Similar results have been obtained for convection-diffusion systems under the diffusive scaling [START_REF] Lions | Diffusive limits for finite velocity Boltzmann kinetic models[END_REF][START_REF] Bouchut | Diffusive BGK Approximations for Nonlinear Multidimensional Parabolic Equations[END_REF][START_REF] Lattanzio | Relative entropy in diffusive relaxation[END_REF][START_REF] Aregba-Driollet | Diffusive kinetic explicit schemes for nonlinear degenerate parabolic systems[END_REF][START_REF] Jin | Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations[END_REF][START_REF] Gosse | Space localization and well-balanced scheme for discrete kinetic models in diffusive regimes[END_REF]. Originally, they presented continuous velocities, see [START_REF] Perthame | Kinetic Formulation of Conservation Laws, Oxford Lecture Series in Mathematics and its Applications 21[END_REF], but later on discrete velocities BGK models inspired by the relaxation method have been introduced, see [START_REF] Mascia | Twenty-eight years with Hyperbolic Conservation Laws with Relaxation[END_REF] for a survey. In the spirit of the relaxation approximations, the main advantage of discrete velocities BGK models is to deal with semilinear systems, see [START_REF] Natalini | A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws[END_REF][START_REF] Brenier | On a relaxation approximation of the incompressible Navier-Stokes equations[END_REF][START_REF] Hachicha | Approximations hyperboliques des équations de Navier-Stokes[END_REF]. Now, let us present our main result. We prove the strong convergence in the Sobolev spaces, for any interval of time [0, T ], T > 0, of the vector-BGK model presented in (1.3) to the incompressible Navier-Stokes equations on the two-dimensioanl torus. To achieve this result, the novelty relies in using local in time H s -estimates from a previous work, see [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF], combined with the L 2 -relative entropy estimate and the standard interpolation theorem. More precisely, part of the results of [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF] provides uniform (in ε) estimates of Gronwall type in the Sobolev spaces, which hold in [0, T * ], where T * > 0 is depending on a fixed constant M > 0 and on the norm of the initial data. These local bounds guarantee the existence, the minimality and dissipative property of the kinetic entropy, i.e. a convex entropy for (1.3), see [START_REF] Bouchut | Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF]. Next, the relative entropy allows us to get a precise rate of convergence of the solutions to our model to the Navier-Stokes equations, which holds for t ∈ [0, T * ], see Theorem 3.2. Thus, the interpolation theorem for Sobolev spaces applied to the relative entropy estimate provides a bound for the solutions to our system which is much more precise than the previous pessimistic Gronwall type estimates. This is the key point in order to close the argument and to prove the strong convergence for all times of the solutions to (1.3) to (1.1), together with the global in time boundedness of the approximating solution itself, in Theorem 3.1. In particular, Lemma 2.6 plays a crucial role in quantifying the dissipation term coming from the entropy inequality. At the best of our understanding, the expansions in Lemma 2.6 are the only way to establish the relative entropy inequality when, as in our case, the explicit dependency of the kinetic entropy on the singular parameter is not known. We point out that we start from initial data in (2.9) that are small perturbation of the Maxwellians and, thanks to the uniform bounds, in the end we prove that everything is done in a bounded set of the densities. This local setting perfectly fits the framework described in [START_REF] Bouchut | Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF]. The relative entropy method, [START_REF] Dafermos | Stability of motions of thermoelastic fluids[END_REF][START_REF] Diperna | Uniqueness of solutions to hyperbolic conservation laws[END_REF], represents an efficient mathematical tool for studying stability and limiting process and it is based on a direct calculation of the relative entropy between a dissipative solution and a conservative smooth solution for the considered system, which provides a remarkable stability estimate. Far from being complete, we collect here a pair of references for hydrodynamic limits [START_REF] Golse | La méthode de l'entropie relative pour les limites hydrodynamiques de modèles cinétiques[END_REF][START_REF] Saint-Raymond | Hydrodynamic limits: some improvements of the relative entropy method[END_REF]. In the context of singular hyperbolic scaled systems, we remind to [START_REF] Tzavaras | Relative Entropy in Hyperbolic Relaxation[END_REF]. Let us point out that this procedure has been successfully applied to the vector-BGK model considered in this paper and presented below (1.3) to prove its convergence to the isentropic Euler equations under the hyperbolic scaling, see [START_REF] Sepe | Convergence of a BGK approximation of the isentropic Euler equations[END_REF]. Again, the relative entropy in hyperbolic relaxation has been used for one-dimensional discrete velocities Boltzmann schemes, see [START_REF] Berthelin | From Discrete Velocity Boltzmann Equations to Gas Dynamics Before Shocks[END_REF], while in the multidimensional case the question in this context seems to be open. On the other hand, the relative entropy method in diffusive relaxation is of course a more delicate issue, being the diffusive limit an order more precise approximation of the starting system in the Chapman-Enskog expansion, see [START_REF] Saint-Raymond | From the BGK model to the Navier-Stokes equations[END_REF]. Besides hydrodynamic limits of the Boltzmann equation, our main reference in this framework is [START_REF] Lattanzio | Relative entropy in diffusive relaxation[END_REF]. In this paper, the authors apply the relative entropy method to the equations of compressible gas dynamics with friction under the diffusive scaling, so obtaining precise estimates coming from the entropy of the limit hyperbolic system. However, in our case, further complications are due to the fact that the explicit dependency of the kinetic entropy on the singular parameter is not known for our model (1.3). The BGK framework in [START_REF] Bouchut | Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF] only guarantees the existence of such an entropy, whose expression is defined by means of the inverse function theorem. This difficulty requires a better understanding of the dissipative terms provided by the entropy inequality in diffusive relaxation, and new ideas are needed with respect to the existing works, for instance [START_REF] Lattanzio | Relative entropy in diffusive relaxation[END_REF][START_REF] Berthelin | From Discrete Velocity Boltzmann Equations to Gas Dynamics Before Shocks[END_REF]. The paper is organized as follows. In Section 2 we introduce the vector-BGK model and provide some preliminary results. Section 3 is devoted to the relative entropy inequality and the strong convergence of the model for all times, in the Sobolev spaces. In the last part of this section we also show the global in time boundedness of the solutions to our model.

Presentation of the model, formal limit, and intermediate results.

First, we aim at providing a relative entropy inequality for a vector-BGK model approximating the two-dimensional incompressible Navier-Stokes equations. After, this inequality will allow us to extend for long times the local convergence for smooth solutions achieved in [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF]. Let us introduce the setting that will be taken into account hereafter. Our approximating vector-BGK model has been presented in (1.3), together with a list of the main properties. We point out that, in order to get consistency with the incompressible Navier-Stokes equations, the Maxwellian functions M i (w ε ), i = 1, • • • , 5, need to satisfy the following compatibility conditions:

• 5 i=1 M i (w ε ) = w ε ; • 5 i=1 λ ij M i (w ε ) = A j (w ε ), j = 1, 2, with A j in (2.
2), with discrete velocities λ 1 = (λ, 0), λ 2 = (0, λ), λ 3 = (-λ, 0), λ 4 = (0, -λ), λ 5 = (0, 0), where λ is a positive constant value.

We provide here the explicit expressions of the Maxwellian functions (2.1)

M 1,3 (w ε ) = aw ε ± A 1 (w ε ) 2λ , M 2,4 (w ε ) = aw ε ± A 2 (w ε ) 2λ , M 5 (w ε ) = (1 -4a)w ε , (2.2) A 1 (w ε ) =    q ε 1 (q ε 1 ) 2 ρ ε + P (ρ ε ) q ε 1 q ε 2 ρ ε    , A 2 (w ε ) =    q ε 2 q ε 1 q ε 2 ρ ε (q ε 2 ) 2 ρ ε + P (ρ ε )    , (2.3) P (ρ ε ) = ((ρ ε ) 2 -ρ2 ) 2ρ ,
where ρ > 0 is constant value, and the following constraint has to be satisfied in order to get consistency with respect to (1.1), see [START_REF] Carfora | A discrete kinetic approximation for the incompressible Navier-Stokes equations[END_REF].

Assumptions 2.1. Let us assume

(2.4) a = ν 2λ 2 τ , 0 < a < 1 4 ,
where ν is the viscosity coefficient in (1.1). Moreover, we also take the parameter λ > 0, whose lower bound is defined in [[6], Assumption 2], big enough. This is necessary in order to:

• guarantee the positivity of the symmetrizer in [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF];

• satisfy the sub-characteristic condition, i.e. the positivity of the spectrum of the Jacobian matrices of the Maxwellians, see Section 2.2 and [START_REF] Bouchut | Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF][START_REF] Natalini | Convergence to equilibrium for the relaxation approximations of conservation laws[END_REF] for a detailed discussion.

The change of variables introduced in [6],

(2.5)

w ε = 5 i=1 f ε i , m ε = λ ε (f ε 1 -f ε 3 ), ξ ε = λ ε (f ε 2 -f ε 4 ), k ε = f ε 1 + f ε 3 , h ε = f ε 2 + f ε 4 .
allows us to recover the consistency with respect to (1.1) in a simple way at the formal level. This way, the vector-BGK model (1.3) reads:

(2.6)

               ∂ t w ε + ∂ x m ε + ∂ y ξ ε = 0, ∂ t m ε + λ 2 ε 2 ∂ x k ε = 1 τ ε 2 ( A1(w ε ) ε -m ε ), ∂ t ξ ε + λ 2 ε 2 ∂ y h ε = 1 τ ε 2 ( A2(w ε ) ε -ξ ε ), ∂ t k ε + ∂ x m ε = 1 τ ε 2 (2aw ε -k ε ), ∂ t h ε + ∂ y ξ ε = 1 τ ε 2 (2aw ε -h ε ).
Hereafter, we will drop the apex ε where there is no ambiguity. Moreover, we denote by M i (w) := f i the solutions to system (1.3) after taking the limit under the diffusive scaling. The relaxation formulation (2.6) of the system gives (2.7)

m = λ ε (f 1 -f 3 ) := λ ε (M 1 (w) -M 3 (w)) = A 1 (w) ε -τ λ 2 ∂ x k + O(ε 2 ), ξ = λ ε (f 2 -f 4 ) := λ ε (M 2 (w) -M 4 (w)) = A 2 (w) ε -τ λ 2 ∂ y h + O(ε 2 ), k = f 1 + f 3 = M 1 (w) + M 3 (w) = 2aw + O(ε 2 ), h = f 2 + f 4 = M 2 (w) + M 4 (w) = 2aw + O(ε 2 ).
Recalling that, from Assumptions 2.1 ν = 2aτ λ 2 , formally we get

∂ t w + ∂ x A 1 (w) ε + ∂ y A 2 (w) ε = ν∆w + O(ε 2 ). More explicitly, from the expressions of w, A 1 (w), A 2 (w) in (1.4)-(2.2), ∂ t   ρ - ρ ερu 1 ερu 2   + ∂ x   ρu 1 ερu 2 1 + ρ 2 -ρ2 2 ρε ερu 1 u 2   + ∂ y   ρu 2 ερu 1 u 2 ερu 2 2 + ρ 2 -ρ2 2 ρε   = ν∆   ρ - ρ ερu 1 ερu 2   + O(ε 2 ).
Dividing the last two lines by ε, this yields

∂ t (ρ -ρ) + ∇ • u = ν∆(ρ -ρ) + O(ε), ∂ t (ρu) + ∇ • (ρu ⊗ u) + ∇(ρ 2 -ρ2 ) 2 ρε 2 = ν∆(ρu) + O(ε),
which is the compressible approximation to the incompressible Navier-Stokes equations provided by the scaled isentropic Euler equations. Now we find an expression of the formal limit in terms of the original kinetic variables (1.3). The limit solution is obtained by solving the linear system (2.7) in the unknowns

M i (w), i = 1, • • • , 5, so providing (2.8) M 1 (w) = M 1 (w) -aελτ ∂ x w, M 2 (w) = M 2 (w) -aελτ ∂ y w, M 3 (w) = M 3 (w) + aελτ ∂ x w, M 4 (w) = M 4 (w) + aελτ ∂ y w, M 5 (w) = M 5 (w).
In order to avoid further complications due to the initial layer, in our convergence proof the two-dimensional vector-BGK model is endowed with the following initial data:

(2.9)

f ε i (0, x) = M i (ρ, ερū 0 ), i = 1, • • • , 5
, where u 0 is in (1.2) and ρ is a positive constant value.

Preliminary results.

Here we collect some preliminary results, which hold for local times, essentially due to our previous work [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF]. Let us start with the following remark.

Remark 2.2. We discuss some differences between [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF] and our current setting.

• In [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF], the compressible pressure

P (ρ ε ) in (2.3) is linear. More precisely, from [[6], (10)], P (ρ ε ) = ρ ε -ρ.
In this paper, we consider the case of a quadratic pressure P (ρ ε ) in (2.3). A simple remark shows that, from (2.3),

P (ρ ε ) = (ρ ε ) 2 -ρ2 2ρ = 2ρ(ρ ε -ρ) + (ρ ε -ρ) 2 2ρ = (ρ ε -ρ) + (ρ ε -ρ) 2 2ρ .
Thus, the estimates in [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF] still hold here: the quadratic pressure only provides an additional quadratic term in the fifth and the ninth line of the nonlinear vector N (w + w) in [ [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF], [START_REF] Jin | The relaxation schemes for system of conservation laws in arbitrary space dimensions[END_REF]]. These supplementary quadratic terms can be handled exactly as the other ones in the energy estimates in [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF]. However, we point out that the same argument holds exactly in the same way for a general compressible pressure

P (ρ ε ) = k γ-1 [(ρ ε ) γ -ργ ], γ > 1, k[ρ ε log(ρ ε ) -ρlog(ρ)], γ = 1,
where k is a positive constant value. • In [ [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF], ( 18)-( 19)], we consider a translated version of the relaxation system (2.6). Of course this is an equivalent formulation of the approximating model, and since the translation vector (ρ, 0, 0) in [ [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF], [START_REF] Demasi | Incompressible Navier-Stokes and Euler Limits of the Boltzmann equation[END_REF]] is constant in t and x, most of the energy estimates in [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF] can be used here.

• A further change of variables, involving the dissipative constant right sym- metrizer Σ in [[6], ( 28
)] is defined in [[6], (30)
]. However, here the energy estimates from [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF] are expressed in terms of the original relaxation variables Taking into account Remark 2.2, we state some results that will be applied below. Hereafter, we denote by T ε the maximum time of existence of the solution to the semilinear vector-BGK approximation (1.3) with initial data (2.9), see [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF]. Of course T ε could depend on ε. In the following, we recall and adapt some results from [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF], showing that there exist ε 0 and a fixed and positive time T * , independent of ε and depending on the Sobolev norm of the initial data, such that, for ε ≤ ε 0 , some local in time H s -estimates on the solutions to the approximating system hold uniformly with respect to ε. In this context, we consider the constant vector (ρ, 0, 0) and the translated variables (2.10)

w * (t, x) = w(t, x) -(ρ, 0, 0), k * (t, x) = k(t.x) -2a(ρ, 0, 0), h * (t, x) = h(t.x) -2a(ρ, 0, 0).
Lemma 2.3. Consider the vector-BGK system (1.3) with initial data (2.9), and u 0 in (1.2) beloging to H s (T 2 ), for s > 3. Then, the following estimates hold true.

(2.11) w * (t) 2 s + ε 2 ( m(t) 2 s + ξ(t) 2 s ) + k * (t) 2 s + h * (t) 2 s + T 0 1 ε 2 w * (θ) 2 s + m(θ) 2 s + ξ(θ) 2 s + 1 ε 2 ( k * (θ) 2 s + h * (θ) 2 s ) dθ ≤ cε 2 ( u 0 2 s + ∇u 0 2 s ) + c(|ρ| L ∞ t L ∞ x , |u| L ∞ t L ∞ x ) T 0 w * (θ) 2 s + ε 2 ( m(θ) 2 s + ξ(θ) 2 s ) dθ + c(|ρ| L ∞ t L ∞ x , |u| L ∞ t L ∞ x ) T 0 k * (θ) 2 s + h * (θ) 2 s dθ, t < T ε .
(2.12)

w * (t) 2 s + ε 2 ( m(t) 2 s + ξ(t) 2 s ) + k * (t) 2 s + h * (t) 2 s ≤ cε 2 ( u 0 2 s + ∇u 0 2 s )e c(|ρ| L ∞ t L ∞ x , |u| L ∞ t L ∞ x )t , t < T ε . (2.13) ∂ t w * (t) 2 s-1 + ε 2 ( ∂ t m(t) 2 s-1 + ∂ t ξ(t) 2 s-1 ) + ∂ t k * (t) 2 s-1 + ∂ t h * (t) 2 s-1 ≤ cε 2 ( u 0 2 s-1 + ∇u 0 2 s-1 + ∇ 2 u 0 2 s-1 )e c(|ρ| L ∞ t L ∞ x , |u| L ∞ t L ∞ x )t , t < T ε .
Moreover, there exists ε 0 , M and T * < T ε fixed such that, for ε ≤ ε 0 ,

(2.14) |ρu(t)| ∞ ≤ M, |ρ(t) -ρ| ∞ ≤ εM, t ∈ [0, T * ],
(2.15)

|ρ(t)| ∞ ≤ ρ + εM, |u(t)| ∞ ≤ M ρ + εM , t ∈ [0, T * ].
(2.16)

T 0 |ρ(t) -ρ| ∞ dt ≤ c(M )ε 2 , T ∈ [0, T * ].
Proof. We discuss each result separately.

L ∞ t L ∞ x , •) on |ρ| L ∞ t L ∞
x is a consequence of the quadratic pressure in (2. • For a fixed constant M > M 0 := ρ u 0 s+1 , let us define (2.17)

T * := sup t∈[0,T ε ) |ρ(t) -ρ| ∞ ε + |ρu(t)| ∞ ≤ M .
The Sobolev embedding theorem applied to (2.12) yields, thanks to the definition of w * in (2.10),

(2.18)

|ρ(t) -ρ| ∞ ε + |ρu(t)| ∞ ≤ cM 0 e c(|ρ| L ∞ t L ∞ x , |u| L ∞ t L ∞ x )t , t ≤ T * .
The

Kinetic entropies and the relative entropy.

Here we recall the definition and the conditions that assure the existence of a kinetic entropy for a discrete velocities BGK model, see [START_REF] Bouchut | Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF] for a detailed discussion. Let E be a non-empty set of convex entropies for a given limit system. Assume also that E is separable. A general BGK model under the diffusive scaling reads as follows

(2.19) ∂ t f i + λ i ε • ∇ x f i = 1 ε 2 (M i (u ε ) -f i ), i = 1, • • • , L, where L ≥ d, for i = 1, • • • , L, f i (t, x) = (f 1 i , • • • , f N i ) : R × R d → R N , λ i = (λ 1 i , • • • , λ d i ), M i (u ε ) = (M 1 i , • • • , M N i ) : R N → R N ,
and, under precise consistency conditions, see [8, 13, 1, 2, 10] for a detailed discussion,

u ε = L i=1 f i ∈ R N
is the approximating vector field to the solution to the limit system. An important feature of these approximations is the existence, under some reasonable assumptions, of a kinetic entropy. Set D i := {M i (u) : u ∈ U}. Definition 2.4. A kinetic entropy for system (2. [START_REF] Diperna | Uniqueness of solutions to hyperbolic conservation laws[END_REF]) is a convex function

H(f) = L i=1 H i (f i ), with H i : D i → R, such that, for η(u) ∈ E, • (E1) H(M (u)) = η(u) for every u ∈ U, • (E2) H(M (u f )) ≤ H(f), where u f := L i=1 f i ∈ U, f i ∈ D i .
Such a property provides an energy inequality which gives robustness for the scheme. Indeed, it is easy to see that, multiplying the BGK system (2. [START_REF] Diperna | Uniqueness of solutions to hyperbolic conservation laws[END_REF]) by ∇ f H(f), the minimality (E2) together with the convexity property, provide the following entropy inequality

(2.20) ∂ t H(f) + Λ ε • ∇ x H(f) = 1 ε 2 ∇ f H(f) • (M (u) -f) ≤ 0,
which means that, according with the definition given in [START_REF] Hanouzet | Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy[END_REF], the kinetic entropy H(f) is dissipative. More precisely, properties (E1)-(E2) under the hypothesis of [ [START_REF] Bouchut | Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF], Thm.

2.1] assure that, for any η(u) ∈ E, defining the projector P such that

(2.21) Pf = L i=1 = u, then (2.22) η(u) = min Pf=u H(f) = H(M (u)).
In this context, the Gibbs principle for relaxation and, in particular, [ [START_REF] Tzavaras | Relative Entropy in Hyperbolic Relaxation[END_REF], Prop. 2.1], imply that (2.23) ∇ f H(M (u)) ⊥ Ker(P).

Since f -M (u) ∈ Ker(P), the convexity property of H(f) together with condition (2.23) allow us to get the following inequality:

(2.24)

∇ f H(f) • (f -M (u)) ≤ -c|f -M (u)| 2 , c = c(|f| ∞ ),
meaning that the kinetic entropy H(M (u)) is strictly dissipative, according to the definition given in [START_REF] Hanouzet | Global Existence of Smooth Solutions for Partially Dissipative Hyperbolic Systems with a Convex Entropy[END_REF]. According to the theory developed by Bouchut [START_REF] Bouchut | Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF], the existence of a kinetic entropy for system (1.3) is subjected to the existence of a convex entropy for the limit solution to (1.3) under the hyperbolic scaling. The hyperbolic parameter of the vector-BGK approximation (1.3) is represented by τ and the limit equations approximated by (1.3) in the vanishing parameter of the hyperbolic scaling τ are the isentropic Euler equations. The convergence of the hyperbolic-scaled system is guaranteed by the structural properties of our vector-BGK model listed before, see [START_REF] Carfora | A discrete kinetic approximation for the incompressible Navier-Stokes equations[END_REF], while a rigorous proof is provided in [START_REF] Sepe | Convergence of a BGK approximation of the isentropic Euler equations[END_REF]. A convex entropy for the limit equation in hyperbolic scaling, i.e, the isentropic Euler equations, is given by

(2.25) η(w ε ) = 1 2 |q ε | 2 ρ ε + k(ρ ε ) 2 .
We can immediately state the following result.

Proposition 2.5. Consider the vector-BGK approximation (1.3) under Assumptions 2.1 and emanating from smooth initial data (2.9). Then there exists a kinetic entropy H(f ε ) = The relative entropy can be seen as a perturbation of the kinetic entropy near to the equilibrium represented by the solution to the limit system. A precise definition in the context of hyperbolic relaxation is provided in [START_REF] Tzavaras | Relative Entropy in Hyperbolic Relaxation[END_REF]. For diffusive relaxation, we will use the following

(2.26) H(f| f) = H(f) -H(M( w)) -∇ f H(M( w)) • (f -M( w)) = i H i (f i ) -H i (M i ( w)) -∇ fi H i (M i ( w)) • (f i -M i ( w)),
where H(f) is in Definition 2.4, and M( w) = (M i ( w)) i=1,••• ,5 are the perturbed Maxwellians in (2.8), evaluated in the solution w = (ρ, ερū) to the incompressible Navier-Stokes equations (1.1).

2.3.

Quantifying the dissipation. The aim of this part is to characterize and to quantify the dissipative terms resulting from the relative entropy estimate. Hereafter, we will drop the apex ε when there is no ambiguity. We start with two preliminary lemmas.

Lemma 2.6. Let η(w) be defined in (2.25). Let H(f) = 5 i=1 H i (f i ) be a kinetic entropy associated with the vector-BGK model in (1.3), such that H(M (w)) = η(w). Then the following entropy expansion is satisfied:

1 ε 2 T 0 5 i=1 ∇ fi H i (f i ) • (M i -f i ) dt dx dy = - T 0 ∇ 2 w η(w) 2aλ 2 τ • (m - A 1 (w) ε ) • (m - A 1 (w) ε ) dt dx dy - T 0 ∇ 2 w η(w) 2aλ 2 τ • (ξ - A 2 (w) ε ) • (ξ - A 2 (w) ε ) dt dx dy - T 0 ∇ 2 w η(w) 2aε 2 τ • (k -2aw) • (k -2aw) dt dx dy - T 0 ∇ 2 w η(w) 2aε 2 τ • (h -2aw) • (h -2aw) dt dx dy - T 0 ∇ 2 w η(w) (1 -4a)τ ε 2 • (4aw -(k + h)) • (4aw -(k + h)) dt dx dy + O(ε 3 ).
Proof. First of all, Proposition 2.5 guarantees the existence of a kinetic entropy for (1.3), such that

H(M (w)) = η(w) in (2.25), ∇ fi H i (M i (w)) = ∇ w η(w), i = 1, • • • , 5.
We point out that the spectrum of the Jacobian matrices of the Maxwellians in (2.1) is positive provided that the parameter a in the expressions (2.1) is positive and λ > 0 is big enough (Assumptions 2.1). This remark, together with the bounds in (2.15), assure the existence of a kinetic convex and dissipative entropy for our system, thanks to [[8], Theorem 2.1], as stated in Proposition 2.5. Notice that in the course of our computations, the densities f ε remain in a bounded set, close enough to the hyperbolic equilibrium. Now we consider the following expansion (2.27)

1 ε 2 5 i=1 ∇ fi H i (f i ) • (M i -f i ) = 1 ε 2 5 i=1 ∇ fi H i (M i ) • (f i -M i ) + 1 ε 2 5 i=1 ∇ 2 fi H i (M i ) • (f i -M i ) • (f i -M i ) + O( |f i -M i | 3 ε 2 ) = - 1 ε 2 5 i=1 ∇ 2 fi H i (M i ) • (f i -M i ) • (f i -M i ) + O( |f i -M i | 3 ε 2 ).
where the first term vanishes thanks to the orthogonality property [[40], Proposition 2.1]. For i = 1, • • • , 4, the first term of the last equality reads

- 1 ε 2 T 0 ∇ 2 fi H i (M i ) • (f i -M i ) • (f i -M i ) dt dx dy = - 1 ε 2 T 0 ∇ 2 H i (aw ± A i (w) 2λ ) • (f i -M i ) • (f i -M i ) dt dx dy.
Note that, from (1.4)-(2.2) and Lemma 2.3,

w =   ρ ερu 1 ερu 2   =   O(1) O(ε) O(ε)   , A 1 (w) =   ερu 1 ε 2 ρu 2 1 + ρ 2 -ρ2 2 ρ ε 2 ρu 1 u 2   =   O(ε) O(ε 2 ) O(ε 2 )   , A 2 (w) =   ερu 1 ε 2 ρu 1 u 2 ε 2 ρu 2 2 + ρ 2 -ρ2 2 ρ   =   O(ε) O(ε 2 ) O(ε 2 )   .
This way,

∇ fi H(M i (w)) = ∇ fi H aw ± A i (w) 2λ = ∇ fi H(aw) + O(ε).
Moreover, from [START_REF] Bouchut | Construction of BGK Models with a Family of Kinetic Entropies for a Given System of Conservation Laws[END_REF], it is also known that

∇ fi H(M i (w)) = ∇ w η(w).
Differentiating again the previous equivalent expressions,

(2.28)

∇ 2 fi H i (aw) = 1 a ∇ 2 w η(w) + O(ε).
Thus, the last equality yields

- 1 ε 2 T 0 ∇ 2 fi H i (aw ± A i (w) 2λ ) • (f i -M i ) • (f i -M i ) dt dx dy ≤ - 1 ε 2 T 0 1 a ∇ 2 w η(w) • (f i -M i ) • (f i -M i ) dt dx dy + c(|w| L ∞ t L ∞ x ) ε |f i -M i | 2 L ∞ t L ∞ x . Now, from (1.3)-(2.5),
(2.29)

M 1 -f 1 ε 2 = ∂ t f 1 + λ ε ∂ x f 1 = 1 2 (∂ t k + ∂ x m) + 1 2λε (ε 2 ∂ t m + λ 2 ∂ x k), M 3 -f 3 ε 2 = ∂ t f 3 - λ ε ∂ x f 3 = 1 2 (∂ t k + ∂ x m) - 1 2λε (ε 2 ∂ t m + λ 2 ∂ x k), M 2 -f 2 ε 2 = ∂ t f 2 + λ ε ∂ y f 2 = 1 2 (∂ t h + ∂ y ξ) + 1 2λε (ε 2 ∂ t ξ + λ 2 ∂ y h), M 4 -f 4 ε 2 = ∂ t f 4 - λ ε ∂ y f 4 = 1 2 (∂ t h + ∂ y ξ) - 1 2λε (ε 2 ∂ t ξ + λ 2 ∂ y h).
Lemma 2.3 and the previous equalities imply that

c(|w| L ∞ t L ∞ x ) ε |f i -M i | 2 L ∞ t L ∞ x = O(ε 3 ),
and so, by using the change of variables (2.5),

- 1 ε 2 T 0 4 i=1 ∇ fi H i (f i ) • (M i -f i ) dt dx dy = - 1 ε 2 T 0 4 i=1 1 a ∇ 2 w η(w) • (M i -f i ) • (M i -f i ) dt dx dy + O(ε 3 ) = T 0 ∇ 2 w η(w) 2aλ 2 τ • (m - A 1 (w) ε ) • (m - A 1 (w) ε ) dt dx dy - T 0 ∇ 2 w η(w) 2aλ 2 τ • (ξ - A 2 (w) ε ) • (ξ - A 2 (w) ε ) dt dx dy - T 0 ∇ 2 w η(w) 2aε 2 τ • (k -2aw) • (k -2aw) dt dx dy - T 0 ∇ 2 w η(w) 2aε 2 τ • (h -2aw) • (h -2aw) dt dx dy + O(ε 3 ).
The expansion

- 1 ε 2 T 0 H i (f 5 ) • (M 5 -f 5 ) dt dx dy = - T 0 ∇ 2 w η(w) (1 -4a)τ ε 2 • (4aw -(k + h)) • (4aw -(k + h)) dt dx dy + O(ε 3 )
is obtained in analogous way.

Lemma 2.7. Consider the limit solution M i , for i = 1, • • • , 4, in (2.8). Then

(2.30) ∇ fi H i (M i ) = ∇ fi H i (M i ) ∓ aελτ ∇ 2 fi H i (M i )∂ xj w + O(ε 3 ) = ∇ w η( w) ∓ λετ ∇ 2 w η( w)∂ xj w + O(ε 3 ), j = 1, 2.
Proof. The proof follows by Taylor expansions and (2.28), in the spirit of Lemma 2.6.

3.

Relative entropy estimate for the vector-BGK model. Our main result is stated here. Theorem 3.1. Consider the vector-BGK model in (1.3) for the two-dimensional incompressible Navier-Stokes equations in (1.1) on [0, +∞) × T 2 , endowed with a kinetic entropy H(f ε ), whose existence and properties are given by Lemma 2.6. Let ū = (ū 1 , ū2 ), ∇ P be a smooth velocity field and pressure satisfying the incompressible Navier-Stokes equations (1.1) on [0, +∞)×T 2 and {f ε } be a family of smooth solutions to (1.3) and emanating from smooth initial data u 0 in (1.2) and f 0 = (f i (0, x)) i=1,••• ,5 in (2.9). Then, defining w ε = i f ε i = (ρ ε , ερ ε u ε ), the following estimate holds for any T > 0 and for ε ≤ ε 0 , where ε 0 is fixed and it depends on

M 0 = ρ u 0 s+1 , sup t∈[0,T ] ρ ε (t) -ρ s ε + u ε (t) -ū(t) s ≤ cε 1 2 -δ ,
with s > 3, 0 < s < s and δ := s-s 2s . Moreover, for ε ≤ ε 0 , the solutions (ρ ε , u ε ) to the approximating system (1.3) are globally bounded in time, and for ε → 0,

∇((ρ ε ) 2 -ρ2 ) ε 2 ∇ P in L ∞ t H s-3 x .
The global in time convergence proof is based on the use of the relative entropy inequality, which is stated and proved here. Theorem 3.2. Under the hypothesis of Theorem 3.1, let T * be defined in (2.17). Then the relative entropy method provides the following estimate:

sup t∈[0,T * ] ρ ε (t) -ρ 0 ε + ρ ε u ε (t) -ρū(t) 0 ≤ c √ ε.
Proof. We start by recalling the definition of the relative entropy in (2.26),

H(f| f) = H(f) -H(M) -∇ f H(M) • (f -M) = i H i (f i ) -H i (M i ) -∇ fi H i (M i ) • (f i -M i ),
where the limit solutions

M i = M i (ρ, ερū), i = 1, • • • , 5, are in (2.8
), ρ is a constant density, ū is the smooth solution to (1.1), and the associated entropy-flux is given by

Q(f| f) = λ ε   H 1 (f 1 ) -H 3 (f 3 ) -(H 1 (M 1 ) -H 3 (M 3 )) H 2 (f 2 ) -H 4 (f 4 ) -(H 2 (M 2 ) -H 4 (M 4 ))   - λ ε   ∇ f1 H 1 (M 1 )(f 1 -M 1 ) -∇ f3 H 3 (M 3 )(f 3 -M 3 ) ∇ f2 H 2 (M 2 )(f 2 -M 2 ) -∇ f4 H 4 (M 4 )(f 4 -M 4 )   .
Hereafter, we adopt the following notation, H i := H i (M i ).

ROBERTA BIANCHINI

Now we proceed to get the desired inequality.

T 0 ∂ t H(f| f) + ∇ x • Q(f| f) dt dx dy = T 0 ∂ t H(f) + λ ε ∂ x (H 1 (f 1 ) -H 3 (f 3 )) + λ ε ∂ y (H 2 (f 2 ) -H 4 (f 4 )) dt dx dy - T 0 ∂ t H(M) + λ ε ∂ x (H 1 (M 1 ) -H 3 (M 3 )) dt dx dy - T 0 λ ε ∂ y (H 2 (M 2 ) -H 4 (M 4 )) dt dx dy - T 0 ∂ t (∇ f1 H 1 (M 1 )(f 1 -M 1 ) + ∇ f2 H 2 (M 2 )(f 2 -M 2 )) dt dx dy - T 0 ∂ t (∇ f3 H 3 (M 3 )(f 3 -M 3 ) + ∇ f4 H 4 (M 4 )(f 4 -M 4 )) dt dx dy - T 0 ∂ t (∇ f5 H 5 (M 5 )(f 5 -M 5 )) dt dx dy - T 0 λ ε ∂ x (∇ f1 H 1 (M 1 )(f 1 -M 1 ) -∇ f3 H 3 (M 3 )(f 3 -M 3 )) dt dx dy - T 0 λ ε ∂ y (∇ f2 H 2 (M 2 )(f 2 -M 2 ) -∇ f4 H 4 (M 4 )(f 4 -M 4 )) dt dx dy = I 1 + I 2 + I 3 + I 4 .
First of all, I 1 is already estimated in Lemma 2.3. Now, let us consider I 2 .

The following expansions are based on Lemma 2.7.

- T 0 ∂ t (H 1 + H 2 + H 3 + H 4 + H 5 ) dt dx dy - T 0 λ ε ∂ x (H 1 -H 3 ) + λ ε ∂ y (H 2 -H 4 ) dt dx dy = - T 0 (∇ w η( w) -aελτ ∇ 2 f1 H 1 ∂ x w)(∂ t M 1 + λ ε ∂ x M 1 ) dt dx dy - T 0 (∇ w η( w) + aελτ ∇ 2 f3 H 3 ∂ x w)(∂ t M 3 - λ ε ∂ x M 3 ) dt dx dy - T 0 (∇ w η( w) -aελτ ∇ 2 f2 H 2 ∂ y w)(∂ t M 2 + λ ε ∂ y M 2 ) dt dx dy - T 0 (∇ w η( w) + aελτ ∇ 2 f4 H 4 ∂ y w)(∂ t M 4 - λ ε M 4 ) dt dx dy - T 0 ∇ w η( w) • ∂ t M 5 dt dx dy - T 0 ∇ f1 H 1 (∂ t M 1 + λ ε ∂ x M 1 ) -∇ f3 H 3 (∂ t M 3 - ε ∂ x M 3 ) dt dx dy - T 0 ∇ f2 H 2 (∂ t M 2 + λ ε ∂ y M 2 ) -∇ f4 H 4 (∂ t M 4 - λ ε ∂ y M 4 ) dt dx dy + O(ε 3 ) = - T 0 ∇ w η( w) • ∂ t (M 1 + M 2 + M 3 + M 4 + M 5 ) dt dx dy - T 0 ∇ w η( w)[ λ ε ∂ x (M 1 -M 3 ) + λ ε ∂ y (M 2 -M 4 )] dt dx dy + 2aτ λ 2 T 0 (∇ 2 w η( w) • ∂ x w)∂ x w + (∇ 2 w η( w) • ∂ y w)∂ y w dt dx dy + O(ε 3 ) = - T 0 ∇ w η( w)[∂ t w + ∂ x A 1 ( w) ε + ∂ y A 2 ( w) ε -ν(∂ xx w + ∂ yy w)] dt dx dy + T 0 τ ∇ 2 w η( w) 2aλ 2 • (ε 2 ∂ t m + λ 2 ∂ x k) • (ε 2 ∂ t m + λ 2 ∂ x k) dt dx dy + T 0 τ ∇ 2 w η( w) 2aλ 2 • (ε 2 ∂ t ξ + λ 2 ∂ y h) • (ε 2 ∂ t ξ + λ 2 ∂ y h) dt dx dy + O(ε 3 ).
Remark 3.3. Notice that the last equalities follow by adding and subtracting terms of order (ε

2 ∂ t m) • ∂ x k, (ε 2 ∂ t ξ) • ∂ y h, |ε 2 ∂ t m| 2 , |ε 2 ∂ t ξ| 2 , where ∂ x k = 2a∂ x w = 2a∂ x   ρ ερū 1 ερū 2   = O(ε), ∂ y h = 2a∂ y w = O(ε), ε 2 ∂ t m = ε 2 ∂ t [ A 1 ( w) ε -ν∂ x w] = ε 2 ∂ t   ρū 1 ερū 2 1 + ε P ερū 1 ū2   -ν∂ x w = O(ε 2 ), ε 2 ∂ t ξ = ε 2 ∂ t [ A 2 ( w) ε -ν∂ y w] = ε 2 ∂ t   ρū 2 ερū 1 ū2 ερū 2 2 + ε P   -ν∂ y w = O(ε 2 ).
This way, every remainder term is O(ε 3 ).

Next, we consider I 3 .

I 3 = - T 0 ∂ t [∇ f1 H 1 (M 1 )(f 1 -M 1 ) + ∇ f2 H 2 (M 2 )(f 2 -M 2 )] dt dx dy - T 0 ∂ t [∇ f3 H 3 (M 3 )(f 3 -M 3 ) + ∇ f4 H 4 (M 4 )(f 4 -M 4 )] dt dx dy - T 0 ∂ t [∇ f5 H 5 (M 5 )(f 5 -M 5 )] dt dx dy = - T 0 ∂ t [∇ w η( w) • (w -w)] dt dx dy + ελτ T 0 ∂ t [∇ 2 w η( w) • ∂ x w • (f 1 -f 3 -(M 1 -M 3 ))] dt dx dy + ελτ T 0 ∂ t [∇ 2 η( w) • ∂ y w • (f 2 -f 4 -(M 2 -M 4 ))] dt dx dy + O(ε 3 ) = - T 0 ∂ t [∇ w η( w) • (w -w)] dt dx dy + ε 2 τ T 0 ∂ t [∇ 2 w η( w) • ∂ x w • (m - A 1 ( w) ε + 2aλ 2 τ ∂ x w)] dt dx dy + ε 2 τ T 0 ∂ t [∇ 2 w η( w) • ∂ y w • (ξ - A 2 ( w) ε + 2aλ 2 τ ∂ y w)] dt dx dy + O(ε 3 ) = T 0 ∇ 2 w η( w) • [∂ x A 1 ( w) ε + ∂ y A 2 ( w) ε -2aλ 2 τ ∂ xx w -2aλ 2 τ ∂ yy w] • (w -w) + ∇ w η( w) • [∂ x m + ∂ y ξ -∂ x A 1 ( w) ε -∂ y A 2 ( w) ε + 2aλ 2 τ ∂ xx w + 2aλ 2 ∂ yy w] + τ ∇ 2 w η( w) • ∂ x w • (ε 2 ∂ t m + λ 2 ∂ x k) -τ λ 2 ∇ 2 w η( w) • ∂ x w • ∂ x k + τ ∇ 2 w η( w) • ∂ y w • (ε 2 ∂ t ξ + λ 2 ∂ y h) -τ λ 2 ∇ 2 w η( w) • ∂ y w • ∂ y h dt dx dy + O(ε 3 ).
It remains to deal with the last term.

I 4 = - T 0 λ ε ∂ x [∇ f1 H 1 (M 1 )(f 1 -M 1 ) -∇ f3 H 3 (M 3 )(f 3 -M 3 )] dt dx dy - T 0 λ ε ∂ y [∇ f2 H 2 (M 2 )(f 2 -M 2 ) -∇ f4 H 4 (M 4 )(f 4 -M 4 )] dt dx dy = - T 0 λ ε ∂ x (∇ w η( w) -ελτ ∇ 2 w η( w)∂ x w)(f 1 -M 1 ) -(∇ w η( w) + ελτ ∇ 2 w η( w)∂ x w)(f 3 -M 3 ) dt dx dy - T 0 λ ε ∂ y (∇ w η( w) -ελτ ∇ 2 w η( w)∂ y w)(f 2 -M 2 ) -(∇ w η( w) + ελτ ∇ 2 w η( w)∂ y w)(f 4 -M 4 ) dt dx dy + O(ε 3 ) = - T 0 λ ε ∂ x [∇ w η( w) • ((f 1 -f 3 ) -(M 1 -M 3 ))] dt dx dy - T 0 λ ε ∂ y [∇ w η( w) • ((f 2 -f 4 ) -(M 2 -M 4 ))] dt dx dy + λ 2 τ T 0 ∂ x [∇ 2 w η( w) • ∂ x w • (f 1 + f 3 -(M 1 + M 3 ))] dt dx dy + λ 2 τ T 0 ∂ y [∇ 2 w η( w) • ∂ y w • (f 2 + f 4 -(M 2 + M 4 ))] dt dx dy + O(ε 3 ) = - T 0 ∂ x [∇ w η( w) • (m - A 1 ( w) ε + 2aτ λ 2 ∂ x w)] dt dx dy - T 0 ∂ y [∇ w η( w) • (ξ - A 2 ( w) ε + 2aτ λ 2 ∂ y w)] dt dx dy + λ 2 τ T 0 ∂ x [∇ 2 w η( w) • ∂ x w • (k -2a w)] dt dx dy + λ 2 τ T 0 ∂ y [∇ 2 w η( w) • ∂ y w • (h -2a w)] dt dx dy + O(ε 3 ) = - T 0 ∇ w η( w) • [∂ x m + ∂ y ξ - A 1 ( w) ε -∂ y A 2 ( w) ε + 2aτ λ 2 ∂ xx w + 2aτ λ 2 ∂ yy w] dt dx dy - T 0 ∇ 2 w η( w) • ∂ x w • (m - A 1 (w) ε ) dt dx dy - T 0 ∇ 2 w η( w) • ∂ x w • ( A 1 (w) ε - A 1 ( w) ε ) dt dx dy -2aτ λ 2 T 0 ∇ 2 w η( w) • ∂ x w • ∂ x w dt dx dy - T 0 ∇ 2 w η( w) • ∂ y w • (ξ - A 2 (w) ε ) dt dx dy - T 0 ∇ 2 w η( w) • ∂ y w • ( A 2 (w) ε - A 2 ( w) ε ) dt dx dy -2aτ λ 2 T 0 ∇ 2 w η( w) • ∂ y w • ∂ y w dt dx dy + λ 2 τ T 0 ∇ 2 w η( w) • ∂ x w • (∂ x k -2a∂ x w) + ∇ 2 w η( w) • ∂ xx w • (k -2a w) dt dx dy + λ 2 τ T 0 ∇ 2 w η( w) • ∂ y w • (∂ y h -2a∂ y w) + ∇ 2 w η( w) • ∂ yy w • (h -2a w) dt dx dy + λ 2 τ T 0 ∇ 3 w η( w)(∂ x w) 2 (k -2a w) + ∇ 3 w η( w)(∂ y w) 2 (h -2a w) dt dx dy + O(ε 3 ) = - T 0 ∇ w η( w) • [∂ x m + ∂ y ξ - A 1 ( w) ε -∂ y A 2 ( w) ε ] dt dx dy -2aτ λ 2 T 0 ∂ xx w + ∂ yy w dt dx dy + T 0 ∇ 2 w η( w) • ∂ x w • τ (ε 2 ∂ t m + λ 2 ∂ x k) dt dx dy - T 0 ∇ 2 w η( w) • ∂ x w • ( A 1 (w) ε - A 1 ( w) ε ) dt dx dy -4aτ λ 2 T 0 ∇ 2 w η( w) • ∂ x w • ∂ x w dt dx dy + T 0 ∇ 2 w η( w) • ∂ y w • τ (ε 2 ∂ t ξ + λ 2 ∂ y h) dt dx dy - T 0 ∇ 2 w η( w) • ∂ y w • ( A 2 (w) ε - A 2 ( w) ε ) dt dx dy -4aτ λ 2 T 0 ∇ 2 w η( w) • ∂ y w • ∂ y w dt dx dy + λ 2 τ T 0 ∇ 2 w η( w) • ∂ x w • ∂ x k + ∇ 2 w η( w) • ∂ y w • ∂ y h dt dx dy + λ 2 τ T 0 ∇ 2 w η( w) • ∂ xx w • (k -2aw) + 2a∇ 2 w η( w) • ∂ xx w • (w -w) dt dx dy + λ 2 τ T 0 ∇ 2 w η( w) • ∂ yy w • (h -2aw) + 2a∇ 2 w η( w) • ∂ yy w • (w -w) dt dx dy + λ 2 τ T 0 ∇ 3 w η( w)[(∂ x w) 2 (k -2a w) + (∂ y w) 2 (h -2a w)] dt dx dy + O(ε 3 ).
As an intermediate step, let us look at the sum

I 3 + I 4 = 2τ T 0 ∇ 2 w η( w) • ∂ x w • (ε 2 ∂ t m + λ 2 ∂ x k) dt dx dy + 2τ T 0 ∇ 2 w η( w) • ∂ y w • (ε 2 ∂ t ξ + λ 2 ∂ y h) dt dx dy - T 0 ∇ 2 w η( w) • ∂ x w • ( A 1 (w) ε - A 1 ( w) ε - A 1 ( w) ε (w -w)) dt dx dy - T 0 ∇ 2 w η( w) • ∂ y w • ( A 2 (w) ε - A 2 ( w) ε - A 2 ( w) ε (w -w)) dt dx dy + λ 2 τ T 0 ∇ 2 w η( w) • [∂ xx w • (k -2aw) + ∂ yy w • (h -2aw)] dt dx dy -4aτ λ 2 T 0 ∇ 2 w η( w) • [∂ x w • ∂ x w + ∂ y w • ∂ y w] dt dx dy + λ 2 τ T 0 ∇ 3 w η( w) • [(∂ x w) 2 (k -2a w) + (∂ y w) 2 (h -2a w)] dt dx dy + O(ε 3 ).
We analyse each line separately.

• The first one can be written as

τ 2aλ 2 T 0 ∇ 2 w η( w) • (ε 2 ∂ t m + λ 2 ∂ x k) • (ε 2 ∂ t m + λ 2 ∂ x k) dt dx dy + O(ε 3 ).
• Similarly for the second line.

• The third/fourth lines are equivalent to

|∇ 2 w η( w)| L ∞ t L ∞ x T 0 |w -w| 2 dt dx dy + O(ε 3 ).
• The fifth line can estimated by

c 1 (|∇ 2 w η( w)| L ∞ t L ∞ x ) T 0 ε 2 |∂ xx w| 2 + ε 2 |∂ yy w| 2 dt dx dy + c 2 (|∇ 2 w η( w)| L ∞ t L ∞ x ) T 0 |k -2aw| 2 ε 2 + |h -2aw| 2 ε 2 dt dx dy,
where the first term is O(ε 4 ), while the second one is absorbed by the dissipation in I 1 .

• The sixth term can be written as

- τ aλ 2 T 0 ∇ 2 w η( w) • (ε 2 ∂ t m + λ 2 ∂ x k) • (ε 2 ∂ t m + λ 2 ∂ x k) dt dx dy - τ aλ 2 T 0 ∇ 2 w η( w) • (ε 2 ∂ t ξ + λ 2 ∂ y h) • (ε 2 ∂ t ξ + λ 2 ∂ y h) dt dx dy + O(ε 3 ).
• The last term presents the following form

λ 2 τ T 0 ∇ 3 w η( w)(∂ x w) 2 (k -2a w) + ∇ 3 w η( w)(∂ y w) 2 (h -2a w) dt dx dy = λ 2 τ T 0 ∇ 3 w η( w)(∂ x w) 2 (k -2aw) -2a∇ 3 w η( w)(∂ x w) 2 (w -w) dt dx dy + λ 2 τ T 0 ∇ 3 w η( w)(∂ y w) 2 (h -2aw) -2a∇ 3 w η( w)(∂ y w) 2 (w -w) dt dx dy ≤ c(|∇ 2 w η( w)| L ∞ t L ∞ x ) T 0 |w -w| 2 + |k -2aw| 2 ε 2 + |h -2aw| 2 ε 2 dt dx dy + O(ε 3 ),
where the right-hand side is controlled by using the dissipation coming from I 1 .

Remark 3.4. Denoting by µ i (∇ 2 w η(w)), µ i (∇ 2 w η( w)) the eigenvalues of ∇ 2 w η(w), ∇ 2 w η( w) respectively, by simple calculations one gets that

T 0 |µ i (∇ 2 w η(w(t))) -µ i (∇ 2 w η( w(t)))| ∞ dt ≤ c T 0 |ρ(t) -ρ| ∞ dt = O(ε 2 ),
where the last equality follows from Lemma 2.3. Thus, we can write

1 2aλ 2 T 0 (∇ 2 w η(w) • (ε 2 ∂ t m + λ 2 ∂ x k)) • (ε 2 ∂ t m + λ 2 ∂ x k) dt dx dy + 1 2aλ 2 T 0 (∇ 2 w η(w) • (ε 2 ∂ t ξ + λ 2 ∂ y h)) • (ε 2 ∂ t ξ + λ 2 ∂ y h) dt dx dy = 1 2aλ 2 T 0 (∇ 2 w η( w) • (ε 2 ∂ t m + λ 2 ∂ x k)) • (ε 2 ∂ t m + λ 2 ∂ x k) dt dx dy + 1 2aλ 2 T 0 (∇ 2 w η( w) • (ε 2 ∂ t ξ + λ 2 ∂ y h)) • (ε 2 ∂ t ξ + λ 2 ∂ y h) dt dx dy + O(ε 3 ).
Now we consider the total sum, given by

I 1 + I 2 + I 3 + I 4 ≤ |∇ 2 w η( w)| L ∞ t L ∞ x T 0 |w -w| 2 dt dx dy - τ 2aλ 2 T 0 ∇ 2 w η( w) • (ε 2 ∂ t m + λ 2 ∂ x k) • (ε 2 ∂ t m + λ 2 ∂ x k) dt dx dy - τ 2aλ 2 T 0 ∇ 2 w η( w) • (ε 2 ∂ t ξ + λ 2 ∂ y h) • (ε 2 ∂ t ξ + λ 2 ∂ y h) dt dx dy + τ aλ 2 T 0 ∇ 2 w η( w) • (ε 2 ∂ t m + λ 2 ∂ x k) • (ε 2 ∂ t m + λ 2 ∂ x k) dt dx dy + τ aλ 2 T 0 ∇ 2 w η( w) • (ε 2 ∂ t ξ + λ 2 ∂ y h) • (ε 2 ∂ t ξ + λ 2 ∂ y h) dt dx dy - c(|∇ 2 w η(w)| L ∞ t L ∞ x (1 -1 δ )) 2aτ ε 2 T 0 |k -2aw| 2 + |h -2aw| 2 dt dx dy - c(|∇ 2 w η(w)| L ∞ t L ∞ x ) (1 -4a)τ ε 2 T 0 |4aw -(k + h)| 2 dt dx dy + O(ε 3 ).
The Gronwall inequality, together with the definition of w in (1.4), yields the following estimate (3.1) sup

t∈[0,T * ] ρ(t) -ρ 0 ε + ρu(t) -ρū(t) 0 ≤ c √ ε,
where the local time T * is defined in (2.17). Now we prove Theorem 3.1.

Proof. We start by using the interpolation properties of Sobolev spaces, see [START_REF] Taylor | Partial differential equations III[END_REF], for 0 < s < s and t ∈ [0, T * ], which gives (3.2) ρu(t) -ρū(t) s ≤ ρu(t) -ρū(t)

(cεM 0 e c(|ρ| L ∞ t L ∞ x ,|u| L ∞ t L ∞ x )t ) s /s .
Recalling the definition of T * in (2.17) and taking M = 4M 0 , estimate (3.7) implies that there exists ε 0 fixed such that, for ε ≤ ε 0 and t ≤ T * ,

|u(t)| ∞ ≤ M 0 + cε 1 2 -δ < 2M 0 = M 2 ,
for 0 < δ = s 2s < 1 2 . Similarly, for t ≤ T * , (3.8)

|ρ(t) -ρ| ∞ ε + |ρu(t)| ∞ ≤ M 0 + cε 1 2 -δ < 2M 0 = M 2 .
Now let us assume T * < T ε . Then, by definition (2.17),

|ρ(T * ) -ρ| ∞ ε + |ρu(T * )| ∞ = 4M 0 = M.
On the other hand, estimate (3.8) implies that there exists a fixed ε 0 , depending on M 0 and small enough such that, for ε ≤ ε 0 ,

|ρ(T * ) -ρ| ∞ ε + |ρu(T * )| ∞ ≤ M 0 + cε 1 2 -δ < 2M 0 .
Now, by contradiction one gets that T * ≥ T ε for ε ≤ ε 0 , where T * is independent of ε. As a consequence, for ε ≤ ε 0 the solutions (ρ ε , u ε ) to the approximating system evaluated in T ε are bounded. This way, the Continuation Principle, see [START_REF] Majda | Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables[END_REF], implies that they are globally bounded in time. Moreover, since the uniform bounds in Lemma 2.3 are based on the L ∞ t L ∞ x boundedness of (ρ ε , u ε ), it turns out that they hold globally in time for ε ≤ ε 0 . In the end, we proved:

• the global in time existence and uniform boundedness of (ρ ε , u ε ) in H s (T 2 ) for a fixed ε ≤ ε 0 depending on M 0 ; • the strong convergence in [0, T ], for any T > 0, of the solutions (ρ ε , u ε ) to the approximating system (1.3) to the solutions (ρ, ū) to the incompressible Navier-Stokes equations in H s (T 2 ), for 0 < s < s and s > 3; • the rate of this strong convergence. Finally, the convergence to the gradient of the limit incompressible pressure ∇P N S in (1.1) is discussed in details in [START_REF] Bianchini | Convergence of a vector-BGK approximation for the incompressible Navier-Stokes equations, to appear on Kinetic and Related Models[END_REF].

  (2.5) to avoid further complications. The explicit change of variables is written in [[6], (78)].

5 i=1

 5 H i (f ε i ) for system(1.3), satisfying the properties listed in Definition 2.4 in a neighbourhood of the Maxwellians M (w ε ) in (2.1), with η(w ε ) in (2.25).Proof. First of all, the local in time estimates in Lemma 2.3 provide the boundedness of the densities f ε . This fact, together with Assumptions 2.1 and [[8], Theorem 2.1], allow us to prove the positivity of the spectrum of the Jacobian matrices of the Maxwellians (2.1). The result follows from [[8], Theorem 2.1].

  3), see Remark 2.2, and the estimates of the nonlinear term in [[6], Lemma 4.2]. • By applying Gronwall's inequality to (2.11), one gets (2.12). • Estimate (2.13) follows from [[6], Proposition 3 and (30)].

  where the last inequality follows by• the H s -bound of the solution to the incompressible Navier-Stokes equations on the two-dimensional torus, i.e. ρū(t) s ≤ ρu 0 s ≤ M 0 ;• the Gronwall inequality applied to estimate (2.11),ρu(t) s ≤ cM 0 e c(|ρ| L ∞ ∞ ≤ M 0 + cε s-s 2s (M 0 + cM 0 e c(|ρ| L ∞

						t L ∞ x ,|u| L ∞ t L ∞ x )t ) s /s .
	Similarly,
		|ρ(t) -ρ| ∞ ≤ c S ρ(t) -ρ s
	(3.4)		≤ c S ρ(t) -ρ	1-s /s 0	ρ(t) -ρ s /s s
			≤ cε	3(s-s ) 2s	(cεM 0 e c(|ρ| L ∞ t L ∞ x ,|u| L ∞ t L ∞ x )t ) s /s ,
	i.e.
	(3.5)	|ρ(t)| ∞ ≤ ρ + cε	3(s-s ) 2s	(cεM 0 e c(|ρ| L ∞ t L ∞ x ,|u| L ∞ t L ∞ x )t ) s /s .
	Now, since
			u -ū =	1 ρ	(ρu -ρū) +	ū ρ	(ρ -ρ),
	then from (3.4)-(3.3)-(3.4),
	|u(t) -ū(t)| ∞ ≤	1 ρ		cε	s-s
	(3.6)
			+ cε	3(s-s ) 2s	(cεM 0 e c(|ρ| L ∞ t L ∞ x ,|u| L ∞ t L ∞ x )t ) s /s ,
	i.e.,
	|u(t)| ∞ ≤ M 0 +	1 ρ	cε	s-s 2s (M 0 + cM 0 e c(|ρ| L ∞ t L ∞ x ,|u| L ∞ t L ∞ x )t ) s /s
	(3.7)	+ cε	3(s-s ) 2s		1-s /s 0	ρu(t) -ρū(t) s /s s
				≤ cε	s-s
						t L ∞ x ,|u| L ∞ t L ∞ x )t .
	Taking s big enough, the Sobolev embedding theorem yields
	(3.3)

2s (M 0 + cM 0 e t|u| L ∞ t L ∞ x ) s /s , |ρu(t) -ρū(t)| ∞ ≤ c S ρu(t) -ρū(t) s ≤ cε s-s 2s (M 0 + cM 0 e c(|ρ| L ∞ t L ∞ x ,|u| L ∞ t L ∞ x )t ) s /s ,

and so

|ρu(t)| 2s (M 0 + cM 0 e c(|ρ| L ∞ t L ∞ x ,|u| L ∞ t L ∞ x )t ) s /s

Acknowledgements. The author is grateful to Roberto Natalini, for his constant support, and to Laure Saint-Raymond, for useful discussions.


Anne-Sophie Bargnoux

Fernando Vetromile

Nils Kuster

Julie Barberet

Anne-Marie Dupuy

Jean Ribstein

Georges Mourad

Jean-Paul Cristol
email: jp-cristol@chu-montpellier.fr

Pierre Fesler

Markers of bone remodeling are associated with arterial stiffness in renal transplanted subjects

Keywords: Arterial stiffness, Osteoprotegerin, Renal transplantation, Vitamin D

Background Bone-vessel interaction in chronic renal failure remains poorly understood and could be driven by bone remodeling factors including osteoprotegerin (OPG), fibroblast growth factor 23 (FGF23), parathormone and vitamin D. Only few data are available in renal transplantation. The aim of this study was to investigate the relationship between bone remodeling factors and large artery function in renal transplant patients. Methods 89 renal transplant patients were enrolled in this cross-sectional study. Carotid to femoral pulse wave velocity (PWV) and central augmentation index (AIx) were determined as an estimation of large artery function. Blood samples were collected for measurement of vascular risk markers. Independent predictors were identified by multivariate linear regression through backward feature selection using Akaike's information criteria. Results At multivariate analysis, age (p \ 0.001) and systolic arterial pressure (p = 0.003) were significantly associated with PWV but not AIx. In addition, both elevated blood concentrations of 1.25(OH) 2 vitamin D (p = 0.013) and OPG (p = 0.047) were still significantly related to high PWV. Conclusions These results underline that age and mean arterial pressure are the main determinants of PWV following renal transplantation. Among bone remodeling biomarkers, plasma OPG and active vitamin D were the strongest determinants of arterial stiffness.

Introduction

Arterial stiffness, as measured by pulse wave velocity (PWV), is an important risk factor for cardiovascular events and has been recently identified as a strong predictor of all-cause mortality in kidney transplant patients [START_REF] Mitchell | Pulse wave velocity predicts mortality in renal transplant patients[END_REF]. Increased arterial stiffness has been associated with numerous traditional risk factors including age, blood pressure and diabetes [START_REF]Determinants of pulse wave velocity in healthy people and in the presence of cardiovascular risk factors: 'establishing normal and reference values[END_REF] as well as with non traditional risk factors such as inflammation [START_REF] Mceniery | C-reactive protein is associated with arterial stiffness in apparently healthy individuals[END_REF], albuminuria or reduced glomerular filtration rate (GFR) [START_REF] Hermans | Estimated glomerular filtration rate and urinary albumin excretion are independently associated with greater arterial stiffness: the Hoorn Study[END_REF]. In addition, disorders of mineral and bone metabolism related to chronic kidney disease (CKD-MBD) [START_REF] London | Mineral metabolism and arterial functions in end-stage renal disease: potential role of 25-hydroxyvitamin D deficiency[END_REF] could contribute to the arterial remodeling and medial calcification leading to arterial stiffness with elevated pulse pressure (PP) and faster PWV [START_REF] Briet | Arterial stiffness and pulse pressure in CKD and ESRD[END_REF]. Bone-vessel interaction in chronic renal failure could be driven by bone remodeling factors including osteoprotegerin (OPG), fibroblast growth factor 23 (FGF23), parathormone (PTH) and vitamin D. However, few data are available in renal transplantation. In renal transplant patients, PWV remains higher than in healthy subjects [START_REF] Verbeke | Arterial stiffness and wave reflections in renal transplant recipients[END_REF] blood was drawn for the determination of plasma radioactivity and hematocrit. The measured GFR (mGFR) value was the average of four measurements.

Blood pressure measurements

In strictly standardized conditions, arterial pressure was measured in the supine position, after a 10-min period of rest every 3-5 min with an automatic device (Dinamap 845 XT, Critikon, Chatenay Malabry, France). The value used in this study was the average of the mean values obtained during the four clearance periods. Pulse pressure (PP) was considered as the difference between the systolic and the diastolic blood pressure (SBP and DBP), and mean arterial pressure (MAP) was calculated by the formula MAP = (SBP ? 2DBP)/3 [START_REF] Vetromile | Early high pulse pressure is associated with graft dysfunction and predicts poor kidney allograft survival[END_REF].

Pulse wave velocity

Femoral, carotid and radial pressure waveforms were recorded by applanation tonometry using the SphygmoCor system (AtCor Medical, Sydney, Australia). From carotid and femoral waveforms, wave transit time was calculated using the R wave of a simultaneously recorded electrocardiogram as a reference frame and the foot of the waves determined with intersecting tangent algorithm. Transit distance was calculated as 80 % of the direct carotidfemoral distance and carotid-femoral PWV was calculated as transit distance divided by transit time [START_REF] Van Bortel | Expert consensus document on the measurement of aortic stiffness in daily practice using carotid-femoral pulse wave velocity[END_REF].

Pulse wave analysis

The central aortic waveform and the augmentation index (AIx) were derived by applying a validated transfer function to recordings of the arterial pressure waves at the radial artery. The AIx was calculated as augmented pressure/central pulse pressure.

Laboratory measurements

While patients were in the supine position and before the start of GFR measurement, blood samples were collected and centrifuged for routine parameter measurements, then the remaining supernatant was stored at -80 °C for later processing of FGF23 and OPG. Enzymatic creatinine and high-sensitivity C-reactive protein (hs-CRP) were measured using Randox reagents (Randox, Mauguio, France) on an Olympus apparatus (Olympus, Rungis, France). Plasma glucose, albumin, calcium and phosphate were assessed on an Olympus apparatus. Serum total cholesterol (TC), highdensity lipoprotein cholesterol (HDL) and triglyceride (TG) levels were measured by an enzymatic method (Architect; Abbott Park, IL, USA). Glycohemoglobin (Hba1c) was despite restoration of renal function and thus elimination of toxins which have a role in promoting vascular remodeling.

The purpose of this study was therefore to evaluate the relationship between biomarkers of vascular risk (CKD, inflammation, malnutrition, mineral metabolism disorders and particularly bone disease markers such as OPG) and large artery function in renal transplant patients.

Methods

Subjects

Eighty-nine transplanted patients from the Montpellier university nephrology transplantation unit were enrolled in this cross-sectional study. As part of our regular patient follow-up and quality assurance process, the patients were admitted in the morning, after an overnight fast, to our outpatient clinic for renal hemodynamic and function tests. All reported investigations were carried out in accordance with the principles of the Declaration of Helsinki as revised in 2000 (http://www.wma.net/e/poling/17-ce.html); informed consent was obtained from all patients before participating in the study. In accordance with French law, the study was registered at the Ministe `re de l'Enseignement Supe ´rieur et de la Recherche with the number DC-2008-417 after it was approved by our institution's ethical committee.

Clinical evaluation

Detailed medical history including age, gender, weight, height, cause of CKD, time spent in dialysis, diabetes mellitus, hypertension, immunosuppressive regimen, medications (vitamin D, phosphate binders, calcimimetics), and presence of atherosclerotic cardiovascular disease was recorded. Existence of hypertension was defined by brachial blood pressure higher than or equal to 140/90 mmHg and/or by current antihypertensive treatment. Presence of atherosclerotic cardiovascular disease was defined by the presence of at least one of the three following manifestations: coronary heart disease, cerebrovascular disease, and peripheral atherosclerotic disease [START_REF] Bargnoux | Evolution of coronary artery calcifications following kidney transplantation: relationship with osteoprotegerin levels[END_REF].

Determination of renal function

Glomerular filtration rate was measured by urinary clearance of technetium-labeled diethylene-triamino-pentaacetic acid ( 99m Tc-DTPA) using the constant infusion technique, as previously described [START_REF] Vetromile | Early high pulse pressure is associated with graft dysfunction and predicts poor kidney allograft survival[END_REF]. Briefly, after the induction of water diuresis and a 90-min equilibration period, four 20-to 30-min urine collections were obtained by spontaneous voiding. At midpoint of each clearance period, measured by the HPLC method (HA 8140 Ò ; Menarini Diagnostic, Rungis, France). The ratio of urinary concentration of albumin (immunoturbidimetric method) and creatinine (Jaffe method) was calculated as a surrogate measurement of the albumin excretion rate [albumin to creatinine ratio (ACR) in mg/mmol]. 25(OH) vitamin D and 1.25(OH) 2 vitamin D were measured by radioimmunoassay (Immunodiagnostic Systems, Boldon, UK). Intact PTH was measured by immunoradiometric assay (N-Tact PTH SP IRMA Kit, DiaSorin, Stillwater, MN, USA). Serum OPG was determined by enzyme-linked immunosorbent assay (ELISA) (MicroVue OPG kit; Quidel, San Diego, CA, USA). Plasma FGF23 was measured with an ELISA detecting both intact FGF23 and C-terminal fragments (Immutopics International Inc., San Clemente, CA, USA).

Statistical analysis

Continuous data are reported as median and interquartile range (IQR) and categorical data as percentage for descriptive purposes. Since the distribution of PWV values in our population was skewed, logarithmic transformation (base 2) was used for PWV values prior to modeling. Predictors of PWV were identified using least squares linear regression. Covariates significantly associated with PWV at the 0.15 level after adjustment for age and gender were proposed for multivariate analysis. For multivariate model building, a backward automated variable selection procedure based on the Akaike information criterion (AIC) was used. Significance level was set to p \ 0.05. Statistical analysis was performed using R 2.15 software (R Foundation for Statistical Computing, Vienna, Austria).

Results

Baseline characteristics

Clinical and biological characteristics of the 89 renal transplant patients at baseline are reported in Table 1. The median age was 51 years and the gender ratio was 1.96 (59 males/30 females). CKD causes were as follows: glomerulonephritis (n = 15), polycystic kidney disease (n = 18), diabetic nephropathy (n = 2), hypertensive nephropathy (n = 2), unknown cause (n = 21) and other (n = 31). Most patients were hypertensive, few diabetics were included, and half of the group never smoked. Past history of cardiovascular disease was found in 1 in 5 patients. Median time since transplantation was 1 year (IQR 3.4-12.1 months). Immunosuppressive protocols included a sequential quadruple therapy regimen. On the day of inclusion, maintenance immunosuppression was as follows: 92.1 % of patients were on calcineurin inhibitors (29.2 % cyclosporine and 62.9 % tacrolimus), 7.9 % were on mammalian target of rapamycin (mTOR) inhibitors (sirolimus/everolimus), 92.1 % on mycophenolic acid compounds, 4.5 % on azathioprine, and 30.3 % were steroid free.

Discussion

In the present study involving renal transplant recipients, the relationship between established as well as newly proposed biomarkers of vascular risk and the extent of arterial stiffness based on PWV was evaluated. As reported in numerous other clinical conditions, age and blood pressure were found to be strong determinants of arterial stiffness in renal transplantation [START_REF]Determinants of pulse wave velocity in healthy people and in the presence of cardiovascular risk factors: 'establishing normal and reference values[END_REF][START_REF] Briet | Arterial stiffness and pulse pressure in CKD and ESRD[END_REF]. We extended these findings by demonstrating that PWV was also independently related to bone remodeling biomarkers such as OPG and 1.25(OH) 2 vitamin D. Cardiovascular disease (CVD) is the leading cause of death in renal transplant patients. Since traditional risk factors may not solely explain the excess risk of CVD, this may suggest another pathophysiology in these populations. Medial arterial calcification (MAC) is recognized to cause arterial stiffness and subsequently could contribute to cardiovascular morbidity and mortality [START_REF] Cianciolo | Importance of vascular calcification in kidney transplant recipients[END_REF]. MAC is mainly associated with CKD-induced mineral metabolism disorders but also with age and diabetes. However, we did not observe any association between diabetes mellitus or CKD stages and PWV in our study. At univariate analysis, median PWV was similar in non-diabetics [8.21 (7.39-9.13) m/s] and diabetic transplanted patients [8.42 (8.00-9.42) m/s] (p = 0.32, Wilcoxon test). In addition, only a non-significant trend was observed between CKD stages progression and PWV [7.77 (7.17-8.67) for CKD stages I-II, 8.32 (7.38-9.62) for CKD stage IIIA, 8.63 (7.96-9.87) for CKD stage IIIB, and 8.59 (8.46-10.25) for CKD stages IV-V; p = 0.14, Kruskal-Wallis test]. The absence of any relationship between diabetes or CKD stages and PWV in the present study may be related to the fact that only 10 % of the study population had diabetes and that the majority had only moderate impairment in kidney functions (59 % with stage III CKD; 60 % with ACR \3 mg/mmol). Indeed, the previously observed weak relationship between renal impairment and arterial stiffness could have been attenuated by the narrow range of renal function in our study [START_REF] Hermans | Estimated glomerular filtration rate and urinary albumin excretion are independently associated with greater arterial stiffness: the Hoorn Study[END_REF][START_REF] Sengstock | Dominance of traditional cardiovascular risk factors over renal function in predicting arterial stiffness in subjects with chronic kidney disease[END_REF]. Inflammation was shown to be associated with increased arterial stiffness [START_REF] Hermans | Estimated glomerular filtration rate and urinary albumin excretion are independently associated with greater arterial stiffness: the Hoorn Study[END_REF]. However, the relationship observed at univariate analysis between low grade inflammation and PWV did not reach significance at multivariate analysis. Here again, it should be noted that CRP distribution was quite similar to that of healthy volunteers [START_REF] Delcourt | Pathologies OculairesLie ´es a `l'Age study group (2005) Albumin and transthyretin as risk factors for cataract: the POLA study[END_REF] and that the evaluation of low grade inflammation was altered by active immunosuppression.

Arterial stiffness has been reported to be dependent on the immunosuppressive protocol but the association remains unclear. Stro ´zecki et al. [START_REF] Stro ´zecki | Factors associated with increased arterial stiffness in renal transplant recipients[END_REF] established that use of cyclosporine in kidney transplant recipients leads to increased PWV compared to a tacrolimus-based immunosuppressive protocol. Conversely, in the present study, a close to significant negative association was observed between PWV and cyclosporine. This difference could be partly explained by a shorter transplant follow-up (median time 1 vs. [2 years) and a higher proportion of patients under tacrolimus (63 vs. 43 %) compared to Stro ´zecki et al. [START_REF] Stro ´zecki | Factors associated with increased arterial stiffness in renal transplant recipients[END_REF]. Of note, cyclosporine has been previously associated with a lower incidence of glucose abnormalities and use of hypoglycemic medication than tacrolimus at 6 months post-transplantation, both conditions related to increased arterial stiffness [START_REF] Vincenti | Results of an international, randomized trial comparing glucose metabolism disorders and outcome with cyclosporine versus tacrolimus[END_REF].

Abnormalities of bone and mineral metabolism have been recognized as an important component of arterial dysfunction in CKD patients [START_REF] Cianciolo | Importance of vascular calcification in kidney transplant recipients[END_REF][START_REF] Savica | An update on calcium metabolism alterations and cardiovascular risk in patients with chronic kidney disease: questions, myths and facts[END_REF][START_REF] Jb | Osteoporosis and adynamic bone in chronic kidney disease[END_REF]. However, the intricacy of mechanisms involved in arterial stiffness development between vitamin D, secondary hyperparathyroidism, FGF23, bone remodeling regulators such as OPG/ receptor activator of NF-B ligand (RANKL) and Wnt/ LRP5/sclerostin remain poorly understood [START_REF] Evenepoel | Laboratory abnormalities in CKD-MBD: markers, predictors, or mediators of disease?[END_REF] and could ligand (TRAIL) [START_REF] Evrard | SN joined working group on vascular calcifications (2015) Vascular calcification: from pathophysiology to biomarkers[END_REF]. Alternatively, it has been reported that OPG exhibits some potential effects that may contribute to vascular injury [START_REF] Van Campenhout | Osteoprotegerin, vascular calcification and atherosclerosis[END_REF]. For example, it has been shown that treatment of apolipoprotein E (ApoE)-deficient mice with recombinant OPG resulted in an increased collagen content of the vascular media [START_REF] Toffoli | Osteoprotegerin promotes vascular fibrosis via a TGF-b1 autocrine loop[END_REF]. Excess synthesis of bone extracellular matrix by osteoblasts could lead to arterial stiffness without any mineralization. In addition, OPG could modulate inflammation and endothelial dysfunction [START_REF] Doherty | Molecular, endocrine, and genetic mechanisms of arterial calcification[END_REF].

The largest studies on evolution of vascular calcification after renal transplantation show that renal transplantation slows but does not halt the progression of both coronary (CAC) and aortic calcification (AoC) compared to what observed in hemodialysis [START_REF] Cianciolo | Importance of vascular calcification in kidney transplant recipients[END_REF]. In line with these data, we have previously shown, in a cohort of 76 patients, that the CAC score progressed in 26.3 % of patients 1 year after renal transplantation [START_REF] Bargnoux | Evolution of coronary artery calcifications following kidney transplantation: relationship with osteoprotegerin levels[END_REF]. Interestingly, we observed in the ''non-progression'' group, a subgroup of patients (14.5 %) who had a reduction in CAC score, and the pre-transplant level of OPG predicted in part progression of calcification after renal transplantation [START_REF] Bargnoux | Evolution of coronary artery calcifications following kidney transplantation: relationship with osteoprotegerin levels[END_REF]. Since OPG has been associated with progression of vascular calcification, it may be questioned whether normalization of OPG, as one of the promotors of MAC, cannot be associated with regression of pre-existing calcification. However, given the relatively short period of time for follow-up post-transplant, it may be difficult to draw conclusions about the reversibility, and long-term follow-up is needed. Circulating OPG may also reflect a new balance between bone turn-over and vascular production which could be affected by immunosuppression [START_REF] Hofbauer | Effects of immunosuppressants on receptor activator of NF-kappaB ligand and osteoprotegerin production by human osteoblastic and coronary artery smooth muscle cells[END_REF][START_REF] Sasaki | Glucocorticoid decreases circulating osteoprotegerin (OPG): possible mechanism for glucocorticoid induced osteoporosis[END_REF]. However, no data are currently available concerning the relative action of cyclosporine and tacrolimus on the OPG/RANKL system.

Our study has some limitations. Firstly, the relatively small sample size of this cross-sectional study may have prevented some associations from being statistically significant. Secondly, known risk factors of MAC leading to arterial stiffness, e.g. diabetes, and advanced CKD stages, were not adequately represented in our population, which may have attenuated the associations. Finally, FGF23 and OPG were analyzed as biomarkers but their pathophysiological role could not be evaluated without an assessment of their corresponding receptor/ligand Klotho or RANKL.

Conclusion

In conclusion, we have demonstrated that OPG and active vitamin D are associated with arterial stiffness in the early post-transplant period following renal transplantation. This complex relationship with bone-vascular disease in kidney recipients remains incompletely understood, since it be further complicated by the use of calcineurin inhibitors in the context of post-transplantation bone disease. Contrary to previous observations in CKD patients [START_REF] London | Mineral metabolism and arterial functions in end-stage renal disease: potential role of 25-hydroxyvitamin D deficiency[END_REF][START_REF] Chitalia | Vitamin D deficiency and endothelial dysfunction in non-dialysis chronic kidney disease patients[END_REF], we did not find any relationship between 25(OH) vitamin D and PWV in the present study, but a clear, independent and positive association between 1.25 (OH) 2 vitamin D and PWV. It is noteworthy that in dialysis children a U-shaped curve has been evidenced between active vitamin D and vascular calcification [START_REF] Shroff | A bimodal association of vitamin D levels and vascular disease in children on dialysis[END_REF]. In addition, data from human studies and animal models suggest a biphasic cardiovascular effect of active vitamin D [START_REF] Razzaque | The dualistic role of vitamin D in vascular calcifications[END_REF] which could be partly related to FGF23 action on 1-alpha and 1-24 hydroxylase.

The level of FGF23 rises as renal function declines [START_REF] Morena | Bone biomarkers help grading severity of coronary calcifications in non dialysis chronic kidney disease patients[END_REF] and reaches maximal values in dialysis patients. FGF23 remains elevated for some months after renal transplantation as compared with subjects of similar GFR and diminishes after 1 year [START_REF] Evenepoel | Recovery of hyperphosphatoninism and renal phosphorus wasting one year after successful renal transplantation[END_REF]. Several studies have reported an association between FGF23 and endothelial dysfunction, vascular calcifications, arterial stiffness and/or outcome in CKD and dialysis patients [START_REF] Gutie ´rrez | Fibroblast growth factor 23 and left ventricular hypertrophy in chronic kidney disease[END_REF][START_REF] Yilmaz | FGF-23 and vascular dysfunction in patients with stage 3 and 4 chronic kidney disease[END_REF] via Klotho-dependent or independent pathways. However a recent study by Scialla et al. [START_REF] Scialla | Fibroblast growth factor 23 is not associated with and does not induce arterial calcification[END_REF] has challenged this point of view. In this translational study including 1,501 CKD patients, no relationship was found between FGF23 and arterial calcification. These features have been recently confirmed in hemodialysis patients [START_REF] Lee | Biomarkers associated with vascular and valvular calcification in chronic hemodialysis patients[END_REF]. Our results are in agreement with these observations and previous data from our group suggesting that, in a cohort of CKD stage I to V, FGF23 was associated with high, not moderate, calcification, in contrast to OPG [START_REF] Morena | Bone biomarkers help grading severity of coronary calcifications in non dialysis chronic kidney disease patients[END_REF].

Evidence supports the assumption that the OPG/ RANKL complex cytokine network, a key factor in the regulation of bone metabolism, may be expressed and implicated in vascular pathophysiology. Indeed, OPG has been proposed as a protective factor for vascular calcium deposition regarding data from animal models [START_REF] Doherty | Molecular, endocrine, and genetic mechanisms of arterial calcification[END_REF]. However, epidemiological studies [START_REF] Nybo | The capability of plasma osteoprotegerin as a predictor of cardiovascular disease: a systematic literature review[END_REF] support the notion that elevated levels of OPG are associated with vascular risk in both general and high-risk populations including CKD patients. We previously demonstrated that OPG was a strong predictor of coronary artery calcification during the progression of CKD as well as during dialysis [START_REF] Bargnoux | Evolution of coronary artery calcifications following kidney transplantation: relationship with osteoprotegerin levels[END_REF][START_REF] Morena | Bone biomarkers help grading severity of coronary calcifications in non dialysis chronic kidney disease patients[END_REF]. In agreement with results in CKD [START_REF] Scialla | Correlates of osteoprotegerin and association with aortic pulse wave velocity in patients with chronic kidney disease[END_REF] and dialysis patients [START_REF] Speer | Serum osteoprotegerin level, carotid-femoral pulse wave velocity and cardiovascular survival in haemodialysis patients[END_REF], OPG is shown to predict PWV in post-transplant renal patients. In this context, it may be questioned whether an increase in OPG level represents a deleterious effect of OPG in the vascular calcification process or a compensatory self-defensive mechanism against factors that promote vascular calcification. Indeed, elevation of OPG levels in response to vascular calcification could block the bone remodeling process in vascular tissue through binding to RANKL and neutralize the pro-apoptotic actions of tumor necrosis factor (TNF)-related apoptosis-inducing probably involves renal and extra-renal factors such as inflammation, immunosuppressive regimens, preexisting bone disease and vascular calcifications.

Table 1

 1 Characteristics of transplant patients at baseline (n = 89) Older age was significantly associated with higher PWV and AIx (p \ 0.001 for both). Gender was significantly associated with higher AIx (p \ 0.01) but not PWV (p = 0.095). After adjustment for these two variables (

	Parameter	Value
	Gender, male (%)	66.3
	Age (years)	50.6 (40.6-60.3)
	Body mass index (kg/m 2 )	23.8 (21.4-25.5)
	Time spent in dialysis (months)	27 (8-43)
	Smoking status (% of never smoked)	53.9 %
	Diabetes mellitus	9 %
	Hypertension	89.9 %
	History of cardiovascular disease	22.5 %
	Cold ischemia time (min)	1080 (832-1352)
	Delayed graft function (%)	12.4
	Rejection episodes (%)	13.5
	Systolic blood pressure (mmHg)	128 (120-136)
	Diastolic blood pressure (mmHg)	75 (70-82)
	Mean arterial pressure (mmHg)	93 (88-100)
	Arterial pulse pressure (mmHg)	53 (45-60)
	Pulse wave velocity, PWV (m/s)	8.3 (7.4-9.1)
	Augmentation index, AIx (%)	24 (16-32)
	Glomerular filtration rate (ml/min/1.73 m 2 )	57.3 (43.2-68.8)
	Albumin to creatinine ratio, ACR (mg/mmol)	1.7 (0.9-4.1)
	Total cholesterol (mmol/l)	5.34 (4.57-6.07)
	Non HDL cholesterol (mmol/l)	3.88 (3.22-4.75)
	C-reactive protein (mg/l)	1.3 (0.6-2.8)
	Glycemia (mmol/l)	5.3 (4.8-5.7)
	Glycohemoglobin, Hba1c (%)	5.8 (5.3-6.1)
	Albumin (g/l)	44 (42-45)
	Corrected calcium (mmol/l)	2.48 (2.43-2.56)
	Phosphorus (mmol/l)	0.83 (0.71-0.93)
	Parathyroid hormone (pg/ml)	72 (47-120)
	1.25 (OH) 2 vitamin D (pg/ml)	52 (40-68)
	25(OH) vitamin D (ng/ml)	22.9 (17.2-29.5)
	Osteoprotegerin (pmol/l)	4.4 (3-5.9)
	FGF23 (RU/ml)	116 (89-153)
	Medications
	Vitamin D	53.9 %
	Calcium-based phosphate binders	3.4 %
	Calcimimetics	4.5 %
	Values are described as proportions for categorical variables and
	median (interquartile range) for quantitative variables
	HDL high-density lipoprotein

Table 2

 2 Among the biomarkers of vascular risk studied, age-sex adjusted logistic regression analysis clearly identified high OPG (p = 0.017), high 1.25(OH) 2 vitamin D (p = 0.007) and elevated CRP (p = 0.048) as the only variables associated with increased PWV (Table2).

		population. By
		contrast, no significant relationship between PWV and
	), systolic BP (p = 0.002), MAP (p = 0.018) and	calcium (p = 0.939), phosphorus (p = 0.35), FGF23
	PP (p = 0.046) were significantly associated with higher	(p = 0.523), PTH (p = 0.503) or 25(OH) vitamin D
	PWV but not smoking habit (p = 0.413), diabetes mellitus	(p = 0.874) was observed. In addition, no relationship was
	(p = 0.801), time since transplantation (p = 0.843), time	found between AIx and the studied parameters (data not
	spent on dialysis prior to transplantation (p = 0.721),	shown).
	mGFR (p = 0.193), vitamin D therapy (p = 0.22) or his-	A multivariate linear regression analysis was carried out
	tory of cardiovascular disease (p = 0.477) in our	to identify independent predictors of PWV. The model

Table 2

 2

	Predictors of log-transformed pulse wave velocity	Parameter	Beta coefficient	p value
		Age (years)	0.012 (0.008 to 0.016)	<0.001
		Gender (female)	0.109 (-0.21 to 0.239)	0.105
		Body mass index (kg/m 2 )	-0.008 (-0.024 to 0.007)	0.293
		Time spent in dialysis (months)	0 (-0.002 to 0.003)	0.721
		Time since transplantation (months)	0.001 (-0.011 to 0.013)	0.843
		Smoking status (yes)	-0.047 (-0.159 to 0.065)	0.413
		Diabetes mellitus	-0.025 (-0.216 to 0.167)	0.801
		History of cardiovascular disease	0.049 (-0.086 to 0.184)	0.477
		Cold ischemia time (min)	-0.003 (-0.01 to 0.004)	0.383
		Delayed graft function (%)	0.055 (-0.108 to 0.218)	0.511
		Rejection episodes (%)	-0.053 (-0.211 to 0.105)	0.513
		Systolic blood pressure (mmHg)	0.008 (0.003 to 0.012)	0.002
		Diastolic blood pressure (mmHg)	0.004 (-0.002 to 0.01)	0.194
		Mean arterial pressure (mmHg)	0.008 (0.002 to 0.015)	0.018
		Arterial pulse pressure (mmHg)	0.005 (0 to 0.01)	0.046
		Glomerular filtration rate (ml/min/1.73 m 2 )	0.002 (-0.001 to 0.005)	0.193
		Albumin to creatinine ratio, ACR (mg/mmol)	003 (-0.001 to 0.008)	0.180
		Total cholesterol (mmol/l)	0.016 (-0.023 to 0.056)	0.419
		Non HDL cholesterol (mmol/l)	0.023 (-0.018 to 0.064)	0.278
		Log C-reactive protein (mg/l)	0.032 (0.001 to 0.063)	0.048
		Glycemia (mmol/l)	0.009 (-0.056 to 0.075)	0.784
		Glycohemoglobin, Hba1c (%)	-0.056 (-0.142 to 0.03)	0.204
		Albumin (g/l)	0.012 (-0.008 to 0.032)	0.232
		Corrected calcium (mmol/l)	0.012 (-0.29 to 0.314)	0.939
		Phosphorus (mmol/l)	-0.119 (-0.368 to 0.129)	0.35
		Parathyroid hormone (pg/ml)	0 (-0.001 to 0.001)	0.503
		1.25 (OH) 2 vitamin D (pg/ml)	0.003 (0.001 to 0.005)	0.007
		25(OH) vitamin D (ng/ml)	0 (-0.005 to 0.006)	0.874
		Osteoprotegerin (pmol/l)	0.042 (0.008 to 0.075)	0.017
		Log FGF23 (RU/ml)	-0.023 (-0.093 to 0.047)	0.523
		Cyclosporine	-0.109 (-0.225 to 0.008)	0.07
		Tacrolimus	0.077 (-0.033 to 0.187)	0.174
		Sirolimus/everolimus	0.06 (-0.139 to 0.259)	0.556
		Medications with vitamin D	-0.068 (-0.174 to 0.039)	0.217
		Bold values indicate significant p value
		HDL high-density lipoprotein

Table 3

 3 Predictors of logtransformed pulse wave velocity at multivariate analysis

	Parameter	Beta coefficient	p value a
	Age (years)	0.009 (0.005 to 0.014)	\0.001
	1.25 (OH) 2 vitamin D (pg/ml)	0.002 (0.001 to 0.004)	0.013
	Osteoprotegerin (pmol/l)	0.033 (0.001 to 0.064)	0.047
	Systolic blood pressure (mmHg)	0.007 (0.003 to 0.012)	0.003
	Cyclosporine treatment	-0.112 (-0.218 to -0.006)	0.041
	mGFR measured glomerular filtration rate
	a Adjustment for gender, diabetes, mGFR, and time spent in dialysis prior to transplantation

Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest.

Informed consent Informed consent was obtained from all individual participants included in the study.

Research involving human participants All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.


Maxime Scrinzi
email: maxime.scrinzi@yahoo.fr

Spatio-temporal analysis of settlement from the Iron Age to the High Middle Ages in southern France: the Vidourle Valley

Introduction

The Vidourle River runs south from the Cevennes hills for around 95km before debouching into a lagoon to the east of Montpellier on the Mediterranean coast of southern France. It traverses a range of different geographical units-hills, foothills, basins, coastal plain, lagoon, garrigue and scrub-draining a catchment area of 800km² (Figures 1 &2). Various archaeological projects carried out over the past fifty years throughout the valley have identified 832 settlements dated between the seventh century BC and the tenth century AD. This significant dataset is the basis for analysing population dynamics throughout the valley across seventeen centuries [START_REF] Scrinzi | Archéologie de la vallée du Vidourle[END_REF]. In what ways did settlement evolve between the Early Iron Age and AD 1000? Are there differences in the human occupation and exploitation of the upper and lower Vidourle valley? And does the river influence the choice of settlement location? All of these questions fall into a more general theme concerning the spatial and temporal analysis of settlement, in accordance with the Archaeomedes programme that forms our methodological model (Van der Leeuw et al. 2003).

Results

The relationship between coast and hinterland, geographically different but linked by the river, is at the core of our analysis. For each major chronological phase, our research highlights similarities in settlement forms across the entire valley; from the concentration of populations within protohistoric oppida and medieval castra and villae, through to the networks of cities and dispersed settlements of classical antiquity, all areas of the watershed share the same settlement distribution and dynamic. Beneath this apparent uniformity, however, there are significant differences in the population density of the coastal plain and inland areas, the latter being much lower. Why? A first response to this question concerns the Mediterranean Sea and its possibilities for commercial and cultural exchange; another response focuses on the natural resources and favourable soils that influenced the development of agricultural and crafting activities. Even if full understanding of settlement in the high valley zone is hindered by vegetation that makes these areas difficult to survey, the levels of settlement remain, in all cases, well below those documented at lower elevations. Indeed, the scarcity of settlement dispersal in the cultivated areas of the high valley indicates limited investment in the hinterland; here, population appears to have remained largely concentrated.

Our settlement database allows us to establish the number of sites occupied in each century, providing a reasonably accurate assessment of settlement trends (Figure 3). The results fit well with knowledge of the wider settlement context of southern France identified in various comparative studies [START_REF] Trément | Archéologie d'un paysage. Les étangs de Saint-Blaise (Bouches-du-Rhône)[END_REF]Van der Leeuw et al. 2003;[START_REF] Vial | Les Volques Arécomiques et le Languedoc oriental protohistorique. Étude d'une entité ethno-politique préromaine[END_REF]. Each major chronological phase demonstrates a unique picture: population mainly grouped within oppida and ports during protohistoric periods; the development of agricultural facilities scattered across the countryside during Antiquity, with urban centres typically located on strategic heights; and settlement on the coastline, and in the hills and plains of the middle valley during the high Middle Ages (Figures 4,5 & 6). Despite these long-term differences, the river and its tributaries continuously structured human occupation of the valley. As a strategic passageway from the coast to the interior, a supply of fresh water and a fertile alluvial plain, the Vidourle and its tributaries have animated and guided the locational choices of its past inhabitants. These demonstrate a desire to be close to the river for convenience and in order to control it; in parallel, the hills and elevated areas provided security against flooding, particularly for urban areas, and the possibility for wider territorial control. As an axis of penetration into the hinterland, the Vidourle was also a trade route between the interior and the sea (Scrinzi 2014: 331-95). Analysis of long-term settlement and land use in the Vidourle valley provides a model for the eastern Languedoc. How does it compare with the wider Mediterranean context? Is it typical of the settlement systems encountered in the river valleys of Spain, Italy and North Africa? Or is each Mediterranean valley different? The results from investigations in the Vidourle valley stand as a starting point for a wider research programme extending into neighbouring regions and countries, with the aim of establishing a methodological and analytical framework for the study of settlement in the river valleys of the western Mediterranean.

Figure 1 .

 1 Figure 1. Location of the Vidourle valley (© M. Scrinzi)

Figure 2 .

 2 Figure 2. Research situation in the study area, cities and different landscapes (© S. Sanz & M. Scrinzi)

Figure 3 .

 3 Figure 3. Number of newly created sites and site abandonments per century between the seventh century BC and the tenth century AD (© M. Scrinzi)

Acknowledgements

This paper has been supported by the Labex ARCHIMEDE from 'Investissement d'Avenir' programme ANR-11-LABX-0032-01. I wish to thank John Bintliff, Nathalie Caballero and Cyrille Rieau for their help with the translation of this paper into English.