A.-C Boulanger
email: céline.boulanger@inria.fr

M.-O Bristeau
email: marie-odile.bristeau@inria.fr

J Sainte-Marie
email: jacques.sainte-marie@inria.fr

ANALYTICAL SOLUTIONS FOR THE FREE SURFACE HYDROSTATIC EULER EQUATIONS

Keywords: Incompressible Euler equations, analytical solutions, free surface flows, entropic shocks subject classifications. 35Q30, 35Q35, 76D05

In this paper we propose a large set of analytical solutions (FRESH-ASSESS) for the hydrostatic incompressible Euler system in 2d and 3d. These solutions mainly concern free surface flows but partially free surface flows are also considered. These analytical solutions can be especially useful for the validation of numerical schemes.

Introduction

Models arising in fluid dynamics often based on the Navier-Stokes equations are generally difficult to analyze both at the mathematical or numerical level. As a consequence the derivation of efficient numerical schemes and their validations are complex. A way to circumvent this difficulty is to have access to analytical solutions of the considered problem and to confront the simulated solutions with them. Even if this approach is not a complete validation of the numerical schemes, it enables to have a good knowledge of the behavior of the discrete schemes in front of typical and complex situations particularly when the family of analytical solutions is large enough. Moreover it makes possible the study of the convergence order of a simulated solution towards the reference one.

Starting from the incompressible Navier-Stokes equations, much work has been done in the derivation of relatively simple mathematical models for shallow water flows and long non-linear water waves. For these models, analytical solutions are often available. Thus the Saint-Venant system [START_REF] Barré De Saint-Venant | Théorie du mouvement non permanent des eaux avec applications aux crues des rivières et à l'introduction des marées dans leur lit[END_REF] that is well suited for the modeling of river hydraulics has been widely studied by a lot of authors. In a chronological order, the most famous contributions are: analytical solutions for the dam break situation by Ritter [START_REF] Ritter | Die fortpflanzung der wasserwellen[END_REF], the nonlinear wave propagation in Shallow water open channels by Stoker [START_REF] Stoker | The formation of breakers and bores the theory of nonlinear wave propagation in shallow water and open channels[END_REF], solutions of the Saint-Venant system for a flow respectively over a ridge and in parabolic bowl by Houghton [START_REF] Houghton | Nonlinear shallow fluid flow over an isolated ridge[END_REF] and Thacker [START_REF] Thacker | Some exact solutions to the non-linear shallow-water wave equations[END_REF]. MacDonald et al. [START_REF] Macdonald | Comparisons of some steady state Saint-Venant solvers for some test problems with analytic solutions[END_REF][START_REF]Steady open channel test problems with analytic solutions[END_REF] have proposed a wide family of analytical solutions with bottom friction, hydraulic jumps and various open-channel cross-sections. Likewise for the Korteweg-de Vries equations [START_REF] Korteweg | On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves[END_REF] and related systems -generally corresponding to potential flows [START_REF] Levi-Civita | Détermination rigoureuse des ondes permanentes d'ampleur finie[END_REF] such as the Boussinesq system [START_REF] Boussinesq | Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire[END_REF], the Green-Naghdi model [START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF], various analytical solutions of soliton type are available [START_REF] Kruskal | Exact invariants for a class of nonlinear wave equations[END_REF][START_REF] Lax | Integrals of nonlinear equations of evolution and solitary waves[END_REF][START_REF] Bonneton | Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes[END_REF].

Unfortunately, the previously mentioned models correspond to averaged -along a spatial coordinate -or irrotational approximations of the Navier-Stokes equations. In this paper we propose various analytical solutions for the incompressible hydrostatic Euler equations in the following situations: stationary flows, transient flows, free (or partially free) surface, external forcing, simplified viscosity terms. These solutions are denoted hereafter by the acronym FRESH-ASSESS corresponding to FREE Surface Hydrodynamics Analytical SolutionS for Euler SystemS. The analytical solutions are available in 2d and 3d for a large class of bottom geometry. Some of these solutions admit entropic shocks that are characterized.

The main concern of the authors is to exhibit reference solutions that can be used for the validation of numerical codes. In the case of a regular flow, the solution is analytic. For non smooth solutions e.g. with shocks, the solution is obtained via the resolution of a simple ODE. If the proposed analytical solutions correspond to physical phenomena, they do not correspond to useful situations for practical use of the models. The analytical solutions are obtained as in MacDonald et al. [START_REF] Macdonald | Comparisons of some steady state Saint-Venant solvers for some test problems with analytic solutions[END_REF][START_REF]Steady open channel test problems with analytic solutions[END_REF], starting from a given water depth and an admissible velocity field, we derive the corresponding topography profile.

The paper is organized as follows. Starting from the incompressible and hydrostatic Euler equations (Sec. 2), we present in Sec. 3 the 2d FRESH-ASSESS solutions in various configurations. The 3d extension of these solutions is given in Sec. 4. In Sec. 5, the various analytical solutions are illustrated in typical situations.

The Euler equations

We start from the 3d hydrostatic and incompressible free surface Euler equations [START_REF] Brenier | Homogeneous hydrostatic flows with convex velocity profiles[END_REF][START_REF] Grenier | On the derivation of homogeneous hydrostatic equations[END_REF]

∂p ∂z = -g, (2.4)
where the z axis represents the vertical direction. We consider this system for

t > 0, (x,y) ∈ Ω, z b (x,y) ≤ z ≤ η(x,y,t),
where Ω is a bounded domain of R 2 , η(x,y,t) represents the free surface elevation, u = (u,v,w) T the velocity vector. The gravity acceleration is denoted g and p is the fluid pressure. The bottom geometry is z b so the water depth is H = ηz b , see Fig. 2.1. Equation (2.1) represents the incompressibility (mass conservation equation) and Eqs. (2.2),(2.3) are respectively the momentum equilibria of forces along axis x and y. Because of the hydrostatic assumption, the momentum equilibrium of forces along the vertical axis reduces to (2.4).

The system (2.1)-(2.4) is completed with boundary conditions. Classically at the free surface we have the kinematic boundary condition

∂η ∂t + u s ∂η ∂x + v s ∂η ∂y -w s = 0, (2.5)
where the subscript s denotes the value of the considered quantity at the free surface.

The dynamical condition at the free surface takes into account the equilibrium with the atmospheric pressure and we have where p a = p a (x,y,t) is a given function corresponding to the atmospheric pressure. At the bottom, the kinematic boundary condition corresponds to a no-penetration condition

p s = p a (x,y,t), (2.6)
u b ∂z b ∂x + v b ∂z b ∂y -w b = 0, (2.7)
where the subscript b denotes the value of the considered quantity at the bottom. The boundary condition (2.6) coupled with (2.4) gives p = p a + g(ηz), so the hydrostatic Euler system (2.1)-(2.4) can be written under the form

∂H ∂t + ∂ ∂x η z b u dz + ∂ ∂y η z b v dz = 0, (2.8) w = - ∂ ∂x z z b u dz - ∂ ∂y z z b v dz, (2.9
∂ ∂t η z b (E + p a ) dz + ∂ ∂x η z b u E + g(η -z) + gz dz + ∂ ∂y η z b u E + g(η -z) + gz dz = H ∂p a ∂t , (2.12)
with

E = u 2 2 + v 2 2 + gz.
(2.13)

2d analytical solutions

For the sake of simplicity, we first restrict our presentation to 2d analytical solutions. The considered horizontal domain is Ω = [0,L x ]. The 3d extension of the proposed solutions is given in Sec. [START_REF] Bonneton | Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes[END_REF].

In this section, we first propose stationary solutions for the system (2.1)-(2.4). The regularity of the solutions is examined in paragraphs 3.3 and 3.4. Then we exhibit analytical solutions for the Euler system in the following situations : external forcing, viscosity terms, reactive tracer, transient solutions, a partially free surface.

Main idea

We first consider the system (2.8)- (2.11) where all the variables are independent of y. Stationary solutions of this system have to fulfill

∂ ∂x η z b u dz = 0, (3.1) w = - ∂ ∂x z z b u dz, (3.2)
u ∂u ∂x + w ∂u ∂z + g ∂η ∂x = - ∂p a ∂x . (3.3)
Except in paragraphs 3.7.1 and 3.8.1, we assume in this section p a = Cst. From (3.3), it comes that

u ∂u ∂x + w ∂u ∂z , (3.4)
has to be independent of z and this is the most difficult requirement to satisfy. The underlying idea for the derivation of the analytical solutions is the following. To satisfy (3.1), we seek for u under the form

u = α ϕ ′ (z -z b ) ϕ(H) , (3.5)
with ϕ(0) = 0, α = Cst. Then from (3.2), w is given by

w = -α ∂ ∂x ϕ(z -z b ) ϕ(H) , (3.6)
and replacing the previous expressions for u and w in

(3.3) implies (3.4) is independant of z when (ϕ ′ (z -z b )) 2 -ϕ(z -z b )ϕ ′′ (z -z b ) = Cst,
that is ensured for

ϕ(z) = sin(z). (3.7) π β 0 C α,β,lim F α,β (H) H F α,β,c H c Fig. 3.1. Function H → F α,β (H).

General continuous stationary solutions

The following proposition gives smooth analytical solutions for the system (3.1)-(3.3). Proposition 3.1. If α, β and z b are three real constants and H 0 a non negative real C 1 (Ω) function such that sin(βH 0 (x)) = 0 for all x ∈ Ω then u = u α,β , w = w α,β and z b respectively given by

z b = z b -H 0 - α 2 β 2 2g sin 2 (βH 0 ) , (3.8)
u α,β = αβ sin(βH 0 ) cos(β(z -z b )), (3.9)
w α,β = αβ ∂z b ∂x cos(β(z -z b )) sin(βH 0 ) + ∂H 0 ∂x sin(β(z -z b ))cos(βH 0 ) sin 2 (βH) , ( 3
u = u α,β = αβ sin(βH) cos(β(z -z b )), (3.11)
and from (3.6), w is given by

w = w α,β = αβ ∂z b ∂x cos(β(z -z b )) sin(βH) + ∂H ∂x sin(β(z -z b ))cos(βH) sin 2 (βH) , (3.12)
then Eq. (3.3) reduces to

∂ ∂x α 2 β 2 2sin 2 (βH) + g(H + z b ) = 0, (3.13)
and the definition (3.8) for z b with H = H 0 completes the proof. The previous proposition is only valid for smooth solutions. The regularity and uniqueness of the solutions and their derivation are discussed in the two following paragraphs.

Continuity of the solutions

Here we adopt a slightly different strategy considering we have a given topography z b (x), x ∈ Ω = [0,L x ], given values for α > 0 and β and a boundary value for H namely, H lim = H(0) or H lim = H(L x ). H(x) will be given by the solution of a simple ODE. We assume u satisfies (3.11), w satisfies (3.12) and it remains to solve (3.13) to obtain H.

We introduce the function

F α,β (x) : x → gx + α 2 β 2 2sin 2 (βx) ,
and we point out that F α,β is not one-to-one (see Fig. 3.1). When considering smooth solutions -and with the notations of Fig. 3.1 -the resolution of (3.13) is equivalent to find for all x ∈ Ω a solution to

F α,β (H(x)) + gz b (x) = F α,β (H lim ) + gz b,lim . (3.14)
If the bottom profile is enough regular, Eq. (3.13) can be written under the form

g -α 3 β 2 cos(βH) sin 3 (βH) ∂H ∂x = -g ∂z b ∂x ,
This previous equation remains well defined until the quantity

g -α 3 β 2 cos(βH) sin 3 (βH) , (3.15
H c = arcsinr 2 c β , and
F α,β,c = gH c + α 2 β 2 2r 4 c , (3.17)
where r 2 c is the unique real solution of the polynomial equation

P (X) = g 2 X 3 + α 4 β 6 X -α 4 β 6 = 0, (3.18)
with

r c = 3 √ 12α 4 β 2 6g - 3 √ 12α 4 β 2 + 3 9g + 12α 4 β 6 + 81g 2 2 3 9g + 12α 4 β 6 + 81g 2 .
Since we have P (0) ≤ 0 and P (1) ≥ 0, we remark that 0 ≤ r c ≤ 1.

The previous results allow to write the following proposition Proposition 3.2. If for any x ∈ Ω, the problem

F α,β (H(x)) = F α,β (H lim ) + g(z b,lim -z b (x)) = C α,β,lim -gz b (x) ≥ 0, (3.19)
has a solution H(x) ≥ 0 then (H,u α,β ,w α,β ) are solutions of (3.1)- (3.3). This result can also be expressed in the following way: if the problem (3.19) admits a solution H c in x = x c or equivalently if

F α,β,c ≤ C α,β,lim ,
then it admits a solution for all x ∈ Ω. And it is easy to prove that if

z b ∈ C k (Ω), k ≥ 0 then (H,u α,β ) ∈ C k (Ω) and w α,β ∈ C k-1 (Ω).
Let Ω c be the subset of Ω containing the points x for which the problem (3.19) has no solution. We are now interested in situations where Ω c = ∅.

Discontinuous solutions

Let us consider now discontinuous solutions. We define the operator [A] = A + -A -, where the superscripts + and -define the variables right after and before the discontinuity.

Stationary shocks of the system (2.12), (3.1) and (3.

g 2 H 2 + α 2 β 2 H 2sin(βH) 2 + α 2 β 2 cos(βH) sin(βH) = 0, (3.20) α 3 β 2 2sin 2 (βH) + gα(H + z b ) = D, (3.21)
since u = u α,β and with D ≤ 0.

If Ω c = Ω, the parameters α,β and the boundary condition H lim are not compatible so we assume Ω c = Ω. Without loss of generality we assume H lim = H(L x ) and α ≥ 0. We also suppose here ) that is equivalent to (3.18) and is given by (3.17). Over the interval [0,x c ], the solution will be continuous and hence

z b ∈ C 1 (Ω) with a unique x c ∈ • Ω=]0,Lx[ such that z b,c = max x∈Ω z b (x), ( 3
α 3 β 2 2sin 2 (βH) + gα(H + z b ) x=xc x=0 = 0.
Over the interval [x c ,L x ], we have

α 3 β 2 2sin 2 (βH) + gα(H + z b ) x=xc x=Lx = α(F α,β,c + gz b,c -F α,β (H lim ) -gz b,
α 2 β 2 2sin 2 (βH c ) + g(H c + z b,c ) = α 2 β 2 2sin 2 (βH s,l ) + g(H s,l + z b,s ), (3.25) α 2 β 2 2sin 2 (βH lim ) + g(H lim + z b,lim ) = α 2 β 2 2sin 2 (βH s,r ) + g(H s,r + z b,s ), (3.26)
where z b,s is the bottom topography at the abscissa of the shock and H s,l , H s,r are the water depth at the left and right of the shock. If the solutions of the system (3.24)-(3.26) are continuous, this means the outflow boundary condition can not be satisfied. In this case, the torrential regime takes place from x = x c to x = L x .

In practice, the solution H(x) is obtained as follows. Starting from the value H lim at abscissa x lim , we calculate the solution of the algebraic equation (3.14) or equivalently the ODE (3.15). If no singularity appears, the problem is solved. If for one abscissa, Eq. (3.20) can not be solved then x c has to be determined by (3.22),(3.23). Then (3.24)-(3.26) allow to build the whole solution over Ω.

An illustration of the analytical solutions obtained in the situations with and without shocks is given in paragraphs 5.1 and 5.2.

Solutions with external forcing

We consider the same hydrostatic Euler system with free surface as previously but we add an external forcing and a viscosity term (in the vertical direction). The governing equations of the considered system are Even if the situations encountered in this paragraph -with an equilibrium between the external forcing and the viscosity -are different from those of the preceding paragraph, the resulting water depth and velocity fields are similar and we do not present any illustration for this situation.

∂ ∂x η z b u dz = 0, ( 3

Coupling with a reactive tracer

We consider in this paragraph the stationary hydrostatic Euler system governed by (3.1)-(3.2) with p a = Cst and coupled with an advection-reaction type equation of the form

u ∂T ∂x + w ∂T ∂z = T 0 -T τ . (3.30)
The equation (3.30) models for example the transport and the reaction of a tracer. In order to build solutions for the system (3.1)-(3.2) and (3.30) we use the analytical solutions proposed in paragraph 3.2. Thus, since the only remaining constraint for the choice of functions T , T 0 and τ is to satisfy (3.30), there exists a lot of solution for this problem. We present hereafter two distinct choices.

Let x → H 0 (x) be a continuous, non negative function defined on Ω and we choose the bottom topography z b under the form defined by (3.8).

First choice

Suppose T 0 = T 0 (x) and τ = τ (x) are defined by τ = sin(βH 0 (x)) 2 αβ 2 cos(bH 0 (x) ∂H0 ∂x , and

T 0 = αb γ(z -z b (x)) sin(bH 0 (x)) , with γ = γ(x) = 1 cotan(x)-csc(x) .
Then the velocity fields u α,β , w α,β defined by (3.9) and (3.10) for H = H 0 and T defined by

T (x,z) = -γ(z -z b (x))u α,β (x,z),
are solutions of (3.1)-(3.2) coupled with (3.30).

Second choice

Similarly, the velocity fields u α,β , w α,β defined by (3.9) and (3.10) for H = H 0 and T defined by

T (x,z) = e -(H(x)-(z-z b (x)))
with T 0 = 0 and

1 τ = αβ cos(β(z -z b )) sin(βH) 1 - tan(β(z -z b )) tan(βH) ∂H ∂x ,
are solutions of (3.1)-(3.2) coupled with (3.30).

Transient analytical solutions

The stationary solutions presented in the preceding paragraphs allow to validate spatial discretization techniques for the considered equations. In order to evaluate the accuracy of time schemes, transient solutions are necessary. In this paragraph we give two examples of time dependent analytical solutions.

First situation

We now consider the hydrostatic Navier-Stokes system defined by

∂ ∂x η z b u dz = 0, (3.31) w = - ∂ ∂x z z b u dz, (3.32) ∂u ∂t + u ∂u ∂x + w ∂u ∂z + g ∂η ∂x = - ∂p a ∂x + ν ∂ 2 u ∂z 2 , (3.33)
where p = p a is the atmospheric pressure at the free surface. We assume the velocity fields u,w can vary w.r.t. time t whereas the water depth H = H(x) only depends on the spatial coordinate x.

As in paragraph 3.2, let x → H 0 (x) be a continuous, non negative function defined on Ω, we choose the bottom topography z b under the form (3.8) and the external pressure p a is given by

p a = - α 2 β 2 2sin 2 (βH 0 ) 1 -e -νβ 2 t .
Then the velocity fields

ũα,β = αβ sin(βH 0 ) cos(β(z -z b ))e -νβ 2 t , wα,β = αβe -νβ 2 t sin 2 (βH 0 (x)) sin(β(z -z b (x)))cos(βH 0 (x)) ∂H 0 ∂x + cos(β(z -z b (x))) ∂z b ∂x sin(βH 0 (x))
and the water depth H 0 satisfy the equations (3.31)- (3.33). This situation is very similar to what has been proposed in paragraph 3.2, the contribution of the horizontal and vertical velocities decreasing exponentially is replaced by the effects of the external pressure p a . These analytical solutions are not illustrated.

The system (3.31)-(3.33) can admit discontinuous solutions but this would imply a discontinuous atmospheric pressure p a that has no physical meaning.

Analytical solution with a travelling wave

In this paragraph, we consider that the bottom topography z b = z b (x,t) can vary with respect to time and p a = Cst. Let us consider the system

∂H ∂t + ∂ ∂x η z b u dz = 0, (3.34) w = - ∂ ∂x z z b u dz, (3.35) ∂u ∂t + u ∂u ∂x + w ∂u ∂z + g ∂η ∂x = f x , (3.36)
with f x given by

f x = - ∂z b ∂t ∂u ∂z .
The energy balance associated with this system is

∂ ∂t η z b E dz + ∂ ∂x η z b u E + g(η -z) + gz dz = - 1 2 ∂z b ∂t u 2 s -u 2 b , with E = u 2 2 + gz. If H 0 (x)
is a non negative quantity such that the bottom topography is given by (3.8) then H(x,t) = H 0 (xc 0 t), u α,β (x,z,t) given by

u α,β (x,z,t) = c 0 + αβ sin(βH) cos(β(z -z b )),
and w obtained from (3.10) are solutions of (3.34)-(3.36).

Interactions with structures

We are not only interested in situations of free surface flows, so we consider now situations without free surface or where only a part of the water surface is free.

2d analytical solutions with partially free surface

Such a situation can be obtained when the atmospheric pressure term p a depends on the flow characteristics H,u,w. This corresponds to the case where the flow interacts with a lid. In such a situation, the closure of the system is ensured with a lid displacement law of the form

p a = K(x) √ η - √ η m √ η m + , (3.37)
with η = H + z b , η m = η m (x) a given function and K(.) a real nonnegative function. Such an expression for p a mimics the interaction between a deformable lid and the free surface. For more informations about the origins and the mechanical justifications of formula (3.37), the reader can refer to [START_REF] Pullan | An anatomically based model of transient coronary blood flow in the heart[END_REF]. Thus, if the water depth H satisfies the ODE

∂ ∂x α 2 β 2 2sin 2 (βH) + gH -K(x) √ η - √ η m √ η m 1 H+z b ≥ηm = -g ∂z b ∂x , (3.38)
then H and u α,β ,w α,β given by (3.9) and (3.10) are solutions of the considered problem. In particular, if H 0 (x) is any non-negative function then H = ηz b with z b given by (3.8) and

η = K(x)(H 0 + z b ) α 2 β 2 2sin 2 (βH0) + g(H 0 + z b ) + K 2 ,
satisfies (3.38).

Analytical solutions in cylindrical coordinates

We now consider the cylindrical representation of the incompressible Euler equations for an axisymmetric flow with the assumption of no tangential velocity. The gravity is also neglected. We denote u x and u r the velocities along the x axis and radial coordinate respectively. The continuity equation reads

1 r ∂ru r ∂r + ∂u x ∂x = 0, (3.39)
and the momentum equations reduces to

u r ∂u x ∂r + u x ∂u x ∂x + ∂p ∂x = 0, (3.40) ∂p ∂r = 0. (3.41)
Suppose the flow takes place in a deformable pipe parallel to the x axis and of radius R(x). The kinematic boundary condition at r = R(x) ensuring the fluid does not leave the pipe is given by Let us consider the velocity field u x given by u x = αβ cosh(βr 2 ) sinh(βR(x) 2 ) , then all the solutions R(x) of

∂R ∂t + ∂R ∂x u x | r=R -u r | r=R = 0. ( 3
∂ ∂x α 2 β 2 2sinh 2 (βR(x) 2 ) + K(x)(R(x) γ -R m (x) γ ) = 0, (3.45)
are solutions of the system (3.39)-(3.44). In particular if R 0 (x) is any non-negative function then R(x) defined by

R(x) = α 2 β 2 2K(x)sinh 2 (βR 0 (x) 2 ) + R 0 (x) γ 1 γ
, is solution. It has to be noticed that whatever the choices for the positive functions K(x) and R m (x), all the solutions of (3.45) are necessarily continuous and no shock can occur. An illustration of the situations proposed here is given in paragraph 5.3.1.

Analytical solutions in 3d

In this paragraph, we propose a 3d extension of the analytical solutions obtained in paragraph 3.2.

From (2.8)-(2.11), stationary solutions of the 3d Euler system with free surface have to fulfill

∂ ∂x η z b u dz + ∂ ∂y η z b v dz = 0, (4.1)
u ∂u ∂x + v ∂u ∂y + w ∂u ∂z + g ∂η ∂x = ∂p a ∂x , (4.2)
w = - ∂ ∂x z z b u dz - ∂ ∂y z z b v dz, (4.4)
where we have considered p a = 0. Let us introduce the velocity fields

u α1,β = α 1 β sin(βH) cos(β(z -z b )), (4.5)
v α2,β = α 2 β sin(βH) cos(β(z -z b )), (4.6)
then we clearly have Using (4.4), we obtain

w α1,α2,β = β sin 2 (βH) sin(β(z -z b ))cos(βH) α 1 ∂H ∂x + α 2 ∂H ∂y +cos(β(z -z b ))sin(βH) α 1 ∂z b ∂x + α 2 ∂z b ∂y , (4.7)
and the momentum equations along x and y (4.

∂ ∂x gH + α 2 1 β 2 2sin 2 (βH) + ∂ ∂y α 1 α 2 β 2 2sin 2 (βH) = -g ∂z b ∂x , (4.8)
∂ ∂x

α 1 α 2 β 2 2sin 2 (βH) + ∂ ∂y gH + α 2 2 β 2 2sin 2 (βH) = -g ∂z b ∂y , (4.9)
where H is the unknown. Therefore a necessary condition for the system (4.8)-(4.9) to admit a solution, is the water depth H being of the form

H(x,y) = H 0 (α 1 x + α 2 y), or H(x,y) = H 0 (α 2 x -α 1 y).
Only the second choice leads to non trivial solutions and thus for any non negative function H 0 (x), the bottom topography given by

z b (x,y) = z b -H 0 (α 1 x + α 2 y) - (α 2 1 + α 2 2 )β 2 2g sin 2 (βH 0 (α 1 x + α 2 y)) , (4.10)
and the velocity fields u, v and w respectively given by (4.5), (4.6) and (4.7) are solutions of (4.1)-(4.4). These 3d analytical solutions are illustrated in paragraph 5.4.

Illustrative examples

For some of the models investigated above, illustrations of the exhibited analytical solutions are given. In the following except for paragraphs 5.3.2 and 5.4, we always consider the situation of a L x = 20 meter-long basin with constant width and vertical shores. Since the initial and boundary conditions do not depend on the variable y and the bottom geometry satisfies z b = z b (x), it is a 2d (xz) problem.

Stationary continuous solutions

In this paragraph we present typical analytical solutions for the problem examined in paragraph 3.2. Let us consider the bottom geometry z b (x) is defined by (3.8) where H 0 (x) is given by

H 0 (x) = 1 2 + π 2 + α 10 tan -1 N L x x - L x 3 -e -N L 2 x (x-3 4Lx ) 2
, with N = 50, α = 3.5 and β = 1. The flow is fixed at inflow (left boundary) with q in = α and the water height is prescribed at outflow (right boundary) with H out = H 0 (L x ).

Following the results of paragraph 3.2, the water depth H = H 0 and the velocity fields u,w given by (3.9),(3.10) are solutions of the considered problem. This situation where no shock occurs, is illustrated by Fig.

Stationary solutions with a shock

Now we consider the situation decribed in paragraph 3.4. The bottom geometry is given by z b (x) = 0.2e -0.5(x-8) 2 -0.4e -0.5(x-12) 2 .

( We use the topography given by (5.1). As for Fig. 5.2, we choose α = 0.65 and β = 2.5. The external pressure p a is given by (3.37) with K(x) = 100x and η m (x) = min(1.2,0.2 + 0.04(x -14) 2 ).

Analytical solutions in cylindrical coordinates

Analytical solutions in 3d

In this paragraph we give an example of the 3d analytic solutions obtained in Section 4.

In this 3d situation, the water depth is given by H(x,y) = H 0 (α 1 x + α 2 y) = 1 + 0.3sin(β(α 1 x + α 2 y)),

with α 1 = 1,α 2 = 2 and β = 1. The bottom geometry is given by (4.10). The bottom profile with 3090 vertices and 5978 triangles is depicted in Fig. 5.5. The corresponding surface level and velocity fields u, v and w have been calculated from (4.5)-(4.7) and are presented in Fig. (5.6).

Conclusion

We have proposed analytical solutions for the 2d and 3d incompressible and hydrostatic Euler system. The FRESH-ASSESS solutions have been obtained in various situations: free surface, partially free surface, shocks, transient regime,. . . and can be derived for a large class of geometrical domains.

The main interest of these analytical solutions is to allow the validation of numerical schemes. The authors have already checked that the numerical schemes they have proposed [START_REF] Audusse | Approximation of the hydrostatic Navier-Stokes system for density stratified flows by a multilayer model. Kinetic interpretation and numerical validation[END_REF][START_REF] Audusse | A multilayer Saint-Venant system with mass exchanges for Shallow Water flows. Derivation and numerical validation[END_REF] for the resolution of the incompressible and hydrostatic Euler system with free surface are able to recover the analytical solutions proposed here.

Fig. 2 . 1 .

 21 Fig. 2.1. Notations: water height H(x,y,t), free surface η(x,y,t) and bottom z b (x,y).

. 10 )

 10 satisfy the 2d stationary Euler equations (3.1)-(3.3) with η = H 0 + z b . Proof. Following (3.5),(3.7), we propose to choose u under the form

  .22) ∂z b ∂x x=xc = 0. (3.23) If Ω c = ∅ then the problem (3.19) has no solution at least for x = x c ∈ Ω c and the solution will be discontinuous. From (3.23), for x = x c the solution H of g -α 2 β 3 cos(βH) sin(βH) with g -α 2 β 3 cos(βH) sin(βH) 3 = 0, would give that a continuous solution exists over Ω that is contradictory with the assumption x c ∈ Ω c . So H c = H(x c ) necessarily satisfies (3.16

  .42) Equation (3.41) gives p = p(x) and as in paragraph 3.8.1, the system (3.39)-(3.42) is closed with a wall displacement law of the formp(x) = K(x)(R(x) γ -R m (x) γ ),where K(.) and R m (.) are two non negative given functions and γ is a positive constant.Using (3.39) and (3.42), the mass conservation in the pipe readsR(x) 0 ru x dr = Cst,(3.43)and the radial velocity u r is given by

u

  α1,β dz = α 1 , and η z b v α2,β dz = α 2 .

  5.1.

Fig. 5 . 1 .

 51 Fig. 5.1. Surface level of the continuous solution and (a) horizontal velocity, (b) vertical velocity, (c) velocity norm, (d) velocity field (arrow).

. 1 )

 1 With this geometry and α = 0.65, β = 2.5 and H(L x ) = 0.6, the solution is discontinuous (cf.(3.19)) and governed by (3.20)-(3.21). The shock is characterized by (3.24)-(3.26) and the vertical velocity w is obtained from(3.10). The obtained solution is depicted in Fig. 5.2. 5.3. Interactions with structures 5.3.1. Analytical solutions with partially free surface The analytical solutions proposed in paragraph 3.8.1 are illustrated here in Fig. 5.3.

- L x 2 2 .Fig. 5 . 2 .

 252 Fig. 5.2. Surface level of the solution with shock and (a) horizontal velocity, (b) vertical velocity and (c) velocity norm, (d) velocity field (arrow).

Fig. 5 . 3 .

 53 Fig. 5.3. Surface level of the solution with partially free surface (a) horizontal velocity u, (b) vertical velocity w and (c) velocity norm (with arrows).

Fig. 5 . 4 .

 54 Fig. 5.4. Deformation of the cylinder and the velocity field (a) velocity norm, (b) slice along z with the velocity ux.

Fig. 5 . 5 .

 55 Fig. 5.5. 3d bottom geometry.

Fig. 5 . 6 .

 56 Fig. 5.6. Surface level of the 3d solution and (a) and (b) horizontal velocities u and v and (c) vertical velocity w.

  is a volumetric source term and the atmospheric pressure p a is supposed to be constant. The following proposition holds Proposition 3.3. If we choose f x = βνu and β ≥ 0 -that corresponds to an energy supply for the system -then for any bottom profile z b of the form(3.8), the quantities u = u α,β , w = w α,β given by (3.9) and (3.10) with H = H 0 are solutions of (3.27)-(3.29).

										.27)
	∂u ∂x	+	∂w ∂z	= 0,			(3.28)
	u	∂u ∂x	+ w	∂u ∂z	+ g	∂η ∂x	= f x + ν	∂ 2 u ∂z 2 ,	(3.29)
	where f x

Acknowledgement. The authors want to thank Emmanuel Audusse and Benoît Perthame for helpful discussions that have allowed to greatly improve the paper.


Diego Chamorro

Pierre-Gilles Lemari É-Rieusset

Real Interpolation method, Lorentz spaces and refined Sobolev inequalities

Keywords: Lorentz spaces, refined Sobolev inequalities, real interpolation spaces, Besov spaces

In this article we give a straightforward proof of refined inequalities between Lorentz spaces and Besov spaces and we generalize previous results of H. Bahouri and A. Cohen [2]. Our approach is based in the characterization of Lorentz spaces as real interpolation spaces. We will also study the sharpness and optimality of these inequalities.

Introduction

This paper is a generalization of recent results by H. Bahouri and A. Cohen [START_REF] Bahouri | Refined Sobolev inequalities in Lorentz spaces[END_REF] on Lorentz spaces and refined Sobolev inequalities. Let us recall the setting: if q > 1, the classical Sobolev inequalities f L p ≤ C f Ẇ s,q with 1/p = 1/q -s/n, had been refined by P. Gérard, Y. Meyer and F. Oru [START_REF] Gérard | Inégalités de Sobolev précisées, Équations aux Dérivées Partielles[END_REF] using a Besov space in the right-hand side of the inequality:

f L p ≤ C f q/p Ẇ s,q f 1-q/p Ḃs-n/q,∞ ∞ . (1)
Similarly the inequality f L p ≤ C f Ḃs,q q for q ≥ 1 may be refined by

f L p ≤ C f q/p Ḃs,q q f 1-q/p Ḃs-n/q,∞ ∞ . (2)
In the previous formula, and for all the following theorems and inequalities, since α, β > 0, we will say that f ∈ Ḃα,p0 are bounded: this way we will choose a natural representation in such spaces.

In [START_REF] Bahouri | Refined Sobolev inequalities in Lorentz spaces[END_REF], H. Bahouri and A. Cohen show that it is possible to improve the estimate f L p,q ≤ C f Ḃs,q q where L p,q is a Lorentz space, into the following inequality f L p,q ≤ C f q/p Ḃs,q q f 1-q/p Ḃs-n/q,∞ q

and they prove that this estimate is sharp since it is not possible to replace the norm Ḃs-n/q,∞ q above by a weaker Besov norm Ḃs-n/q,∞ r with r > q. They also ask the following question: it is possible to improve the inequality f L p,r ≤ C f Ḃs,q q into f L p,r ≤ C f q/p Ḃs,q r f 1-q/p Ḃs-n/q,∞ r when r = q?

The proof they gave for (3) involves difficult estimations of Lorentz norms and could not easily be extended to the case r = q.

In this article we are going to provide an elementary proof of (3), show that the inequality ( 4) is valid and we will also study a wider family of related inequalities. The main idea is, in the spirit of [START_REF] Lemarié-Rieusset | Recent developments in the Navier-Stokes problem[END_REF], to make a systematic use of the characterization of Lorentz spaces as interpolation spaces and to never use the traditional definition of Lorentz spaces as Orlicz spaces.

Our first theorem deals with the two first inequalities given above in a quite general form. As we shall see in the proof given in section 3, this is just a variant of Hedberg's inequality [START_REF] Hedberg | On certain convolution inequalities[END_REF].

Theorem 1 Let α, β > 0 and q 0 , q 1 ∈ [1 + ∞]. Let θ = α
α+β ∈ (0, 1) and let1 p = 1-θ q0 + θ q1 . Then there exists a constant C 0 such that, for every

f ∈ Ḟ α,q0 ∞ ∩ Ḟ -β,q1 ∞ (R n ), we have |f (x)| ≤ C 0 sup j∈Z 2 jα |∆ j f (x)| 1-θ sup j∈Z 2 -jβ |∆ j f (x)| θ (5)
In particular, we get :

f L p ≤ C 0 f 1-θ Ḟ α,q 0 ∞ f θ Ḟ -β,q 1 ∞ (6)
Let us show briefly how to obtain the two first inequalities from this theorem. We observe that, for 0 < s < n/q and for 1 < q < +∞, we have Ẇ s,q = Ḟ s,q

2 ⊂ Ḟ s,q ∞ ⊂ Ḟ s-n q ,∞ ∞
. Thus, if we set α = s > 0, -β = s -n/q < 0, q 0 = q and q 1 = +∞, we deduce from (6) the inequality (1). In the same way, for 0 < s < n/q and 1 ≤ q ≤ +∞, we have Ḃs,q

q = Ḟ s,q q ⊂ Ḟ s,q ∞ ⊂ Ḟ s-n q ,∞ ∞
and we get inequality (2) from inequality [START_REF] Frazier | A discrete transform and decompositions of distribution spaces[END_REF].

Note also that we obtain by similar arguments the following useful inequality:

f L p ≤ C 0 f 1-θ Ḃα,q 0 q 0 f θ Ḃ-β,q 1 q 1 (7) with 1 p = 1-θ q0 + θ q1 and θ = α α+β ∈]0, 1[.
Our next result studies inequalities (3) and ( 4). The proof of this theorem reduces to a few lines once we have in mind the characterization of Lorentz spaces as interpolation spaces.

Theorem 2 Let α, β > 0, q 0 , q 1 ∈ [1 + ∞] with q 0 = q 1 . Let θ = α α+β ∈]0, 1[, 1 p = 1-θ q0 + θ q1 and let r ∈ [1, +∞]. If f ∈ Ḃα,q0 r ∩ Ḃ-β,q1 r (R n ) then f ∈ L p,r (R n ) and we have f L p,r ≤ C 0 f 1-θ Ḃα,q 0 r f θ Ḃ-β,q 1 r . (8)
In the expression above we have the same index r in Lorentz and Besov spaces, so the the next step is to deal with general Besov spaces Ḃα,q0 r0 and Ḃ-β,q1 r1 and to try to control a Lorentz norm in L p,r . For this we define θ = α α+β ∈]0, 1[ and we investigate the validity of the inequality

f L p,r ≤ C 0 f 1-θ Ḃα,q 0 r 0 f θ Ḃ-β,q 1 r 1 . (9)
A scaling argument 1 gives us that necessarily we have 1 p = 1-θ q0 + θ q1 . For the index r, we will have a similar condition as it is explained in the next theorem.

It is important to observe that inequality (9) can be studied from the point of view of interpolation theory by the following equivalent problem: Ḃα,q0 r0 , Ḃ-β,q1 r1 θ,1 ⊂ L p,r . However the interpolation between these spaces with these parameters is a delicate issue as it is explained in [START_REF] Krepkogorskii | Interpolation in Lizorkin-Triebel and Besov spaces[END_REF] and, to the best of our knowledge, it was not treated before.

The next result studies the validity of inequality [START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF] in some particular cases.

Theorem 3 Let α, β, r * > 0 and θ = α α+β with 1 p = 1-θ q0 + θ q1 and 1 r * = 1-θ r0 + θ r1 . 1) For f ∈ Ḃα,q0 r0 ∩ Ḃ-β,q1 r1 (R n ) and if r > r * , we have f ∈ L p,r (R n ) and f L p,r ≤ C f 1-θ Ḃα,q 0 r 0 f θ Ḃ-β,q 1 r 1
.

2) Moreover, this inequality is valid for r = r * in the following cases : a) r = r 0 = r 1 , b) r 0 = q 0 and r 1 = q 1 , c) 1 < p ≤ 2 and r * = p.

3) Finally, the condition r ≥ r * is sharp.

As we shall see in section 3, theorems 2 & 3 are obtained by direct interpolation. However, it is possible to go one step further with the following results:

Theorem 4 Let α, β > 0, q 0 , q 1 ∈ [1 + ∞], q 0 < q 1 . Let θ = α α+β ∈]0, 1[ and let 1 p = 1-θ q0 + θ q1 . Let q 0 ≤ r 0 ≤ r 1 ≤ q 1 and let 1 r = 1-θ r0 + θ r1 . Then we have f L p,r ≤ C 0 f 1-θ Ḃα,q 0 r 0 f θ Ḃ-β,q 1 r 1
.

As we shall see, theorem 4 can be obtained by the use of inequalities ( 7) and ( 8) following the ideas given in theorem 2.

Note now that in all these inequalities we have 1 p = 1-θ q0 + θ q1 for 1 ≤ q 0 < q 1 ≤ +∞ and a similar condition for r, r 0 and r 1 : namely 1 r = 1-θ r0 + θ r1 . We also assumed the following relationship between these parameters:

q 0 ≤ r 0 ≤ r 1 ≤ q 1 .
A more general result is given with the next theorem:

Theorem 5 Let α, β > 0, q 0 , q 1 ∈ [1 + ∞], q 0 = q 1 . Let θ = α α+β ∈]0, 1[ and let 1 p = 1-θ q0 + θ q1 . If p ≤ 2, we define I = (x, y) ∈ [0, 1] 2 / x(1 -θ) + θy = 1 p = [(x 0 , y 0 ), (x 1 , y 1 )]
with x 0 < y 0 and x 1 < y 1 , then if r 0 and r 1 satisfy

x 0 ≤ 1 r0 ≤ 1 r1 ≤ y 0 or y 1 ≤ 1 r1 ≤ 1 r0 ≤ x 1 , and if 1 r = 1-θ r0 + θ r1 then we have f L p,r ≤ C 0 f 1-θ Ḃα,q 0 r 0 f θ Ḃ-β,q 1 r 1 .
Finally, all these inequalities are sharp in the sense that it is not possible to remove the condition 1 r = 1-θ r0 + θ r1 .

The plan of the paper is the following: in section 2 we recall some facts about Lorentz and Besov spaces and we will pay a special attention to the interpolation definition of Lorentz spaces. In section 3 we will give the proofs and finally, in section 4 we will treat the part 3) of theorem 3 where we will study the sharpness of these inequalities.

Functional spaces and real interpolation method

For Besov and Triebel-Lizorkin spaces we will use the characterization based on a Littlewood-Paley decomposition. We start with a nonnegative function ϕ ∈ D(R n ) such that ϕ(ξ) = 1 over |ξ| ≤ 1/2 and ϕ(ξ) = 0 if |ξ| ≥ 1. Let ψ be defined as ψ(ξ) = ϕ(ξ/2) -ϕ(ξ). We define the operators S j and ∆ j in the Fourier level by the formulas

S j f (ξ) = ϕ(ξ) f (ξ) and ∆ j f (ξ) = ψ(ξ) f (ξ). The distribution ∆ j f is called the j-th dyadic block of the Littlewood-Paley decomposition of f . If lim j→-∞ S j f = 0 in S ′ (R n ), then the equality f = j∈Z ∆ j f is called
the homogeneous Littlewood-Paley decomposition of f . Definition 2.1 For 1 ≤ p, q ≤ +∞ and s ∈ R, we define the homogeneous Besov spaces as the set of distributions Ḃs,p

q (R n ) = f ∈ S ′ (R n ) : f Ḃs,p q < +∞ , where f Ḃs,p q =   j∈Z 2 jsq ∆ j f q L p   1/q
with the usual modifications when q = +∞. For 1 < p < +∞, 1 ≤ q ≤ +∞ and s ∈ R, we define in the same way the homogeneous Triebel-Lizorkin spaces by

Ḟ s,p q (R n ) = f ∈ S ′ (R n ) : f Ḟ s,p q < +∞ with f Ḟ s,p q =   j∈Z 2 jsq |∆ j f | q   1/q L p
with the usual modifications when q = +∞.

Note that the quantities • Ḃs,p q and • Ḟ s,p q are only semi-norms since for j ∈ Z we have ∆ j P = 0 for all polynomials P .

We turn now to Lorentz spaces which are a generalization of the Lebesgue spaces. For (X, µ) a measurable space, they are usually defined in terms of the distribution and rearrangement functions d f (t) and f * (s) given by the formulas

d f (t) = µ({x : |f (x)| ≥ t}) and f * (s) = inf{t : d f (t) ≤ s},
where µ(A) denotes the measure of a set A. Then for 1 ≤ p < +∞ and 1 ≤ r ≤ +∞, the Lorentz spaces L p,r (X, µ) are traditionally defined in the following way

L p,r (X, µ) = f : X -→ R : f L p,r < +∞ where f L p,r = +∞ 0 s 1 p f * (s) r ds s 1/r , (10)
with the usual modifications when r = +∞, which corresponds to the weak-L p spaces. With this characterization is not complicated to see that we have L p,p = L p and that for r 0 < r 1 we have the embedding L p,r0 ⊂ L p,r1 . However, the previous formula is not very useful since it depends on the rearrangement function f * and we will use a more helpful characterization which is given in the lines below.

We recall now some classical results from interpolation theory concerning the real interpolation method. See [START_REF] Bergh | Interpolation spaces. An introduction[END_REF] for a detailed treatment. If A 0 and A 1 are two Banach spaces which are continuously embedded into a common topological vector space V , if 0 < θ < 1 and 1 ≤ r ≤ +∞, then the real interpolation space [A 0 , A 1 ] θ,r may be defined in the following way: f ∈ [A 0 , A 1 ] θ,r if and only if f ∈ V and f can be written in

V as f = j∈Z f j , with f j ∈ A 0 ∩ A 1 and (2 -jθ f j A0 ) j∈Z ∈ ℓ r , (2 j(1-θ) f j A1 ) j∈Z ∈ ℓ r . This space is normed with f [A0,A1] θ,r = inf f = fj j∈Z 2 -jθr f j r A0 1/r + j∈Z 2 j(1-θ)r f j r A1 1/r (11)
For f = j∈Z f j and ρ > 0, with ρ = 1, we have the following inequality that will be very helpful in the sequel:

f [A0,A1] θ,r ≤ C ρ,θ,r j∈Z ρ -jθr f j r A0 (1-θ)/r j∈Z ρ j(1-θ)r f j r A1 θ/r . ( 12
)
An important property of the real interpolation method is the reiteration theorem:

Proposition 2.1 1) If θ 0 = θ 1 , we have [A 0 , A 1 ] θ0,r0 , [A 0 , A 1 ] θ1,r1 θ,r = [A 0 , A 1 ] (1-θ)θ0+θθ1,r . (13)
2) If θ 0 = θ 1 , ( 13) is still valid if 1 r = 1-θ r0 + θ r1 . We saw with the expression [START_REF] Krepkogorskii | Interpolation in Lizorkin-Triebel and Besov spaces[END_REF] how to define Lorentz spaces L p,r (X, µ) for 1 < p < +∞, 1 ≤ r ≤ +∞ as an Orlicz space. However, it will be simpler to use their characterization as real interpolates of Lebesgue spaces [START_REF] Bergh | Interpolation spaces. An introduction[END_REF]: Proposition 2.2 (Lorentz spaces as interpolation spaces)

1) For 1 < p < +∞, 1 ≤ r ≤ +∞ L p,r = [L 1 , L ∞ ] θ,r with θ = 1 - 1 p . (14)
2) For p 0 = p 1 , we have

[L p0 , L p1 ] θ,r = [L p0,r0 , L p1,r1 ] θ,r = L p,r with 1 p = 1 -θ p 0 + θ p 1 . ( 15
)
3) In the case p 0 = p 1 = p we have

[L p,r0 , L p,r1 ] θ,r = L p,r if 1 r = 1 -θ r 0 + θ r 1 . (16)
Of course, ( 15) and ( 16) are consequences of ( 14) through the reiteration theorem. In this paper, we shall use decompositions [START_REF] Lemarié-Rieusset | Recent developments in the Navier-Stokes problem[END_REF] and estimates ( 12) when we deal with functions in Lorentz spaces, and we will mainly consider the cases (X, µ) = (R n , λ) where λ is the Lebesgue measure, or (X, µ) = (Z, µ) with µ the counting measure.

In the case of Lorentz spaces, we can use decomposition [START_REF] Lemarié-Rieusset | Recent developments in the Navier-Stokes problem[END_REF] with an useful extra property (see [START_REF] Lemarié-Rieusset | Recent developments in the Navier-Stokes problem[END_REF]) :

Lemma 2.1 Let 1 < p < +∞, 1 ≤ r ≤ +∞.
Then there exists a constant C 0 such that every f ∈ L p,r (X, µ) can be decomposed as f = j∈Z f j where

• (2 -j(p-1)/p f j L 1 ) ℓ r + (2 j/p f j L ∞ ) ℓ r ≤ C 0 f L p,r • the f j have disjoint supports : if j = k, f j f k = 0.
Inequality ( 12) is very useful to provide an upper bound for the Lorentz norm of f and it will be systematically used here. In order to get a lower bound, we shall use the following duality result :

Lemma 2.2 Let 1 < p < +∞, 1 ≤ r ≤ +∞.
Then there exists a constant C 0 such that for every f ∈ L p,r (X, µ) and every g ∈ L p p-1 , r r-1 (X, µ), we have f g ∈ L 1 (X, µ) and

f g dµ ≤ C 0 f L p,r g L p p-1 , r r-1 .
Remark 2.1 As we shall see in the proofs given in the section below, the characterization of Lorentz and Besov spaces based on real interpolation is a useful tool since the problem we are dealing with can be studied in terms of weighted sequences. See [START_REF] Bergh | Interpolation spaces. An introduction[END_REF], [START_REF] Devore | Interpolation of Besov spaces[END_REF] or [START_REF] Krepkogorskii | Interpolation in Lizorkin-Triebel and Besov spaces[END_REF] for more details concerning the interpolation of Besov spaces.

3 Refined inequalities: the proofs Proof of theorem 1. We just write

A α (x) = sup j∈Z 2 jα |∆ j (x)| and A β (x) = sup j∈Z 2 -jβ |∆ j (x)| to obtain |f (x)| ≤ j∈Z |∆ j f (x)| ≤ j∈Z min 2 -jα A α (x), 2 jβ A β (x) .
We define j 0 (x) as the largest index such that 2 jβ A β (x) ≤ 2 -jα A α (x) and we write

|f (x)| ≤ j≤j0(x) 2 jβ A β (x) + j>j0(x) 2 -jα A α (x) ≤ CA α (x) β α+β A β (x) α α+β , (17)
thus, inequality (5) is proved. In order to obtain [START_REF] Frazier | A discrete transform and decompositions of distribution spaces[END_REF], it is enough to apply Hölder inequality in the expression above since we have θ = α α+β and 1 p = 1-θ q0 + θ q1 .

Remark 3.1 Inequality (17) is a little more precise than Hedberg's inequality [START_REF] Adams | Function spaces and potential theory[END_REF][START_REF] Hedberg | On certain convolution inequalities[END_REF] : if f = I α g(x) where I α is a Riesz potential2 with 0 < α < n and if g ∈ Ḃ-β,∞ ∞ , then, if M g is the Hardy-Littlewood maximal function of g, we have A α (x) ≤ CM g (x) and A β (x) ≤ C g Ḃ-β,∞ ∞ . Thus, we find easily the refined Sobolev inequality [START_REF] Adams | Function spaces and potential theory[END_REF]. See more details in [START_REF] Oru | Rôle des oscillations dans quelques problèmes d'analyse non linéaire[END_REF][START_REF] Chamorro | Improved Sobolev inequalities and Muckenhoupt weights on stratified Lie groups[END_REF].

Proof of theorem 2. We start picking p 0 and p 1 such that 1 ≤ q 0 < p 0 < p < p 1 < q 1 ≤ +∞ with 2 p = 1 p0 + 1 p1 . We have then 1 pi = 1-ai q0 + ai q1 with 0 < a i < 1 and i = 0, 1. We write

∆ j f L p i ≤ ∆ j f 1-ai L q 0 ∆ j f ai L q 1 = 2 jα ∆ j f L q 0 1-ai 2 -jβ ∆ j f L q 1 ai 2 j -α(1-ai)+βai .
Recalling that 1 p = 1-θ q0 + θ q1 and θ = α α+β we have -α(1 -a 0 ) + βa 0 = α(1 -a 1 ) -βa 1 . Thus, noting ρ = 2 -2[α(1-a0)-βa0] > 0 and using the Hölder inequality we obtain

j∈Z ρ -jr/2 ∆ j f r L p 0 ≤ f r(1-a0) Ḃα,q 0 r f ra0 Ḃ-β,q 1 r and j∈Z ρ jr/2 ∆ j f r L p 1 ≤ f r(1-a1) Ḃα,q 0 r f ra1 Ḃ-β,q 1 r .
From this, and applying proposition 2.2, we deduce that if f ∈ Ḃα,q0

r ∩ Ḃ-β,q1 r (R n ) then f ∈ [L p0 , L p1 ] 1 2
,r = L p,r . Furthermore, using inequality [START_REF] Lemarié-Rieusset | Ondelettes et bases hilbertiennes[END_REF] we finally have:

f L p,r ≤ C ρ,r f 1-θ Ḃα,q 0 r f θ Ḃ-β,q 1 r Proof of theorem 3.
1) Case r > r * : With no loss of generality, we may assume that q 0 < q 1 and we fix ε > 0 such that

1 q 1 < 1 p -ε( 1 q 0 - 1 q 1 ) = 1 p 1 < 1 p + ε( 1 q 0 - 1 q 1 ) = 1 p 0 < 1 q 0 .
The proof follows essentially the same ideas used in the previous theorem. Indeed, we have, for γ j = 2 jα ∆ j f L q 0 and η j = 2 -jβ ∆ j f L q 1 , and for ǫ 0 = 1 and ǫ 1 = -1,

∆ j f L p i ≤ ∆ j f 1-θ+ǫiε L q 0 ∆ j f θ-ǫiε L q 1 = γ 1-θ+ǫiε j η θ-ǫiε j 2 -jǫiε(α+β) .
As r 0 = r 1 , we can only say that (γ 1-θ+ǫiε

j η θ-ǫiε j ) j∈Z ∈ ℓ ρi where 1 ρi = 1-θ+ǫiε r0 + θ-ǫiε r1 .
We may use inequality ( 12), but we get only that f ∈ [L p0 , L p1 ] 1/2,ρ = L p,ρ with ρ = max(ρ 0 , ρ 1 ), and satisfies inequality (9) with r = ρ. However, we may choose ε as small as we want, and thus ρ as close to r * as we want; thus f satisfies [START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF] for every r > r * .

2) Case r = r * : a) if r = r 0 = r 1 : this case was treated in theorem 2. b) if r 0 = q 0 and r 1 = q 1 : This is a direct consequence of (6) since we have f Ḃα,q

i q i = f Ḟ α,q i q i ⊂ f Ḟ α,q i ∞ and f Ḃ-β,q i q i = f Ḟ -β,q i q i ⊂ f Ḟ -β,q i ∞
, we obtain [START_REF] Gérard | Inégalités de Sobolev précisées, Équations aux Dérivées Partielles[END_REF]. c) Case 1 < p ≤ 2 and r * = p: We just write

∆ j f L p ≤ ∆ j f 1-θ L q 0 ∆ j f θ L q 1 = (2 α ∆ j f L q 0 ) 1-θ (2 -jβ ∆ j f L q 1 ) θ
and get by Hölder inequality:

f Ḃ0,p p ≤ C f 1-θ Ḃα,q 0 r 0 f θ Ḃ-β,q 1 r 1 .
We then use the embedding Ḃ0,p p ⊂ L p = L p,p , which is valid for p ≤ 2.

Proof of theorem 4. We see how direct interpolation has given us theorem 3, but we only obtained partial results for theorems 4 and 5. Indeed, in theorem 4, we want a positive result for q 0 ≤ r 0 ≤ r 1 ≤ q 1 but thus far we have proven the result only for (r 0 , r 1 ) = (q 0 , q 1 ) and for q 0 ≤ r 0 = r 1 ≤ q 1 . To complete the proof of theorem 4, we must reiterate interpolations to those new estimates.

This will be done through the following lemma :

Lemma 3.1 Let α, β > 0, q 0 , q 1 ∈ [1 + ∞], q 0 < q 1 . Let θ = α α+β ∈]0, 1[ and let 1 p = 1-θ q0 + θ q1 . 1) If q 0 ≤ r 0 ≤ q 1 and let 1 r = 1-θ r0 + θ q1 , then we have f L p,r ≤ C 0 f 1-θ Ḃα,q 0 r 0 f θ Ḃ-β,q 1 q 1 . ( 18
)
2) If q 0 ≤ r 1 ≤ q 1 and let 1 r = 1-θ q0 + θ r1 , then:

f L p,r ≤ C 0 f 1-θ Ḃα,q 0 q 0 f θ Ḃ-β,q 1 r 1 . ( 19
)
Proof of the lemma 3.1. We only prove the first inequality, as the proof for the second one is similar. Since f ∈ Ḃα,q0 r0 , noting λ j = 2 jα ∆ j f L q 0 we have (λ j ) j∈Z ∈ ℓ r0 . Thus, using lemma 2.1 for the interpolation

ℓ r0 = [ℓ q0 , ℓ q1 ] η,r , (20)
with 1 r0 = 1-η q0 + η q1 , we see that we have a partition

Z = k∈Z Z k such that 2 -kη j∈Z k λ q0 j 1 q 0 ℓ r 0 + 2 k(1-η) j∈Z k λ q1 j 1 q 1 ℓ r 0 ≤ C λ j ℓ r 0 (21) Moreover since f ∈ Ḃ-β,q1 q1
we have

j∈Z k 2 -jβq1 ∆ j f q1 L q 1 1/q1 k∈Z ∈ ℓ q1 . Let us note α k = j∈Z k 2 -jβq1 ∆ j f q1 L q 1 1/q1 , β k = 2 -kη j∈Z k λ q0 j 1 q 0 , γ k = 2 k(1-η) j∈Z k λ q1 j ) 1 q 1 and f k = j∈Z k ∆ j f .
We apply now inequality [START_REF] Gérard | Inégalités de Sobolev précisées, Équations aux Dérivées Partielles[END_REF] and theorem 2 to obtain

f k L p ≤ C A f k 1-θ Ḃα,q 0 q 0 f k θ Ḃ-β,q 1 q 1 ≤ Cα θ k β 1-θ k 2 kη(1-θ) (22)
and

f k L p,q 1 ≤ C B f k 1-θ Ḃα,q 0 q 1 f k θ Ḃ-β,q 1 q 1 ≤ Cα θ k γ 1-θ k 2 -k(1-η)(1-θ) . (23)
Since we have f = k∈Z f k , with these two inequalities at hand, and using [START_REF] Lemarié-Rieusset | Ondelettes et bases hilbertiennes[END_REF], we find that f ∈ [L p , L p,q1 ] η,r with 1 r = 1-η p + η q1 . But, since 1 r0 = 1-η q0 + η q1 and 1 p = 1-θ q0 + θ q1 , we obtain [L p , L p,q1 ] η,r = L r with 1 r = 1-θ r0 + θ q1 .

Remark 3.2 Note that we use twice interpolation arguments: first in estimate (21) and then with inequalities ( 22) and (23) in order to obtain f ∈ [L p , L p,q1 ] η,r .

Once this lemma is proved, it is enough to reapply similar arguments to obtain theorem 4. Indeed, since we have q 0 < r 0 < r 1 < q 1 , we start using ℓ r0 = [ℓ q0 , ℓ r1 ] η,r0 instead of (20) and we obtain a partition

Z = k∈Z Z k such that 2 -kη j∈Z k λ q0 j 1 q 0 ℓ r 0 + 2 k(1-η) j∈Z k λ r1 j 1 r 1 ℓ r 0 ≤ C λ j ℓ r 0 with 1 r0 = 1-η q0 + η r1
and where the sequence (λ j ) j∈N with λ j = 2 jα ∆ j f L q 0 belongs to

ℓ r0 since f ∈ Ḃα,q0 r0 . Since f ∈ Ḃ-β,q1 r1 we have j∈Z k 2 -jβq1 ∆ j f q1 L q 1 1/q1
k∈Z ∈ ℓ q1 , and we note again

α k = j∈Z k 2 -jβq1 ∆ j f q1 L q 1 1/q1 , β k = 2 -kη j∈Z k λ q0 j 1 q 0 , γ k = 2 k(1-η) j∈Z k λ q1 j ) 1 q 1 and f k = j∈Z k ∆ j f
. Next, we only need to apply (19) and [START_REF] Hedberg | On certain convolution inequalities[END_REF] instead of ( 22) and (23) to obtain

f k L p,s ≤ C A f k 1-θ Ḃα,q 0 q 0 f k θ Ḃ-β,q 1 r 1 ≤ Cα θ k β 1-θ k 2 kη(1-θ)
where 1 s = 1-θ q0 + θ r1 , and

f k L p,r 1 ≤ C B f k 1-θ Ḃα,q 0 r 1 f k θ Ḃ-β,q 1 r 1 ≤ Cα θ k γ 1-θ
Finally, we have via inequality [START_REF] Lemarié-Rieusset | Ondelettes et bases hilbertiennes[END_REF] that f ∈ [L p,s , L p,r1 ] η,r with 1 r = 1-η s + η r1 . To conclude, we use the fact that 1 s = 1-θ q0 + θ r1 and 1 r0 = 1-η q0 + η r1 in order to obtain that f ∈ L p,r with 1 r = 1-θ r0 + θ r1 .

Proof of theorem 5. Similarly, in theorem 5, we want a positive result on the triangles 1/y 0 ≤ r 1 ≤ r 0 ≤ 1/x 0 and 1/x 1 ≤ r 0 ≤ r 1 ≤ 1/y 1 , and we have already obtained that the theorem is true for 1/y 0 ≤ r 1 = r 0 ≤ 1/x 0 and 1/x 1 ≤ r 0 = r 1 ≤ 1/y 1 as well as for (1/x 0 , 1/y 0 ) and (1/x 1 , 1/y 1 ). To complete the proof of theorem 5, we must reiterate interpolations to those new estimates. This will be achieved with the following lemma.

Lemma 3.2 Let Σ be the set of points (x, y) ∈ [0, 1] × [0, 1] such that, for z * = (1 -θ)x + θy, we have the inequality

f L p,1/z * ≤ C x,y f 1-θ Ḃα 0 ,q 0 1/x f θ Ḃ-β,q 1 1/y . and let A, B, C ∈ [0, 1] × [0, 1] such that [A, B] is horizontal (y A = y B ) and [A, C] is vertical (x A = x C
). Then :

• if A ∈ Σ and B ∈ Σ then [A, B] ⊂ Σ, • if A ∈ Σ and C ∈ Σ then [A, C] ⊂ Σ, • if A ∈ Σ and [B, C] ⊂ Σ, then the triangle ABC is contained in Σ.
Proof of the lemma 3.2. The proof of this inequality follows closely the ideas of lemma 3.1. We give the details here for the sake of completness.

We begin with the case of M ∈ [A, B]: set x A , z 0 , ρ such that z 0 = (1 -θ)x A + θρ and x B , z 1 , ρ such that z 1 = (1 -θ)x B + θρ. Then, if we define x M such that x M = (1 -η)x A + ηx B , we must show that we have the inequality

f L p,1/z ≤ C 0 f 1-θ Ḃα,q 0 1/x M f θ Ḃ-β,q 1 1/ρ with z = (1 -θ)x M + θρ.
• Since f ∈ Ḃα,q0 1/xM , we have that (λ j ) j∈Z ∈ ℓ 1/xM , with λ j = 2 jα ∆ j f L q 0 . Moreover, by hypothesis we have

x M = (1 -η)x A + ηx B , so we can write ℓ 1/xM = [ℓ 1/xA , ℓ 1/xB ] η,1/xM . Thus, with lemma 2.1 we obtain a partition Z = k∈Z Z k such that 2 -kη j∈Z k λ 1/xA j xA ℓ 1/x M + 2 k(1-η) j∈Z k λ 1/xB j xB ℓ 1/X M ≤ C λ j ℓ 1/x M . We will note β k = 2 -kη j∈Z k λ 1/xA j xA , γ k = 2 k(1-η) j∈Z k λ 1/xB j xB and f k = j∈Z k ∆ j f . • Since f ∈ Ḃ-β,q1 1/ρ , we can write α k = j∈Z k 2 -jβ1/ρ ∆ j f 1/ρ L q 1 ρ k∈Z ∈ ℓ 1/ρ . Now, since z 0 = (1 -θ)x A + θρ and z 1 = (1 -θ)
x B + θρ we can apply theorem 4 to the functions f k to obtain:

f k L p,1/z 0 ≤ C A f k 1-θ Ḃα,q 0 1/x A f k θ Ḃ-β,q 1 1/ρ ≤ Cα θ k β 1-θ k 2 kη(1-θ) f k L p,1/z 1 ≤ C B f k 1-θ Ḃα,q 0 1/x B f k θ Ḃ-β,q 1 1/ρ ≤ Cα θ k γ 1-θ k 2 -k(1-η)(1-θ)
From these two estimates we deduce, since f

= k∈Z f k , that f ∈ [L p,1/z0 , L p,1/z1 ] η,1/z with z = (1 -η)z 0 + ηz 1 .
But by hypothesis we have z 0 = (1 -θ)x A + θρ and z 1 = (1 -θ)x B + θρ, so we find that z = (1 -θ)x M + θρ.

We have proven that we may interpolate along horizontal lines and the vertical case is totally similar. To finish the proof of the lemma suppose that A ∈ Σ and [B, C] ⊂ Σ and take a point

P ∈ Σ. Write P ∈ [M, N ] where [M, N ] is horizontal, M ∈ [A, C] and N ∈ [B, C].
Then we find that M ∈ Σ by vertical interpolation between A and C, and that P ∈ Σ by horizontal interpolation between M and N . Thus, Lemma 3.2 is proven, and this finishes the proof of theorem 5.

Sharpness and optimality of the inequalities.

In this section, we adapt Bahouri and Cohen's example [START_REF] Bahouri | Refined Sobolev inequalities in Lorentz spaces[END_REF] to the general case. Their idea is to use the analysis of the regularity of chirps by Jaffard and Meyer [START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF]. They express their example in terms of wavelets, as it is easy to estimate Besov norms in wavelet bases [START_REF] Lemarié-Rieusset | Ondelettes et bases hilbertiennes[END_REF][START_REF] Meyer | Wavelets and operators[END_REF]. They pick just one wavelet per scale and adjust its support so that they are able to compute the Lorentz norm of the sum. As we shall see, their example can be extended to the general case but we will not use wavelets, but atoms for Besov spaces [START_REF] Frazier | A discrete transform and decompositions of distribution spaces[END_REF][START_REF] Triebel | Theory of function spaces II[END_REF], as we shall use only upper estimates for Besov norms. We consider a function ω such that it is supported in the ball B(0, 1), is

C N (R n ) and R n
x γ ω(x) dx = 0 for all γ ∈ N n with |γ| < N , for some N such that N > max(|α|, |β|). Then we have, for all 1 ≤ q ≤ +∞, 1 ≤ r ≤ +∞ and |s| < N , j∈Z k∈Z n λ j,k ω(2 j x -k) Ḃs,q r ≤ C j∈Z 2 jsr 2 -jnr/q k∈Z n |λ j,k | q r/q 1/r .

(24)

For our example, we shall fix some X ∈ R, some Y ∈ R and some γ ∈ R and we define where the K j are finite sets such that

• the supports of the functions (ω(2 j x -k)) j1≤j≤jL,k∈Kj are disjoint each one from each other

• K j is a set of cardinal j with 2 δj ≤ A j ≤ 2 δ(j+1) (which is possible if δj 1 ≥ 0 and δj L ≥ 0).

For every r ∈ [1, +∞], we then have the following estimates (from (24)):

f L Ḃα,q 0 r ≤ C r jL j=j1
2 jr(X+α-n/q0+δ/q0) 1/r f L Ḃ-β,q 1 r ≤ C r jL j=j1

2 jr(X-β-n/q1+δ/q1) 1/r g L Ḃ-α,q 0 /(q 0 -1)

r ≤ C r jL j=j1
2 jr(Y -α-n(1-1/q0)+δ(1-1/q0)) 1/r g L Ḃβ,q 1 /(q 1 -1)

r ≤ C r jL j=j1
2 jr(Y +β-n(1-1/q1)+δ(1-1/q1)) 1/r

We thus fix X, Y, γ such that      X + α -n/q 0 + δ/q 0 = 0 X -β -n/q 1 + δ/q 1 = 0 Y -α -n(1 -1/q 0 ) + δ(1 -1/q 0 ) = 0 (25) Since α -n/q 0 + δ/q 0 = -β -n/q 1 + δ/q 1 , we have as well Y + β -n(1 -1/q 1 ) + δ(1 -1/q 1 ) = 0. This gives that, for all r ∈ [1, +∞] we have

f L Ḃα,q 0 r ≤ C r L 1/r f L Ḃ-β,q 1 r ≤ C r L 1/r
g L Ḃ-α,q 0 /(q 0 -1) r ≤ C r L 1/r g L Ḃβ,q 1 /(q 1 -1) From (25), we have X + Y -n + δ = 0, so that, using lemma 2.2 we obtain:

ω 2 L 2 L ≤ C f L L p,r g L L p p-1 , r r-1 ≤ C ′ f L L p,r L 1-1 r
If we asssume that the interpolation inequality ( 8) is valid, we get that

L 1/r ≤ C f L L p,r ≤ C ′ f L 1-θ Ḃα,q 0 r 0 f L θ Ḃ-β,q 1 r 1 ≤ C ′′ L (1-θ)/r0+θ/r1
Letting L -→ +∞, we find that 1/r ≤ (1-θ)/r 0 +θ/r 1 : we have thus proven the optimality of these inequalities.

q0∩

  Ḃ-β,p1 q1 if f can be writen using the Littlewood-Paley decomposition f = j∈Z ∆ j f and if the seminorms • Ḃα,p 0 q 0 and • Ḃ-β,p 1 q 1

r≤

  C r L 1/rwhere C r does not depend on L. Theorem 2 shows thatf L L p,r ≤ C r L 1/r and g L L p/(p-1),r ≤ C r L 1/r .Moreover, since the supports of the functions ω(2 j x -k) are disjoint, we havef L g L dx = jL j=j1 2 j(X+Y ) k∈Kj 2 -jn ω 2 L 2 ≥ jL j=j1 2 j(X+Y -n+δ) ω 2 L 2

The norms X involved in[START_REF] Jaffard | Wavelet methods for pointwise regularity and local oscillations of functions[END_REF] are homogeneous: λ > 0 → f (λx) X is an homogeneous function of λ.

defined in the Fourier level by Iαg(ξ) = |ξ| -α g(ξ).

-k(1-η)(1-θ) .


Said Harchi

Jean-Philippe Georges

Thierry Divoux

WSN dynamic clustering for oil slicks monitoring

Keywords: wireless sensors network, clustering, mobility

Clustering algorithms for wireless sensor networks are known as application-dependent. In the framework of the oil slicks monitoring in marine environment, this paper describes the modeling of a node implementing a dedicated dynamic partitioning algorithm, using a specific metric for which simulation provides optimal weights for combining various parameters (energy, density, mobility).

I. INTRODUCTION

Initiated following a recommendation adopted at the United Nations Conference on the Human Environment (Stockholm, 1972), the CDI (Center for Documentation and Information) and the WMO (World Metrological Organization) launched a monitoring program of marine pollution by hydrocarbons. This program is part of the Integrated Global Ocean Station System (IGOSS). It can advantageously exploit the technology of wireless sensor networks (WSN). Indeed, the drift of oil slicks on the surface of seawater, their consistency and their size could be monitored continuously and in real time by several wireless sensors deployed on these slicks.

These sensors have a storage unit, a data processing unit, various units of measurement, a battery supply and a radio communication interface. Once deployed on the oil slicks, these wireless sensors self-organize into distributed subnetworks to collect data and transmit them to a data processing station called Base Station. Such a system can be beneficial (continuous monitoring and in real-time) compared for example to the classical plane flyover. However, although this type of network is increasingly used in various applications [START_REF] Yick | Wireless sensor network survey[END_REF], [START_REF] Regan | A demonstration of wireless sensing for long term monitoring of water quality[END_REF], there are still challenges that could be to build a consistent information system based on this type of network.

The major challenge to use WSN in our application is to cope with the dynamics of the system due to:

• The mobility of oil slicks which induces a topological evolution of the network. Indeed, oil tends to spread immediately to the water surface and form slicks which, in theory, are circles whose diameter increases eight times in the hour following the spill, and then five times before the end of the week. Then, the wind, waves and the Coriolis force (due to Earth rotation) contribute to the stretch, displacement and fragmentation, and finally to their decomposition into droplets. Therefore, it is obvious that the future of slicks in the medium and long term is very difficult to predict.

• The mobility of nodes. Indeed, under the influence of factors such as wind or wave, nodes can move inside the oil slicks. Moreover, depletion of batteries or failure of nodes also change the network topology. Even if a node might slightly travel with a different speed than the oil slicks, it is assumed that a set of nodes will be representative of an oil slick location.

Thus, the network dynamics caused by these two types of mobility (slicks and nodes) can lead to a total or partial loss of connectivity for one or more nodes leading to an inconsistency of the global information system.

The architecture of this information system for the oil slicks surveillance using a WSN comprises three levels: level 3 consists of an infrastructure for processing data collected at level 2. This second level is composed of mobile collectors which periodically poll cluster heads defined at level 1. This last level represents the partitioned network on which we focus in this paper by describing a model of a sensor node, using the Opnet environment. This model implements a partitioning (clustering) algorithm adapted to a dynamic network. The choice of the partitioning strategy is dictated by the nature of our system, since the nodes will be grouped naturally on the oil slick. Finally, we evaluate the influence of some parameters of the metric on the stability of our network clustering.

This paper is organized as follows: in the next section, we describe the macroscopic model of the sensor node that we developed in the Opnet environment. In section III, we present the clustering algorithm which meets the challenge "dynamic network". In section IV, we discuss about the metric we have to implement in the previous clustering algorithm. In section V, we illustrate through simulation, the usefulness of our model and discuss the adequate choice of some parameters of the chosen metric. We conclude this work and present future research directions in the last section.

II. NODE MODELING

In order to evaluate the impacts of the clustering parameters for such oil slicks monitoring application, we decided to use the Opnet Modeler network simulation tool. However, only models of basic protocols like the IEEE 802.15.4 are available for this tool. Based on these models, we are proposing a complete sensor node model. The main development consists in adding a network layer implementing the clustering. Thus the model implements four layers (Fig. 1). Medium access relies on an unslotted mechanism without beacon. When a node wants to send a message, it starts by testing if the medium is busy. If it is not the case, it sends directly otherwise it waits for a variable period defined in standard (IEEE 802.15.4). In this mode, it is expected that the energy consumption will be optimized and that the node will use the channel only when a message has to be sent.

b) Clustering layer: implements a distributed and hierarchical clustering algorithm. This layer aims at building the clusters, electing cluster heads (CH) and defining the routes that will follow the messages from a member to a CH.

The CH election is based on weights defined for each node. The weight of a node comes from a metric that should be adapted to the application features. It associates the remaining residual energy, density and mobility (defined as the evolution of the average distance between a node and its neighbors). The node with the larger weight is elected as CH. If two nodes have the same weight, the node with smaller mobility is elected.

Fig. 2 shows the finite state machine of the process implemented at the clustering layer in our sensor node model. Its is based on the different states/roles that a node might have.

• init : initialization of state variables • ordinary: when a node is not yet associated to a cluster • member: the node is associated to a cluster (and will send to the CH its measurements) • gateway: the node is at the border of a cluster • cluster head: the node is the cluster head (it receives data from cluster members and transfer them to the collector) c) Application layer: implements a specific process that depends on the node's role in the clustering (cluster head or member). A member node deals only with the level 1 of the infrastructure while a cluster head is related to both levels 1 and 2. For members, this process sends periodically measurements to the CH (the event-triggered is not considered here as a continuous and in real-time monitoring of oil slicks Fig. 2. Finite state machine of the clustering process is expected). For CH, this process listens the data coming from the members plus a special event "visit of a mobile collector" (level 2). When this event occurs, this process will transfer to the collector the data collected.

III. CLUSTERING ALGORITHM

The clustering algorithm uses four broadcast messages:

• INVITE: sent by a cluster head to invite its neighbors to join its cluster • ADHESION: sent by a node to a cluster head (to notify its membership to the cluster and also invite its neighbors to join the cluster) • CLOSE: sent by a gateway to notify that it is joining a cluster and enclose a branch of the cluster tree • HELLO: periodically sent by each node in order to enable the discovery of neighbors Each message contains different fields as the node localization, the metric, the state or the parent of the source. The algorithm is based on two steps: neighbors discovery and cluster head election, and cluster building and maintenance.

A. Neighbors discovery and cluster head election

Initially, nodes start at the ordinary state as shown in Fig. 3. Here, each node sends to its neighbors an HELLO message. When a node receives an HELLO message, it updates its neighbors table by adding or updating the information: the address of the node, its coordinates, the number of neighbors and the value of its metric. These information will be next used to compute the node mobility and density.

This ordinary state is maintained for a time sufficient for the neighbors tables to be consistent. This time is fixed at 2 s in avoid the clustering instability. Then, the node compares its metric to others. If its value is the larger, transition trans O CH is activated and the node moves to the cluster head state (Fig. 2). Otherwise, it waits an INVITE message. Fig. 3. Neighbors discovery and cluster head election for a node (u) that will receive HELLO message from node (v)

B. Cluster building and maintenance

Once a node enters in the cluster head state, it launches the building of the cluster shown in Fig. 4 by sending an INVITE message to its neighbors. When a node receives such message,

• at the ordinary state, it stores the information related to the cluster head in its neighbors table, marks the cluster head node as its parent (for routing). The transition trans O M is activated and it moves to the member state. It sends then an ADHESION message to its neighbors (which are invited to join the cluster). • at the member sate, the message is ignored.

• at the cluster head state, the node compares its metric with the one included in the INVITE message. If it is better, the sender is now considered as its parent, the transition trans CH M is activated, the node shifts to the member state and sends an ADHESION message. Otherwise, the INVITE message is ignored and the node responses by a new INVITE message in order to notify that one of its neighbors has a better metric.

When a node receives an ADHESION message, it updates its neighbors table and:

• at the member or the gateway states, the neighbors with the best metric is chosen as its parent. Depending on wether the parent has changed, an ADHESION or a CLOSE message is sent. • at the ordinary state, the message is stored and the node waits during a given time. After this time, the neighbor which is member of the cluster and has the best metric is selected as parent. Moreover, if all ADHESION messages received are coming from the same cluster, trans O M is activated and the node shifts to the member state. A new ADHE-SION message is sent by the node to its neighbors. otherwise, trans O G is activated: it shifts to the gateway state and sends a CLOSE message.

• at the cluster head state, the message is ignored.

When a node receives a CLOSE message, it updates its neighbors table and stops current transmission ADHESION, INVITE and CLOSE messages.

A. State of the art

Many clustering techniques, centralized or distributed, have been proposed in the literature. They vary according to node deployment, the starting strategy, the election process, the features of the cluster head (CH), the size or diameter of the cluster, the used architecture, the network modeling, etc.. For example, a CH could be elected by the ordinary nodes of a cluster or pre-assigned by the network designer. It could also be an ordinary node or a super-node with fewer resource constraints (energy reserve, radio power, throughput, etc.). The cluster size (number of nodes) may be fixed or variable. In a cluster, any member could be at most 1-hop or k-hops from its CH [START_REF] Abbasi | A survey on clustering algorithms for wireless sensor networks[END_REF], [START_REF] Naruephiphat | Clustering techniques in wireless sensor networks[END_REF].

Given that our network is homogeneous and dynamic, without pre-assigned CH, we will focus on election-based partitioning algorithms. Thus the choice of CHs involves a decision criterion. It is usually a metric or combination of metrics such as: the nodes identifier [START_REF] Ephremides | A design concept for reliable mobile radio networks with frequency hopping signaling[END_REF], the nodes degree (i.e. the number of neighbors at k-hops) [START_REF] Gerla | Multicluster, mobile, multimedia radio network[END_REF], the nodes density, the nodes residual energy, the nodes mobility [START_REF] Basu | A mobility based metric for clustering in mobile ad hoc networks[END_REF], a random function [START_REF] Nagpal | An algorithm for group formation and maximal independent set in an amorphous computer[END_REF], a probabilistic function [START_REF] Bandyopadhyay | An energy efficient hierarchical clustering algorithm for wireless sensor networks[END_REF], or a weighted com-bination of all these elements [START_REF] Chatterjee | Wca: A weighted clustering algorithm for mobile ad hoc networks. cluster computing[END_REF].

Thus, from the literature, it can be highlighted that the DCA algorithm (Distributed Clustering Algorithm) proposed by Basagni [START_REF] Basagni | Distributed clustering for ad hoc networks[END_REF] where the weight of the nodes is defined by the inverse of their velocity. That increases the probability of the least mobile nodes to become a stable CH. A node is elected as CH if it has the maximum weight among the nodes in its neighborhood. The DCA algorithm was designed for networks where nodes are static or with low mobility.

The DMAC algorithm (Distributed and Mobility-Adaptive Clustering) [START_REF] Basagni | Distributed and mobility-adaptive clustering for ad hoc networks[END_REF] represents an evolution of the DCA algorithm that allows the mobility of the nodes during or after the computation of the cluster. It also allows each CH to have up to (k) neighbors, and reduces the number of reassignments by adding a restructuring threshold (h).

In the MOBIC algorithm presented in [START_REF] Basu | A mobility based metric for clustering in mobile ad hoc networks[END_REF] the CH election reflects the relative mobility of nodes. The relative mobility of a node is determined according to its neighbors transmission power. The node with the lowest mobility becomes CH. This heuristics is complex because it requires that each node is able to evaluate the power level of its neighbors to estimate its relative mobility. Moreover, the calculation of this metric does not consider physical phenomena that degrade the signal quality (obstacles, reflection, etc.).

The WCA algorithm (Weight Clustering Algorithm) was proposed in [START_REF] Chatterjee | Wca: A weighted clustering algorithm for mobile ad hoc networks. cluster computing[END_REF]. It uses a weighted sum of several metrics such as the degree, the Euclidean distance, the relative mobility, and the life time of a node as a CH. The node with the lowest weight among its neighbors becomes CH. The weight of a node (u) is defined as follows: weight (u) = αD u + βP u + λM u + σT u where:

• α + β + λ + σ = 1 • D u
is the difference between the degree of the node (u) and a constant M representing the maximum size of a cluster • P u is the sum of the distances between the node (u) and its neighbors. These distances are obtained by means of a locating system such as GPS • M u is the average relative mobility of the node (u) obtained as in the MOBIC algorithm • T u is the life time of a node as a CH WCA does not minimize the cost of maintenance because the partitioning process is restarted when a node is without any CH. The weight calculation requires significant traffic and it uses a GPS system that is expensive and energy consumer.

[13] uses as a metric, the density which is defined by the ratio of the number of links by the number of nodes in the kneighborhood of the considered node. Periodically, each node calculates its density and broadcasts it to its 1-neighborhood. Thus each node is able to compare its own density with that of its neighbors. Then it decides to promote itself as CH if it has the highest density, or to choose as CH its neighbor with the highest density. In case of equality the stability is favored. [START_REF] Lehsaini | A novel cluster-based selforganization algorithm for wireless sensor networks[END_REF] proposes the CSOS algorithm (Cluster-based Self-Organisation Scheme) which builds 2-hop clusters, of similar size. The composite metric is determined as follows:

weight u (t) = αd u,2 (t) + βE u (t) + γM u (t)
where:

• α + β + γ = 1 • d u,2 (t) is the the 2-hops density • E u (t) is the residual energy • M u (t)
is the node mobility [START_REF] Theoleyre | A self-organization structure for hybrid networks[END_REF] constructed a backbone on which are built clusters using a metric defined as:

P stability = ǫ α (1 + ∆) -1 + β (1 + M ) -1
with:

• ǫ is the residual energy of the node • ∆ = |∆ real -∆ opt |: distance to an optimal degree (∆ real : real degree, ∆ opt : degree depending on application) • M is the relative mobility of the node • The parameters α and β depend on the application. This overview shows that we can choose various parameters to partition our network. These parameters can be considered separately or in combination. In addition, each of these parameters may be subject to different definitions including the density that can be seen at 1, 2, 3 or k-hops. Finally, different types of weighting are proposed. Furthermore, these different studies have highlighted the influence of the application.

In this paper we deal with a particular application where the network dynamics will be strong. Also we will propose the definition of a clustering algorithm that is based on a metric combining energy, density and mobility. By modeling this algorithm in the Opnet environment we will demonstrate that it is possible to define the best weights applied to these three parameters in the context of the continuous and real-time oil slicks monitoring.

B. Statement

The metric used is a weighted ponderation of energy, density and node mobility. We choose the one-density, that is to say, calculated to 1-hop, instead of k-density (k > 1) based on the work of Mitton [START_REF] Mitton | Auto-organisation des reseaux sans fil multi-sauts a grande echelle[END_REF]. It showed that the 1-density provides a more robust structure compared to the k-density. In addition to a 1-hop density minimizes information over the network and therefore the bandwidth and the energy consumption. The metric is hence expressed as:

weight u (t) = ǫ u (t) (αd u,1 (t) + (1 -α) M u (t)) (1)
where:

• d u,1 (t) = N u,1 (t -∆t) /N u,2 (t -∆t): the density of node (u) with N u,k (t) the number of k-hop neighbors of node (u) • M u (t)
: the mobility of node (u) at time t which might be relative or absolute • The parameters α depend on the application • ǫ u (t): remaining energy of node (u) at time t. It is computed according to the consumption pattern energy proposed by [START_REF] Jurčík | The ieee 802.15.4 opnet simulation model: Reference guide v2.0[END_REF] for IEEE 802.15.4 MICAZ motes.

In this paper, it is assumed that nodes are able to determine their position, such that we will use the relative mobility defined as:

M u (t) = d u (t) -d u (t -∆t) ∆t with: d u (t) = 1 N u,1 (t) v∈Nu,1(t) dist u,v (t)
where:

• x u (t) , y u (t): the coordinates of node (u) at time t • dist u,v (t): the euclidian distance between (u) and (v) • d u (t): the average euclidian distance between node (u)

and its 1-hop neighbors Equation ( 1) is a generic metric used to compute which node will be elected as the cluster head and which cluster a (member) node will belong to. According to the parameters, it means that the logical topology might change. In this paper, we are not focusing on the residual energy level but we are concerned to see how the parameter α will act for such mobile WSN applications. In particular, the parameter α will be studied in terms of quality of clustering (mainly its stability).

V. ILLUSTRATION

A. Simulations

This section presents simulation results obtained with the Opnet tool and the sensor node model detailed in the previous section. Objective of the simulation is to emphasize the impact of the parameter α of the metric on the clustering quality. This quality is evaluated in terms of number of clusters built, number of cluster head updates during a simulation and life time of the cluster head status on nodes. The varying parameters considered during the simulation are α and the nodes speed from 0, 1 to 6 m/s which corresponds to realistic climate conditions. For example, 0, 2 m/s corresponds to a wind of 40 km/h. The simulated scenario is related to the hazardous deployment of 100 sensors on a square of 1 km 2 . Sensors were placed at the beginning of the simulation on a square of 50 m 2 . Nodes are configured with a radio range of 50 m. The shifting of the nodes is simulated according to the Random Way Point [START_REF] Bettstetter | The node distribution of the random waypoint mobility model for wireless ad hoc networks[END_REF], [START_REF] Camp | A survey of mobility models for ad hoc network research[END_REF] profile. It means that a node remains fixed during a finish time after moving. For moving, trajectory is randomly selected. These two steps are repeated until the end of the simulation. This model might be not very representative of our application, and further works will provide a realistic model. On a node, the neighbors discovery process is periodically run each second, such that the density and the mobility used in the metric will be computed with ∆t = 7 s. In order to face with intermittent communication capacities, the proposed metric consider the whole list of messages received during a window time ∆t. Hence the impact of messages losses will remained limited. Furthermore a node is not anymore considered as a neighbor if no HELLO message is received after 7 s.

Fig. 5 shows for a given value of α = 0.5 and a constant nodes speed of 2 m/s, the evolution of the number of clusters during a simulation of 2 mn. Longer simulations have been launched to visualize the node energy consumption. For instance, after 10 mn, only 0.0016 % of energy was consumed. Number of clusters is evaluated here each second. The dynamic illustrated on Fig. 5 reflects this. Initially, the sensors are close, the graph is connex and hence one or two cluster might be sufficient. Next, sensors will start to move from each other, such that the density of the graph decreases, subnetworks appear and finally, more clusters are defined. Other simulations are next launched in order to illustrate the impact of the weight α.

B. Results

For each configurations (mobility speed and α values), 900 simulations (based on 30 seeds) are launched. Average values are computed with a confidence interval of 95 %. The quality of the metric defined previously is evaluated here in terms of stability of the clustering. When mobile collectors will have to periodically go through the different cluster heads in order to collect their data, it is important that the trajectory of a collector remains as possible (sensors are still moving) like the previous one. It means that inside a cluster, such application are interested in the fact that the same sensor remains the cluster head. Hence, the election rate of cluster heads and the life time of the cluster head status was evaluated trough simulations. Different speeds and different values of the metric weight α were tested. Fig. 6 shows for different speeds, the average number of times a node became head of a cluster for an α value and Fig. 7 shows the average time a node remains cluster time.

Fig. 6 shows that α = 1 reduces the nodes rate election as CH. Similarly, Fig. 7 shows it is better to have α = 1 for a node to maintain its CH status. It means that the higher the speed, the more it damages the stability of the clustering.

However, Fig. 8 shows the interest to adapt α to the speed in order to minimize the number of clusters.

It can be highlighted one more time that for high speed, mobility must be neglected regarding density. But for low and medium speeds, the compromise is interesting because it reduces the number of clusters (Fig. 8) without affecting the stability (Fig. 6 and7). Indeed, for speeds lower than 2 m/s,

VI. CONCLUSION

In case of disaster, the node model proposed in this paper enables to design the monitoring system to be deployed in terms of number of nodes, the measurement period, the clustering dynamics and the metric weights regarding area surface to supervise and climate conditions.

Future works will focus now on the level 2 of the monitoring system dedicated to the data mobile collectors management for which a session protocol has to be developed in order to retrieve the context of theirs previous visits. The goal is to recover a discrete connectivity in order to periodically rebuild a consistent information system.

Fig. 1 .

 1 Fig. 1. Sensor node model

Fig. 4 .

 4 Fig. 4. Determination of the parent of a node (u) when it receives a message from node (v)

5 Fig. 5 .

 55 Fig. 5. Number of clusters during simulation (time in s)

Fig. 6 .Fig. 7 .

 67 Fig. 6. Number of times a node became head of a cluster according to α and the speed

Fig. 8 .

 8 Fig. 8. Number of clusters according to α and the speed

1 :

 1 state ← ordinary 2: SEND(HELLO) 3: if a HELLO message is received then

	4:	update the neighbors table
	5:	if metric(u)>metric(v) then
	6:	state ← cluster head
	7:	else
	8:	wait for an SEND(IN V IT E) message
	9:	end if
	10: end if

  1: update the neighbors table 2: if an IN V IT E message is received then

	3:	if (state = cluster head and metric(u)<metric(v)) or
		state = ordinary then
	4:	parent ← v
	5:	state ← member
	6:	SEND(ADHESION )
	7:	else if state = cluster head then
	8:	SEND(IN V IT E)
	9:	end if
	10: else if an ADHESION message is received then
	11:	if state = member or gateway then
	12:	parent ← neighbor with the best metric
	13:	if parent changed then
	14:	SEND(ADHESION )
	15:	else
	16:	SEND(CLOSE)
	17:	end if
	18:	else if state = ordinary then
	19:	update the neighbors table
	20:	parent ← neighbor with the best metric
	21:	if all ADHESION messages received are coming
		from the same cluster then
	22:	state ← member
	23:	SEND(ADHESION )
	24:	else
	25:	state ← gateway
	26:	SEND(CLOSE)
	27:	end if
	28:	end if
	29: end if
		IV. CLUSTERING METRIC