Bilal Al Taki

Khawla Msheik

Jacques Sainte-Marie

Jacques Sainte

B Al Taki

J Sainte Marie

On the rigid-lid approximation of shallow water bingham model

Keywords: Bingham fluid, Muckenhoupt weights, Variational inequality, Low Froude number AMS subject classification: 35Q30, 76N10, 35B65, 35D35

This paper discusses the well posedness of an initial value problem describing the motion of a Bingham fluid in a basin with a degenerate bottom topography. A physical interpretation of such motion is discussed. The system governing such motion is obtained from the Shallow Water-Bingham models in the regime where the Froude number degenerates, i.e taking the limit of such equations as the Froude number tends to zero. Since we are considering equations with degenerate coefficients, then we shall work with weighted Sobolev spaces in order to establish the existence of a weak solution. In order to overcome the difficulty of the discontinuity in Bingham's constitutive law, we follow a similar approach to that introduced in [G. Duvaut and J.-L. Lions, Springer-Verlag, 1976]. We study also the behavior of this solution when the yield limit vanishes. Finally, a numerical scheme for the system in 1D is furnished.

Introduction

Bingham fluids constitute a crucial topic of study on which many applicable researches are conducted. As an interesting example we suggest the snow that appears in some important natural phenomena such as avalanches. Technically speaking, the avalanche dynamics can be described through different perspectives, such as the center of mass consideration [START_REF] Perla | A two-parameter model of snow-avalanche motion[END_REF] and the density one. In addition, many approaches consider the avalanche to be a deformable body whereas others describe it as a granular material. Not only the dynamics of snow, but also its constitutive behavior can be described according to different points of views such as Newtonian fluid and Bingham fluid. In many literatures, snow is considered to be a non-Newtonian fluid. One of the reasons is that Newtonian fluids adapt rapidly to deform themselves where they reach a negligible depth in an unbounded space, on the other hand, snow will pile in such a case and thus will rest with a finite depth. This can be explained by the fact that upon being in rest state, snow achieves a yield value with a non zero shear stress, and thus a threshold stress value should be exceeded to start deformation. Another reason is the experimental results of the analysis of the avalanche's velocity profile along its depth, which reveals the viscosity dependence of the shear [START_REF] Naaim | Etude expérimentale des écoulements davalanches de neige dense. Mesures et interprétations des profils de vitesse en écoulements quasi permanents et pleinement développés[END_REF] [START_REF] Nishimura | Contribution of viscous forces to avalanche dynamics[END_REF]. These two reasons triggered many authors in the literature to treat snow as a Bingham fluid which is characterized mainly by a non constant stress that depends on viscosity and that may differ in the same body of mass. Hence, one would notice some portions flowing while others moving in a bulk motion as solids. More precisely, at low shear stress, Bingham fluids possess high viscosity, and thus they behave as a shear thinning fluid at low or zero speed. As the shear stress reaches a certain limit-denoted yield stress-one notices a sudden drop in the viscosity, while above the yield stress, snow behaves like a low viscosity liquid.

The need to use reduced models arises from the difficulty that the three dimensional Cauchy momentum equations induce in the analytical and numerical studies, especially if we consider a fee boundary case. In this sequel, the reduced model adopted is the Shallow water-Bingham model that is derived using depth integration of the Navier-Stokes-Bingham system with free surface, taking into account that the horizontal length scale is much greater than the vertical one. Since there is no well posedness result on the initial system and the approximated one, then any mathematical justification of this procedure is still as far as we know an interesting open problem due to the wide range of applicability especially in numerical applications. For instance, in [START_REF] Bresch | Augmented Lagrangian method and compressible viscoplastic flows: applications to shallow dense avalanches[END_REF], [START_REF] Fernández-Nieto | Shallow water equations for power law and Bingham fluids[END_REF], the authors proposed several models of a Shallow water type system for Bingham fluid. The resulting system is given by (1.1) where h(t, x) is the water height, v(t, x) is the horizontal velocity. We denote by hv ⊗ v the matrix with component hv i v j , Fr is the Froude number and σ is the stress tensor given by

∂ t h + div(hv) = 0 ∂ t (hv) + div(hv ⊗ v) -div(σ) +
(1.2) σ =    2µh D(v) + λh div v I + gh D(v) | D(v)| if D(v) = 0, |σ| < gb if D(v) = 0,
where D(v) represents the symmetric part of the velocity gradient given by

D(v) = 1 2 ∇v + ∇ t v).
As mentioned above, the Shallow water Bingham model is still for today an open problem. We remark that, in general, the usual strategy of the existence theory is based on two steps: the first one consists of finding u n , a solution of a suitably chosen approximating problem (a Galerkin approximation for example). The second one amounts to establish uniform estimates in proper spaces and one can then obtains u, a solution to System (1.3)-(1.4), as a "weak" limit of a suitable chosen subsequence of u n . The central difficulty is the passage to the limit in the nonlinear terms and in the discontinuous constitutive law. Actually, the ideas developed in studying the well posedness issue of Shallow water equations (or compressible Navier-Stokes equations) for Newtonian fluids (see [START_REF] Lions | Mathematical topics in fluid mechanics[END_REF], [START_REF] Bresch | Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model[END_REF], [START_REF] Vasseur | Existence of global weak solutions for 3D degenerate compressible Navier-Stokes equations[END_REF], [START_REF] Feireisl | Singular limits in thermodynamics of viscous fluids[END_REF]) don't work for non Newtonian ones. As a forward step in this approach, we study here the rigid lid approximation of such system, that we shall call Bingham lake equations. Bingham lake equations characterizing lake equations decode many natural as well as industrial phenomena. The most interesting example is that of mixed flows of Bingham type fluids (such as petroleum) in closed supply pipes. Two types of flows are exhibited inhere characterizing a transition phase from free surface (i.e when only part of the section of pipe is filled and the pressure is being known: atmospheric pressure), to pressurized flow (i.e when the section of the pipe is full, and the pressure is an unknown). The authors in [START_REF] Bourdarias | A finite volume scheme for a model coupling free surface and pressurised flows in pipes[END_REF] have studied such kind of flows occurring for Newtonian fluids thoroughly. They presented a new model called PFS model, based on coupling the free surface part equations derived from the incompressible Navier Stokes or Euler systems, and the pressurized part equations derived from the Compressible Euler equations. A finite volume discretization have been studied in [START_REF] Bourdarias | A finite volume scheme for a model coupling free surface and pressurised flows in pipes[END_REF], and a kinetic formulation of such models is presented in [START_REF] Bourdarias | A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes[END_REF]. Mathematically, this system can be obtained from (1.1)-(1.2) system by passing to the limit (Fr → 0), where the initial height converges to a non constant function b(x) depending on the space variables only. The obtained model is given by:

(1.3) ∂ t (bu) + div(bu ⊗ u) -div σ + b∇p = bf, div(bu) = 0.
The shear stress σ satisfies the special constitutive law of a Bingham fluid (see [START_REF] Bingham | Fluidity and plasticity[END_REF]) :

(1.4)

σ :=    2µb D(u) + λb div u I + gb D(u) | D(u)| if D(u) = 0, |σ| < g if D(u) = 0.
Here, u(t, x) denotes the velocity vector, p the pressure, f (t, x) the known external force, µ and λ the Lamé viscosity coefficients, g the yield limit. We couple the system (1.3) with the so called Lions boundary conditions given by

(1.5) bu • n = 0 (σ • n) • τ + κ(x)bu • τ = 0 (t, x) ∈ (0, T ) × ∂Ω,
and with the initial data (defined in a weak sence, see Theorem 4.2)

(1.6) u(t, x)| t=0 = u 0 (x), x ∈ Ω.
In (1.5), n and τ denote respectively the unit normal and tangential vectors to the boundary, whereas κ(x) is the curvature of ∂Ω. Lions boundary conditions, the particular case of Navier boundary condition that were first used by Navier in 1872, regard that there is a stagnant layer of fluid close to the wall allowing a fluid to slip, and the slip velocity is proportional to the shear stress.

Since the singular term D(u)/| D(u)| is not always defined, numerical and mathematical obstacles appear, which forced many authors to develop new formulations of the problem in order to tackle the difficulty. One of the approaches used was introduced in [START_REF] Bulíček | On unsteady flows of implicitly constituted incompressible fluids[END_REF] for an unsteady flow of an incompressible fluid having its Cauchy stress given implicitly and relating both the symmetric part of the velocity gradient D(u) and the deviatoric part of the Cauchy stress. The authors regard the stress tensor σ as a new variable (along with the density and the velocity vector), and this results in a dissipation rate in the form of a Young function depending on D(u) and σ t . The second approach, that we will adopt in the sequel, was initiated by G. Duvaut and J. L. Lions [START_REF] Duvaut | Inequalities in mechanics and physics[END_REF] who replaced the original system by a variational inequality to get rid of the singularity.

In this paper, we will first adopt the methodology of variational inequality used in [START_REF] Duvaut | Inequalities in mechanics and physics[END_REF] to prove the existence of weak solution of the incompressible Bingham fluid confined to a shallow basin with a varying bottom topography but with some changes in the nature of spaces used. Again recalling that we are dealing with a degenerate bathymetry b(x), meaning that b(x) may vanish on the boundary, we will prove that the solution exists in some weighted Sobolev space where the weight is assumed to be a Muckenhoupt type. For more details about these spaces, we refer the reader to [START_REF] Muckenhoupt | Weighted norm inequalities for the Hardy maximal function[END_REF]. Then, we will discuss the behavior of solution when when g tends to zero. It is good to mention that the vanishing viscosity limit (µ tends to zero) of Bingham fluids is studied by J.-L. Lions [START_REF] Lions | Remarks on some nonlinear evolution problems arising in Bingham flows[END_REF] in a bounded open set of R 2 without bottom topography. However, an improvement of weak regularity on the solution is required in order to achieve the convergence on the nonlinear term. Yet, as long as we can't improve the regularity of our solution due to the degeneracy of our equations, then we are not able to study such limit in our case.

The paper consists of 6 parts. In the next part (Section 2), we will introduce the spaces that we prove the solution in and some preliminary results related to the functional analysis of the problem. In addition, we will accomplish the equivalence between the initial system and the variational inequality introduced in Section 3. Then comes Section 4 where we exhibit the existence of solution via a Galerkin method. In Sections 5, we study the behavior of the solution when g tends to zero. Finally, in Section 6, a numerical scheme is implemented in 1D verifying our theoretical results.

Functional spaces

Before we start the analysis, and since we are going to work with weighted Sobolev spaces, let us first give a brief definition of some of the spaces and preliminaries that we shall need in the sequel, especially the constraints on the weight b and the domain Ω. Domain: (I) For an integer m, 1 ≤ m ≤ 2, we set Q m = (0, 1) m . We assume that there exists a bi-Lipschitz mapping

B : Q 2 → Ω, such that B(Q 1 ) = ∂Ω.
Weight: (II) We define a space function b(x), locally integrable and belonging to the Muckenhoupt class A q . Generally speaking, for a weight of Muckenhoupt type, the definition of trace operator is well defined. More precisely, one can check that if b ∈ A q , we have u ∈ W 1,q b (Ω) → W 1,q loc (Ω) and hence there is a linear trace operator γ 0,b : W 1,q loc (Ω) → L 1 loc (Ω). Though it is well defined, yet we lack characterization of such trace regarding a general Muckenhoupt weight. That's why we restrict ourselves in what follows to a more specific weight that provides a characterization of the boundary terms. Its expression is given in a neighborhood of the boundary V (∂Ω) by

(2.1) b = ρ α (x), 0 < α < 1/2 ρ(x) = dist(x, ∂Ω) for x ∈ V (∂Ω).
In this situation, the definition of the trace is more accurate. For more details about examples of weights satisfying Muckenhoupt condition, we refer the reader to [START_REF] Farwig | Weighted L q -theory for the Stokes resolvent in exterior domains[END_REF], [START_REF] Heinonen | Nonlinear potential theory of degenerate elliptic equations[END_REF], [START_REF] Turesson | Nonlinear potential theory and weighted Sobolev spaces[END_REF].

Weighted Sobolev spaces: As mentioned in the introduction, we will prove our solution in Weighted Sobolev space V b . We introduce the following weighted spaces:

D b (Ω) = φ ∈ C ∞ 0 (Ω); div(bφ) = 0 in Ω, bφ • n = 0 on ∂Ω , L q b (Ω) = φ ; Ω |φ| q b dx <∞ , (L q b (Ω)) = L q b (Ω) with 1 q + 1 q = 1 and b = b -1 q-1 , H b (Ω) = φ; φ ∈ L 2 b (Ω), div(bφ) = 0, bφ • n = 0 on ∂Ω , V b (Ω) = φ; φ ∈ H 1 b (Ω), div(bφ) = 0, bφ • n = 0 on ∂Ω .
For more details about Muckenhoupt classes and Weighted Sobolev spaces we refer the reader to [START_REF] Muckenhoupt | Weighted norm inequalities for the Hardy maximal function[END_REF] and [START_REF] Taki | Viscosity effect on the degenerate lake equations[END_REF].

Trace: As mentioned above in (2.1), we will adopt a special choice of Muckenhopt type weights that provide a good characterization of the trace. In fact, this result is proved by A. Nekvinda via the following theorem.

Theorem 2.1. ([Theorem 2.8, [START_REF] Nekvinda | Characterization of traces of the weighted Sobolev space W 1,p (Ω, d M ) on[END_REF]]). Suppose that Hypothesis (I) holds. Then for b = ρ α (x), -1 < α < q -1, there exists a unique bounded linear operator

T 1,q b ( ∂Ω) : W 1,q b (Ω) → W 1-1+α q ,q ( ∂Ω), such that T 1,q b ( ∂Ω)(u) = u | ∂Ω .
As a consequence of this theorem, we remark that if

f ∈ H 1 ρ α (Ω), 0 < α < 1 the trace of f is well defined and belong to L 2 ( ∂Ω) → L 2 b ( ∂Ω)
. This helps us define the boundary integrals coming from the Navier boundary conditions.

Throughout this paper, we assume that the domain Ω satisfies Hypothesis (I), and b satisfies Hypothesis (II), and any other restrictions would be specified .

Notation Remark: Let us fix some notations which will be used throughout the sequel:

• D(u) : D(v) = 2 i,j=1 D i,j (u) D i,j (v). • ∇b ⊗ u := ( ∂ j b u i ) 1≤i,j≤2 .
• We say that u is a b-divergence free or u satisfies the b-incompressibility condition if div(bu) = 0.

Remark 2.1. We should remark that in the sequel of the study, we will need to have q = 3 2 , i.e the weight b should belong to A 3 2 . In fact, this choice is for the sake of bounding the nonlinear term (u • ∇)u in lemma 4.1. Notice that in studying the Stokes problem, we just need to take q = 2 which is necessary to have a weighted version of Poincaré (or Korn's inequality). The reader can refer to lemma 4.1 in [START_REF] Taki | Viscosity effect on the degenerate lake equations[END_REF] for a detailed explanation of such choice.

Variational Inequality

Following Duvaut and Lions in [START_REF] Duvaut | Inequalities in mechanics and physics[END_REF], we will derive in this section a variational inequality and prove (at least formally) that the resolution of this variational inequality is equivalent to solve the problem (1.3)-(1.4) in a weak sense (solution of variational problem). Indeed, let us establish first the variational formulation of system (1.3)- (1.4). For this sake, we consider a set V of functions v : Ω → R 2 of enough regularity and such that div(bv) = 0 in Ω and bv • n = 0 in ∂Ω. Then, we multiply (1.3) by (v -u) and integrate in space. We get

Ω ∂ t u • (v -u) b dx + Ω div(bu ⊗ u) div(bu)•u+bu•∇u •(v -u) dx - Ω div σ • (v -u) dx + Ω ∇p • (v -u) b dx = Ω f • (v -u) b dx.
Accounting for the Navier boundary conditions and the relation div(bv) = 0, we can write

Ω ∂ t u • (v -u) b dx + Ω (u • ∇)u • (v -u) b dx + Ω σ : D(v -u) dx - ∂Ω σ • n • (v -u) ds - Ω p div(bv -bu) =0 dx + ∂Ω p (bv -bu) • n =0 ds = Ω f • (v -u) b dx. Thus (3.1) Ω ∂ t u • (v -u) b dx + Ω (u • ∇)u • (v -u) b dx + 2µ Ω D(u) : D(v -u) b dx + λ Ω div u div(v -u) b dx + Ω g D(u) | D(u)| : D(v -u) b dx + ∂Ω κu • (v -u) b ds = Ω f • (v -u) b dx.
In equation (3.1), the ratio D(u) | D(u)| is not always defined. The physical understanding of this term indicates that it should be interpreted when D(u) = 0 as "any trace-free symmetric matrix with norm less or equal to one". In the mathematical language such quantity is called "multivalued". This fundamental difficulty makes the approximation of the problem (3.1) already a complex challenge , and has motivated a large literature, see [START_REF] Duvaut | Inequalities in mechanics and physics[END_REF], [START_REF] Bulíček | On unsteady flows of implicitly constituted incompressible fluids[END_REF] and references therein. Below, we follow the procedure in [START_REF] Duvaut | Inequalities in mechanics and physics[END_REF] to derive a "formal" equivalent form of the variational formulation (3.1). Indeed, regarding the Cauchy-Schwartz inequality

D(f ) : D(g) = i,j D i,j (f ) D i,j (g) ≤ | D(f )|| D(g)|, as D(u) = 0, we have Ω D(u) : D(v -u) | D(u)| b dx = Ω 1 | D(u)| D(u) : D(v) - 1 | D(u)| D(u) : D(u) b dx ≤ Ω 1 | D(u)| | D(u)|| D(v)| - 1 | D(u)| | D(u)| 2 b dx ≤ Ω | D(v)| -| D(u)| b dx.
Hence we get

Ω f • (v -u) b dx ≤ Ω ∂ t u • (v -u) b dx + Ω u • ∇u • (v -u) b dx + 2µ Ω D(u) : D(v -u) b dx + λ Ω div u div(v -u) b dx + g Ω (| D(v)| -| D(u)|) b dx + ∂Ω κu • (v -u) b ds for all v ∈ V . (3.2)
By now we have proved that if (u, p) is a formal solution of (1.3), then it would satisfy (3.2). Reciprocally, assuming that u is a solution of the variational inequality (3.2) such that D(u) = 0 a.e. in Ω. Letting θw = v -u in the previous inequality, one gets

Ω f • (θw) b dx ≤ Ω ∂ t u • (θw) b dx + Ω (u • ∇)u • θw b dx + 2µ Ω D(u) : D(θw) b dx + λ Ω div u div(θw) b dx + g Ω (| D(u + θw)| -| D(u)|) b dx + ∂Ω κu • (θw) b ds for all w ∈ V.
Now, as we divide by θ we get:

(3.3) Ω f • w b dx ≤ Ω ∂ t u • w b dx + Ω u • ∇u • w b dx + 2µ Ω D(u) : D(w) b dx + λ Ω div u div w b dx + g Ω 1 | D(u)| (D(u) : D(w) S b dx + ∂Ω κu • w b ds.
Concerning the term S, we have in fact:

(3.4) S = lim θ→0 g θ Ω (| D(u + θw)| -| D(u)|) b dx.
On the other hand, we have

| D(u + θw)| 2 -| D(u)| 2 = i,j (D i,j (u + θw)) 2 - i,j (D i,j (u)) 2 = i,j (D i,j (u)) 2 + 2θ D i,j (u) D i,j (w) + θ 2 (D i,j (w)) 2 -(D i,j (u)) 2 .
Hence

S = lim θ→0 g θ Ω | D(u + θw)| 2 -| D(u)| 2 | D(u + θw)| + | D(u)| b dx = g Ω lim θ→0 1 | D(u)| + | D(u + θw)| × lim θ→0 i,j 2θ D i,j (u) D i,j (w) b dx + θ 2 D i,j (w) θ b dx = g Ω 1 | D(u)| D(u) : D(w) b dx.
By changing w into -w, we find the opposite inequality of (3.3), and so we get

Ω ∂ t (bu) • w dx + Ω bu • ∇u • w dx + 2µ Ω b D(u) : D(w) dx + λ Ω b div u div w dx + g Ω b | D(u)| D(u) : D(w) dx + ∂Ω κu • w b ds = Ω bf • w dx.
Thus, we establish formally the equivalence between (3.2) and (1.3).

Let us now precise our definition of weak solution of System (1.3)-(1.4).

Definition 3.1. We say that u is a solution of system (1.3)-(1.4) equipped with boundary and initial conditions given in (1.5)-(1.6) if u satisfies the following regularity

u ∈ L ∞ (0, T ; H b ) ∩ L 2 (0, T ; V b ), ∂ t (bu) ∈ L 2 (0, T ; V b ),
and the following variational inequality holds

(3.5) Ω f • (v -u) b dx ≤ Ω ∂ t u • w b dx + Ω u • ∇u • (v -u) b dx + 2µ Ω D(u) : D(v -u) b dx + λ Ω div u div(v -u) b dx + g Ω (| D(v)| -| D(u)|) b dx + ∂Ω κu • (v -u) b ds for all v ∈ V b .
Remark 3.1. The well definition of the right hand side will be discussed in the sequel as we take u and v ∈ V b (mainly in lemma 4.1). It is good to mention here that for left hand side term in (3.5), one should pay attention for the regularity of bf, hence assuming the least regularity possible, i.e in the dual of V b , we can replace the integral form by the dual representation: bf, v -u V b ,V b . Though, this in fact doesn't affect the computations. Without loss of generality, we will assume in the sequel that bf is in

(L 2 b (Ω)) = L 2 b -1 (Ω), or equivalently f in L 2 b (Ω).

Main Results

We state in this section the existence result of problem (3.5). The presence of b in the diffusion operator and the "b-incompressibility condition" make the weighted Sobolev spaces the ambient ones to prove existence within. In what follows, we introduce a priori estimates concerning the non linear term in lemma 4.1, and which will later on serve the well definition of the integral forms. Next, we exhibit in theorem 4.2 the existence and uniqueness results. We will rely in the proof on a Galerkin approximation technique. Several operators will be used in the latter method for which different properties will be given in lemma 4.1.

∈ L ∞ (0, T ; H b ) ∩ L 2 (0, T ; V b ) and v ∈ V b , we have: u ∈ L 2 (0, T ; L 6 b (Ω)), (u • ∇)u ∈ L 1 (0, T ; L 3/2 b (Ω)), (u • ∇)u • v ∈ L 1 (0, T ; L 1 b (Ω))
, and there exists a positive real number C independent of u and v such that

||u|| L 2 (0,T ;L 6 b (Ω)) ≤ C||∇u|| L 2 (0,T ;L 2 b (Ω)) , T 0 Ω (u • ∇)u b dx ≤ C ||∇u|| 2 L 2 (0,T ;L 2 b (Ω)) , T 0 Ω (u • ∇)u • v b dx ≤ C ||∇u|| 2 L 2 (0,T ;L 2 b (Ω)) ||∇v|| L 2 b (Ω) .
We define the following linear operators A and B such that

Au ∈ L 2 (0, T ; V b ) and Bu ∈ L 2 (0, T ; V b ),
where A and B are given by

Au, v V b ,V b := a(u, v) = 2µ Ω D(u) : D(v) b dx + λ Ω div u div v b dx + ∂Ω κu • v b ds, Bu, v V b ,V b := b(u, u, v) = Ω (u • ∇)u • v b dx,
and b satisfies b(u, u, u) = 0.

Proof. Here we just want to prove that Au ∈ L 2 (0, T ; V b ) and Bu ∈ L 2 (0, T ; V b ). The other properties are proved in [START_REF] Taki | Viscosity effect on the degenerate lake equations[END_REF]. Indeed, For u ∈ L ∞ (0, T ; H b ) ∩ L 2 (0, T ; V b ) and v ∈ V b , we estimate using Hölder's inequality

| Au, v V b ,V b | = 2µ Ω D(u) : D(v) b dx + λ Ω div u div v b dx, + ∂Ω κu • v b ds ≤ C u V b v V b .
Since u ∈ L 2 (0, T ; V b ), thus by duality we get

Au ∈ L 2 (0, T ; V b ).
As for the operator B, we estimate the nonlinear term using Hölder and Gagliardo-Nirenberg inequalities as follows

| Bu, v V b ,V b | = Ω (u • ∇)u • v b dx = - Ω (u • ∇)v • u b dx ≤ C u 2 L 4 b (Ω) ∇v L 2 b (Ω) ≤ C u L 2 b (Ω) ∇u L 2 b (Ω) ∇v L 2 b (Ω) ≤ C u H b u V b v V b . Since u H b u V b ∈ L 2 (0, T ),

thus by duality we get

Bu ∈ L 2 (0, T ; V b ).

Remark 4.1. Notice that, in [Proposition 3.1, [START_REF] Taki | Viscosity effect on the degenerate lake equations[END_REF]] , the first author proved a weighted version of Korn's inequality. However, some constraints on the weight b and the domain Ω were needed. More precisely, the author proved that if we exclude that case when the domain Ω is a disc, b is radial and not identically zero on the boundary, then we have

D(u) L 2 b (Ω) ≥ u H 1 b for all u in V b .
Nevertheless, he showed also that these assumptions are not required when studying the evolution problem. That is why in our case we won't suppose such assumptions. For more details, the reader is referred to [START_REF] Taki | Viscosity effect on the degenerate lake equations[END_REF].

The main result of this paper is given below.

Theorem 4.2. (Existence of weak solution). We suppose that f and u 0 are the applied force and the initial datum given such that f lies in L 2 (0, T, L 2 b (Ω)) and u 0 belongs H b . Assume that b satisfies Hypothesis (II), and that κ(x) is in L ∞ (Ω), then there exists a unique vector field u such that

(4.1) u ∈ L 2 (0, T ; V b ) ∂ t (bu) ∈ L 2 (0, T ; V b ) satisfying Ω f • (v -u) b dx ≤ Ω ∂ t u • (v -u) b dx + Ω u • ∇u • (v -u) b dx + g Ω (| D(v)| -| D(u)|) b dx +2µ Ω D(u) : D(v -u) b dx + λ Ω div u div(v -u) b dx + ∂Ω κu • (v -u) bdx for all v ∈ V b ,
and the initial condition is defined in a weak sense:

Ω u • v b dx (0) = Ω u 0 • v b dx for all v ∈ V b .
Proof. In the same spirit of Lions and Duvaut's approach, our proof will be composed of three main steps. Let us denote by the operator j the integral form:

j(ψ) = g Ω | D(ψ)| b dx.
(1) Step1: Regularizing j

In attempt to approximate the problem, we start first by regularizing the operator j which is derived from Bingham's singular term.

Let ∈ (0, 1). We approximate j by a differentiable functional

j (ψ) = g 1 + Ω (| D(ψ)|) 1+ b dx.
In fact, j (ψ) is well defined since as we deal with a bounded domain and the fact that D(ψ) is in L 2 b (Ω), we infer that j (ψ)<∞. The Gateaux differential of j (•) along v is given by

D v (j (w)) := lim τ →0 j (w + τ v) -j (w) τ = g 1 + Ω D v (| D(w)| 1+ ) b dx. Then D v (| D(w)| 1+ ) = D v ((| D(w)| 2 ) 1+ 2 ) = 1 + 2 (| D(w)| 2 ) -1 2 D v (| D(w)| 2 ) = 1 + 2 | D(w)| -1 × 2 D(w) : D(v) from (3.4). Therefore (4.2) D v (j (w)) = g Ω | D(w)| -1 D(w) : D(v) b dx < + ∞.
In fact, we claim that

| D(w)| -1 D(w) belong to L 2 (0, T ; H b ) T 0 Ω | D(w)| -1 × D(w) b dx dt ≤ T 0 Ω | D(w)| b dx dt < ∞.
We adopt the notation in [START_REF] Duvaut | Inequalities in mechanics and physics[END_REF] (4.3)

(j (w), v) = g Ω | D(w)| -1 D(w) : D(v) b dx.
It is good to mention here that j (w) ∈ L 2 (0, T, V b ) for all w in V b . More precisely

T 0 (j (w), v) dt = T 0 Ω g| D(w)| -1 D(w) : D(v) b dx dt ≤ C T 0 (D(w)) L 2 b (Ω) D(v) L 2 b (Ω) ≤ C w L 2 (0,T ;V b ) v L 2 (0,T :V b ) ≤ C.
We have the following property Proof. By Hölder's inequality, we have first

T 0 j(w ) dt = T 0 Ω | D(w )| b dx dt ≤ T 0 Ω | D(w )| 1+ b dx 1 1+ Ω b dx 1+ dt ≤ C T 0 Ω | D(w )| 1+ b dx 1 1+ dt ≤ C T 0 Ω | D(w )| b dx 1 1+ dt ≤ C T 0 j (w ) dt 1 1+
.

Thus T 0 j(w ) dt 1+ ≤ C T 0 j (w ) dt.
As v -→ j(v) is lower semi continuous for the weak topology of L 2 (0, T ; V b ) (from the definition of weak convergence), then

T 0 j(w) dt ≤ lim inf T 0 j(w ) dt.
Therefore, combining the above we get our result

T 0 j(w) ≤ lim inf T 0 j (w ).
We render the problem into a new approximated one corresponding to j such that:

Ω ∂ t u • v b dx + Ω (u • ∇)u • v b dx + 2µ Ω D(u) : D(v) b dx + λ Ω div u • div v b dx + ∂Ω κ(x)u • v b ds + (j (u ), v) = Ω f • (v -u) b dx ∀v ∈ V b , (4.4)
where the representation (j (•), •) stands for (4.3).

(

) 2
Step 2: Seeking for a sequence (u m ) via Galerkin approximation Consider the canonical isomorphism ∧ : V b -→ V b with {w 1 , ..., w m , ..} being the set of unit eigenfunctions of the operator ∧, Define now the space V m = Span{w 1 , . . . w m }, where {w 1 , . . . , w m } is a free and total family in H b (it is permissible to choose the set as that since V b is separable). In fact this isomorphism is proven for weighted Sobolev spaces by F röhlich in bounded domains, check for instance [START_REF] Fröhlich | The Stokes operator in weighted L q -spaces. II. Weighted resolvent estimates and maximal L p -regularity[END_REF]. Upon projecting our system on V m , and according to the Cauchy theory in a finite space, we can construct a solution u m := u m in V m for the approximated variational inequality (4.4)

Ω ∂ t u m • w j b dx + Ω (u m • ∇u m ) • w j b dx + 2µ Ω D(u m ) : D(w j ) b dx + λ Ω div u m div w j b dx + ∂Ω κ(x)u m • w j b ds + (j (u m ), w j ) = Ω f • w j b dx, 1 ≤ j ≤ m, (4.5)
with u m (0) being the projection of u 0 on V m , and u m being defined in a space interval [0, t m ]. Since

u m = m i=0 (u m , w i ) V b w i ,
we multiply (4.5) by (u m , w j ) V b and sum over j, we get:

Ω ∂ t u m • u m b dx + Ω (u m • ∇)u m • u m b dx + 2µ Ω | D(u m )| 2 b dx + λ Ω | div u m | 2 b dx + ∂Ω κ(x)|u m | 2 b ds + (j (u m ), u m (t)) dx = Ω f • u m b dx. (4.6)
From Lemma 4.1, we deduce that the second term is equal to zero, and since

(j (u), u) = Ω g|D(u)| -1 D(u) : D(u) b dx = Ω |D(u)| +1 dx ≥ 0.
Notice that using Hölder's and Young's inequalities, we can estimate the term on the right hand of equation (4.6) as follows

Ω f • u m b dx ≤ f L 2 b (Ω) u m L 2 b (Ω) ≤ η f 2 L 2 b (Ω) + C(η) u m 2 V b ,
for some arbitrary η > 0. Now, after choosing η sufficiently small, we can deduce using weighted Korn's inequality from Remark 4.1 that

d dt u m 2 L 2 b (Ω) + u m 2 H 1 b (Ω) ≤ C f 2 L 2 b (Ω) .
Hence, we deduce

(4.7) u m ∈ L ∞ (0, T ; H b ) ∩ L 2 (0, T ; V b ),
and u m remains in a bounded set of L ∞ (0, T ; H b ) and L 2 (0, T ; V b ) uniformly with respect to m. The next step is to prove (bu m ) ∈ L 2 (0, T ; V b ) uniformly, i.e for all m, (bu m ) remains in a bounded set of L 2 (0, T ; V b ). Introduce the projection operator P m : V b -→ V m . Since Au m , Bu m and j (u m ) belong to L 2 (0, T ; V b ), then, we can write from (4.5)

Ω ∂ t (bu m ) • w j dx = bf -Au m -Bu m -j (u m ), w j V b ,V b ∀1 ≤ j ≤ m.
Since we have P m ( ∂ t (bu m )) = ∂ t (bu m ), then we can write

∂ t (bu m ) = P m (bf -Au m -Bu m -j (u m )).
Therefore, we get

b∂ t u m = P m (bf -Au m -Bu m -j (u m )) in D (0, T, V m ), := P m (k m ). Since k m is bounded in L 2 (0, T ; V b ), then what remains to show is that P m k m V b ≤ k m V b .
Since ∧ is an isomorphism from V b to V b , then we can deduce that λ 1 2 w j constitute an orthogonal basis of V b for the norm X V b = ∧ -1 X V b . Thus, we get that

P m k m V b = m j=1 (k m , λ 1 2 w j ) L 2 b ≤ m j=1 (k m , λ 1 2 w j ) L 2 b ≤ k m V b .
Finally, we have our aimed result

(4.8) bu m ∈ L 2 (0, T ; V b ).
As a result of (4.7) and (4.8), we can deduce that up to a sequence (denoted again u m ), we have (4.9)

u m * u in L ∞ (0, T ; H b ), u m u in L 2 (0, T ; V b ), bu m bu in L 2 (0, T ; V b ).

Now, in order to use compactness result we must apply a fractional estimate technique to have

||τ h u m -u m || L 2 (0,T -h;L 2 b (Ω)) converges to zero.
Since it is a technical step, though, we refer the reader to [START_REF] Taki | Viscosity effect on the degenerate lake equations[END_REF] for a detailed proof. Thus we are able to use the compactness result and deduce that u m → u strongly in L 2 (0, T, H b ). Now we are concerned in the convergence of terms in (4.5):

(3) Step 3. Passage to the limit in m:

Now, we will prove the convergence of the operators as m → 0.

• Convergence of j (u m ) j (u m ) is bounded uniformly in L 2 (0, T, V b ), hence j (u m ) ξ in L 2 (0, T, V b ).
• Convergence of the linear operator A As for the operator A given by

Au m , v V b ,V b = Ω 2µ D(u m ) : D(v) + λ div u m div v b dx + ∂Ω κu m • v b ds, we have in fact that u m u in L 2 (0, T ; V b ), then ∂ i u j ∂ i (u ) j in L 2 (0, T ; L 2 b (Ω))
, and hence

T 0 Ω D(u m ) : D(v) b dx dt → T 0 Ω D(u ) : D(v) b dx dt, and
T 0 Ω div u m div v b dx dt → T 0 Ω div u div v b dx dt.
Before we treat the convergence on the boundary term, let us prove the following lemma.

Lemma 4.4. Let u and v be two sufficiently smooth vectors such that bu • n = 0 and bv • n = 0, then

(4.10) (v • ∇(bu)) • n = -κbu • v.
Proof. Since we have bu

• n = 0 on ∂Ω, then 0 = ∂ ∂τ (bu • n) = ∂(bu) ∂τ • n + bu • ∂n ∂τ = (τ • ∇(bu)) • n + κbu • τ.
Recall that κ is the curvature of ∂Ω where we have ∂n ∂τ

:= dn ds = κu • τ.
But v is parallel to τ , so (4.10) follows by linearity. Now back to prove the convergence of the boundary term, we notice first using (4.10) and the following identity

∇(bu) : ∇ t v = ∂ i (bu j ) ∂ j v i = ∂ j (v i ∂ i (bu j )) = div(v • ∇(bu))
that we can write

∂Ω κ(u m -u ) • v b ds = - ∂Ω (v • ∇(b(u m -u ))) • n ds = - Ω div(v • ∇(b(u m -u ))) dx = - Ω ∇ t v : ∇(b(u m -u )) dx = - Ω ∇ t v : ∇(u m -u ) b dx + Ω ∇ t v : ∇b b ⊗ (u m -u ) b dx.
We easily remark that the first term on the right hand side of the above equality converges to zero because of the weak convergence of u m to u in V b . For the second term, we need to use a Hardy's type inequality to show that

∇b b ⊗ (u m -u ) is uniformaly bounded in L 2 b (Ω).
Indeed for b = ρ α , we have ∇b/b ∼ 1/ρ, then (see [START_REF] Edmunds | Weighted Hardy inequalities[END_REF])

Ω ∇b b ⊗ (u m -u ) 2 b dx = Ω 1 ρ (u m -u ) 2 b dx ≤ C Ω |∇(u m -u )| 2 b dx.
Thus we get the convergence of the boundary term.

• Convergence of the trilinear term B For the trilinear term, we remark that

b(u m , u m , v) = -b(u m , v, u m ).
Using the fact that

H 1 b (Ω) → L 6 b (Ω) → L 4 b (Ω) (since Ω is bounded, and b ∈ A 3 2 [19]
), and due to the boundedness and strong convergence results proved for u m , we obtain

| T 0 Ω (u m • ∇)v • u m b dx - Ω (u • ∇)v • u b dx dt| = | T 0 Ω (u m -u ) • ∇v • u m + Ω u • ∇v • (u m -u ) b dx dt| ≤ T 0 u m L 4 b ∇v L 2 b u m -u L 4 b dt + T 0 u L 4 b ∇v L 2 b u m -u L 4 b dt ≤ c T 0 ∇v H b u m V b + u V b ∇u m L 2 b + ∇u L 2 b 1 2 u m -u 1 2 L 2 b dt ≤ c u m -u 1 2 L 2 (0,T,L 2 b ) -→ 0 m→0 . • Convergence of the integral containing u m Since we have bu m bu in L 2 (0, T, V b ), thus T 0 Ω ∂ t u m • v b dx dt -→ T 0 Ω ∂ t u • v b dx dt ∀v ∈ V b .

• Convergence of the initial condition

The space V b can be viewed as

V b = D b (Ω) ||.|| H 1 b (Ω)
Now, we choose the orthonormal base of V b such that for all j, we have w j ∈ D b (Ω). In this case,

Ω (u m • ∇)u m • w j b dx is bounded in L 2 (0, T ) since (bu m • ∇)u m is bounded in L 2 (0, T ; (L 1 (Ω)) 2 ). Moreover we have a(u m , w j ) + b(u m , u m , w j ) ∈ L 2 (0, T ).
Thus Equation (4.5) shows that (4.11)

∂ t u m , w j b is bounded in L 2 (0, T ),
and hence Ω u m .w j b dx is bounded in H 1 (0, T ). On the other hand, we know that H 1 (0, T ) is compact in C u (0, T ) (see Theorem III.2.34 in [START_REF] Boyer | Mathematical tools for the study of the incompressible Navier-Stokes equations and related models[END_REF]) which yields

Ω u m • w j b dx → Ω u • w j b dx in C u (0, T ).
The above convergence holds also in L 2 (0, T ) (due to strong convergence of u m in L 2 (0, T ; H b )).

In particular

Ω u m • w j b dx (0) → Ω u • w j b dx (0).
Notice that, by definition, we have u m (0) = P m (u 0 ), so we can write (4.12)

Ω (u m (0, x) -P m (u 0 (x))) • w j (x) b dx = 0, Passing to the limit in Equation (4.12) yields to (4.13)

Ω u • w j b dx (0) = Ω u 0 • w j b dx.
On other hand, by virtue of Equation (4.4), we have for all v ∈ V b ,

∂ t Ω u • v b dx = ∂ t u , v b ∈ L 1 (0, T ) which yields that u , v b ∈ C([0, T ]). Since u ∈ L ∞ (0, T ; H b ) and V b is dense in H b , we infer that (see Lemma 1.4 in [28]). u ∈ C([0, T ]; H b -weak) and that u (0) = u 0 in C([0, T ]; H b -weak).
Now, it remains to show that ξ = j (u ). By now we have got

(4.14) T 0 Ω u • v b dx dt + T 0 Bu , v V b ,V b dt + T 0 Au , v V b ,V b dt + T 0 (ξ, v) dt = T 0 Ω f • v b dx dt.
We choose a function φ such that φ in L 2 (0, T, V b ), bφ in L 2 (0, T, V b ). Now, we introduce the following quantity

X m = T 0 bu m -bφ , u m -φ V b ,V b dt + T 0 (j (u m ) -j (φ), u m -φ) dt + T 0 Au m -Aφ, u m -φ V b ,V b dt.
One can show obviously that X m is positive. For instance, we can rewrite X m such that

X m = T 0 (j (u m ), u m ) + Au m , u m V b ,V b + bu m , u m V b ,V b -(j (u m ), φ) -(j (φ), u m -φ) -Au m , φ V b ,V b -Aφ, u m -φ V b ,V b -bu m , φ V b ,V b -bφ , u m -φ V b ,V b dt.
Substituting (4.5) in the above expression

X m = T 0 bf, u m V b ,V b -(j (u m ), φ) -(j (φ), u m -φ) -Au m , φ V b ,V b -Aφ, u m -φ V b ,V b -bu m , φ V b ,V b -bφ , u m -φ V b ,V b dt. Then X m -→ m→0 T 0 bf, u V b ,V b -(ξ, φ) -(j (φ), u -φ) -Au , φ V b ,V b -Aφ, u -φ -bu , φ V b ,V b -bφ , u -φ V b ,V b dt := X .
Again, take v = u in (4. [START_REF] Farwig | Weighted L q -theory for the Stokes resolvent in exterior domains[END_REF], and then substitute this latter in X , we get

X = T 0 bu , u V b ,V b + Au , u V b ,V b + ξ, u V b ,V b -ξ, φ V b ,V b -(j (φ), u -φ) -Au , φ V b ,V b -Aφ, u -φ V b ,V b -bu , φ V b ,V b -bφ , u -φ V b ,V b dt = T 0 bu -bφ , u -φ V b ,V b + Au -Aφ, u -φ V b ,V b + (ξ -j (φ), u -φ) dt.
As X m ≥ 0, then X ≥ 0. Take now φ = u -θψ with θ ∈ R in the expression of X . This yields

X = T 0 bθψ, , θψ V b ,V b + A(θψ), θψ V b ,V b + (ξ -j (u -θψ), θψ) dt ≥ 0.
Dividing by θ we get

T 0 θ bψ, , ψ V b ,V b + θ A(ψ), ψ V b ,V b + (ξ -j (u -θψ), ψ) dt ≥ 0.
Take θ → 0, we obtain

T 0 (ξ -j (u ), ψ) dt ≥ 0.
This is true for all ψ, hence:

ξ = j (u ).
Therefore in this step we have proved the existence of a sequence (u ) such that:

(4.15)                        u in, L ∞ (0, T ; H b ) ∩ L 2 (0, T ; V b ), bu in L 2 (0, T ; V b ), T 0 bu , v V b ,V b + Bu , v V b ,V b + Au , v V b ,V b + (j (u ), v) = T 0 bf, v V b ,V b ∀v ∈ L 2 (0, T ; V b ). (4)
Step 4: Passage to the limit in . From the previous steps, in particular (4.15), we conclude that there exists a subsequence of u , that is still denoted the same, such that

u u in L 2 (0, T, V b ), u * u in L ∞ (0, T, H b ), bu bu in L 2 (0, T, V b ).
Again, by compactness theory we get u → u strongly in L 2 (0, T ; H b ). We introduce now the following quantity

Z = T 0 bu , v -u V b ,V b + Au , v -u V b ,V b + Bu , v -u V b ,V b + j (v) -j (u ) -bf, v -u V b ,V b dt.
Taking v = u in (4.15) (which is allowed since u ∈ L 2 (0, T, V b )), and then substituting in Z , we get

Z = T 0 j (v) -j (u ) -(j (u ), v -u ) dt.
In fact, since j is coercive, the we get Z ≥ 0. Hence

T 0 bu , v V b ,V b + Au , v V b ,V b + Bu , v V b ,V b + j (v) -bf, v -u V b ,V b dt ≥ T 0 bu , u V b ,V b + Au , u V b ,V b + Bu , u V b ,V b 0 +j (u ) dt.
Now as we have u → u in L 2 (0, T, H b ), then there exists a subsequence such that u (T ) → u(T ) in H b . Therefore:

lim inf →0 T 0 bu , v V b ,V b + Au , v V b ,V b + Bu , v V b ,V b + j (v) -bf, v -u V b ,V b dt ≥ 1 2 lim inf →0 u (T ) 2 L 2 b (Ω) - 1 2 u 0 2 L 2 b (Ω) + lim inf →0 T 0 Au , u V b ,V b dt + lim inf →0 T 0 j (u ) dt.
Due to the fact that all terms on the right hand side are lower semi continuous (norms and linear continuous maps are lower semi continuous), and using lemma 4.3, we obtain

T 0 bu , v V b ,V b + Au, v V b ,V b + Bu, v V b ,V b + j(v) -bf, v -u V b ,V b ≥ 1 2 u(T ) 2 L 2 b (Ω) - 1 2 u 0 2 L 2 b (Ω) + T 0 Au, u V b ,V b dt + T 0 j(u) dt.
Finally, we get (4.16)

T 0 bu , v -u V b ,V b + Au, v -u V b ,V b + Bu, v -u V b ,V b + j(v) -j(u) -bf, v -u V b ,V b dt ≥ 0 ∀v ∈ L 2 (0, T ; V b ).
The last step is to prove that the above inequality holds not in integral form but almost everywhere in (0, T ), in particular, we have to prove

(4.17) bu (t),v -u(t) V b ,V b + Au(t), v -u(t) V b ,V b + Bu(t), v -u(t) V b ,V b + j(v) -j(u(t)) ≥ bf, v -u(t) V b ,V b dt, t ∈ (0, T ), v ∈ L 2 (0, T ; V b ).
Proving that in fact allows us tp prove the existence of solution for system (3.5). Let us fix a random function v ∈ V b , and t 0 ∈ (0, T ). Define

Θ j = t 0 - 1 j , t 0 + 1 j ⊂ (0, T ) j large enough. We introduce v = v t ∈ Θ j u(t) t ∈ [0, T ]\Θ j .
Hence taking v = v, inequality (4.16) becomes

Θj bu + Au + Bu -bf, v V b ,V b dt X + Θj j(v) -j(u) -bu + Au -bf, u V b ,V b dt Y ≥ 0.
Using the Lebesgue theorem in the differentiation of set functions, we have

X -→ j→0 bu (t 0 ) + Au(t 0 ) + Bu(t 0 ) -bf, v V b ,V b , and
Y -→ j→0 bu (t 0 ) + Au(t 0 ) -bf, u(t 0 ) V b ,V b + j(u(t 0 )) -j(v).
But t 0 is arbitrary in (0,T), thus we get

bu (t), v -u(t) V b ,V b + Au(t), v -u(t) V b ,V b + Bu(t), v -u(t) V b ,V b + j(v) -j(u(t)) ≥ bf, v -u(t) V b ,V b , (4.18)
for all t ∈ (0, T ), for all v ∈ L 2 (0, T ; V b ). By this we end the proof of existence.

Uniqueness of solution. let u 1 , u 2 be two solutions of the variational inequality (4.18). We take v = u 2 (t) (resp v = u 1 (t)) as a test function in the variational inequality satisfied by u 1 (resp. u 2 ). Let U = u 1 -u 2 . Now, if we add both inequalities we get

Ω ∂ t U • U b dx + 2µ Ω | D(U )| 2 b dx + λ Ω | div(U )| 2 b dx - Ω (u 1 •∇) u 1 •U b dx + Ω (u 2 •∇) u 2 •U b dx ≤ 0.
Therefore, using Remark 4.1, and the fact that

Ω (u • ∇)v • v b dx = 0 for all u, v in V b , we get (4.19) 1 2 d dt Ω |U | 2 b dx + 2C 1 µ U 2 V b ≤ Ω (u 1 •∇) u 1 •U b dx - Ω (u 2 •∇) u 2 •U b dx = Ω (u 1 •∇)U • U b dx + Ω (u 1 •∇) u 2 •U b dx - Ω (u 2 •∇) u 2 •U b dx = Ω (U • ∇) u 2 •U b dx,
where C 1 is the constant resulting from Korn's inequality. We estimate the term on the right hand side of Inequality (4.19) using Weighted Gagliardo-Nirenberg inequality and Young's inequality as follows (4.20)

Ω (U • ∇) • u 2 •U b dx ≤ U 2 L 4 b (Ω) ∇u 2 L 2 b (Ω) ≤ C ∇u 2 L 2 b (Ω) U V b U L 2 b (Ω) ≤ C ∇u 2 L 2 b (Ω) 1 2C 1 µ U 2 L 2 b (Ω) + C 1 µ 2 U 2 V b .
From Inequality (4.19) and estimate (4.20), we deduce that 1 2

d dt U | 2 L 2 b (Ω) ≤ C ∇u 2 L 2 b (Ω) U 2 L 2 b (Ω)
. which readily ensures that U = 0 whence u 1 -u 2 = 0 by applying a Gronwall's type inequality. Hence, the proof of uniqueness of solution is finished.

Integrating in time gives

t 0 bu g , v V b ,V b + a(u g , v) + b(u g , u g , v) + j(v) -j(u g ) -bf, v -u g V b ,V b dt ≥ 1 2 u g (t) 2 L 2 b (Ω) - 1 2 u g (0) 2 L 2 b (Ω) + t 0 a(u g , u g ) dt.
Using the fact that RHS is lower semi continuous Therefore, for all t in (0, T ), we have

(5.2) lim inf g→0 t 0 bu g , v V b ,V b + a(u g , v) + b(u g , u g , v) -bf, v -u g V b ,V b dt ≥ lim inf g→0 1 2 u g (t) 2 L 2 b (Ω) - 1 2 u 0 2 L 2 b (Ω) + t 0 a(u g , u g ) dt .
Due to the convergence of the LHS terms and the last term in the RHS of (5.2) (following the same strategy as in the proof of theorem (4.2) in section 4), and due to the lower semi-continuity of the norm operator for the topology of L 2 b , we obtain

t 0 bu , v V b ,V b + a(u, v) + b(u, u, v) -bf, v -u V b ,V b dt ≥ 1 2 u(t) 2 L 2 b (Ω) - 1 2 u 0 2 L 2 b (Ω) + t 0 a(u, u) dt = 1 2 t 0 d dt u, u L 2 b (Ω) dt + t 0 a(u, u) dt.
Thus, for all v ∈ L 2 (0, T, V b ), and for all t in (0, T ), we get

t 0 bu , v -u V b ,V b + a(u, v -u) + b(u, u, v -u) dt ≥ t 0 bf, v -u V b ,V b dt.
Again, using Lebesgue theory in the differentiation of sets functions, we obtain for all v ∈ L 2 (0, T ;

V b ) (5.3) bu , v -u V b ,V b + a(u, v -u) + b(u, u, v -u) ≥ bf, v -u V b ,V b a.e in [0, T ]. If we suppose that φ = ±(u -v) ∈ L 2 (0, T ; V b ), then substituting v in (5.3) yields bu , φ V b ,V b + a(u, φ) + b(u, u, φ) = bf, φ V b ,V b a.e in [0, T ],
which means that u satisfies the weak formulation of the viscous Lake system. Thus we end the proof.

Numerical Scheme

In this section, we propose in the one dimensional case a numerical scheme for the approximation of the studied model i.e. the system (1.3)-(1.4).

6.1. Semi-discrete scheme. The system (1.3)-(1.4) can be rewritten as follows

∂ t X + ∂ x F (X) -b ∂ x p = bf, ∂ x X = 0, where X = (bu), F (X) = bu 2 .
Notice that in the 1d case, the definition of σ given by (1.4) reduces to (6.1)

σ :=    2µb ∂ x u + gb ∂ x u | ∂ x u| if ∂ x u = 0, |σ| < bg, if ∂ x u = 0.
For the time discretisation, we denote t n = k≤n ∆t k where the time steps ∆t k will be precised later though a CFL condition. Following [START_REF] Chorin | Numerical solution of the Navier-Stokes equations[END_REF], we use an operator splitting technique resulting in a two step scheme

X n+1/2 -X n ∆t n = -∂ x F (X n ) + bf n , (6.2) X n+1 -X n+1/2 ∆t n = b ∂ x p n+1 , (6.3)
where the quantity X n+1 satisfies the divergence free constraint (6.4) ∂ x (X n+1 ) = 0.

The system (6.2)-( 6.4) has to be completed with suitable boundary conditions that will be precised later, see paragraph 6.4. More precisely, the prediction step (6.2) consists in the resolution of advection diffusion equation i.e.

X n+1/4 = X n -∆t n ∂ x F (X n ) + ∆t n ∂ x µb ∂ x X n+1/4 b , (6.5) X n+1/2 = X n+1/4 + ∆t n ∂ x (bσ n+1/4 ) + ∆t n bf n , (6.6) where σn+1/4 = g ∂xu n+1/4 | ∂xu n+1/4 | if ∂ x u n+1/4 = 0 |σ n+1/4 | ≤ g else
Notice that the definition of σn+1/4 will be precised hereafter. Notice also that in order to avoid a too restrictive CFL condition we propose an implicit discretisation of the linear viscosity term.

Concerning the correction step (6.3)-(6.4), inserting (6.4) into (6.3) gives the elliptic equation governing the pressure p n+1 under the form (6.7) ∂ x b ∂ x p n+1 = -1 ∆t n ∂ x X n+1/2 . Thus, the numerical approximation of (1.3)-(1.4) consists in the numerical resolution of Eqs. (6.5)-(6.6), together with (6.7) and (6.3). 6.2. Discrete scheme. To approximate the solution (X, p) T of the system (1.3)-(1.4), we use a combined finite volume/finite element framework. We assume that the computational domain is discretized with I nodes x i , i = 1, . . . , I. We denote C i the cell (x i-1/2 , x i+1/2 ) of length ∆x i = x i+1/2 -x i-1/2 with x i+1/2 = (x i + x i+1 )/2. We denote X n i = (b i u n i ) with

X n i ≈ 1 ∆x i Ci X(x, t n )dx,
the approximate solution at time t n on the cell C i . The pressure p is discretized on a staggered grid (in fact the dual mesh if we consider the 2d case) 1/2 = (u n 1 + q in (t n )/b 1 )/2 and assuming b 0 = b 1 > 0. For the parabolic part of Eq. (6.8) we use Dirichlet boundary conditions defined by q in (t n ).

p n i+1/
Since the inflow q in (t n ) is prescribed, it is natural to assume Neumann boundary at the entry for p in Eq. (6.11) i.e. ∂ x p| 0 (t n+1 ) = 0. Now we consider the boundary at the exit of the domain i.e. at abscissa x I+1/2 . Assuming Neumann boundary conditions, we define As mentioned in paragraph 6.4, the inflow is prescribed at the entry x = 0 with q in (t) = 2 + sin(2πt/T ), the simulations are carried out over the time interval (0, T ) with T = 20 seconds.

Lemma 4 . 1 .

 41 Let Ω be an open bounded Lipschitz domain of R 2 , and b satsifying Hypothesis (II). Then, for u

Lemma 4 . 3 .

 43 If w converges weakly to w in L 2 (0, T ; V b ), then

  , and since | D(u g )| and | D(v)| are bounded in L 1 (0, T ; H b ), then lim g→0 j(v) = lim g→0 Ω g| D(v)| b dx = 0, and lim g→0 j(u g ) = lim g→0 Ω g| D(u g )| b dx = 0.

F n I+1/ 2 6 . 5 ./ 2 ,

 2652 = u n I X n I ,and ∂ x X| I+1/2 (t n+1/4 ) = 0 with b I+1 = b I for the parabolic part. In this context, it is convenient to assume Dirichlet boundary at the exit for p in Eq. (6.11) i.e. p| I+1 (t n+1 ) = 0. For a more complete justification of the choices for the boundary conditions, the reader can refer to[START_REF] Aïssiouene | A two-dimensional method for a dispersive shallow water model[END_REF]. Simulation results. We present now some simulations results for the model (1.3)-(1.4) with the numerical scheme (6.8)-(6.11) where we have chosen the source term f f = g ∂ x b, mimicking the effects of the slope over the fluid rheology.We consider a fluid domain defined by x ∈ [0, x max ] with x max = 20 meters and I = 500 nodes and a domain profile (see Fig.1) defined by b(x) = 1the fluid is initially at rest i.e. u 0 i = 0, ∀i ∈ I.

Figure 1 .

 1 Figure 1. Chosen profile for b(x).

Figure 2 .Figure 3 .

 23 Figure 2. (a) variations of the velocity u and (b) variations of the pressure p in the fluid domain at time t = T /2.

  = x i+1 -x i .Hence, the discrete (in space and time) version of Eqs. (6.5)-(6.6),(6.7) is given by

	with u n
								2 ≈	1 ∆x i+1/2	xi+1 xi	p(x, t n )dx,
	with ∆x i+1/2 X i n+1/4	= X n i -	∆t n ∆x i	F n i+1/2 -F n i-1/2
	(6.8)			+µ	∆t n ∆x 2 i	b i+1/2 b i+1	X i+1 n+1/4	-	b i+1/2 + b i-1/2 b i	X i n+1/4	+	b i-1/2 b i-1	X i-1 n+1/4	,
	(6.9)	X i n+1/2	= X i n+1/4	+	∆t n ∆x i	b i+1/2	σn+1/4 i+1/2 + τ n i+1/2--b i-1/2	σn+1/4 i-1/2 + τ n i-1/2+	,
	(6.11)	X n+1 i	= X i n+1/2	+	∆t n ∆x n+1/2 i+1	-X i n+1/2	,

i b i p n+1 i+1/2 -p n+1 i-1/2 , (6.10) b i+1 p n+1 i+3/2 -(b i+1 + b i )p n+1 i+1/2 + b i p n+1 i-1/2 = -∆x i ∆t n X

Newtonian fluids as a limit of Non-Newtonian fluids

Viscous lake systems for Newtonian fluids are asymptotically derived systems according to two successive approximations that are characterized by the smallness of non dimensional parameters. The first is the rigid lid approximation assuming that the typical deviation of the top of the fluid's surface from the mean level is much smaller than the typical depth. This smallness can be regarded also as a Froude number due to the dynamics of the physical case at hand. The second approximation is the shallow water approximation characterizing the ansatz of the smallness of the typical depth compared to the typical horizontal length, see for instance [START_REF] David Levermore | A shallow water model with eddy viscosity for basins with varying bottom topography[END_REF]. Nevertheless, in this section, we will prove the existence of a weak solution of the viscous lake system by passing to the zero limit of g (yield stress limit) in the variational inequality satisfied by the weak solution of Bingham system (1.3). Roughly speaking, one can naively conclude that the fluid will behave as a viscous liquid once g vanishes, yet for the mathematical justification we need to set on a rigorous proof. So, we will prove the following theorem.

Theorem 5.1. Let Ω a bounded Lipschitz domain and b satisfying Hypothesis (II). Consider the Bingham model introduced in section 3. We suppose that all the parameters therein are fixed except for the the yield stress limit g which is assumed to be independent of other parameters and rendered to vary (consequently taken to zero). Denote by u g the solution of system (3.5) proved in section 4. Then, there exits

where u is a weak solution of the viscous Lake system.

Proof. Let's recall the definition of the weak solution of viscous lake equations:

) is said to be a weak solution of the viscous lake equations if it satisfies

2), we find that the weak solution u g satisfies 1 2

Thus, u g remains in a bounded set with respect to g in L ∞ (0, T ; H b ) ∩ L 2 (0, T ; V b ). In addition, from previous section, we have

Following the same compactness technique used in previous section, we get u g -→ u ∈ L 2 (0, T ; H b ), and the convergence of the corresponding operators follows as in section 4. Hence u g satisfies

with (6.12) b i+1/2 = min{b i , b i+1 }, and F n i+1/2 is a numerical flux accounting for the advection term bu 2 and any classical flux e.g. full upwind, Rusanov. . . can be used. In the simulation results given at the end of this section, an upwind formula has been used and having the form (6.13)

i+1/2-and τ n+1/4

i-1/2+ correspond to discretisation using the apparent topography technique (see [START_REF] Bouchut | Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources[END_REF]) of the source term f and we have

It remains to define the quantities σn+1/4 i±1/2 in Eq. ( 6.9) and we use the strategy proposed by Bouchut [4, paragraph 4.12.1]. The definition of σ given by (6.1) has to be understood as multivalued: σ can be any value in [-bg, bg] when ∂ x u = 0. When ∂ x u = 0, Eqs. (6.5)-(6.6) become

The formula (6.14) is consistent with the definition (6.1) because |σ

, with b ε i = max{b i , ε} and 0 < ε 1.

6.4. Boundary conditions. Boundary conditions have to be defined for Eqs. (6.8) and (6.11) at both side of the domain. And we have to face two difficulties

• Eq. (6.8) contains an hyperbolic part and a parabolic part whereas Eq. (6.11) is an elliptic equation,

• the boundary conditions applied to (6.11) have to be consistent with those applied (6.8).

The proposed solution has been adapted from [START_REF] Aïssiouene | A two-dimensional method for a dispersive shallow water model[END_REF] but notice that other solutions can be investigated since the coupling of the boundary conditions between a hyperbolic step and a parabolic/elliptic step is far from being obvious.

Let us consider the boundary at the entry of the domain i.e. at abscissa x 1/2 . We assume the inflow is prescribed typically b 0 u n 0 = q in (t n ) where q in (t n ) is a given quantity then the definition (6.13) can be used to define F n 1/2 under the form


Jinzeng Wang

Carole Daiguebonne

Yan Suffren

Thierry Roisnel

Stéphane Freslon

Guillaume Calvez

Kevin Bernot

Olivier Guillou

A new

come

ABSTRACT.

Reactions

KEYWORDS.

Lanthanide coordination polymers -Crystal structure -Luminescent propertiesazoxybenzene-3,3',5,5'-tetracarboxylic acid -Thermal analysis

HIGHLIGTHS.

First azoxybenzene-3,3',5,5'-tetracarboxylate lanthanide coordination polymers Excitation and emission wavelengths in the IR region Structural characterization of a series of lanthanide coordination polymers

INTRODUCTION

Lanthanide-based coordination polymers arouse increasing interest because of their various potential applications [START_REF] Cui | Photonic functional metal-organic frameworks[END_REF] such as lighting and display [START_REF] Bünzli | Rising stars in science and technology : Luminescent lanthanide materials[END_REF][START_REF] Li | Luminescence modulation, white light emission and energy transfer in a family of lanthanide Metal-Organic Frameworks based on a planar -conjugated ligand[END_REF], thermometric probes, [START_REF] Rocha | Lanthanide organic framework luminescent thermometers[END_REF][START_REF] Rao | A highly sensitive mixed lanthanide metal organic framework self calibrated luminescent thermometer[END_REF][START_REF] Yang | A family of doped lanthanide metal organic frameworks for wide-range temperature sensing and tunable white light emission[END_REF][START_REF] Khudoleeva | Terbium-europium fluorides surface modified with benzoate and terephthalate anions for temperature sensing : Does sensitivity depends on the ligand ?[END_REF] chemical sensing [START_REF] Bünzli | Lanthanide luminescence for biomedical analyses and imaging[END_REF][START_REF] Dang | Lanthanide metal-organic frameworks showing luminescence in the visible and near infrared regions with potential for acetone sensing[END_REF] or fight against counterfeiting [START_REF] Guillou | A long journey in lanthanide chemistry : from fundamental crystallogenesis studies to commercial anti-counterfeiting taggants[END_REF][START_REF] Gooch | Taggant materials in forensic science[END_REF][START_REF] Andres | A new counterfeiting feature relying on invisible luminescent full color images printed with lanthanide-based Inks[END_REF][START_REF] Wang | Anticounterfeiting quick response code with emission color of invisible Metal-Organic Frameworks as encoding information[END_REF], for instance. Our group, which is involved in that field for more than twenty years, has synthesized numerous lanthanide-based coordination polymers, playing with both the nature of the metallic nodes [START_REF] Calvez | Lanthanide-based hexanuclear complexes and their use as molecular precursors[END_REF][START_REF] Calvez | Unprecedented lanthanide containing coordination polymers constructed from hexanuclear molecular building blocks :{[Ln6O(OH)8](NO3)2(bdc)(Hbdc)2,2NO3,H2bdc}n[END_REF] or of the ligands [START_REF] Fan | A family of lanthanide based coordination polymers with boronic acid as ligand[END_REF], targeting new interesting porosity [START_REF] Kerbellec | New lanthanide based coordination polymers with high potential porosity[END_REF][START_REF] Kustaryono | New family of porous lanthanide-containing coordination polymers : [Ln2(C2O4)3(H2O)6] with Ln = La -Er[END_REF], magnetic [START_REF] Calvez | Sterically-induced synthesis of 3d-4f one-dimensional compounds: a new route towards 3d-4f Single Chain Magnets[END_REF][START_REF] Yi | Rational organization of lanthanidebased SMMs dimers into three dimensional networks[END_REF] or optical [START_REF] Badiane | High britness and easy color modulation in lanthanide-based coordination polymers with 5-methoxyisophthalate as ligand: Toward emission colors additive strategy[END_REF][START_REF] Freslon | Brightness and color tuning in a series of lanthanide-based coordination polymers with benzene 1,2,4,5-tetracarboxylic acid as ligand[END_REF][START_REF] Freslon | Influence of photo-induced electron transfer on lanthanide-based coordination polymers luminescence : A comparison between two pseudo-isoreticular molecular networks[END_REF][START_REF] Fan | Heteronuclear lanthanide-based coordination polymers exhibiting tunable multiple emission spectra[END_REF] properties. Because of the unique optical properties of the lanthanide ions [START_REF] Eliseeva | Rare earths : jewels for functionnal materials of the future[END_REF][START_REF] Eliseeva | Lanthanide luminescence for functionnal materials and biosciences[END_REF][START_REF] Bünzli | Basics of lanthanide photophysics[END_REF] much work has been devoted to the modulation of the intensity and color of the luminescence of lanthanide-based coordination polymers [START_REF] Haquin | Color and Brightness tuning in heteronuclear lanthanide teraphthalate coordination polymers[END_REF][START_REF] Bünzli | On the design of highly luminescent lanthanide complexes[END_REF][START_REF] Le Natur | Coordination polymers based on hexanuclear rare earth complexes : Toward independant luminescence brightness and color emission[END_REF][START_REF] Dang | Tunable emission based on lanthanide(III) metal organic frameworks : an alternative approach to white light[END_REF]. One of the challenge in that field is to produce intense white light [START_REF] Li | Luminescence modulation, white light emission and energy transfer in a family of lanthanide Metal-Organic Frameworks based on a planar -conjugated ligand[END_REF][START_REF] Yang | A family of doped lanthanide metal organic frameworks for wide-range temperature sensing and tunable white light emission[END_REF][START_REF] Da Luz | Controlling the energy transfer in lanthanide-organic frameworks for the production of white-light emitting materials[END_REF][START_REF] Lekshmi | Lanthanide complex-derived white-light emitting solids: A survey on design strategies[END_REF]. The red and green components can be easily obtained using luminescence of the Eu 3+ and Tb 3+ ions respectively [START_REF] Zhang | Coordination polymers for energy transfer: Preparations, properties, sensing applications, and perspectives[END_REF]. The blue component is more difficult to produce because it is absent from trivalent lanthanide ions emission colors. Alternatively, this blue emission can be achieved by ligand phosphorescence [START_REF] Li | Luminescence modulation, white light emission and energy transfer in a family of lanthanide Metal-Organic Frameworks based on a planar -conjugated ligand[END_REF][START_REF] Yang | Linear dependance of photomuminescence in mixed Ln-MOFs for color tunability and barcode application[END_REF]. Therefore, we have decided to investigate lanthanide coordination polymers based on azoxybenzene-3,3',5,5'-tetracarboxylic acid ligand (Scheme 1). Indeed this planar ligand presents an extended conjugated -system that could induce blue phosphorescence.

Additionally, because of its low energy excited states, efficient ligand-to-metal energy transfer is not expected, which is supposed to be favorable to ligand emission. At last its four carboxylate functions are appropriate for coordinating lanthanide ions [START_REF] Karraker | Coordination of trivalent lanthanide ions[END_REF] and, to the best of our knowledge, there is, to date, no example of lanthanide coordination polymer based on this ligand.

Scheme 1. Schematic representation of azoxybenzene-3,3',5,5'-tetracarboxylic acid hereafter referred as H4aobtc.

EXPERIMENTAL SECTION.

Synthesis of the ligand azoxybenzene-3,3',5,5'-tetracarboxylic acid (H4aobtc).

Nitro-isophthalic acid (C8H5NO6) was purchased from Aldrich (98%) and used without any further purification. The ligand was synthesized on the basis of procedures that have previously been reported (Scheme 2) [START_REF] Wang | Design, synthesis and photochemical behavior of polybenzylesters dendrimers with azobenzene groups throughout their architecture[END_REF][START_REF] Wang | Metal-organic frameworks based on double-bon-coupled diisophthalate linkers with High hydrogene and methane uptakes[END_REF][START_REF] Ebel | Studien in der pentaasteranreihe[END_REF]. Precipitation occurred and the yellow precipitate was filtered. It was then dissolved in 50 mL of an aqueous solution of NaOH (1 mol.L -1 ) and the resulting solution was filtered to remove insoluble residues. Then, pH of the filtrate was adjusted to pH ≈ 3 using an aqueous solution of HCl (3 mol.L -1 ). Precipitation of a yellow solid occurred that was filtered and dried in air. IR spectrum (Figure S1) shows characteristic bands for protonated carboxylic function (1680 cm -1 for the ν(C=O) stretching mode), for N=N-O groups in azoxy compounds (1300 cm -1 and 1440 cm -1 for the νs symmetric and νas antisymmetric stretching modes, respectively) and finally for C-OH groups and water (broad band at 3350 cm -1 for the ν(O-H) stretching modes) [START_REF] Nakamoto | Infrared and raman spectra of inorganic and coordination compounds Part A: Theory and applications in inorganic chemistry[END_REF]. IR spectrum [Y(Haobtc)(H2O)2•2H2O]∞ (Figure S1) is similar to the one of the ligand except that it shows characteristic bands for deprotonated carboxylate function (1380 and 1590 cm -1 ) as well as for protonated carboxylic function (1680 cm -1 ) which suggests that at least one carboxylate group is protonated in the crystal structure [START_REF] Nakamoto | Infrared and raman spectra of inorganic and coordination compounds Part A: Theory and applications in inorganic chemistry[END_REF].

Synthesis of [Ln(Haobtc)(H2O)2•2H2O]∞ with

Elemental analyzes of [Ln(Haobtc)(H2O)2•2H2O]∞ with Ln = Nd -Er plus Y are listed in Table S1.

Single crystal X-ray diffraction.

A single crystal of [Y(Haobtc)(H2O)2•2H2O]∞ was mounted on a Bruker D8 Venture.

Crystal data collection was performed with MoKα radiation (λ = 0.70713 Å) at 150 K. Crystal structure was solved by direct methods using the SIR97 program [START_REF] Altomare | EXPO: a program for full powder pattern decomposition and crystal structure solution[END_REF] and then refined with full matrix least-squares methods based on F 2 (SHELX97) [START_REF] Sheldrick | SHELXL : High-Resolution Refinement[END_REF] with the aid of WINGX program [START_REF] Farrugia | WinGX suite for smallmolecule single-crystal crystallography[END_REF]. All non-hydrogen atoms were refined anisotropically using the SHELXL program. Hydrogen atoms bound to the organic ligand were located at ideal positions.

Hydrogen atoms of water molecules were not located. Absorption correction were performed using WinGX program facilities [START_REF] Farrugia | WinGX suite for smallmolecule single-crystal crystallography[END_REF][START_REF] Sluis | BYPASS: an Effective Method for the Refinement of Crystal Structures Containing Disordered Solvent Regions[END_REF]. Full details of the X-ray structure determination of the crystal structure have been deposited with the Cambridge Crystallographic Data Center under the depositary number CCDC-1871493. Crystal and final structure refinement data are listed in Table 1. Powder X-ray diffraction.

Experimental diagrams have been collected with a Panalytical X'pert Pro diffractometer equipped with a X'Celerator detector. Typical recording conditions were 45 kV, 40 mA for CuK ( = 1.542Å) in / mode. Calculated pattern were produced using the Mercury and WinPLOTR software programs [START_REF] Roisnel | A Window Tool for Powder Diffraction Patterns Analysis[END_REF][START_REF] Roisnel | WinPLOTR : a windows tool for powder diffraction pattern analysis[END_REF][START_REF] Macrae | Mercury CSD 2.0 -New features for the visualization and investigation of crystal structures[END_REF].

Thermal analysis.

Thermal analysis was performed with a Perkin-Elmer Pyris-Diamond analyzer in a platinum crucible between room temperature and 1000°C under N2 atmosphere with a heating rate of 5°C.min -1 . The compound was maintained at 1000°C under air atmosphere for one hour to insure complete combustion.

Optical measurements.

Solid and aqueous solution UV-visible absorption spectra have been recorded with a

Perkin Elmer Lambda 650 spectrometer equipped with a 60 mm integrated sphere.

Solid state luminescence measurements have been performed on a Horiba Jobin-Yvon

Fluorolog III fluorescence spectrometer equipped with a Xe lamp 450 W, a UV-Vis photomultiplier (Hamamatsu R928, sensitivity 190-860 nm) and an infrared-photodiode cooled by liquid nitrogen (InGaAs, sensitivity 800-1600 nm) at room temperature. The visible phosphorescence of the ligand and the infrared emission of the Nd(III) were measured directly on powder samples introduced in quartz capillary tubes or on powder samples pasted on copper plates with a silver lacquer. The phosphorescence of the ligand has been measured at 77 K inside a small Dewar that contained liquid nitrogen. Appropriate filters were used to remove the residual excitation laser light, the Rayleigh scattered light and associated harmonics from spectra. All spectra were corrected for the instrumental response function.

The lifetime for the phosphorescence was measured directly with Horiba Jobin-Yvon Fluorolog III fluorescence spectrometer coupled with an additional TCSPC module (Time-Correlated-Single-Photon-Counting) and a 320 nm pulsed Delta-Diode.

FT-IR spectra were obtained from solid samples with a Perkin Elmer Frontier FT-IR spectrometer equipped with a MIR-ATR module between 4000 cm -1 and 650 cm -1 .

RESULTS AND DISCUSSION.

Description of the crystal structure of [Y(Haobtc)(H2O)2•2H2O]∞.

The crystal structure contains only one independent Y 3+ ion. This ion is linked to eight oxygen atoms that form a slightly distorted dodecahedron. Two out of these oxygen atoms belong to coordination water molecules. The six other belong to carboxylate groups from five different ligands: one according to a bidentate mode and the four other according to a monodentate one. The crystal structure presents only one independent ligand that is coordinated to five lanthanide ions (see Figure 2). One out of the four carboxylic functions is protonated. This is supported by the IR spectrum (Figure S1) that shows the characteristic peak of a protonated carboxylic acid (1680 cm -1 ). In the ligand, oxygen atoms bound to the nitrogen atoms have 0.5 occupancy factors. Surprisingly, this crystal structure is very close to that of a Dy 3+ -based coordination polymer with azobenzene-3,3',5,5'-tetracarboxylic acid (H4abtc) as ligand that has been recently reported (CCDC-1855292) [START_REF] Castells-Gil | Implementation of slow magnetic relaxation in a SIM-MOF through a structural rearrangement[END_REF]. This crystal structure can be described as the superimposition of double-planes that spread perpendicular the a axis. In the double-plane molecular motif, there are four different

Ln-Ln distances depending on the ligand coordination: 5.452 Å between two lanthanide ions linked by the same carboxylate group, 10.052 Å or 8.997 Å between lanthanide ions linked by two different carboxylate groups bound to the same benzene ring and 8.996 Å between lanthanide ions linked by two carboxylate groups bound to two different benzene rings (See Figure 3). Between two adjacent double-plans, the shortest Ln-Ln distance is 6.689 Å. The crystal structure presents channels, that spread along the a and b axes, in which crystallization water molecules are located. These crystallization water molecules are strongly bound to the molecular framework via a hydrogen-bonds network that involve crystallization and coordination water molecules, oxygen atoms of the carboxylate groups and of the azoxy groups of the ligands (Figure 4 and Table S2).

Thermal analysis of [Y(Haobtc)(H2O)2•2H2O]∞.

Thermal analysis of the coordination polymer (Figure 5) shows a first weight loss before 200°C (14%) that can be attributed to the departure of the four crystallization and coordination water molecules (calc. 13.7%). Then, the anhydrous compound [Y(Haobtc)]∞ is stable between 200°C and 350°C. Above this temperature the decomposition of the ligand occurs. At 1000°C, the yttrium oxide is obtained. This result is confirmed by the powder X-ray diffraction diagram of the residual solid (Figure S3) that can be indexed by JCPDS-05-0574. effect [START_REF] Weissman | Intramolecular energy transfer -The fluorescence of complexes of europium[END_REF]. Commonly, energy of the first excited triplet state is estimated on the basis of the first edges of the phosphorescence spectrum of the Gd 3+ -derivative. This spectrum has been recorded at 77 K (Figure S4). The lowest wavelength emission edge of the spectrum is about 360 nm (27780 cm -1 ) which does not favor an efficient ligand-to-metal energy transfer according to Latva's empirical rules [START_REF] Latva | Correlation between the lowest triplet state energy level of the ligand and lanthanide luminescence quantum yields[END_REF] that are commonly evoked for Tb 3+ -and Eu 3+ -based compounds. Moreover, luminescence decay of [Gd(Haobtc)(H2O)2•2H2O]∞ has been measured. It is bi-exponential with 1 = 3.50(2) µs and 2 = 18.9(4) µs. These lifetimes are very short for phosphorescence that is three to four orders of magnitude smaller than what is usually observed for lanthanide-based coordination polymers. [START_REF] Badiane | Lanthanide based coordination polymers with a 4,5-dichlorophthalate ligand exhibiting highly tunable luminescence : Toward luminescent bar codes[END_REF] Additionally, except for the Nd 3+ -derivative (Figure 6), no luminescence is observed by direct excitation of the lanthanide ion. This can be related to the optical properties of the ligand. Indeed, absorption spectrum of the Gd 3+ -based compound has been recorded (Figure S5). It shows that the ligand absorbs up to 550 nm (18180 cm -1 ) which prevents direct excitation of lanthanide ions that present absorption bands below 400 nm such as Eu 3+ , Tb 3+

or Dy 3+ , for instance. Furthermore, we observe a significant overlap between the absorption band (singlet states) and the phosphorescence band (triplet states). This indicates that energy levels of the triplet are lower than the energy levels of the different lanthanides that emit in the visible (Sm, Eu, Tb, Dy). So, a back transfer probably occurs and quenches any luminescence of the rare earth in the visible.

On the opposite, the Nd 3+ -derivative emission spectrum shows sizeable luminescence in the IR domain: the energy of the excited state of the Nd(III) (11400 cm -1 ) is lower than the energy state of the ligand and the IR emission occurs without significant back transfer. The three classical emissions for the Nd(III) are observed at 878, 1059 and 1338 nm (assigned 4 F3/2 → 4 I9/2, 4 F3/2 → 4 I11/2 and 4 F3/2 → 4 I13/2, respectively), under direct excitation at 586 nm ( 4 I9/2 → 2 G7/2, 2 G5/2), 747 nm ( 4 I9/2 → 4 F7/2, 4 S3/2) and 805 nm ( 4 I9/2 → 4 F5/2, 2 H9/2) (Figure 6).

CONCLUSION AND OUTLOOKS

A series of lanthanide-based coordination polymers with azoxybenzene-3,3',5,5'-tetracarboxylic acid as ligand has been synthesized and structurally characterized. Compounds that constitute this family don't show luminescence in the visible region. However the Nd 3+ -derivative shows sizeable emission band in the IR region, under excitation wavelengths comprised between 586 nm and 805 nm that are compatible with potential applications in biological media.

  Haobtc)(H2O)2•2H2O]∞ with Ln = Nd-Er plus Y. The crystal structure has been solved on the basis of the Y 3+ -derivative. It crystallizes in the triclinic system, space group P-1 (n°2) with the following cell parameters: a = 6.6890(18) Å, b = 10.052(3) Å, c = 13.879(4) Å,  = 75.756(9)°,  = 77.551(9)°,  = 83.964(9)° and Z = 2. The crystal structure is two-dimensional (2D). Thermal properties and luminescent properties of the Nd 3+ -containing compound have been studied.

Scheme 2 .

 2 Scheme 2. Schematic representation of the synthetic route for H4aobtc.

  Ln = Nd -Er plus Y. 0.2 mmol of a lanthanide nitrate (Ln(NO3)3.6H2O), 0.1 mmol (37.4 mg) of H4aobtc, 0.8 mL of aqueous HCl (12.5 mol.L -1 ), 4 mL of water and 8 mL of acetonitrile were put in a 24 mL Parr autoclave. Parr autoclave was heated at 110°C during four days. Then autoclave was cooled down at a cooling rate of 3°C per hour. The obtained microcrystalline powder was filtered and dried at ambient air. Single crystals suitable for X-ray diffraction crystal structure determination were obtained with the yttrium-derivative (Figure1).

Figure 1 .

 1 Figure 1. Picture of a single crystal of [Y(Haobtc)(H2O)2•2H2O]∞

Figure 2 .

 2 Figure 2. Projection views of coordination modes of the Y 3+ ion (left) and of the Haobtc 3- ligand (right) in [Y(Haobtc)(H2O)2•2H2O]∞.

Figure 3 .

 3 Figure 3. Lanthanide distances in a double-plane molecular motif (top), between two adjacent double-planes (bottom right) and depending on the ligand coordination modes (bottom left) in [Y(Haobtc)(H2O)2•2H2O]∞.

Figure 4 .

 4 Figure 4. Hydrogen-bonds network (light blue broken lines) in [Y(Haobtc)(H2O)2•2H2O]∞.

Figure 5 .

 5 Figure 5. Thermal analysis of [Y(Haobtc)(H2O)2•2H2O]∞.

Figure 6 .

 6 Figure 6. Solid state excitation and emission spectra of [Nd(Haobtc)(H2O)2•2H2O]∞ measured at room temperature under different excitation wavelengths in the visible and the IR.

Table 1 .

 1 Crystal and final structure refinement data for

	[Y(Haobtc)(H2O)2•2H2O]∞
	Molecular formula	YC16H15N2O13
	System	triclinic
	Space group (No.)	P-1 (2)
	a (Å)	6.6890(18)
	b (Å)	10.052(3)
	c (Å)	13.879(4)
	α (°)	75.756(9)
	β (°)	77.551(9)
	γ (°)	83.964(9)
	V (Å 3 )	881.9(4)
	Z	2
	Formula weight (g.mol -1 )	532.21
	Dcalc (g.cm -3 )	2.004
	µ (mm -1 )	3.391
	R (%)	9.03
	RW (%)	20.79
	GoF	1.145
	N° CCDC	1871493

ACKNOWLEDGEMENTS

The China Scholarship Council Ph.D. Program, a cooperation program with the French UT & INSA, is acknowledged for financial support.

SUPPORTING INFORMATION

Infra

TABLE OF CONTENT GRAPHIC TABLE OF CONTENT

HIGHLIGTHS.

First azoxybenzene-3,3',5,5'-tetracarboxylate lanthanide coordination polymers;

Excitation and emission wavelengths in the IR region;

Structural characterization of a series of lanthanide coordination polymers.