C Bourdarias

M Gisclon

S Junca

Some mathematical results on a system of transport equations with an algebraic constraint describing fixed-bed adsorption of gases

Keywords: boundary conditions, systems of conservation laws, Godunov scheme 1991 MSC: 35L65, 35L67, 35Q35

This paper deals with a system of two equations which describes heatless adsoption of a gaseous mixture with two species. When one of the components is inert, we obtain an existence result of a weak solution satisfying some entropy condition under some simplifying assumptions. The proposed method makes use of a Godunov-type scheme. Uniqueness is proved in the class of piecewise C 1 functions.

Introduction

Heatless adsorption is a cyclic process for the separation of a gaseous mixture, called "Pressure Swing Adsorption" cycle. During this process, each of the d Preprint submitted to Elsevier Science species (d ≥ 2) simultaneously exists under two phases, a gaseous and movable one with concentration c i (t, x) (0 ≤ c i ≤ 1), or a solid (adsorbed) other with concentration q i (t, x), 1 ≤ i ≤ d. Following Ruthwen (see [START_REF] Ruthwen | Principles of adsorption and adsorption processes[END_REF] for a precise description of the process) we can describe the evolution of u, c i , q i according to the following system, where C = (c 1 , • • • , c d ):

∂ t c i + ∂ x (u c i ) = A i (q i -q * i )(C), (1)
∂ t q i + A i q i = A i q * i (C) t ≥ 0, x ∈ (0, 1), (2)
with suitable initial and boundary data. In ( 1)-( 2) the velocity u(t, x) of the mixture has to be found in order to achieve a given pressure (or density in this isothermal model)

d i=1 c i = ρ(t), (3)
where ρ represents the given total density of the mixture. The experimental device is realized so that it is a given function depending only upon time.

The function q * i is defined on (R + ) d , depends upon the assumed model and represents the equilibrium concentrations. Its precise form is usually unknown but is experimentally obtained. Simple examples of such a function are for instance the linear isotherm q * i = K i c i , with K i ≥ 0 and the Langmuir isotherm

q * i = Q i K i c i 1 + d j=1 K j c j
, with K i ≥ 0, Q i > 0 (see for instance [START_REF] Bourdarias | Approximation of the solution to a system modeling heatless adsorption of gases[END_REF], [START_REF] James | Convergence results for some conservation laws with a reflux boundary condition and a relaxation term arising in chemical engineering[END_REF], [START_REF] Ruthwen | Principles of adsorption and adsorption processes[END_REF])

The right hand side of ( 1)- [START_REF] Bourdarias | Approximation of the solution to a system modeling heatless adsorption of gases[END_REF] rules the matter exchange between the two phases and quantifies the attraction of the system to the equilibrium state: it is a pulling back force and A i is the "velocity" of exchange for the species i.

A component with concentration c k is said to be inert if A k = 0 and q k = 0.

A theoretical study of the system (1)-( 2)-(3) was presented in [START_REF] Bourdarias | Sur un système d'edp modélisant un processus d'adsorption isotherme d'un mélange gazeux. (french) [on a system of p.d.e. modelling heatless adsorption of a gaseous mixture[END_REF] and a numerical approach was developed in [START_REF] Bourdarias | Approximation of the solution to a system modeling heatless adsorption of gases[END_REF]. Let us point out that one of the mathematical interests of the above model is its analogies and differences compared 2 to variousother classical equations of physics or chemistry. First, when d = 1 (and eventually with A i = 0) this model shares a similar structure withconservation laws under the form

∂ t ρ + ∂ x (ρ u(ρ)) = 0, ∂ x u(ρ) = F (ρ)
where u(ρ) has an integral dependance upon ρ , while in scalar conservation laws u depends upon ρ. In [START_REF] Bourdarias | Sur un système d'edp modélisant un processus d'adsorption isotherme d'un mélange gazeux. (french) [on a system of p.d.e. modelling heatless adsorption of a gaseous mixture[END_REF] both BV and L ∞ theory are developed for this model, but oscillations can propagate thus differing from Burger's example (see Tartar [START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF],Lions, Perthame, Tadmor [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related questions[END_REF]).

Secondly, when the coefficients A i tend to infinity (instantaneous equilibrium)

we get formally

q i -q * i = - 1 A i ∂ t q i → 0
and the equations ( 1)-( 2) reduce to

∂ t (c i + q * i (C)) + ∂ x (u c i ) = 0, i = 1, • • • , d. (4)
Joined to (3), the system of conservation laws (4) generalizes the system of chromatography which has been intensively studied (see [START_REF] James | Sur la modélisation mathématique des équilibres diphasiques et des colonnes de chromatographie[END_REF][START_REF] Rhee | On the theory of multicomponent chromatography[END_REF] for the Langmuir isotherm) whereas the system (1)-(2) enters more in the field of relaxation systems (see for instance Jin and Xin [START_REF] Jin | The relaxing schemes for systems of conservation laws in arbitrary space dimensions[END_REF], Katsoulakis and Tzavaras [START_REF] Katsoulakis | Contractive relaxation systems and the scalar multidimensional conservation law[END_REF]). Actually the system of chromatography corresponds, like in (4), to instantaneous adsorption, but the fluid speed is a constant u(t, x) = u. One may consult James [START_REF] James | Sur la modélisation mathématique des équilibres diphasiques et des colonnes de chromatographie[END_REF] for anumerical analysis and the relationships with thermodynamics, Canon and James [START_REF] Canon | Resolution of the Cauchy problem for several hyperbolic systems arising in chemical engineering[END_REF] in the case of the Langmuir isotherm. In [START_REF] James | Convergence results for some conservation laws with a reflux boundary condition and a relaxation term arising in chemical engineering[END_REF], James studied a system closely related to (1)- [START_REF] Bourdarias | Approximation of the solution to a system modeling heatless adsorption of gases[END_REF] in which the speed is constant and the coefficients A i are equal to 1/ε, where ε is a small parameter. Using compensated compactness, he proved, under some assumptions on the flux, that the solution of this system converges, as ε → 0, to a solution to a system of quasilinear equations similar to (4) satisfying a set of entropy inequalities. The extension of his method to (4) with constraint (3) seems not straightforward and is still an open problem.

In this paper, we deal with the system of equations ( 4)-( 3) with two components (d = 2), one adsorbable with concentration c 1 and one inert with concentration c 2 . Moreover in(3) we assume that ρ ≡ 1, which is not really restrictive from a theoretical point of view. Then, the corresponding system of transport equations writes:

∂ t (c 1 + q * 1 (c 1 , c 2 )) + ∂ x (u c 1 ) = 0, ( 5
) ∂ t c 2 + ∂ x (u c 2 ) = 0, (6)
with the algebraic constraint

c 1 + c 2 = 1. (7)
Notice that we seek positive solutions (c 1 , c 2 ), thus, in view of (7), c 1 , c 2 must satisfy 0 ≤ c 1 , c 2 ≤ 1. Adding ( 5) and ( 6), we get, thanks to (7) :

∂ t q * 1 (c 1 , c 2 ) + ∂ x u = 0.
In the sequel we set c := c 2 and h(c

) = -q * 1 (c 1 , c 2 ) = -q * 1 (1 -c, c), thus our
purpose is to study the system ( 5)-( 6)-( 7) under the form:

               ∂ t c + ∂ x (uc) = 0, ∂ t h(c) -∂ x u = 0, (8)
supplemented by initial and boundary values:

                           c(0, x) = c 0 (x) ∈ [0, 1], x > 0, c(t, 0) = c b (t) ∈ [0, 1], t > 0, u(t, 0) = u b (t), t > 0. (9)
We assume in (9) an influx boundary condition, i.e. ∀t > 0, u b (t) > 0. We choose ]0, +∞[ instead of ]0, 1[ as spatial domain for the sake of simplicity.

In order to investigate some properties of the function h we look at some commonly used isotherm ( [START_REF] Douglas Le | Fixed-bed adsorption of gases: Effect of velocity variations on transition types[END_REF]). For linear isotherm, we have:

q * 1 := K 1 c 1 with K 1 > 0, then h ′ (c) := dh dc > 0 ( 10
)
and h ′′ = 0. For the binary Langmuir isotherm wich is:

q * 1 = Q 1 K 1 c 1 1 + K 1 c 1 + K 2 c 2 , with K 1 > 0, Q 1 > 0, K 2 ≥ 0, we have also h ′ > 0, and h ′′ (c) := d 2 h dc 2 ≥ 0 if K 2 < K 1 (actually K 2 = 0 if
the second species is inert). For the so called BET isotherm defined by

q * 1 = QKc 1 (1 + Kc 1 -(c 1 /c s ))(1 -(c 1 /c s )) , Q > 0, K > 0, c s > 0,
we have still h ′ > 0 but no longer h" ≥ 0. Nevertheless the function h ′ + ch", first derivative of H(c) := 1 + ch ′ (c) remains nonnegative for a convenient choice of the parameters (but unfortunately not in all the physically relevant situations). In this first simplified approach we will assume [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related questions[END_REF] and

H ′ (c) ≥ 0. ( 11
)
Single-component adsorption is of course of a poor physical meaning, but must be understood as a preliminary theoretical study.

The paper is organized as follows. In section 2, we give some results for smooth solutions. These results suggest us an entropy condition. In section 3, we give solutions for the Riemann Problem satisfying such an entropy condition. In section 4, we use a Godunov scheme to construct an approximate weak solution of problem ( 8)-( 9) and we give some useful bounds. Next, in section 5, we obtain an existence theorem for a weak solution of problem ( 8)-( 9). Lastly, in section 6, the uniqueness is obtained in the class on piecewise C 1 functions.

Smooth solutions

Proposition 2.1 For smooth solutions, the system [START_REF] Jin | The relaxing schemes for systems of conservation laws in arbitrary space dimensions[END_REF] with the initial boundary conditions [START_REF] Katsoulakis | Contractive relaxation systems and the scalar multidimensional conservation law[END_REF] becomes:

∂ t c + ∂ x [α(t)F (c)] = 0, t, x > 0, ( 12
) c(0, x) = c 0 (x), x > 0, ( 13
) c(t, 0) = c b (t), t > 0, (14)
with α(t) = u b (t) exp(g(c b (t))) > 0, F (c) = c exp(-g(c)) > 0, where

g ′ (c) = h ′ (c) H(c) , H(c) = 1 + ch ′ (c) ( 15
)
and necessarily

u(t, x) = α(t) exp(-g(c(t, x))) > 0, t, x > 0. ( 16
)
Moreover, under asumption ( 10)- [START_REF] Rhee | On the theory of multicomponent chromatography[END_REF] we have

F ′ > 0 > F ′′ .
Notice that g and F depend only on h ′ , but α depends also on boundaries values u b , c b . The maximum principle is valid for c but not for u: see for instance Figure 6 .

Proof : since c and u are smooth, we can apply the chain rule formula. So, the second equation of ( 8) can be rewritten

∂ x u = h ′ (c)∂ t c, then, with the first equation, ∂ x u = -h ′ (c)∂ x (uc) and we get (∂ x u)(1 + ch ′ (c)) = -uh ′ (c)∂ x c.
Finally, with the notations introduced in (15) we have:

∂ x u = -u h ′ (c) (∂ x c)/H(c) = -u ∂ x (g(c)). ( 17
)
For a fixed t > 0, the function x → u(t, x) is the unique solution of the ordinary linear differential equation (17) with the "initial" condition u(t, 0) = u b (t) > 0.

Explicitely, we have:

u(t, x) = u b (t) exp(g(c b (t)) -g(c(t, x)), then u(t, x) is
positive for all x. Replacing u in the first equation of (8) we get [START_REF] Ruthwen | Principles of adsorption and adsorption processes[END_REF]. Now, a direct computation gives us:

F ′ (c) = exp(-g(c))/H(c), F ′′ (c) = - exp(-g(c)) H 2 (c) (H ′ (c) + h ′ (c)). ( 18
)
and thanks to the hypothesis [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related questions[END_REF] and [START_REF] Rhee | On the theory of multicomponent chromatography[END_REF] we have F ′ > 0 and F " < 0: the flux in the scalar conservation law ( 12) is strictly concave.

Theorem 2.1 (Global smooth solution) Assume ( 10)- [START_REF] Rhee | On the theory of multicomponent chromatography[END_REF].

If u b ∈ C 1 ([0, +∞[, ]0, +∞[), if c b , c 0 ∈ C 1 ([0, +∞[, [0, 1]
) satisfy the following compatibility conditions at the corner:

c b (0) = c 0 (0), c ′ b (0) + u b (0)c ′ 0 (0) + h ′ (c b (0))c ′ b (0)c 0 (0) = 0
and if c ′ 0 ≤ 0 ≤ c ′ b , then the system ( 8)-( 9) admits one and only one smooth solution:

(c, u) ∈ C 1 ([0, +∞[ t ×[0, +∞[ x , [0, 1]) × C 1 ([0, +∞[ t ×[0, +∞[ x , ]0, +∞[). Moreover: ∀t > 0, ∂ x c(t, x) ≤ 0, u(t, x) > 0, ∂ x u(t, x) ≥ 0.
We deduce from this result an entropy condition for shockwaves:

(EC) "c increases through a shock".

For smooth solutions, the active gas desorbs and u increases to evacuate gases.

Notice that the same theorem is true for continuous solutions with only one compatibility condition at thecorner: c b (0) = c 0 (0) and replacing the sign of the derivative of the concentrations on the boundary by monotonicity conditions. Figure 5 shows a (non global) smooth solution which produces a shock wave in finite time.

Proof : for a smooth solution we can use the last proposition, so

∂ t c + α(t) F ′ (c) ∂ x c = 0.
Using the characteristic curve defined by:

∂X ∂s (s, t, x) = α(s) F ′ (c(s, X(s, t, x))), X(t, t, x) = x, (19)
we get

∂ ∂s c(s, X(s, t, x)) = 0. ( 20
)
Thus, c is constant along the characteristic curve (19), i.e. c(s, X(s, t, x)) = c(t, x), and X writes:

X(s, t, x) = x + F ′ (c(t, x)) s t α(z) dz. (21)
To construct a solution, we need only to construct all characteristic curves issuing from the boundary and verify that no characteristic curves cross each other, see [START_REF] Smoller | Shock Waves and Reaction-Diffusion Equations[END_REF] p. 241-244, or [START_REF] Fritz | Nonlinear wave equations, Formation of singularities[END_REF], i.e. we need to satisfy: β := ∂ x X(s, t, x)) > 0.

Differentiating (19) with respect to x, we get:

∂β ∂s (s, t, x) = α(s)F ′′ (c(s, X(s, t, x)))∂ x c(s, X(s, t, x)β, β(t, t, x) = 1.
On the other hand we have ∂ x c(s, X(s, t, x)) = ∂ x c(t, x), then for s > t:

∂β ∂s (s, t, x) = [α(s) × F "(c(t, x)) × ∂ x c(t, x)] β(s, t, x), β(t, t, x) = 1.
Since F ′′ (c) < 0 and α(s) > 0, the sufficient way to keep β positive is:

∀(t, x) ∂ x c(t, x) ≤ 0. Since ∂ x c is constant along any characteristic curve,
it suffices to satisfy this condition on the boundary. For characteristic curves issuing from {t = 0}, this last condition becomes ∂ x c(0, x) = c ′ 0 (x) ≤ 0. For characteristic curves issuing from {x = 0}, remark that on x = 0, thanks to the equation ( 12), we have

∂ t c(t, 0) = -α(t) F ′ (c(t, 0)) ∂ x c(t, 0). Since F ′ (c) > 0 and α(t) > 0 we need to have ∂ t c(t, 0) = c ′ b (t) > 0.

Riemann Problem

It is well known (see for instance Dafermos [4], Serre [START_REF] Serre | Systèmes de lois de conservation I[END_REF], Smoller [START_REF] Smoller | Shock Waves and Reaction-Diffusion Equations[END_REF]) that in the context of hyperbolic systems of conservation laws, the life span of smooth solutions is finite even when the initial/boundary data are smooth. For the system studied in this paper, it will be the case if for instance the monotonicity conditions c ′ 0 ≤ 0 ≤ c ′ b are not satisfied, thus we have to deal with weak solutions. In order to get a general existence result via the construction of a sequence of approximate solutions, we are going to adapt the Godunov scheme to the system (8): the first step is the resolution of the Riemann problem.

We are thus looking for a weak solution of the following Riemann problem:

∂ t c + ∂ x (uc) = 0, ∂ t h(c) -∂ x u = 0, ∀x > 0 c(0, x) = c + , ∀t > 0 c(t, 0) = c -and u(t, 0) = u -, (22)
with c -, c + ∈ [0, 1] and u -> 0. By symmetry, we search a selfsimilar so-

lution, i.e. : c(t, x) = C(z), u(t, x) = U(z) with z = x t > 0.
Recall that from Theorem2.1 we proposed the following (EC) entropy condition for shock waves: c increases through a shock. Then, if c -> c + , we find a continuous solution. To have a global smooth solution, we find necessarily a decreasing solution thanks to Theorem 2.1 and if c -< c + , we find a shockwave.

Proposition 3.1 (Rarefaction wave)

Assume ( 10)- [START_REF] Rhee | On the theory of multicomponent chromatography[END_REF]. If c -> c + , the only smooth selfsimilar solution of( 22) is such that

                           C(z) = c -, 0 < z < z -, dC dz = - G(C) z , z -< z < z + C(z) = c + , z + < z, , (23)
where

G(c) = H(c) h ′ (c) + H ′ (c) > 0, z -= u - H(c -) > 0, ( 24
)
z + is defined by the equation C(z + ) = c + , u + = z + H(c + )
, and U is given by:

                           U(z) = u -, 0 < z < z -, U(z) = z H(C(z)), z -< z < z + , U(z) = u + z + < z. (25)
So, along a rarefaction wave, c decreases, u increases, z -< u -, and z + < u + .

Notice that the computations of z + and u + need the resolution of an O.D.E.

Figure 2 shows a desorption step corresponding to a rarefaction wave arising from a discontinuity at (t = 0, x = 0).

Proof : setting C ′ (z) = dC dz and U ′ (z) = dU dz
, we get from( 8)

-z C ′ + (U C) ′ = 0, (26)
U ′ = -z h ′ (C) C ′ . ( 27
) U(z) = z H(C(z)). (28)
We are looking for a simple wave, so C ′ = 0 on (z -, z + ). From (28), z -is defined by z -= u -/H(c -) and we have to find z + and u + .

From ( 28) and ( 26) we get

C ′ (z) G(C) = - 1 z . Let φ(C) = C c - 1 G(s)
ds. Thanks to the hypothesis ( 10)-( 11) we have G > 0 and, for C < c -, we have φ(C) < 0. But

d dz φ(C(z)) = φ ′ (C) C ′ (z) = C ′ (z) G(C) = - 1 z . Then φ(C(z)) = ln( z - z ) because φ(c -) = 0. Finally, φ(C(z + )) = ln( z - z + ) and z + = z -exp(-φ(c + )
). Now, using again (28) we get u + .

Proposition 3.2 (Shockwave)

Assume ( 10)- [START_REF] Rhee | On the theory of multicomponent chromatography[END_REF]. If c -< c + , the only weak selfsimilar solution of ( 22) is

C(z) =                c -if 0 < z < s, c + if s < z , U(z) =                u -if 0 < z < s, u + if s < z , (29)
where u + is defined by

u + = u - [c] + c -[h] [c] + c + [h] , (30)
and where the speed s of the shock satisfies

0 < s = - [u] [h] = [uc] [c] = u - [c] [c] + c + [h] = u + [c] [c] + c -[h] < u + < u - ( 31
)
with the classical notations for the jumps.

Thanks to the Rankine-Hugoniot condition, this is the only weak monotonic solution with only one jump, i.e. c and u are monotonic functions. So, through a shock wave, c increases, u decreases but remains positive. The speed of the shock is proportional to u -and lower than the fluid velocity u. Notice the difference with a strictly hyperbolic 2 × 2 system. Here we have three data: c -, c + , u -and two unknowns: u + , s. In the hyperbolic case for two shocks, we have four data: c -, c + , u -, u + and four unknowns: c 0 , u 0 , s 1 , s 2 . Figure 3 shows an adsorption step corresponding to a shock wave arising from a discontinuity at (t = 0, x = 0). See also Fig. 4 for the junction of two shocks.

Proof : we cannot find a smooth solution since G > 0 and c should decrease, by (23). Let be ν = (ν t , ν x ) a normal vector to the shock line. The Rankine-

Hugoniot conditions write ν t [c] + ν x [u c] = 0, ν x [u] = ν t [h(c)].
We have

[c] = 0 thus [h(c)] = 0 and ν x = 0. Then the slope s of the shock line satisfies

s = [u c] [c] = - [u] [h] . Then from [u] [c] + [uc] [h] = 0 we get u + u - = [c] + c -[h] [c] + c + [h] ( 32
)
and all results follow. Lemma 3.1 Assume ( 10)- [START_REF] Rhee | On the theory of multicomponent chromatography[END_REF]. For the solution of the Riemann Problem (22) given in propositions 3.1 and 3.2 we have the following estimate:

| ln(u + ) -ln(u -)| ≤ γ |c + -c -|,
where γ is a true constant depending only on the h function.

Proof : if the solution of the Riemann Problem ( 22) is a rarefaction wave then, by proposition (3.1), we have:

0 < u + u - = z + H(c + ) z -H(c -) < z + z - , since c -> c
dD dz = - β z , z -≤ z < z + , D(z -) = c -, C(z) ≤ D(z) on (z -, z + ).
explicitely D and z 0 :

D(z) = c --β ln(z/z -), z 0 = z -exp(|c + -c -|/β). Then, it suffices to take γ 1 = 1 β = 1 min c∈[0,1]

G(c)

. If the solution of the Riemann Problem ( 22) is a shock wave then, by proposition(3.2) and equality (32), we have:

0 < u + u - = [c] + c -[h] [c] + c + [h] = S(c -, c + ), c -< c + .
The function S is smooth and positive on Ω = {(c -, c + ), 0 ≤ c -< c + ≤ 1}.

On the diagonal we have c -= c + and S ≡ 1, therefore we verify that ln(S) is a smooth function on Ω, vanishing on the diagonal. Then, there exists γ 2

such that | ln(u + ) -ln(u -)| ≤ γ 2 |c + -c -|. Finally lemma (3.1) holds with γ = max(γ 1 , γ 2 ).

Godunov Scheme

We adapt the classical Godunov scheme for hyperbolic systems to the system of adsorption [START_REF] Jin | The relaxing schemes for systems of conservation laws in arbitrary space dimensions[END_REF]. Let be T > 0, X > 0 fixed. For a fixed integer N we set ∆x = X N + 1 and ∆t = T M + 1

, where M is an integer depending upon N and will be choosen later to satisfy a CFL-type condition. We are going to build an approximate solution (c N , u N ) of ( 8) on (0, T ) × (0, X).

For i = 0, • • • , N and j = 0, • • • , M we denote by B i,j the box B i,j = [t j , t j+1 [×[x i , x i+1 [, where
x i = i∆x, t j = j∆t. We use also midle mesh (x i+1/2 = x i + ∆x/2, t j+1/2 = t j + ∆t/2). We discretize the initial boundary values as follows:

c N (0, x) = c N (0, x i+1/2 ) := 1 ∆x x i+1 x i c 0 (x) dx, x i < x < x i+1 , c N (t, 0) = c N (t j+1/2 , 0) := 1 ∆t t j+1 t j c b (t) dt, t j < t < t j+1 , u N (t, 0) = u N (t j+1/2 , 0) := 1 ∆t t j+1 t j u b (t) dt, t j < t < t j+1 .
where 0 ≤ i ≤ N and 0 ≤ j ≤ M. Assume that, for a given i, we have given c

N (t j , x) = c + on [x i , x i+1 [, c N (t, x i ) = c -and u N (t, x i ) = u -on [t j , t j+1 [, then:
(1) if c -< c + (shock) we compute s and u + according to (31) and (30).

Thanks to the CFL condition and (29) we get c N (t, x i+1 ) = c + , u N (t, x i+1 ) = u + on [t j , t j+1 [ and we define c N (t j+1 , x) on [x i , x i+1 [ as the mean value of the solution of the Riemann problem, that is:

c N (t j+1 , x) = c N (t j+1 , x i+1/2 ) := λ s c -+ (1 -λ s) c + with λ = ∆t ∆x
(2) if c -> c + (rarefaction wave) we compute z -by (24). Then, z + is computed as the unique solution of C(z + ) = c + with C defined through(23).

U is defined by (25) with u + = z + H(z + ). As in the preceding case we have

c N (t, x i+1 ) = c + , u N (t, x i+1 ) = u + on [t j , t j+1 [ and we define c N (t j+1 , x)
on [x i , x i+1 [ as the mean value of the solution of the Riemann problem.

Using for instance the trapezoid rule we get :

c N (t j+1 , x) = c N (t j+1 , x i+1/2 ) := λ z -+ z + 2 c -+ (1 -λ z -+ z + 2 ) c + .
Notice that we could proceed as well by columns before rows, (i beforej). To ensure the CFL condition (33), we need to control sup u. Therefore, by lemma 3.1, we have to control the total variation in space of c for all time. Recall that, for any function v defined on (a, b):

T V (v, (a, b)) = sup n k=0 |v(z k+1 ) -v(z k )| ; n ∈ N, a < z 0 < • • • < z n+1 < b = sup b a v(z)φ ′ (z)dz ; φ ∈ C ∞ c (a, b), |φ| ≤ 1 and v ∈ BV (a, b) if and only if T V (v, (a, b)) < +∞.
In the following lemmas, we prove that this scheme is well defined and we give some useful bounds.

Lemma 4.1 Let be γ the constant defined in lemma 3.1. If the CFL condition is fulfilled, then, for all t ∈ (0, T ):

T V [ln(u(t, .)), (0, X)] ≤ γ T V [c(t, .), (0, X)].
scheme and the monotonicity of c and u on each box (see remark 3.1).

Let us define the total variation of initial-boundary concentration by:

T V (c b , c 0 ) := T V (c b , (0, T ))+T V (c 0 , (0, X))+ sup 0<t<T,0<x<X |c b (t)-c 0 (x)|. ( 34
)
Lemma 4.2 If the CFL condition is fulfilled, then, for all N ≥ 0:

sup 0<t<T T V [c N (t, .), (0, X)] ≤ T V (c b , c 0 ).
Proof : by monotonicity of the solution of the Riemann problem under the CFL condition (see remark 3.1) we have, for all t ∈ (t j , t j+1 ) and all t ∈

(t j , t j+1 ): T V [c N (t, .), (x i , x i+1 )] = |c N (t j+1/2 , x i ) -c N (t j+1/2 , x i+1 )|. Therefore, we have: T V [c N (t j+1/2 , .), (0, X)] = N i=0 |c N (t j+1/2 , x i+1 ) -c N (t j+1/2 , x i )|.
In particular, in the lower row, we obtain:

T V [c N (t 1/2 , .), (0, X)] = |c N (t 1/2 , 0) -c N (0, x 1/2 )| + N i=1 |c N (0, x i-1/2 ) -c N (0, x i+1/2 )| ≤ |c N (t 1/2 , 0) -c N (0, x 1/2 )| + T V (c 0 (.), (0, X)).
By induction, we get easily

T V [c N (t j+1/2 , .), (0, X)] ≤ |c N (t j+1/2 , 0) -c N (t j , x 1/2 )| +T V [c N (t j-1/2 , .), (0, X)].
Since c N (t j , x 1/2 ) is between c N (t j-1/2 , 0) and c N (t j-1 , x 1/2 ) we have

|c N (t j+1/2 , 0) -c N (t j , x 1/2 )| ≤ |c N (t j+1/2 , 0) -c N (t j-1/2 , 0)| +|c N (t j-1/2 , 0) -c N (t j-1 , x 1/2 )|.
Then, we get:

T V [c N (t j+1/2 , .), (0, X)] ≤ j k=1 |c N (t k+1/2 , 0) -c N (t k-1/2 , 0)| + T V [c N (∆t/2, .), (0, X)] ≤ j k=1 |c N (t k+1/2 , 0) -c N (t k-1/2 , 0)| + |c N (t 1/2 , 0) -c N (0, x 1/2 )| +T V (c 0 (.), (0, X)) ≤ T V (c b (.), (0, t j+1 )) + |c N (t 1/2 , 0) -c N (0, x 1/2 )| + T V (c 0 (.), (0, X)) ≤ T V (c b , c 0 ). Lemma 4.3 Let be λ = u b ∞ × exp(γ T V (c b , c 0 )) > 0. If λ∆t < ∆x, then
the CFL condition is fulfilled.

Proof : we proceed by induction. Let (H i,j ) the following hypothesis: on R i,j = (0, t j+1 ) × (0, x i+1 ) we have

0 < u ≤ sup 0<t<t j+1 (u b (t) exp(γ T V [c(t, .), (0, x i+1 )])) . (35)
Since λ ≥ u b ∞ , (H 0,j ) is satisfied for all j. We have to show that for j from 0 to M, if (H i,j ) is true and i < N then (H i+1,j ) is also true. To this purpose we need only to prove that u satisfies inequality (35) on B i+1,j .

If (H i,j ) is true, the CFL condition is fulfilled on rectangle R i,j , then

u -:= u(t j+1/2 , x i+1 ) ≤ sup 0<t<t j+1 (u b (t) exp(γ T V [c(t, .), (0, x i+1 )])) .
Solving the Riemann problem on B i+1,j , we get u

+ ≤ u -exp(γ |c + -c -|) thanks to lemma 3.1. Then, sup B i+1,j u ≤ sup 0<t<t j+1 (u b (t) exp(γ T V [c(t, .), (0, x i+1 )])) × exp(γ |c + -c -|) ≤ sup 0<t<t j+1 (u b (t) exp(γ T V [c(t, .), (0, x i+2 )])) .
Therefore, H i+1,j is true. Finally, we have

u ≤ λ = u b ∞ exp(γ T V (c b , c 0 ))
and the CFL condition holds.

Denote by ceil(x) the lowest integer bigger than x. We can fix M as follows:

M = 1 + ceil λT ∆x = 1 + ceil λ T X (N + 1) . ( 36
)
and the CFL condition is then satisfied. Notice that M∆x ∼ λT and ∆x ∆t → λ

as N → ∞.

Lemma 4.4 Let be L > 0, f ∈ BV (0, L), f = 1 L L 0 f (x)dx, or f = f (0 + ) + f (L -) 2 , then L 0 |f (x) -f |dx ≤ L × T V (f, (0, L)).
We skip the proof of this rather classical lemma.

Lemma 4.5 Let be λ 1 = sup N ∈N ∆x ∆t < ∞, then for any 0 ≤ s < t < T the sequence (c N ) satisfies:

X 0 |c N (t, x) -c N (s, x)| dx ≤ 2λ 1 T V (c b , c 0 ) (|t -s| + 2∆t). ( 37
)
Proof : we recall that CFL condition is fulfilled. First, we work on B i,j ,

t j ≤ s 1 < s 2 < t j+1
. By monotonicity with respect to time of c N on each box, we have:

x i+1 x i |c N (s 2 , x) -c N (s 1 , x)| dx ≤ x i+1 x i |c N (t j+1 , x -0) -c N (t j , x)| dx ≤ ∆x |c N (t j+1 , x i ) -c N (t j , x i+1/2 )| = ∆x T V (c N (t j+1/2 , .), (x i , x i+1 )).
Since ∆x ≤ λ 1 ∆t, after summation with respect to i, we get:

X 0 |c N (s 2 , x) -c N (s 1 , x)|dx ≤ ∆x T V (c N (t j+1/2 , .), (0, X)) ≤ λ 1 ∆t T V (c b , c 0 ).
Otherwise, on t = t j , there is a jump, but by Lemma 4.4:

x i+1 x i |c N (t j , x) -c N (t j -0, x)| dx = x i+1 x i |c N (t j , x 1/2 ) -c N (t j -0, x)| dx ≤ ∆x T V (c N (t j+1/2 , .), (x i , x i+1 )).
Summing over i, we get

X 0 |c N (t j , x) -c N (t j -0, x)| dx ≤ ∆x T V (c N (t j+1/2 , .), (0, X)) ≤ λ 1 ∆t T V (c b , c 0 ).
For any 0 ≤ s < t < T , let be j := min{i, s ≤ t i }, k := max{l, t j+l ≤ t}, and s ≤ t j < t j+1 < • • • < t j+k ≤ t. By convention t -1 = 0, so we have

|t j+k+1 -t j-1 | ≤ |t -s| + 2∆t and X 0 |c N (t, x) -c N (s, x)| dx ≤ k l=-1 X 0 |c N (t j+l+1 , x) -c N (t j+l , x)| dx ≤ 2λ 1 T V (c b , c 0 )(|t -s| + 2∆t). Lemma 4.6 Assume that the CFL condition is fulfilled, that u b ∈ L ∞ (0, T ),
(u N ) is bounded inL ∞ ((0, T ) × (0, X)) and in L ∞ (0, T ; BV (0, X)). Further- more: inf N inf (0,T )×(0,X) u N > 0 and sup N u N ∞ ≤ u b ∞ exp(γ T V (c b , c 0 )).
Proof : solving the Riemann problem, u + > 0 follows from u -> 0 and we Let be X > 0, T > 0. Assume ( 10)- [START_REF] Rhee | On the theory of multicomponent chromatography[END_REF] and that c 0 ∈ BV (0, X), c b ∈

have u N > 0 on (0, T ) × (0, X) . If u b ∈ L ∞ (0,
BV (0, T ), u b ∈ L ∞ (0, T ), satisfying 0 ≤ c 0 , c b ≤ 1 and inf 0<t<T u b (t) > 0.
Then the system ( 8)-( 9) admits a weak solution given by Godunov scheme.

Furthermore, c and u satisfy:

c ∈ L ∞ ((0, T ) × (0, X)) ∩ L ∞ ((0, T ); BV (0, X)), ( 38
) c ∈ Lip(0, T ; L 1 (0, X)), ( 39
) c ∈ BV ((0, T ) × (0, X)), (40) u ∈ L ∞ ((0, T ) × (0, X)) ∩ L ∞ ((0, T ); BV (0, X)), (41)
with the following bounds:

X 0 c(t, x)dx ≤ X 0 c 0 (x)dx + u b ∞ t 0 c b (s) ds, ( 42
) 0 ≤ min(inf c b , inf c 0 ) ≤ c ≤ max(sup c b , sup c 0 ) ≤ 1, (43) c L ∞ ((0,T ),BV (0,X)) ≤ T V (c b , c 0 ), (44) u L ∞ ((0,T )×(0,X)) ≤ u b ∞ exp(γ T V (c b , c 0 )), (45) inf [0,T ]×[0,X] u > 0. ( 46
)
(γ is the constant defined in lemma 4. [START_REF] Canon | Resolution of the Cauchy problem for several hyperbolic systems arising in chemical engineering[END_REF]

and depending only on the hfunction)

Proof : let be (c N , u N ) N the sequence constructed in section 4. We are going to prove that a subsequence of (c N , u N ) N converges towards a weak solution (c, u) of ( 8)-( 9), satisfying the estimates (38) to (46).

First step : convergence of c N , u N , u N c N up to a subsequence.

By lemma 4.2, the sequence (c N ) is bounded in L ∞ ((0, T ); BV (0, X)). Furthermore, by lemma 4.5, we obtain a classical compactness argument on (c N ) (see [START_REF] Smoller | Shock Waves and Reaction-Diffusion Equations[END_REF]). Then, up to a subsequence, (c N ) converges to c in L 1 ((0, T )×(0, X))

and a.e. Then c satisfies the same bounds i.e. ( 38), ( 39), ( 43) and (44) hold, in particular c verifies the maximum principle.

By lemma 4.6, the sequence (u N ) is bounded in L ∞ , then, up to a subsequence, (u N ) converges weakly to u in L ∞ weak -⋆. By the same lemma, the sequence

(∂ x u N ) is bounded L ∞ t M 1 x , dual from L 1 t C 0 x , then there exists v ∈ L ∞ t M 1 x such that (∂ x u N ) converges weakly to v in L ∞ t M 1
x weak -⋆. But the weak limit is unique then 45), (46) hold. Now, we can pass to the limit in the nonlinear term u N c N because the sequence (u N ) converges weakly to u in L ∞ weak -⋆ and the sequence (c N ) converges strongly to c in L 1 .

∂ x u = v and u ∈ L ∞ t BV x . Furthermore we have u L ∞ ≤ lim inf N u N L ∞ < +∞, u L ∞ t BVx ≤ lim inf N u N L ∞ t BVx < +∞, inf u ≥ inf N u N > 0 and (41), (
Second step : we show that (c, u), obtained in the previous step is a weak solution of ( 8)- [START_REF] Katsoulakis | Contractive relaxation systems and the scalar multidimensional conservation law[END_REF]. Recall that (c, u) is a weak solution of ( 8)-( 9) on (0, T )×(0, X)

if and only if, for any smooth functions φ, ψ ∈ C ∞ c ((-∞, T ) × (-∞, X)): T 0 X 0 (c ∂ t φ + (cu) ∂ x φ)(t, x) dx dt + X 0 c 0 (x) φ(0, x) dx (47)
+ T 0 u b (t)c b (t) φ(t, 0) dt = 0, (48)
T 0 X 0 (h(c) ∂ t ψ -u ∂ x ψ)(t, x) dx dt + X 0 h(c 0 (x)) ψ(0, x) dx (49)
- T 0 u b (t) ψ(t, 0) dt = 0. (50)
We are going to prove that (c, u) satisfies (47). A similar proof works to obtain(49). By construction, (c N , u N ) is a weak solution of (8) on each box B i,j and, thanks to the fulfilledCFL condition, is also a weak solution on each row (t j , t j+1 ) × (0, X). Theproblem is only on line t = t j , 0 < j ≤ M and t = 0, x = 0 for the discretisation of the initial boundaryvalue [START_REF] Katsoulakis | Contractive relaxation systems and the scalar multidimensional conservation law[END_REF]. So, for any φ, we have:

X 0 (c N ∂ t φ + c N u N ∂ x φ)(t, x)dxdt + X 0 c N (0, x)φ(0, x)dx + T 0 u N (t, 0)c N (t, 0)φ(t, 0)dt = -J N , where J N = M j=1 X 0 (c N (t j , x + 0) -c N (t j , x -0)) φ(t j , x) dx.
In order to prove that (c, u) satisfies (47), thanks to the results of the first step, we have just to show that J N → 0. We can rewrite J N under the form

J N = M j=1 N i=0
J i,j where

J i,j = ∆x 0 c N (t j , x i+1/2 ) -c N (t j , x i + y) φ(t j , x i + y) dy, and
c N (t j , x i+1/2 ) = 1 ∆x ∆x 0 c N (t j , x i + y) dy. Since ∆x 0 c N (t j , x i+1/2 ) -c N (t j , x i + y) φ(t j , x i ) dy = 0, we write φ(t j , x i + y) = φ(t j , x i ) + (φ(t j , x i + y) -φ(t j , x i )).
We have

|φ(t j , x i + y) -φ(t j , x i )| ≤ ∂ x φ ∞ ∆x because 0 ≤ y ≤ ∆x.
Thanks to lemma4.4, we have also

|J i,j | = ∆x 0 c N (t j , x i+1/2 ) -c N (t j , x i + y) (φ(t j , x i + y) -φ(t j , x i )) dy ≤ ∂ x φ ∞ ∆x ∆x 0 c N (t j , x i+1/2 ) -c N (t j , x i + y) dy ≤ ∂ x φ ∞ (∆x) 2 T V (c(t j , .), (x i , x i + ∆x))
Therefore,

|J N | ≤ M j=1 ∂ x φ ∞ (∆x) 2 T V (c N (t j , .), (0, X)) ≤ ∂ x φ ∞ T V (c b , c 0 ) M ∆x × ∆x thus, if M ≤ T ∆t , we have |J N | ≤ T ∂ x φ ∞ T V (c b , c 0 ) ∆x ∆t × ∆x.
Since ∆x ∆t → λ when N → ∞, J N converges towards 0. Lastly we get easily (42) by integrating ( 8) over [0, t] × [0, X] and using the positivity of u and c.

Since (c, u) is a weak solution of (8) we have ∂ x u = ∂ t h(c) and, thanks to the estimate on ∂ x u, we get ∂ t h(c) ∈ L ∞ ((0, T ); M 1 (0, X)). We have h ′ > 0, then c = h -1 (h(c)) and the chain rule formula in BV gives

∂ t c = (h -1 ) ′ ∂ t h(c) ∈ L ∞ M 1
x . Then ∂ t c and ∂ x c lie in M 1 ((0, T )×(0, X)) and finally c ∈ BV ((0, T )× (0, X)), which is (40).

We have now strong trace results.

Proposition 5.1 The functions c and u satisfy initial boundary conditions ( 9) strongly.

Proof : the function c belongs to BV ((0, T ) × (0, X)), then admits a strong trace on {t = 0} and {x = 0}. But c is a weak solution of ( 8), ( 9), then admits also a weak trace on the boundary. By uniqueness of traces, c satisfies the initial boundary conditions ( 9) strongly. On the other hand, u belongs to L ∞ ((0, T ) × (0, X)) ∩ L ∞ ((0, T ); BV (0, X)), then admits a strong trace v(t) in {x = 0} defined for a.e. t ∈ (0, T ). We have u(t, x) → v(t) for a.e.

t when x → 0 + and v ∈ L ∞ (0, T ) with v L ∞ t ≤ u L ∞ t,
x , thus, thanks to the Lebesgue's theorem, u admits v as strong trace on {x = 0} in L 1 (0, T ):

lim x→0 + T 0 | ũ(t, x) -v(t) | dt = 0,
where ũ is defined for a.e. t ∈ [0, T ] and all

x ∈ [0, X] as the mean value ũ(t, x) = u(t, x -0) + u(t, x + 0) 2 .

Uniqueness

We study the uniqueness problem for weak entropic solutions in some class of piecewise smooth functions. More precisely we denote by

C 1 p ([0, T ]×[0, X], R 2 ) (C 1 p in brief) the set of functions (c, u) : [0, T ] × [0, X] → R 2 such
that there exists a finite number of continuous and piecewise C 1 curves outside of which (c, u) is C 1 and across which (c, u) has a jump discontinuity. In the sequel, we consider weak solutions (c, u) ∈ C 1 p of ( 8)-( 9) in (0, T ) × (0, X), with piecewise smooth initial and boundary data, satisfying the entropy condition (EC) and our usual assumptions ( 10)-( 11) on h.

We restrict ourselves to the piecewise smooth case since we do not have a weak formulation for the entropy condition (EC). Formally we can expect to obtain such a condition as for hyperbolic PDEs, but it is still an open problem.

Nevertheless, this case is relevant in most practical cases and involve global solutions with shock waves and contact discontinuities.

Theorem 6.1 Let be T, X > 0. Let be u b : [0, T ] → R + , c b : [0, T ] → [0, 1], c 0 : [0, X] → [0, 1] some piecewise C 1 functions. Assume inf [0,T ]
u b (t) > 0 and ( 10), [START_REF] Rhee | On the theory of multicomponent chromatography[END_REF]. Then there exists at most one weak C 1 p solution (c, u) of the system ( 8)-( 9) satisfying the entropy condition (EC), the maximum principle ( 43) and (46). Lemma 6.1 Any shock curve across which c has a non zero jump admits a parametrization t → x(t).

Proof : let be ν = (ν t , ν x ) a normal of the shock line. Since (c, u) is a weak solution, it satisfies the Rankine Hugoniot condition and we get ν x = 0 and lemma 6.1 holds .

Remark: in the case where [c] = 0 and [u] = 0, the solution admits a contact discontinuity. We can easily obtain such a solution by considering for instance the following set of initial boundary data: c 0 ≡ a, c b ≡ a, u b = u 1 for 0 < t < t * and u b = u 2 for t * < t < T . We have an obvious weak solution defined by c(t, x) ≡ a, u(t, x) ≡ u 1 on (0, t * ) × (0, X) and u(t, x) ≡ u 2 on (t * , T ) × (0, X): the boundary discontinuity of u is linearly propagated. Figure 6 shows an example of such a situation. We define now a "determination zone" Ω = {(t, x), t 0 < t < t 1 , x 1 (t) < x < x 2 (t)} where 0 ≤ t 0 < t 1 < T , x 1 (t) and

x 2 (t) are shock curves. We assume that (c, u) ∈ C 1 (Ω).

Lemma 6.2 The characteristics curves lying in Ω satisfy

0 < dX ds (s, t, x) = u H(c) ≤ u (51)
Proof : since (c, u) ∈ C 1 (Ω), we have Proof : this proof relies on the entropy condition (EC). Let be s ∈]t 0 , t 1 [ and s → x(s) a shock curve. As usually we definec + = c(s, x(s) + 0), c -= c(s, x(s) -0), u + = u(s, x(s) + 0) and u -= u(s, x(s) -0). It follows from ( 19) that lemma 6.3 reduces to the inequalities α(s)

∂ t c + α(t) F ′ (c)∂ x c = 0
F ′ (c + ) < x ′ (s) < α(s) F ′ (c -).
Consider for instance the fist one: thanks to (31) and (51) it is equivalent to

x ′ (s) = u + [c] [c] + c -[h] > u + H(c + ) . Now we have u + > 0, c + > c -> 0, H(c + ) = 1 + c + h ′ (c + )
> 0 and the assumption [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related questions[END_REF], thus an easy computation leads to

α(s) F ′ (c + ) < x ′ (s) ⇐⇒ c + h ′ (c + ) [c] -c -[h] > 0 ⇐⇒ φ(c -) > 0 where φ is defined by φ(y) = c + h ′ (c + ) (c + -y) -y (h(c + ) -h(y)). We have φ ′ (y) = -(c + h ′ (c + ) -y h ′ (y)) -(h(c + ) -h(y)) = -(H(c + ) -H(y)) -(h(c + ) -h(y)) .
Thanks to [START_REF] Lions | A kinetic formulation of multidimensional scalar conservation laws and related questions[END_REF] and [START_REF] Rhee | On the theory of multicomponent chromatography[END_REF] we have φ ′ (y) < 0 for y < c + , moreover φ(c + ) = 0.

Thus we get φ(c -) > 0 and lemma 6. Proof : let be (t 0 , x 0 ) ∈ [0, T [×[0, X[ and assume that there exists two shock cuves x 1 (s) and x 2 (s) issuing from (t 0 , x 0 ) such that we have for instance x 1 (s) < x 2 (s) locally in time (t 0 < s < t 1 ) and (c, u) smooth in

Z := {(s, ξ) ; t 0 < s < t 1 , x 1 (s) < ξ < x 2 (s)}.
Then we show easily that a backward characteristic line drawn from any point (t, x) ∈ Z enter one of the two shock curves which contradicts lemma 6.3

We prove now the local uniqueness for rarefaction waves.

Lemma 6.5 Let be (t 0 , x 0 ) a point of discontinuity for c(t, x),c + = c(t 0 , x 0 +0) and c -= c(t 0 , x 0 -0). If c -> c + , then there exists an openset U containing (t 0 , x 0 ), there exists t 1 > t 0 such that ( 8)-( 9)admits an unique smooth solution in (]t 0 , t 1 [×]0, X[) ∩ U.

Proof : we assume that x 0 > 0 (the case x 0 = 0 is similar). According to (EC) there is no shock curve passing through (t 0 , x 0 ), thus the solution is smooth in an open setV =]t 0 , t are piecewise C 1 and we get the local uniqueness of the solution for t > t 0 .

-= {(s, ξ) ; t 0 < s < t 1 , x 0 -δ + M (s -t 0 ) < x < X -(s)} and D + = {(s, ξ) ; t 0 < s < t 1 , X + < x < x 0 + δ -M (s -t 0 )}
F (C -, C + ) = U + -U - h(C + ) -h(C -) is C 1 ,
Corollary 6.1 There exists τ > 0 such that the solution is unique on (0, τ ) × (0, X).

Proof : it follows from lemmas 6.5 and 6.6) that for all x 0 ∈ (0, X) there exists δ > 0, there exists τ > 0 such that the solution isunique on (0, τ ) × (x 0 -δ, x 0 + δ). Then we conclude using a mere compacity argument Proof of theorem 6.1: let T * = sup{τ ∈ [0, T ] ; the solution is unique on (0, τ ) × (0, X)} and assume that T * < T . The solution is unique on (0, T * ) × (0, X). By corollary6.1 there exists τ > 0 such that we have uniqueness on (T * , T * + τ ) × (0, X). Then we have uniqueness on (0, T * + τ ) × (0, X), contradicting the assumption. Finally T * = T and theorem 6.1 holds.

Remark 6.2 In the paragraph (2) we showed that, in the case of smooth solutions, c is the solution of the scalar conservation law [START_REF] Ruthwen | Principles of adsorption and adsorption processes[END_REF]. Thus, it is a natural question to wonder if the weak entropic solutions of (12) (in the usual sense) are the same as those of the system ( 8)-( 9) with the entropy condition (EC) (at least in the case of uniqueness). Actually the answer is positive if and only if the function h is linear and increasing, i.e. if and only if the isotherm function is linear (q * (c 1 , c 2 ) = a c 1 with a > 0 or equivalently "propagates at infinite speed". We show the evolution of c and u at the position x = 0.5. Notice that the maximum principle is not valid for u.

Remark 3 . 1

 31 For the Riemann Problem notice that c satisfies the maximum principle. It is very important since c must be in [0, 1]. Notice also that for all t > 0 the functions c(t, •) and u(t, •) are monotonic thanks to (10)-[START_REF] Rhee | On the theory of multicomponent chromatography[END_REF].

  + and c → H(c) is an increasing function. Let be β = min 0≤c≤1 G(c) > 0 and D the upper solution:

Fig. 1 .

 1 Fig. 1. Riemann problem in a box B ij

  inf 0<t<T u b (t) > 0 and that c 0 and c b have bounded variations. Then the sequence

  T ) and if inf 0<t<T u b (t) > 0 then ln(u b ) ∈ L ∞ . Thanks to lemma 4.1 and lemma 4.2, if c 0 and c b have bounded variations, we have sup N sup 0<t<T T V x (ln(u N (t, .))) < +∞ and lemma 4.6 holds. 5 Convergence towards a weak solution Theorem 5.1 (Global large weak solution)

3 holds. Lemma 6 . 4

 64 From each point (t 0 , x 0 ) ∈ [0, T [×[0, X[ emerges at most one shock curve.

h

  (c) = ac -a). Let us briefly justify this claim . For a shock wave connecting (c -, u -) and (c + , u + ), let be σ the speed of the shock given by the Rankine-Hugoniot condition for[START_REF] Ruthwen | Principles of adsorption and adsorption processes[END_REF]: σ = α(t)[F (c)][c] and let be s the corresponding speed for (8)-(9), given by (31). Writting α(t) = u -e g(c -) we get:s = σ ⇐⇒ c + -c - c + -c -+ c + (h(c + ) -h(c -)) = c + e -(g(c + -g(c -)) -c - c + -c - .Setting c -= 0, c + = x and using[START_REF] Tartar | Compensated compactness and applications to partial differential equations[END_REF] we get, after differentiation with respect to x: g(x) = ln(1 + x h ′ (x)). Differentiating again we get finally h ′′ = 0 as a necessary condition. It is very easily shown that this condition is also sufficient. Finally if h(c) = a c + b we have g ′ (c) = a a c + 1 and, up to an additive constant, F (c) = c a c + 1 : the (EC) condition (c increases through a shock) coincides with the Oleinik condition if and only if F is concave, i.e. a > 0.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Development of a shock. The initial concentration is continuous andincreasing, there is no discontinuity at (t = 0, x = 0). Boundary data arec b = 0.2 and u b = 0.5.

  [START_REF] Bourdarias | Sur un système d'edp modélisant un processus d'adsorption isotherme d'un mélange gazeux. (french) [on a system of p.d.e. modelling heatless adsorption of a gaseous mixture[END_REF] [×]x 0 -2 δ, x 0 + δ[ and has no discontinuity point in {t 0 }×]x 0 -2 δ, x 0 [ and in {t 0 }×]x 0 +δ[. Let be X ± the"limiting characteristics" defined for s ≥ t 0 , following (21), by X ± (s) = x 0 + F ′ (c±) Let time t 1 . This last point is easily justified using (21), (53), inf [0,T ] u b (t) > 0 and that F ′ is continuous and strictly decreasing. It follows that the solution cannot be smooth in Z = {(s, ξ) ; t 0 < s < t 1 , X + (s) < ξ < X -(s)}. Using the characteristic lines given by (52), we define the C 1 functions C -and C + respectively on the open sets D

	s	α(τ ) dτ . We de-
	t 0

fine as above the open set Z = {(s, ξ) ; t 0 < s < t 1 , X -(s) < ξ < X + (s)}.

  which both contains Z. Thanks to (16), we associate them two C 1 functions U - and U + . Then the o.d.e.

	dξ ds	= F (C -(s, ξ(s)), C + (s, ξ(s))), ξ(t 0 ) = x 0 , where

  admits locally (on ]t 0 , t 1 [, restricting t 1 if

	[u]	dX ds	=
	[H(c)]

necessary) an unique solution which determines the shock curve. The entropic solution is uniquely defined for (s, ξ) ∈ D, x < ξ(s) or x > ξ(s) by C -or C + respectively. Remark 6.1 If (t 0 , x 0 ) is a point of discontinuity for u but not for c, the entropy condition (EC) implies that there is no shock curve passing through this point. The characteristic lines, locally defined around (t 0 , x 0 ) by

be (t, x) ∈ Z ∩ V and X(s, t, x), t 0 < s ≤ t the associated backward characteristic line. We have lim s→t 0 +0 X(s, t, x) = x 0 because the characteristic lines cannot cross each other, thus x 0 = x -F ′ (c(t, x)) A(t) with A(t) = t t 0 α(s) ds. Since F ′ is strictly decreasing (prop. 2.1) we getc(t, x) = (F ′ ) -1 x -x 0 A(t) and conversely this last formula defines a smooth solution in Z. Along (s, X ± (s)) we have c = c ± and u = u ± . Lastly the solution is defined in an unique way, using the characteristics lines, in V ∩ {X(s, t 0 , x 0 -δ) < x < X -(s) or x > X + (s)} and lemma 6.5 follows.

We prove now the local uniqueness for the shock waves.

. Under the assumption c -< c + , there exists t 1 > t 0 , there exists δ > 0 such that the solution is unique on

} and presents an unique admissible shock curve issuing from (t 0 , x 0 ).

Proof : let be δ > 0 such that x 0 is the only discontinuity point for c(t, x)

in {0}×]x 0 -δ, x 0 + δ[, and X ± defined as in the proof of lemma 6.5 (notice that X + < X -). Let t 1 > t 0 such that the solution of the o.d.e.

and that t 1 -t 0 is small enough to ensure that the characteristic lines issuing respectively from {0}×]x 0 -δ, x 0 [ and {0}×]x 0 , x 0 + δ[ meet each other before

The following results have been obtained with a Langmuir isotherm, using the Godunov scheme presented in section 4. The values of the various parameters, adapted from those in [START_REF] Douglas Le | Fixed-bed adsorption of gases: Effect of velocity variations on transition types[END_REF] are not important: our purpose is to illustrate the phenomena pointed out along the previous study. The bed profiles in the cases of adsorption or desorption steps (Fig. 2 and3) for the Langmuir or the linear isotherm are the same as in [START_REF] Douglas Le | Fixed-bed adsorption of gases: Effect of velocity variations on transition types[END_REF], but, as pointed out in the introduction, the case of the so called BET isotherm is out of our reach under the assumptions ( 10)-( 11). The "small shock" catches the other and merge into a single one. The concentration c of the inert gas evolves towards the steady state c ≡ 0.1.


Hicham Ezzat

Marine Agogué

Mathieu Cassotti

Pascal Le Masson

Benoit Weil

Leadership-driven Ideation: The Cognitive Effects of Directive Feedbacks on Creativity

Keywords: Leadership, Creativity, Ideation, Functional Fixedness, Directive Feedback

Leadership and creativity have usually been viewed as antagonist concepts, compromised between two contradictory variables: control and freedom. There is growing evidence that too much leadership control could kill subordinates' creativity, while in contrary, too much freedom could lead them to chaos and disorder. In the past decades, countless studies suggested that in order for creativity to emerge, leaders should grant more freedom and autonomy to their followers. Our hypothesis is that leaders could foster subordinates' creative ideation capacities by controlling their ideation processes through directive feedbacks. In this study, we explored the influence of directive feedbacks interactively given by a leader at each idea generated by his/her subordinate, throughout a classical creative problem-solving task done online via a distant text conversation. The task consisted of generating as many original solutions as possible that allows that a hen's egg dropped from a height of ten meters does not break. Results confirmed that leaders' directive feedbacks were able to drive and guide subordinates' ideation paths in two distinctive directions, according to leaders' domainrelevant knowledge and vision for creativity.

Introduction

In today's world, creativity is considered a highly appreciated, not to say indispensable element. In almost every occupation, there is a place for a certain level of creativity and innovative thinking required. Creativity of employees has always been seen as the fundamental element for organizations' success and continuous development. Creativity has been described as the ability to generate ideas that are both novel and useful, while innovation extends this definition, and involve taking those creative ideas and carrying them through to implementation [START_REF] Amabile | The social psychology of creativity: A componential conceptualization[END_REF].

However, generating creative ideas is not an easy task, and despite the favourable conditions and necessary resources available in organizations to innovate, most people have been facing serious difficulties to continuously generate creative ideas. Explanations and enlightenments regarding the obstacles to creativity are well known and studied in cognitive sciences' contexts. Various studies in cognitive sciences have underlined the negative role of cognitive biases to creativity. The most famous one is called functional fixedness or fixation effect [START_REF] Jansson | Design fixation[END_REF], which is the fact that some existing knowledge or known solutions are spontaneously activated in individual's minds, therefore constraining the ability of individuals to generate more creative ideas in creativity contexts. This mental block limits the ability of individuals to see certain objects in different ways in spite of solving particular problems.

Recent findings demonstrated that overcoming these cognitive biases to creativity could be made through expansive examples, i.e. examples of ideas and solutions that are outside the fixation effect, and consequently helps increasing the creative generation capacities of individuals [START_REF] Agogué | The Impact of Type of Examples on Originality: Explaining Fixation and Stimulation Effects[END_REF]. Nevertheless, not only cognitive sciences have been interested in overcoming these cognitive biases. In the field of design sciences, several theorists and scholars have identified creativity stimulation techniques [START_REF] Kowaltowski | Methods that may stimulate creativity and their use in architectural design education[END_REF]. For instance innovative design methods like C-K theory [START_REF] Hatchuel | A new approach of innovative Design: an introduction to CK theory[END_REF], and KCP method [START_REF] Hatchuel | Design theory and collective creativity: a theoretical framework to evaluate KCP process[END_REF] are well recognized to help bypassing cognitive biases' effects (particularly functional fixedness) occurring in creative design contexts.

Numeral researches have lately paid attention on the significant role that can be played by leadership for creativity [START_REF] Mumford | How creative leaders think: Experimental findings and cases[END_REF][START_REF] Reiter-Palmon | Leadership and creativity: Understanding leadership from a creative problem-solving perspective[END_REF][START_REF] Sternberg | A propulsion model of creative leadership[END_REF]. Consequently, considering leadership as a key and central factor to foster creativity among subordinates, much effort has been made by management scholars to design new forms of leadership behaviors that could strongly manage and control the necessary conditions for creativity [START_REF] Agbor | Creativity and innovation: The leadership dynamics[END_REF][START_REF] Amabile | Creativity in the R&D laboratory: Center for Creative Leadership[END_REF]. As a matter of fact, leadership researchers showed that creative leaders could manipulate multiple critical contextual factors for creativity stimulation, using among others, leadership tools like: goal-settings, instructions, feedbacks, etc … In this study, we were interested to emphasize this important role leadership could play for creativity stimulation, by exploring the cognitive effects of leadership feedbacks on subordinates' idea generation processes. Feedbacks are defined as the modification or control of a process by its results, in which the output of an action is returned to modify the subsequent action. Our hypothesis is that leadership feedbacks could have a significant role to play to control ideation processes, and most importantly drive it in certain potential directions for creativity.

Leadership: A Major Function for Creativity

Introduction to Leadership

Although leadership has multiple definitions depending on the perception of leadership theorists [START_REF] Stogdill | Handbook of leadership: A survey of the literature[END_REF], it can be generally described as "a process of social influence in which an individual (the leader) is able to enlist the support of others (subordinates or followers) in the accomplishment of a common task" [START_REF] Chemers | An integrative theory of leadership[END_REF].

Leadership has multiple types and styles that can be classified broadly by their behaviors towards their followers and their decision-making styles. It is important to note that a leader is hardly belonging to only one specific type of leadership, but is rather a mixture and combination of different types. Lewin examined for instance three leadership styles related to the relations of power existing between leaders and subordinates, which are: autocratic, democratic and laissez-faire types [START_REF] Lewin | Patterns of aggressive behavior in experimentally created "social climates[END_REF]. Autocratic leaders are easily known by their unilateral and authoritarian decisions, as they command work processes and constrain significantly subordinates' participation. On the contrary, democratic leaders involve subordinates through joint and cooperative decision-making, after facilitating consensus and agreement with their teams, and usually realize tasks collectively.

Finally, laissez-faire leaders (known as the most passive type of leadership style) are recognized by their delegated decision making style, or even sometimes their decision avoidance, and thus usually create a comfortable work climate that could easily lead to chaos and disorder.

Other recent leadership types exist as well in the rich leadership bibliography.

Transactional leaders are characterized by their cold social exchange with subordinates, depending on rewards and punishments as transaction exchanges to ensure followers' fulfillment, based on job descriptions to complete well-defined goals [START_REF] Weber | The theory of economic and social organization[END_REF].

Transactional leadership continues to subsist well among leadership types, and still has a significant place in contemporary leadership theories. Moreover, recognized as an extension of transactional leadership, transformational leadership is more than a leadership style.

Transformational leadership is a process that changes and transforms people, whereby subordinates feel trust, and appreciation towards their leaders. It is recognized today as one of the most promised leadership type in the evolutionary tree leadership theories [START_REF] Van Seters | The evolution of leadership theory[END_REF]. Transformational leaders behave according to the following factors: employing idealized influence, inspirational motivation, intellectual stimulation, and individualized considerations [START_REF] Bass | Choking under monitoring pressure: being watched by the experimenter reduces executive attention[END_REF].

Recently, and among various new leadership types that emerged lately throughout the last decade, new types like servant, rotated, authentic, and virtual or e-leadership (to cite only the most famous ones) also emerged and took their positions among existing leadership traditional types.

Although leadership theorists recognized that many of these types (specifically transactional and transformational leadership styles) could have certain interesting relationship with creativity, there is clear evidence that they were not specifically designed and modeled for creativity stimulation.

A Growing Importance of Creativity in Leadership Theories

Studies on leadership have produced numerous theories all sorted according to a varying number of leadership study eras [START_REF] Bolden | A review of leadership theory and competency frameworks: Centre for Leadership Studies[END_REF][START_REF] Brungardt | The making of leaders: A review of the research in leadership development and education[END_REF][START_REF] Daft | The leadership experience: Cengage Learning[END_REF][START_REF] Van Seters | The evolution of leadership theory[END_REF][START_REF] Yukl | Leadership in organizations: Pearson Education India[END_REF]. Most important study eras are: personality and traits, power and influence leadership, behavioral, situational, contingency, transactional, and transformational leadership eras.

However, in most of these leadership study eras, leadership scholars seem to have directly integrated creativity in the broader and larger concept of efficiency. But recently, being a subject of interest among scholars from different disciplines ranging from neurosciences to management, and as a consequence of its growing importance, leadership theorists and academics have finally began to add the element creativity as an imperative variable of the conceptualization equation of today's leadership definition.

Leadership Competencies: A Focus on Creativity

In 2010, IBM made a survey to more than 1500 Chief Executive Officers from 60 different countries, asking them what are the most important leadership qualities for the next five years. Results revealed that 60% of CEOs considered creativity as the most important quality and competency for leaders in future years [START_REF] Carr | The most important leadership quality for CEOs? Creativity[END_REF].

Leadership competencies are personal skills and behaviors contributing to better subordinate performance. There is no doubt that leaders must have minimum competencies to inspire their subordinates. Besides classical leadership skills as planning, organization and goal setting, literature has highlighted four main categories of non-traditional leadership skills, which are: social, behavioral, emotional, and cognitive competencies [START_REF] Day | Leadership development: A review in context[END_REF]. The ability of a leader to tailor his competencies according to the current situation turns out to be his distinctive and personalized competency. All these competencies are well linked to leader's cognitive competencies, which refers to leader's degree of integration and ability to use knowledge structure to organize, interpret and process big amount of complex information within his/her cognitive space.

Today, creativity is considered as a principal competency and quality for any leader. "Being creative" or "being innovator" for a leader become today a fundamental element.

Very early studies on creative leadership have emphasized the importance of the creativity of leaders themselves, assuming that by being creative, leaders will be having the appropriate vision and guidance methodologies to enhance followers creativity [START_REF] Mumford | How creative leaders think: Experimental findings and cases[END_REF].

Literature Review

Leadership versus Creativity: A Balance between Control and Freedom

As we have previously indicated, creativity and innovation were newly added to today's leadership design. Management scholars argue that in order for creativity to appear, a certain level of autonomy and freedom is needed [START_REF] Amabile | Assessing the work environment for creativity[END_REF][START_REF] Mclean | Organizational culture's influence on creativity and innovation: A review of the literature and implications for human resource development[END_REF]. As a matter of fact, researchers in management tended to believe that the most important aspect of works' environments leading to creativity and innovation is autonomy, described as the degree to which individuals have freedom to decide how to carry out their work task. In this regard, Amabile et al. have identified that, in order for creativity to flourish, leaders should allow followers to decide how to climb a mountain, rather than letting them choose which one to climb [START_REF] Amabile | How to kill creativity[END_REF].

From another point of view, leadership's literature demonstrated that creativity could as well emerge with a certain level of leadership control and guidance. In this regard, leadership should be able and competent enough to stimulate creativity among subordinates.

Literature review on creative leadership has underlined the role played by leaders for creativity [START_REF] Mumford | Leading creative people: Orchestrating expertise and relationships[END_REF]. Prior works have majorly reduced and concentrated leaders' role as facilitators, mentors, or mediators to organizational creativity [START_REF] Woodman | Toward a theory of organizational creativity[END_REF]. However, numerous studies have examined varied factors that can either foster or hinder employees' creativity at individual, group, and organizational levels, and have then introduced the role of creative leaders in this regard.

Literature has shown that leaders can enhance subordinates' creativity by directly or indirectly adapting appropriate contextual factors like group climate, group composition, resources, knowledge management, or even human resources issues [START_REF] Hemlin | Creative knowledge environments[END_REF][START_REF] Shalley | What leaders need to know: A review of social and contextual factors that can foster or hinder creativity[END_REF]. These various contextual factor could be managed and manipulated using leadership tools, among others: leadership goal-settings [START_REF] Carson | Managing Creativity Enhancement Through Goal-Setting and Feedback*[END_REF][START_REF] Locke | A theory of goal setting & task performance[END_REF], leadership instructions [START_REF] Paulus | Effects of quantity and quality instructions on brainstorming[END_REF][START_REF] Runco | Creativity, originality, and appropriateness: What do explicit instructions tell us about their relationships?[END_REF][START_REF] Runco | Explicit Instructions to Be Creative and Original[END_REF][START_REF] Runco | The instructional enhancement of the flexibility and originality scores of divergent thinking tests[END_REF], leadership feedbacks [START_REF] De Stobbeleir | Feedback-Seeking Behavior as a Self-Regulation Strategy for Creative Performance[END_REF][START_REF] Zhou | Feedback valence, feedback style, task autonomy, and achievement orientation: Interactive effects on creative performance[END_REF][START_REF] Zhou | When the presence of creative coworkers is related to creativity: role of supervisor close monitoring, developmental feedback, and creative personality[END_REF][START_REF] Zhou | Can supervisor feedback always promote creativity? The moderating role of employee self-monitoring[END_REF], and even less conventional and usual leadership tools like nonverbal devices [START_REF] Brun | Managing the impacts of non-verbal devices on idea generation: a new challenge for creative leaders[END_REF].

Feedbacks: A Central Leadership Tool to Foster Creativity

Among the long list of tools leaders could use to foster creativity among subordinates (or at least avoid hindering it), there is no doubt that feedbacks maintain a central position.

Feedbacks could be used to regulate and control subordinates' performance in real-time.

Although the concept of feedback itself is widely used in management theories, its definitions vary considerably depending on management theorists perception [START_REF] Ramaprasad | On the definition of feedback[END_REF]. As a basis for improvement and goal attainment, leaders provide subordinates with certain information and reactions about followers' performance to tune it according to their visions.

Very few studies have focused on the close relationship existing between feedbacks and creativity. Indeed, some researchers explored the effect of feedbacks from a relatively broad angle, focusing on the social interaction perspective. Generally, these researches analyzed the exchange of evaluative information on creative performance, arguing that it could have a strong impact on enhancing creative processes [START_REF] De Stobbeleir | Self-regulation of creativity at work: The role of feedback-seeking behavior in creative performance[END_REF]. These studies incorporated feedbacks in the broader concept of interactions existing between employees and supervisors or between employees themselves, underscoring the importance of being exposed to others' ideas and perspectives to boost the generation of creative ideas. Other researches studied for instance feedbacks from a more self-monitoring and regulation perception, noting that it helps regulate individuals' creative performance [START_REF] Zhou | Can supervisor feedback always promote creativity? The moderating role of employee self-monitoring[END_REF]. However, most of the researches that have explored more deeply the question of feedbacks, from a creative ideation perspective, share common views with management scholars encouraging a certain level of autonomy for creativity. In fact, these findings have emphasized that delivering negative and controlling feedbacks to employees could damage their creative performance, while in contrary delivering constructive or developmental feedbacks to employees could have a positive impact on creativity [START_REF] Carson | Managing Creativity Enhancement Through Goal-Setting and Feedback*[END_REF][START_REF] Zhou | Feedback valence, feedback style, task autonomy, and achievement orientation: Interactive effects on creative performance[END_REF][START_REF] Zhou | When the presence of creative coworkers is related to creativity: role of supervisor close monitoring, developmental feedback, and creative personality[END_REF][START_REF] Zhou | Can supervisor feedback always promote creativity? The moderating role of employee self-monitoring[END_REF].

Nevertheless, most of the abovementioned studies have evaluated feedbacks from a relatively large angle, and none of these studies have focused on the real utility of interactive feedbacks, which is regulating the ideation process depending on leader's vision.

Research Question

In line with the above, could leadership feedbacks tune and regulate subordinates' ideation processes according to certain creativity goals and visions? In other words, could leadership feedbacks direct and guide ideation processes towards certain potential directions that could most likely lead to creative outcomes? Are creative leaders capable of driving subordinates' path of ideation in certain directions?

Our general hypothesis is that despite the negative perception of "controlling" feedbacks on creativity as reported by management and creativity literature encouraging more freedom and autonomy for creativity to arise, feedbacks could in contrary enhance employees' creative performance in certain cases, provided that leaders have appropriate visions and domain-relevant knowledge, and this in order to manage appropriately ideation processes in real time.

To test this general hypothesis, we first modeled the ideation process followed by a feedback sub process, in order to use it as a basis of our experiment. We then modeled five different leadership behaviors, depending on two types of directive feedbacks that could guide subordinates' ideas generation paths in two opposite directions, either in fixation or expansion. We then analyzed the obtained results in the subsequent sections. Finally, we ended our paper with the conclusion, limitations, and future works. We first modeled our experimentation as a process of idea generation based on an initial instruction and guideline (leadership goal-setting), where leader clarifies the goals and objectives to his/her subordinates via an instructional process (input). The system is then regulated and tuned according to leader's goal vision attainment or not by subordinates (output), and this via another process of cognitive feedback stimulus given by the leader to influence subordinate's subsequent ideation response.

Modeling Directive Feedbacks using C-K Theory

In line with the above, we were interested in this study to model new approaches of leadership directive feedbacks that could guide subordinates' ideas generation paths exclusively inside a restrictive zone, i.e. a conceptual space associated to the fixation effect;

or in contrary to guide the generation of ideas in an expansive zone (outside the fixation effect zone).

To do that, we chose to model leadership cognitive feedback stimulus as directive feedbacks type using C-K (Concept-Knowledge) theory [START_REF] Hatchuel | CK theory[END_REF], since this theory of cognitive reasoning is well recognized to have strong effects on overcoming cognitive biases' effects occurring in creative design contexts. C-K theory defines two distinct spaces: a space of concept ("C") and a space of knowledge ("K"). The process of design is described as a double expansion of both C and K spaces, and this via four operators as illustrated in Figure 2 below: • C à K: this operator named "conjunctions" pursues for added (or subtracted)

properties in K space to reach propositions having a logical status (true or false).

• K à C: this operator name "disjunctions" in contrary to conjunctions, adds (or subtracts) some properties coming from K space to new concepts in C space having no logical status.

• C à C: this operator expands the C space by adding a new partition to it. This new partition can be "restrictive" if it does not change object's definition or K4Expansion( C4Expansion( attributes, or "expansive" if it transforms object's definition and identity by adding (or removing) unexpected attributes.

• K à K: this operator expands the K space by adding new knowledge basis to it, and indicates the knowledge structures created within concept designs.

Identifying Fixations and Expansions

We tested our experiment using a classical creative task, which consists of designing a process that allows that a hen's egg dropped from a height of ten meters does not break. [START_REF] Agogué | The Impact of Type of Examples on Originality: Explaining Fixation and Stimulation Effects[END_REF] We chose this particular creativity task among others since we have a vast existing database of ideas and solutions of more than thousands subjects from different profiles that have performed this task within the past years. This database reveals that more than 80% of previous participants generated ideas around three main categories of conventional "restrictive" solutions (which are damping the shock, slowing the fall, and protecting the egg). However, only 20% of participants usually generate unconventional "expansive" solutions (for instance: before and after the fall, with a living device, using the intrinsic properties of the environment, etc..) as presented in the C-K diagram in Figure 3.

Using CK theory, we were able to differentiate between restrictive paths of ideas and solutions (dark partitions) and expansive ones as shown in Figure 3 above. By doing this, two distinct sub-spaces could be identified: fixation and expansion zones [START_REF] Agogué | The Impact of Type of Examples on Originality: Explaining Fixation and Stimulation Effects[END_REF].

Subsequently, we modeled two types of feedback stimulus given by leaders by participants at each idea generated by them throughout this creative task, according to Table 1. Those directive feedbacks consisted on stimulating the generation of expansive or restrictive ideas and solutions, by forcing subordinates' idea generation paths into two opposing directions (fixation or expansion) depending on the following stimuli:

• Stimulus 1 ("continue in this path"): This stimulus underscores the supposedly approval and agreement of the leader with the type of ideas generated by followers. We hypothesized that this leadership cognitive feedback stimulus should force the following generation of ideas and solutions to be consistent with the type of ideas and solutions generated by subordinates.

• Stimulus 2 ("search for another path"): This stimulus underscores the supposedly disapproval and disagreement of the leader with the idea generated by followers. We hypothesized that this leadership cognitive feedback stimulus should force the following generation of ideas and solutions to be different and dissimilar from the type of ideas and solutions generated by subordinates.

We considered the third type of stimulus ("I confirm receipt of your idea, and await for the next one") to stand neutral and unbiased vis-à-vis of the two other stimuli, with no major effects on subordinates' ideation response, and in order to be used as a control case for the comparison study of expansion-oriented and fixation-oriented groups.

Participants

Participants (N=90) of the Faculty of Psychology of Paris Descartes University have participated in this study. Subjects were between 17 and 30 years old, with a mean age of 20,5. All subjects recruited to perform this task didn't know the egg's task and haven't done any creativity tasks previously.

Procedure

Participants were recruited from the Faculty of Psychology of Paris Descartes University. They were asked to perform a creative task via an online text conversation with their experimenter. We chose to perform the complete task online via a text (written) chat conversation on Skype to avoid any type of social biases that could appear and affect our experimentation, as a consequence of the physical presence of experimenters face to face in front of participants [START_REF] Bass | Choking under monitoring pressure: being watched by the experimenter reduces executive attention[END_REF]. The task duration was accurately set to 10 minutes for each participant. Participants were randomly divided into five groups (20 subjects in each of the groups 1, 2 and 3; and 15 subjects in each of the groups 4 and 5). Each participant was assigned a different leadership behavior type (neutral case, expansion-oriented or fixation-oriented or independent) depending on the group he/she was randomly assigned to, as shown in Table 1.

Groups Leadership Cognitive Feedback Stimulus

Leadership goal-setting initial instruction consisted of explaining the task guidelines to subordinates ("the aim is to propose the maximum number of original solutions to ensure that a hen's egg dropped from a distance of ten meters does not break."), specifying to them to "be as creative as possible in this task".

Participants of the first group were chatting with a neutral leader, which simply acknowledges reception of ideas generated by subordinates, awaiting the next one. We considered group 1 as a referential case for studying the other four groups.

Participants of the second group were chatting with an expansion-oriented leader trying to push the ideation process towards expansion; if the idea is generated by subordinates in fixation zone, leader asks subordinates to search for another path (stimulus 2), while in contrary if the idea is generated by subordinates in expansion zone, leader supports subordinates to continue in this path (stimulus 1). We considered that this leader behavior should force the stimulation of expansive solutions.

Contrary to group 2, the third group was having a fixation-oriented leader trying to mislead the ideation process towards fixation; if the idea is generated by subordinates in fixation zone, leader supports subordinates to continue in this path (stimulus 1), while in contrary if the idea is generated by subordinates in expansion zone, leader asks subordinates to search for another path (stimulus 2). We considered that this leader behavior should force the stimulation of restrictive solutions.

Group 4 and 5 were having leaders that permanently send them the same stimuli independently from the type of ideas generated: group 4 always sent the stimulus "continue in this path" (stimulus 1), and group 5 the stimulus "search for another path" (stimulus 2), and this throughout the complete task duration. We considered that leader behavior of group 4 should force the generation of similar types of solutions (whether expansive or restrictive), while leader behavior of group 4 should force the generation of dissimilar types of solutions.

Results

Statistical Analysis of Groups' Creativity Performance

Creative people usually exhibit a high ideational fluency, which is the aptitude to come up with many new ideas, high degree of novelty, and also flexibility known as the ability to stimulate variety among new ideas [START_REF] Guilford | Traits of creativity[END_REF]. Divergent thinking has been known to consider three main elements for evaluating a creative ideation process, which are the ideational fluency, ideational originality, and ideational flexibility. In terms of fluency, we computed the mean number of ideas generated by participants in each group. Results showed that participants exposed to expansion-oriented leader proposed slightly more ideas than referential group, while participants of fixation-oriented group where somewhat inferior than group 1 and group 2 in this regard. Groups 4 and 5 were respectively able to generate fewer ideas than other groups.

In terms of flexibility of solutions, we calculated the mean number of different types of ideas generated by participants in each group. In this regard, all groups were below the referential group. Interestingly, the presence of directive feedbacks perturbed the variety of solutions of subordinates.

Finally, we analyzed originality of ideas generated in each group by computing the frequency of occurrence of type of solutions given across all the subjects in each group. In this regard, we found that participants of group 2 generated more original and unique ideas than other groups, while participants of group 3 and 4 generated the lowest number of original and unique ideas. This result indicates the opposite effects of the expansion-oriented leader in overcoming fixation effects among subordinates, and the fixation-oriented leader in considerably forcing subordinates' exploration inside the fixation effects zone. To have a clearer view of the abovementioned facts, we analyzed as well the mean number of ideas and solutions that individuals were able to generate inside the fixation zone (restrictive solutions), and inside the expansion zone, i.e. outside the fixation zone (expansive solutions). Results in Figure 5, show that leader of group 2 and 4 notably forced participants to generate more expansive ideas, and in parallel reduced the number of restrictive ideas; whereas leader of group 3 and group 5 had an impact on enhancing participants' generation of restrictive ideas, and at the same time decreased the number of expansive ideas.

From a general perspective, these results confirm that leaders' involvement in subordinates' ideation processes without clear task domain knowledge and vision is blocking their creative performance (groups 4 and 5), which confirms the point of view of most management scholars that leaders should grant more freedom to subordinates for creativity to emerge. In contrary, if leaders have enough task domain knowledge and vision (groups 2 and 3), they can lead subordinates into two opposite directions, either towards expansion (group 2) or towards fixation (group 3).

Ideas Sequential Analysis

Two State Markov Chain Analysis

Nevertheless, analyzing mean number of solutions in each group in terms of fluency/flexibility/originality and in terms of the ratio fixation/expansion is not enough in this study. For this reason, we analyzed the sequential train of ideas generated by participants of each group, by computing the probabilities of transitions from: fixation to expansion, and vice versa. Moreover, we also computed the probabilities of consecutively remaining in one of the two states, or oscillating between the two states (as shown in Figure 6).

Using these probabilities computations, we were interested to illustrate the effect of leadership feedbacks in each group on the probabilities of transitions from a state of fixation to a state of expansion and vice versa. To do so, we considered our system as a stochastic process representing the evolution of ideation process over time. We could then estimate a two state Markov chain as illustrated in Figure 7. that independently from the idea generated, this leader continuously asked to continue in the same path. Otherwise, leader's feedback stimuli in group 5 forced subordinates to oscillate between states of fixation and expansion.

Timing Analysis

Moreover, we were interested to explore the emergence of expansive and restrictive ideas over time only for leaders of group 2 and 3, and the effects of their stimuli in this regard. To do so, we divided the task duration (10 minutes) into 15 periods of 40 seconds each, in which we computed the mean number of expansive and restrictive ideas generated by participants in each of these periods as shown in Figure 8. We chose to discard the analysis for group 4 and group 5, as their timing results were not necessarily significant and meaningful for this study. Linear regression analyses demonstrates that, comparing to the referential case (leader of group 1), leader of group 2 was able to gradually direct the ideation path towards expansion. In contrary, leader of group 3 somewhat progressively impacted the direction of the ideation path towards fixation over time.

Conclusions, Limitations and Future Works

In this study, we explored the effect of leadership directive feedbacks interactively given on subordinates' creative idea generation capacity, according to five leadership behaviors. We performed this on a creative task where the aim was to propose the maximum number of original solutions to ensure that a hen's egg dropped from a distance of ten meters does not break.

The major finding in this study is that leaders could in a certain case improve, and in most cases obstruct, subordinates' creative idea generation capacities by driving their ideation processes through directive feedbacks. Indeed, we explored the influence of two different types of directive feedbacks given by a leader to his/her subordinate ("continue in this path" or "search for another path"), using an online text conversation in order to avoid any social biases that could be resulted from the presence of the leader in front of participants. Results confirmed that leaders' directive feedbacks were able to: (i) control the fluency and originality of subordinates' ideas generated (ii) drive and guide subordinates' ideation paths in two distinctive states (inside or outside fixation), and this according to leaders' domain-relevant knowledge and vision for creativity.

From a management sciences perspective, these results confirm (i) the point of view of most management scholars and academics that leaders should grant more freedom and autonomy to their followers in order for creativity to emerge, since leadership directive feedbacks without knowledge and vision noticeably obstructed the generation capacity of subordinates; (ii) but at the same time these study showed that ideation processes could be controlled and directed towards potential directions for creativity (expansion zone) via directive feedbacks, using certain type of expansion-oriented leadership behavior (group 2), provided that leaders have minimum knowledge and vision for creativity, i.e. if leaders know the dominant design, and are at least able to identify and recognize the principal categories of restrictive ideas and solutions.

A limitation of our study is its high level of abstraction comparing to a typical leadermember situation, taking into account the complexity of the numerous contextual factors that define leadership conceptual equation (real leader-member contexts with incentives, real hierarchical issues with subordinates of different levels, team-level and organizational-level contextual factors, etc..).

Future works will consist in exploring the cognitive effects of particular directive feedbacks that could be less dependent from the type of ideas generated by subordinates, and this in order to examine if "expert-less" leaders, not necessarily having solid domain-relevant knowledge and well-defined vision for creativity, could also improve the creative generation capacities of followers, by driving their ideation paths in potential directions for creativity.

Fig. 1 :

 1 Fig. 1: Modeling Leader-Member Ideation/Feedback Process

Fig. 2 :

 2 Fig. 2: Concept-Knowledge Diagram Operators

Fig. 3 :

 3 Fig. 3: Identifying Fixation/Expansion Paths in the Egg's Task[START_REF] Agogué | The Impact of Type of Examples on Originality: Explaining Fixation and Stimulation Effects[END_REF]

Fig. 4 :

 4 Fig. 4: Statistical Analysis of Groups' Results

Fig. 5 :

 5 Fig. 5: Mean Number of Solutions inside/outside Fixation

Fig. 6 :Fig. 7 :

 67 Fig. 6: State Transition Mean Probabilities and Standard Deviations

Fig. 8 :

 8 Fig. 8: Linear Regression: Expansive/Restrictive Ideas' Timing Analysis

Generation of Restrictive Idea Generation of Expansive Idea Group 1 (Control case)

		"I confirm receipt of your idea,	"I confirm receipt of your idea,
		and await for the next one"	and await for the next one"
	Group 2	"Continue in this path"	"Search for another path"
	(Expansion-oriented)
	Group 3	"Search for another path"	"Continue in this path"
	(Fixation-oriented)
	Group 4	"Continue in this path"	"Continue in this path"
	(independent type A)
	Group 5	"Search for another path"	"Search for another path"
	(independent type B)

Table 1 :

 1 Directive Feedbacks Stimuli

Acknowledgements

This research was financed by a grant from the French National Research Agency (project ANR-13-SOIN-0004-02 IDéfixE). The authors would also like to thank Clara Jonquières for helping us to perform this experiment.