Loïc Bourdin
email: bourdin.l@univ-pau.fr

Nonshifted calculus of variations on time scales with ∇-differentiable σ

Keywords: Time scale, calculus of variations, Euler-Lagrange equations, Noether's theorem. AMS Classification: 34N05, 39A12, 39A13, 39A10

In calculus of variations on general time scales, an Euler-Lagrange equation of integral form is usually derived in order to characterize the critical points of nonshifted Lagrangian functionals, see e.g. [R.A.C. Ferreira and co-authors, Optimality conditions for the calculus of variations with higher-order delta derivatives, Appl. Math. Lett., 2011]. In this paper, we prove that the ∇-differentiability of the forward jump operator σ is a sharp assumption on the time scale in order to ∇-differentiate this integral Euler-Lagrange equation. This procedure leads to an Euler-Lagrange equation of differential form. Furthermore, from this differential form, we prove a Noether-type theorem providing an explicit constant of motion for Euler-Lagrange equations admitting a symmetry.

Introduction

The time scale theory was introduced by S. Hilger in his PhD thesis [START_REF] Hilger | Ein Masskettenkalkül mit Anwendungen auf Zentrumsmannigfaltigkeiten[END_REF] in 1988 in order to unify discrete and continuous analyses. The general idea is to extend classical theories on an arbitrary non-empty closed subset T of R. Such a subset T is called a time scale. In this paper, it is assumed to be bounded with a = min T and b = max T. Hence, the time scale theory establishes the validity of some results both in the continuous case T = [a, b] and in the purely discrete case T = {a = t 0 < . . . < t N = b}. Moreover, it also treats more general models involving both continuous and discrete time elements, see e.g. [START_REF] Gamarra | Complex discrete dynamics from simple continuous population models[END_REF][START_REF] May | Simple mathematical models with very complicated dynamics[END_REF] for dynamical populations whose generations do not overlap. Another example of application is to consider T = {a} ∪ {a + λ N } with 0 < λ < 1 allowing the time scale theory to cover the quantum calculus [START_REF] Kac | Quantum calculus. Universitext[END_REF].

Since S. Hilger defined the ∆-and ∇-derivatives on time scales, many authors have extended to time scales various results from the continuous or discrete standard calculus theory. We refer to the surveys [START_REF] Agarwal | Basic calculus on time scales and some of its applications[END_REF][START_REF] Agarwal | Inequalities on time scales: a survey[END_REF][START_REF] Bohner | Dynamic equations on time scales[END_REF][START_REF] Bohner | Advances in dynamic equations on time scales[END_REF] of M. Bohner and co-authors. In the continuous case T = [a, b], the operators ∆ and ∇ coincide with the usual derivative operator d/dt, i.e. u ∆ = u ∇ = u where u is the classical derivative of a function u. In the discrete case T = {a = t 0 < . . . < t N = b}, ∆ is the usual forward Euler approximation of d/dt, i.e. ∆u(t k ) = (u(t k+1 )u(t k ))/(t k+1t k ) and similarly, ∇ is the usual backward Euler approximation of d/dt, i.e. ∇u(t k ) = (u(t k )u(t k-1 ))/(t kt k-1 ).

Context in shifted calculus of variations. The pioneering work on calculus of variations on general time scales is due to M. Bohner in [START_REF] Bohner | Calculus of variations on time scales[END_REF]. In particular, he obtains a necessary condition for local optimizers of Lagrangian functionals of type

L(u) = b a L(u σ (τ ), u ∆ (τ ), τ ) ∆τ, (1)
where L : R n × R n × T κ -→ R, (x, v, t) -→ L(x, v, t) is a continuous Lagrangian of class C 1 in its two first variables, u is a C 1,∆ rd (T)-function, u σ = u • σ, u ∆ is the ∆-differential of u and ∆τ denotes the Cauchy ∆-integral defined in [9, p.26]. We refer to Section 2.1 for the precise definitions.

Notation: in the whole paper, the function defined by t -→ ∂L/∂v(u σ (t), u ∆ (t), t), where u ∈ C 1,∆ rd (T), is denoted by ∂L/∂v(u σ , u ∆ , •).

Precisely, M. Bohner characterizes the critical points of L as the solutions of the following ∆ • ∆differential Euler-Lagrange equation, see [START_REF] Bohner | Calculus of variations on time scales[END_REF]Theorem 4.2]:

∂L ∂v (u σ , u ∆ , •) ∆ (t) = ∂L ∂x (u σ (t), u ∆ (t), t). (EL ∆•∆ diff )
Here, the notation ∆•∆ refers to the composition of ∆ with itself in the left-hand term of (EL ∆•∆ diff ). As it is mentioned in [START_REF] Bohner | Calculus of variations on time scales[END_REF], the result of M. Bohner recovers the usual continuous case T = [a, b] (where σ is the identity) where the critical points of Lagrangian functionals of type

L(u) = b a L(u(τ ), u(τ ), τ ) dτ (2)
are characterized by the solutions of the well known continuous Euler-Lagrange equation (see e.g.

[4, p.12]) given by d dt

∂L ∂v (u, u, •) (t) = ∂L ∂x (u(t), u(t), t). (3)
Moreover, the work of M. Bohner in [START_REF] Bohner | Calculus of variations on time scales[END_REF] also recovers the following discrete case T = {a = t 0 < . . . < t N = b} where the critical points of discrete Lagrangian functionals of type

L(u) = N -1 k=0 (t k+1 -t k )L(u(t k+1 ), ∆u(t k ), t k ) (4)
are characterized by the solutions of the well known discrete Euler-Lagrange equation (see e.g. [START_REF] Ahlbrandt | Discrete Hamiltonian systems[END_REF]) given by

∆ ∂L ∂v (u σ , ∆u, •) (t k ) = ∂L ∂x (u(t k+1 ), ∆u(t k ), t k ). ( 5
)
In what follows, we will speak about L (defined in (1)) as a shifted Lagrangian functional in reference to the presence of u σ (instead of u) in its definition. Note that this characteristic has no consequence on the continuous case but let us mention the presence of u(t k+1 ) (instead of u(t k )) in the discrete case. We will see that this difference is important at the discrete level and a fortiori at the time scale one too. In particular, this shifted framework does not cover the variational integrator studied in [START_REF] Hairer | Geometric numerical integration[END_REF][START_REF] Marsden | Discrete mechanics and variational integrators[END_REF]. We refer to the next paragraph for more details.

Since the publication of [START_REF] Bohner | Calculus of variations on time scales[END_REF], the shifted calculus of variations is widely investigated in several directions: with double integral [START_REF] Bohner | Double integral calculus of variations on time scales[END_REF], with higher-order ∆-derivatives [START_REF] Ferreira | Higher-order calculus of variations on time scales[END_REF], with nonfixed boundary conditions and transversality conditions [START_REF] Hilscher | Calculus of variations on time scales: weak local piecewise C 1 rd solutions with variable endpoints[END_REF], with double integral mixing ∆-and ∇-derivatives [START_REF] Malinowska | The delta-nabla calculus of variations[END_REF], with higher-order ∇-derivatives [START_REF] Martins | Calculus of variations on time scales with nabla derivatives[END_REF], etc. We also refer to [START_REF] Hilscher | Weak maximum principle and accessory problem for control problems on time scales[END_REF][START_REF] Hilscher | First order conditions for generalized variational problems over time scales[END_REF] for shifted optimal control problems. Let us mention that shifted variational problems are particularly suitable (in comparison with the nonshifted ones) because of the emergence of a shift in the integration by parts formula on time scales (see [START_REF] Bohner | Dynamic equations on time scales[END_REF]Theorem 1.77

p.28]) given by b a u(τ ) • v ∆ (τ ) ∆τ = u(b) • v(b) -u(a) • v(a) - b a u ∆ (τ ) • v σ (τ ) ∆τ. ( 6
)
Context in nonshifted calculus of variations. As mentioned in the previous paragraph, the shifted calculus of variations on general time scales developed in [START_REF] Bohner | Calculus of variations on time scales[END_REF] does not cover an important discrete calculus of variations. Precisely, recall that the critical points of (nonshifted) discrete Lagrangian functionals of type

L(u) = N -1 k=0 (t k+1 -t k )L(u(t k ), ∆u(t k ), t k ) (7)
are characterized by the solutions of the well known discrete Euler-Lagrange equation (see e.g. [START_REF] Hairer | Geometric numerical integration[END_REF]) given by

∇ ∂L ∂v (u, ∆u, •) (t k ) = t k+1 -t k t k -t k-1 ∂L ∂x (u(t k ), ∆u(t k ), t k ). ( 8
)
Recall that the above discrete Euler-Lagrange equation ( 8) corresponds to the variational integrator obtained and well studied in [START_REF] Hairer | Geometric numerical integration[END_REF][START_REF] Marsden | Discrete mechanics and variational integrators[END_REF]. In particular, it is an efficient numerical scheme for the continuous Euler-Lagrange equation (3) preserving its variational structure and relative properties at the discrete level. In this (nonshifted) discrete case, note the emergence of the composition between the operators ∇ and ∆. In numerical analysis, it is well known that such a composition allows to get a larger order of convergence. Up to our knowledge, only few studies treat on the nonshifted calculus of variations on general time scales, see [START_REF] Cresson | Time scale differential, integral, and variational embeddings of Lagrangian systems[END_REF][START_REF] Ferreira | Optimality conditions for the calculus of variations with higher-order delta derivatives[END_REF][START_REF] Hilscher | A note on the time scale calculus of variations problems[END_REF]. In these papers, the critical points of (nonshifted) Lagrangian functionals of type

L(u) = b a L(u(τ ), u ∆ (τ ), τ ) ∆τ ( 9
)
are characterized by the solutions of the following ∆-integral Euler-Lagrange equation, see e.g. [START_REF] Hilscher | A note on the time scale calculus of variations problems[END_REF]Theorem 4]:

∂L ∂v (u(t), u ∆ (t), t) = σ(t) a ∂L ∂x (u(τ ), u ∆ (τ ), τ ) ∆τ + c. (EL ∆ int )
The objective of this paper is to ∇-differentiate (EL ∆ int ) in order to get a ∇ • ∆-differential Euler-Lagrange equation of type

∂L ∂v (u, u ∆ , •) ∇ (t) = ω(t) ∂L ∂x (u(t), u ∆ (t), t) (10)
that encompasses both the classical continuous case (3) and the (nonshifted) discrete case [START_REF] Bohner | Double integral calculus of variations on time scales[END_REF]. Towards this goal, a difficulty emerges due to the presence of σ in the upper bound of the ∆integral in (EL ∆ int ). In fact, we will exhibit a counter-example (see Example 1) showing that we cannot ∇-differentiate (EL ∆ int ) on general time scales with a general Lagrangian L. It leads us to consider a subclass of time scales. Precisely, we prove that the ∇-differentiability of σ is a sharp assumption on the time scale in order to ∇-differentiate (EL ∆ int ) and to obtain a ∇ • ∆-differential Euler-Lagrange equation of type [START_REF] Bohner | Advances in dynamic equations on time scales[END_REF]. Moreover, in such a case, an interesting phenomena is the direct emergence of this assumption in [START_REF] Bohner | Advances in dynamic equations on time scales[END_REF] since we prove that ω = σ ∇ .

In this paper, we study the consequences of the ∇-differentiability of σ on the structure of T. In particular, we will see how this assumption allows us to apply ∇ on (EL ∆ int ) in order to obtain [START_REF] Bohner | Advances in dynamic equations on time scales[END_REF] with ω = σ ∇ . Note that the ∇-differentiability of σ is not a loss of generality since it is satisfied in the continuous case T = [a, b] (with σ ∇ = 1) and in the discrete case

T = {a = t 0 < . . . < t N = b} (with σ ∇ (t k ) = (t k+1 -t k )/(t k -t k-1 )).
In particular, note that our main result recovers both the usual continuous case (3) and the nonshifted discrete case [START_REF] Bohner | Double integral calculus of variations on time scales[END_REF].

Derivation of Noether-type results. In shifted calculus of variations, we refer to the paper [START_REF] Bartosiewicz | Noether's theorem on time scales[END_REF] studying the existence of a constant of motion for ∆ • ∆-differential Euler-Lagrange equations (EL ∆•∆ diff ). We refer to [START_REF] Martins | Noether's symmetry theorem for nabla problems of the calculus of variations[END_REF] for a similar study with ∇-derivatives. The common strategy is to generalize the celebrated Noether's theorem [START_REF] Kosmann-Schwarzbach | Les théorèmes de Noether[END_REF][START_REF] Noether | Invariant variation problems[END_REF] to time scales. Precisely, under invariance assumption on the Lagrangian L, authors prove that a conservation law can be obtained.

In nonshifted calculus of variations, the nondifferential form of (EL ∆ int ) is an obstruction in order to develop the same strategy. A direct application of our main result is then to provide a Noether-type theorem based on the differential form [START_REF] Bohner | Advances in dynamic equations on time scales[END_REF]. This will be done in Section 4.

Remark 1. For sake of completeness of this introduction, we mention that an Euler-Lagrange equation of differential form in the nonshifted case is obtained in [START_REF] Hilscher | A note on the time scale calculus of variations problems[END_REF]Remark 4]. Precisely, the author characterizes the critical points of L (defined in (9)) as the solutions of the following differential Euler-Lagrange equation:

∂L ∂v (u, u ∆ , •) -µ ∂L ∂x (u, u ∆ , •) ∆ (t) = ∂L ∂x (u(t), u ∆ (t), t). ( 11
)
The advantage of this result is to be valid on every time scale. Nevertheless, this differential form does not directly coincide with the usual discrete Euler-Lagrange equation [START_REF] Bohner | Double integral calculus of variations on time scales[END_REF] and the obtaining of a Noether-type result from this differential form remains an open problem. These observations both give a particular interest to the ∇-differentiation of (EL ∆ int ) and to the ∇ • ∆-differential formulation (10).

Remark 2. Finally, it has to be noted that our whole study is made in terms of Lagrangian functionals involving ∆-integral and ∆-derivative. However, thanks to the duality principle introduced in [START_REF] Caputo | Time scales: from nabla calculus to delta calculus and vice versa via duality[END_REF], all results can be analogously derived for Lagrangian functionals involving ∇-integral and ∇-derivative. This will be done in Section 5.

Organization of the paper. We first give basic recalls on time scale calculus in Section 2.1 and on nonshifted calculus of variations on general time scales in Section 2.2. Section 2.3 is devoted to our main result (Theorem 1). A study on time scales with continuous σ is provided in Section 3.1 and with ∇-differentiable σ in Sections 3.2 and 3.3. The results obtained in Section 3 are instrumental in order to prove Theorem 1. We prove a Noether-type theorem (Theorem 2) in Section 4. In Section 5, we conclude this paper with the analogous results for nonshifted calculus of variations defined in terms of ∇-integral and ∇-derivative.

Nonshifted calculus of variations on time scales with ∇differentiable σ

In this paper, N denotes the set of nonnegative integers, N * denotes the set of positive integers and R + denotes the set of nonnegative reals. We denote by T a bounded time scale with a = min(T), b = max(T) and card(T) ≥ 3. In Section 2.1, we give basic recalls on time scale calculus.

Section 2.2 is devoted to recalls on nonshifted calculus of variations on general time scales developed in [START_REF] Cresson | Time scale differential, integral, and variational embeddings of Lagrangian systems[END_REF][START_REF] Ferreira | Optimality conditions for the calculus of variations with higher-order delta derivatives[END_REF][START_REF] Hilscher | A note on the time scale calculus of variations problems[END_REF]. In particular, the ∆-integral Euler-Lagrange equation (EL ∆ int ) is given as a necessary condition for local optimizers of nonshifted Lagrangian functionals, see Proposition 3.

In Section 2.3, under the assumption of ∇-differentiability of σ, our main result provides a ∇•∆differential Euler-Lagrange equation of type [START_REF] Bohner | Advances in dynamic equations on time scales[END_REF] as a necessary condition for local optimizers of nonshifted Lagrangian functionals, see (EL ∇•∆ diff ) in Theorem 1. We also prove that this assumption is sharp, see Example 1.

Basic recalls on time scale calculus

We refer to the surveys [START_REF] Agarwal | Basic calculus on time scales and some of its applications[END_REF][START_REF] Agarwal | Inequalities on time scales: a survey[END_REF][START_REF] Bohner | Dynamic equations on time scales[END_REF][START_REF] Bohner | Advances in dynamic equations on time scales[END_REF] for more details on time scale calculus. The backward and forward jump operators ρ, σ : T -→ T are respectively defined by

∀t ∈ T, ρ(t) = sup{s ∈ T, s < t} and σ(t) = inf{s ∈ T, s > t}, (12)
where we put sup ∅ = a and inf

∅ = b. A point t ∈ T is
T -→ R + (resp. ν : T -→ R + ) is defined by µ(t) = σ(t) -t (resp. ν(t) = t -ρ(t)) for any t ∈ T.
We set

T κ = T\]ρ(b), b], T κ = T\[a, σ(a)[ and T κ κ = T κ ∩ T κ . Note that T κ κ = ∅ since card(T) ≥ 3. Let us recall the usual definitions of ∆-and ∇-differentiability. A function u : T -→ R n , where n ∈ N * , is said to be ∆-differentiable at t ∈ T κ (resp. ∇-differentiable at t ∈ T κ ) if the following limit exists in R n : lim s→t s =σ(t) u(σ(t)) -u(s) σ(t) -s   resp. lim s→t s =ρ(t) u(s) -u(ρ(t)) s -ρ(t)   . (13)
In such a case, this limit is denoted by u ∆ (t) (resp. u ∇ (t)). Let us recall the following results on ∆-differentiability, see [9, Theorem 1.16 p.5] and [9, Corollary 1.68 p.25]. The analogous results for ∇-differentiability are also valid.

Proposition 1. Let u : T -→ R n and t ∈ T κ . The following properties hold:

1. if u is ∆-differentiable at t, then u is continuous at t. 2. if t ∈ RS and if u is continuous at t, then u is ∆-differentiable at t with u ∆ (t) = u(σ(t)) -u(t) µ(t) . ( 14
)
3. if σ(t) = t, then u is ∆-differentiable at t if and only if the following limit exists in R n :

lim s→t s =t u(t) -u(s) t -s . ( 15
)
In such a case, this limit is equal to u ∆ (t).

Proposition 2. Let u : T -→ R n . Then, u is ∆-differentiable on T κ with u ∆ = 0 if and only if there exists c ∈ R n such that u(t) = c for every t ∈ T.

From Proposition 1 and for every t ∈ RS, note that a function u is ∆-differentiable at t if and only if u is continuous at t. Still from Proposition 1, note that every ∆-differentiable function on T κ is continuous on T.

Recall that a function u is said to be rd-continuous on T if it is continuous at every t ∈ RD and if it admits a left-sided limit at every t ∈ LD, see [9, Definition 1.58 p.22]. We respectively denote by C 0 rd (T) and C 1,∆ rd (T) the functional spaces of rd-continuous functions on T and of ∆differentiable functions on T κ with rd-continuous ∆-derivative. Recall the following results, see [9, Theorem 1.60 p.22]:

• σ is rd-continuous.

• if u ∈ C 0 rd (T), the composition u σ = u • σ is rd-continuous. • if u ∈ C 0 rd (T)
, the composition f • u with any continuous function f is rd-continuous. Let us denote by ∆τ the Cauchy ∆-integral defined in [9, p.26]. For every u ∈ C 0 rd (T κ ), recall that the function U , defined by U (t) = t a u(τ )∆τ for every t ∈ T, is the unique ∆-antiderivative of u (in the sense that U ∆ = u on T κ ) vanishing at t = a, see [9, Theorem 1.74 p.27]. In particular, we have U ∈ C 1,∆ rd (T).

Recalls on nonshifted calculus of variations on general time scales

In this section, we recall some results on nonshifted calculus of variations on general time scales provided in [START_REF] Cresson | Time scale differential, integral, and variational embeddings of Lagrangian systems[END_REF][START_REF] Ferreira | Optimality conditions for the calculus of variations with higher-order delta derivatives[END_REF][START_REF] Hilscher | A note on the time scale calculus of variations problems[END_REF]. Let L be a Lagrangian i.e. a continuous map of class C 1 in its two first variables

L : R n × R n × T κ -→ R (x, v, t) -→ L(x, v, t) (16)
and let L be the following (nonshifted) Lagrangian functional:

L : C 1,∆ rd (T) -→ R u -→ b a L(u(τ ), u ∆ (τ ), τ ) ∆τ. ( 17
)
In this section, our aim is to give a necessary condition for local optimizers of L (with or without boundary conditions at t = a and t = b). In this way, we introduce the following notions and notations:

• C 1,∆ rd,0 (T) = {w ∈ C 1,∆ rd (T), w(a) = w(b) = 0} is called the set of variations of L.
• u ∈ C 1,∆ rd (T) is said to be a critical point of L if DL(u)(w) = 0 for every w ∈ C 1,∆ rd,0 (T). Let us precise that DL(u)(w) denotes the Gâteaux-differential of L at u in direction w.

In particular, if u is a local optimizer of L, then u is a critical point of L. Finally, let us recall the following characterization of the critical points of L, see [START_REF] Cresson | Time scale differential, integral, and variational embeddings of Lagrangian systems[END_REF]Theorem 11] for every t ∈ T κ .

Hence, Proposition 3 provides a necessary condition for local optimizers of L. Precisely, if u is a local optimizer of L, then there exists c ∈ R n such that u satisfies the ∆-integral Euler-Lagrange equation (EL ∆ int ). We refer to Example 1 in Section 2.3 for an application of Proposition 3.

Main result

In this paper, we aim to ∇-differentiate the ∆-integral Euler-Lagrange equation (EL ∆ int ) in order to get a ∇ • ∆-differential one of type [START_REF] Bohner | Advances in dynamic equations on time scales[END_REF]. Precisely, we prove the following result under the assumption of ∇-differentiability of σ.

Theorem 1 (Main result). Let us assume that σ is ∇-differentiable on T κ and let u ∈ C 1,∆ rd (T). Then, u is a critical point of L if and only if u is a solution of the following ∇ • ∆-differential Euler-Lagrange equation:

∂L ∂v (u, u ∆ , •) ∇ (t) = σ ∇ (t) ∂L ∂x (u(t), u ∆ (t), t), (EL ∇•∆ diff )
for every t ∈ T κ κ . Proof. We refer to Propositions 2 and 3 and Corollary 1 in Section 3.3.

Note that this result encompasses both the usual continuous and discrete Euler-Lagrange equations given by ( 3) and ( 8) in Introduction. Indeed, as it is mentioned in Example 4 in Section 3.2, the following properties are satisfied:

• if T = [a, b], σ is ∇-differentiable on T κ with σ ∇ = 1. • if T = {a = t 0 < . . . < t N = b}, σ is ∇-differentiable on T κ with σ ∇ = µ/ν.
Let us prove, from the following simple example, that the assumption of ∇-differentiability of σ is sharp for the validity of Theorem 1 when the Lagrangian L is general (i.e. not specified).

Example 1. Let us consider n = 1, L(x, v, t) = x + v 2 /2 and u ∈ C 1,∆ rd (T) defined by u(t) = t a σ(τ )∆τ for every t ∈ T. Since u satisfies (EL ∆ int ) with c = a, we conclude that u is a critical point of L, see Proposition 3. However, note that ∂L/∂v(u, u ∆ , •) = u ∆ = σ and consequently, the ∇ • ∆-differential Euler-Lagrange equation (EL ∇•∆ diff ) has no sense if σ is not ∇-differentiable. Nevertheless, the following example proves that the ∇-differentiability of σ is not necessary for some Lagrangian (e.g. independent of the variable x).

Example 2. Let us consider n = 1, L(x, v, t) = v 2 /2 and u ∈ C 1,∆ rd (T) defined by u(t) = t for every t ∈ T. Since u satisfies (EL ∆ int ) with c = 1, we conclude that u is a critical point of L, see Proposition 3. Note that u also satisfies (EL ∇•∆ diff ) even if σ is not ∇-differentiable. This phenomena comes from ∂L/∂x = 0 and consequently (EL ∆ int ) is independent of σ and it can be ∇-differentiated.

However, it has to be noted that the independence of the Lagrangian L with respect to the variable x is a very restrictive assumption while the ∇-differentiability of σ is not. We refer to Example 3 for time scales with continuous and noncontinuous σ. We respectively refer to Examples 4 and 5 for time scales with ∇-differentiable and non-∇-differentiable σ.

Time scales with ∇-differentiable σ

In Section 3.1, we study the consequences of the continuity of σ. Then, we study the consequences of the ∇-differentiability of σ in Section 3.2. From these preliminaries, we prove the most important results of this section (Proposition 6 and some corollaries) in Section 3.3. In particular, Corollary 1 is instrumental to prove our main result (Theorem 1).

| | | | | | | | | | -1 0 1 / ∈ LD / ∈ RS 5. If T = [0, 1] ∪ [2, 3], σ is not continuous at 1 ∈ RS ∩ LD. | ∈ RS ∩ LD | | | 0 1 2 3
6. If T is the usual Cantor set (see [9, Example 1.47 p.18]), σ is not continuous at 1/3 ∈ RS ∩ LD.

Remark 3. Let us give a short discussion on the notion of regular time scale originally introduced in [16, Definition 9]. We refer to [START_REF] Bartosiewicz | Control systems on regular time scales and their differential rings[END_REF][START_REF] Mozyrska | A study of diamond-alpha dynamic equations on regular time scales[END_REF] for other applications of this notion. Recall that T is said to be regular if for every t ∈ T, σ • ρ(t) = ρ • σ(t) = t. In particular, for every bounded regular time scale (containing at least two elements), a is necessarily right-dense and b is necessarily left-dense, see [START_REF] Gürses | Integrable equations on time scales[END_REF]Proposition 10]. Hence, all finite time scales (containing at least two elements) are not regular. The regularity of a time scale is then an assumption relatively restrictive. Consequently, we suggest the introduction of the following weakened notion: a time scale is said to be quasi-regular if σ and ρ are continuous on T. Hence, a time scale is quasi-regular if and only if σ • ρ(t) = t for every t ∈ T κ and ρ • σ(t) = t for every t ∈ T κ . Such a weakened notion allows to cover the finite time scales and to preserve the essence of the initial notion.

∇-differentiability of σ

From Lemma 1 and from the nabla version of Proposition 1, we derive the following result.

Proposition 5. The following properties are satisfied:

1. if σ is continuous at t ∈ LS, then σ is directly ∇-differentiable at t with σ ∇ (t) = µ(t)/ν(t).

2. if σ is continuous on T, then σ is ∇-differentiable on T κ if and only if for every t ∈ T κ such that ρ(t) = t, the following limit exists in R:

lim s→t s =t σ(s) -t s -t . (18)
In such a case, this limit is equal to σ ∇ (t).

Proof. Let us prove the first point. From the nabla version of Proposition 1 and since σ is

continuous at t ∈ LS ⊂ T κ , σ is directly ∇-differentiable at t with σ ∇ (t) = σ(t) -σ(ρ(t)) ν(t) = σ(t) -t ν(t) = µ(t) ν(t) , (19)
since σ • ρ(t) = t from Lemma 1.

Let us prove the second point. Since σ is continuous on T, σ is directly ∇-differentiable at every t ∈ LS from the first point. Consequently, σ is ∇-differentiable on T κ if and only if σ is ∇-differentiable at every t ∈ T κ such that ρ(t) = t i.e. if and only if for every t ∈ T κ such that ρ(t) = t, the following limit exists in R:

lim s→t s =ρ(t) σ(s) -σ(ρ(t)) s -ρ(t) = lim s→t s =t σ(s) -σ(t) s -t . (20)
To conclude, it is sufficient to note that if t = a ∈ T κ , then a ∈ RD and σ(a) = a and if t = a, then t ∈ LD and the continuity of σ implies that t / ∈ RS (see Proposition 4) i.e. σ(t) = t. The proof is complete.

If

T = {0} ∪ {±1/2 k , k ∈ N}, then σ is not ∇-differentiable in 0 since 2 k /2 k-1 = 2, 2 k /2 k+1 = 1/2 and 2 = 1/2.
Examples 3, 4 and 5 allow to get a better understanding of the restrictions imposed on a time scale by the ∇-differentiability of σ. Indeed, we conclude that such a time scale has to satisfy the following properties:

• Due to the continuity of σ, no point can be right-scattered and left-dense.

• Due to the ∇-differentiability of σ, the density in a dense point cannot be too weak, in contrary to 1. in Example 5. Secondly, in a left-and right-dense point, the left and the right densities have to be homogeneous with limit equal to 1, as in 8., 10. of Example 4 and in contrary to 2., 3. of Example 5.

Some results and proof of Theorem 1

The most important result of this section is the following one.

Proposition 6. Let u : T -→ R n and let t ∈ T κ κ . If the two following properties are satisfied:

• σ is ∇-differentiable at t; • u is ∆-differentiable at t; then, u σ is ∇-differentiable at t with (u σ ) ∇ (t) = σ ∇ (t)u ∆ (t).
Proof. Since σ is continuous at t, recall that σ • ρ(t) = t from Lemma 1. We distinguish two cases: t ∈ LS and ρ(t) = t.

• Firstly, let us consider that t ∈ LS. Since σ is continuous at t, we have σ ∇ (t) = µ(t)/ν(t), see Proposition 5. If moreover t ∈ RS, then u ∆ (t) = (u(σ(t))u(t))/µ(t) and since t is isolated, u σ is ∇-differentiable at t with

(u σ ) ∇ (t) = u σ (t) -u σ (ρ(t)) ν(t) = u(σ(t)) -u(t) ν(t) = µ(t) ν(t) u ∆ (t) = σ ∇ (t)u ∆ (t). (22)
In the contrary case σ(t) = t, since σ is continuous at t and since u is continuous at t = σ(t), we deduce that u σ is continuous at t ∈ LS. Then, from the nabla version of Proposition 1,

u σ is ∇-differentiable at t with (u σ ) ∇ (t) = u σ (t) -u σ (ρ(t)) ν(t) = u(σ(t)) -u(t) ν(t) = 0, (23)
since σ(t) = t. However, in this case, we have σ ∇ (t) = µ(t)/ν(t) = 0. Consequently, we also retrieve (u σ ) ∇ (t) = σ ∇ (t)u ∆ (t) in this case.

• Secondly, let us consider that ρ(t) = t. Necessarily, t ∈ RD. Indeed, if t = a ∈ T κ , then a ∈ RD. If t = a, then t ∈ LD and since σ is continuous at t and t ∈ T κ , we deduce that t ∈ RD from Lemma 1. Finally, since u is ∆-differentiable at t and since σ is ∇-differentiable at t, we have

(u σ ) ∇ (t) = lim s→t s =t u σ (s) -u σ (ρ(t)) s -ρ(t) = lim s→t s =t σ(s) -t s -t u(σ(s)) -u(t) σ(s) -t = σ ∇ (t)u ∆ (t). (24)
In the previous limit, since σ is continuous at t ∈ LD ∩ RD, we have used that s → t, s = t implies that σ(s) → σ(t) = t, σ(s) = t.

The proof is complete.

From Proposition 6, the following corollary is directly derived.

Corollary 1. Let u : T -→ R n . If the following properties are satisfied:

• σ is ∇-differentiable on T κ ; • u is ∆-differentiable on T κ ;
then, u σ is ∇-differentiable at every t ∈ T κ κ with (u σ ) ∇ (t) = σ ∇ (t)u ∆ (t). From Propositions 2, 3 and Corollary 1, our main result (Theorem 1) is proved. We conclude this section by introducing the following Leibniz formula useful in order to derive a Noether-type theorem (Theorem 2) in Section 4.

Proposition 7 (Leibniz formula). Let u, v : T -→ R n and t ∈ T κ κ . If the following properties are satisfied:

• σ is ∇-differentiable at t; • u is ∆-differentiable at t; • v is ∇-differentiable at t;
then, u σ • v is ∇-differentiable at t and the following Leibniz formula holds:

(u σ • v) ∇ (t) = u(t) • v ∇ (t) + σ ∇ (t)u ∆ (t) • v(t). (25)
Proof. Since σ is continuous at t ∈ T κ , we have σ • ρ(t) = t from Lemma 1. From Proposition 6, u σ is ∇-differentiable at t with (u σ ) ∇ (t) = σ ∇ (t)u ∆ (t). Finally, from the usual Leibniz formula on time scale (see [9, Theorem 1.20 p.8]), we have

u σ • v is ∇-differentiable at t with (u σ • v) ∇ (t) = u σ (ρ(t)) • v ∇ (t) + (u σ ) ∇ (t) • v(t) = u(t) • v ∇ (t) + σ ∇ (t)u ∆ (t) • v(t). ( 26
)
The proof is complete.

Application to a Noether-type theorem

We first review the definition of a one-parameter family of infinitesimal transformations of R n .

Definition 1. Let η > 0. A map Φ is said to be a one-parameter family of infinitesimal transfor-

mations of R n if Φ is a map of class C 2 Φ : [-η, η] × R n -→ R n (θ, x) -→ Φ(θ, x), (27)
such that Φ(0, •) = Id R n .
The action of a one-parameter family of infinitesimal transformations of R n on a Lagrangian allows us to introduce the notion of symmetry for a ∇ • ∆-differential Euler-Lagrange equation (EL ∇•∆ diff ).

Definition 2. Let Φ be a one-parameter family of infinitesimal transformations of R n . A Lagrangian L is said to be invariant under the action of Φ if for every solution u of (EL ∇•∆ diff ) and every t ∈ T κ κ , the map θ -→ L(Φ(θ, u(t)), Φ(θ, u) ∆ (t), t)

has a null derivative at θ = 0. In such a case, Φ is said to be a symmetry of the ∇ • ∆-differential Euler-Lagrange equation (EL ∇•∆ diff ) associated. The most classical examples of invariance of a Lagrangian under the action of a one-parameter family of infinitesimal transformations of R n are given by quadratic Lagrangians and rotations: Example 6. Let us consider n = 2, L(x, v, t) = x 2 + v 2 , η = π > 0 and Φ defined by

Φ : [-π, π] × R 2 -→ R 2 (θ, x 1 , x 2 ) -→ cos(θ) -sin(θ) sin(θ) cos(θ) × x 1 x 2 . (29)
Then, for every u ∈ C 1,∆ rd (T), every (θ, t) ∈ [-π, π] × T κ κ , we have Φ(θ, u) ∆ (t) = Φ(θ, u ∆ (t)) from linearity and continuity of Φ in its two last variables. Consequently, for every u ∈ C 1,∆ rd (T) and every T κ κ , one can easily prove that the map

θ -→ L(Φ(θ, u(t)), Φ(θ, u) ∆ (t), t) (30)
is independent of θ and consequently is constant and has a null derivative at θ = 0.

Finally, on time scales with ∇-differentiable σ, we prove the following Noether-type theorem providing a constant of motion for ∇• ∆-differential Euler-Lagrange equations (EL ∇•∆ diff ) admitting a symmetry.

Theorem 2 (Noether). Let us assume that σ is ∇-differentiable on T κ and let Φ be a oneparameter family of infinitesimal transformations of R n . If L is invariant under the action of Φ, then for every solution u of (EL ∇•∆ diff ), there exists c ∈ R such that

∂L ∂v (u(t), u ∆ (t), t) • ∂Φ ∂θ (0, u σ (t)) = c, (31)
for every t ∈ T κ .

Proof. Let u be a solution of (EL ∇•∆ diff ). Let us differentiate the map given by ( 28) at θ = 0 and let us invert the operators ∆ and ∂/∂θ from the C 2 -regularity of Φ. We obtain for every t ∈ T κ κ :

∂L ∂x (u(t), u ∆ (t), t) • ∂Φ ∂θ (0, u(t)) + ∂L ∂v (u(t), u ∆ (t), t) • ∂Φ ∂θ (0, u) ∆ (t) = 0.

Finally, multiplying this last equality by σ ∇ (t) and using that u is solution of (EL

for every t ∈ T κ κ . From the nabla version of Proposition 2, the proof is complete. Note that this theorem encompasses both the well known Noether's theorems given in the continuous case [4, p.88] and in the (nonshifted) discrete case [START_REF] Hairer | Geometric numerical integration[END_REF]Theorem 6.4]. For an example of application of Theorem 2, one can consider the framework given in Example 6.

The ∇-analogous results

We conclude this paper with the following remark. The whole study made in this paper can be analogously derived for nonshifted calculus of variations with Lagrangian functionals involving a ∇-integral dependent on a ∇-derivative. We refer to the duality principle introduced in [START_REF] Caputo | Time scales: from nabla calculus to delta calculus and vice versa via duality[END_REF] and to [START_REF] Malinowska | A general backwards calculus of variations via duality[END_REF] for an example of application in calculus of variations on general time scales.

Precisely, let us assume that ρ is ∆-differentiable on T κ . Then, the following ∆ • ∇-differential Euler-Lagrange equation on T κ κ :

∂L ∂v (u, u ∇ , •)

∆ (t) = ρ ∆ (t) ∂L ∂x (u(t), u ∇ (t), t) (EL ∆•∇ diff )
characterizes the critical points of the following (nonshifted) Lagrangian functional:

L : C 1,∇ ld (T) -→ R u -→ b a L(u(τ ), u ∇ (τ ), τ ) ∇τ. (35)
In particular, a necessary condition for local optimizers of L is to be a solution of (EL ∆•∇ diff ). Finally, let us assume that L is moreover invariant under the action of a one-parameter family Φ of infinitesimal transformations of R n in the sense that for every solution u of (EL ∆•∇ diff ) and every t ∈ T κ κ , the map θ -→ L(Φ(θ, u(t)), Φ(θ, u) ∇ (t), t)

has a null derivative at θ = 0. Then, for every solution u of (EL ∆•∇ diff ), there exists c ∈ R such that

∂L ∂v (u(t), u ∇ (t), t) • ∂Φ ∂θ (0, u ρ (t)) = c, (37)
for every t ∈ T κ .

Continuity of σ

Let us prove the following characterizations of the continuity of σ at a given point t ∈ T κ . Lemma 1. Let t ∈ T κ . The following properties are equivalent:

1. σ is continuous at t; 2. σ • ρ(t) = t;

t /

∈ RS ∩ LD.

Proof. Let us prove that 1. implies 2.. If t = a ∈ T κ , then a ∈ RD. Then, σ • ρ(a) = σ(a) = a. If t = a, let us assume by contradiction that σ • ρ(t) = t. Necessarily, we have t ∈ RS ∩ LD. Then, let (s k ) ⊂ T be a sequence such that s k < t for any k ∈ N and s k → t. Thus, we have σ(s k ) < t < σ(t) for any k ∈ N and consequently, (σ(s k )) does not tend to σ(t). This is a contradiction with the continuity of σ at t. Let us prove that 2. implies 3..

Let us prove that 3. implies 1.. By contradiction, let us assume that σ is not continuous at t. As a consequence, there exist ε > 0 and a monotone sequence (s k ) ⊂ T such that s k → t and |σ(t)σ(s k )| ≥ ε for every k ∈ N. Firstly, let us assume that (s k ) is decreasing. Then, t ∈ RD and we have t < s k < s k-1 . As a consequence, t = σ(t) < σ(s k ) ≤ s k-1 for any k ∈ N * . It is a contradiction since s k-1 → t. Secondly, let us assume that (s k ) is increasing. As a consequence, t ∈ LD and then t / ∈ RS (see 3.) i.e. σ(t) = t. Finally, we have s k-1 < s k < t and then,

In both cases, we have obtained a contradiction. Note that σ is always continuous at a. Indeed, if a ∈ RD, then a ∈ T κ , a / ∈ RS ∩ LD and Lemma 1 concludes. If a ∈ RS, then a is isolated and thus, σ is continuous at a. This remark and Lemma 1 lead to the following proposition. 1

Let us give some examples of time scale with ∇-differentiable σ.

Example 4.

3. If T = {0} ∪ {z k , k ∈ N} where (z k ) is a decreasing positive sequence tending to 0 and if lim k→∞ z k-1 /z k exists (denoted by ℓ), then σ is ∇-differentiable on T κ . In particular, we have σ ∇ (0) = ℓ. Indeed, let (s k ) ⊂ T be a positive sequence tending to 0. Then, for every k ∈ N, there exists p k ∈ N such that s k = z p k . Since s k → 0, we have p k → +∞. Finally, we obtain

In particular, we have σ ∇ (0) = r.

5. Similarly to 3., we can prove that if T = {0} ∪ {z k , k ∈ N} where (z k ) is an increasing negative sequence tending to 0 and if lim k→∞ z k+1 /z k exists (denoted by ℓ), then σ is ∇differentiable on T κ . In particular, we have σ ∇ (0) = ℓ.

In particular, we have σ ∇ (0) = 1/r.

7. Similarly to 3., we can prove that if T = [-1, 0] ∪ {z k , k ∈ N} where (z k ) is a decreasing positive sequence tending to 0 and if lim k→∞ z k-1 /z k = 1, then σ is ∇-differentiable on T κ .

In particular, we have σ ∇ (0) = 1.

Application

In particular, we have σ ∇ (0) = 1.

9. Similarly to 3., we can prove that if

) is an increasing negative (resp. decreasing positive) sequence tending to 0 and if

In particular, we have σ ∇ (0) = ℓ. Note that, in such a case, we can only have ℓ

for every k ∈ N, then σ is ∇-differentiable on T κ . In particular, we have σ ∇ (0) = 1.

Let us give some examples of time scale with continuous but non-∇-differentiable σ.

Example 5.

1. If T = {0} ∪ {1/k!, k ∈ N}, then σ is not ∇-differentiable in 0 since k!/(k -1)! = k tends to +∞.

2. If T = [-1, 0] ∪ {1/2 k , k ∈ N}, then σ is not ∇-differentiable in 0 since 2 k /2 k-1 = 2 does not tend to 1.


Loïc Bourdin

EXISTENCE OF A WEAK SOLUTION FOR FRACTIONAL EULER-LAGRANGE EQUATIONS. by

Keywords: Fractional Euler-Lagrange equations, existence, fractional variational calculus. AMS Classification: 70H03, 26A33

We derive sufficient conditions ensuring the existence of a weak solution u for fractional Euler-Lagrange equations of the type:

on a real interval [a, b] and where D α -and D α + are the fractional derivatives of Riemann-Liouville of order 0 < α < 1.

1. Introduction 1.1. Context in the fractional calculus. -The mathematical field that deals with derivatives of any real order is called fractional calculus. For a long time, it was only considered as a pure mathematical branch. Nevertheless, during the last two decades, fractional calculus has attracted the attention of many researchers and it has been successfully applied in various areas like computational biology [START_REF] Magin | Fractional calculus models of complex dynamics in biological tissues[END_REF] or economy [START_REF] Comte | Opérateurs fractionnaires en économétrie et en finance[END_REF]. In particular, the first and well-established application of fractional operators was in the physical context of anomalous diffusion, see [START_REF] Zoia | Continuous-time random-walk model of transport in variably saturated heterogeneous porous media[END_REF][START_REF] Zoia | Mass transport subject to time-dependent flow with nonuniform sorption in porous media[END_REF] for example. Let us mention [START_REF] Metzler | The random walk's guide to anomalous diffusion: a fractional dynamics approach[END_REF] proving that fractional equations is a complementary tool in the description of anomalous transport processes. We refer to [START_REF] Hilfer | Applications of fractional calculus in physics[END_REF] for a general review of the applications of fractional calculus in several fields of Physics. In a more general point of view, fractional differential equations are even considered as an alternative model to non-linear differential equations, see [START_REF] Bonilla | Fractional differential equations as alternative models to nonlinear differential equations[END_REF].

For the origin of the calculus of variations with fractional operators, we should look back to 1996-97 when Riewe used non-integer order derivatives to better describe non conservative systems in mechanics [START_REF] Riewe | Nonconservative Lagrangian and Hamiltonian mechanics[END_REF][START_REF] Riewe | Mechanics with fractional derivatives[END_REF]. Since then, numerous works on the fractional variational calculus have been made. For instance, in the same spirit, authors of [START_REF] Cresson | Lagrangian for the convection-diffusion equation[END_REF][START_REF] Cresson | Variational formulations of differential equations and asymmetric fractional embedding[END_REF] have recently derived fractional variational structures for non conservative equations. Furthermore, one can find a comprehensive literature regarding necessary optimality conditions and Noether's theorem, see [START_REF] Bourdin | Formulation of Euler-Lagrange equations for fractional variational problems[END_REF][START_REF] Almeida | A fractional calculus of variations for multiple integrals with application to vibrating string[END_REF][START_REF] Baleanu | Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives[END_REF][START_REF] Bourdin | A continuous/discrete fractional Noether's theorem[END_REF][START_REF] Frederico | A formulation of Noether's theorem for fractional problems of the calculus of variations[END_REF][START_REF] Odzijewicz | Fractional variational calculus with classical and combined Caputo derivatives[END_REF]. Concerning the state of the art on the fractional calculus of variations and respective fractional Euler-Lagrange equations, we refer the reader to the recent book [START_REF] Malinowska | Introduction to the fractional calculus of variations[END_REF].

In the whole paper, we consider a < b two reals, d ∈ N * and the following Lagrangian functional

L(u) = b a L(u, D α -u, t) dt, (1)
where L is a Lagrangian, i.e. a map of the form:

L : R d × R d × [a, b] -→ R (x, y, t) -→ L(x, y, t), (2)
where D α -is the left fractional derivative of Riemann-Liouville of order 0 < α < 1 and where the variable u is a function defined almost everywhere (shortly a.e.) on (a, b) with values in R d . The precise definitions of the fractional operators of Riemann-Liouville will be recalled in Section 2.2. It is well-known that critical points of the functional L are characterized by the solutions of the fractional Euler-Lagrange equation:

∂L ∂x (u, D α -u, t) + D α + ∂L ∂y (u, D α -u, t) = 0, (EL α )
where D α + is the right fractional derivative of Riemann-Liouville, see detailed proofs in [START_REF] Bourdin | Formulation of Euler-Lagrange equations for fractional variational problems[END_REF][START_REF] Baleanu | Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives[END_REF] for example. However, as far as the author is aware and despite particular results in [START_REF] Jiao | Existence of solutions for a class of fractional boundary value problems via critical point theory[END_REF][START_REF] Klimek | Existence -uniqueness result for a certain equation of motion in fractional mechanics[END_REF], no existence result of a solution for (EL α ) exists in a general case.

The aim of this paper is to derive sufficient conditions on L so that (EL α ) admits a weak solution.

Let us note that, in a more general setting, existence results for fractional equations is an emerging field. For instance, there are recent results about existence and uniqueness of solution for a class of fractional evolution equations in [START_REF] Wang | A class of fractional evolution equations and optimal controls[END_REF][START_REF] Zhou | Nonlocal Cauchy problem for fractional evolution equations[END_REF]. Let us enunciate the main result of the paper: Theorem 1. -Let L be a Lagrangian of class C 1 and 0 < (1/p) < α < 1. If L satisfies the following hypotheses denoted by (H 1 ), (H 2 ), (H 3 ), (H 4 ) and (H 5 ):

-

there exist 0 ≤ d 1 ≤ p and r 1 , s 1 ∈ C (R d × [a, b], R + ) such that: ∀(x, y, t) ∈ R d × R d × [a, b], |L(x, y, t) -L(x, 0, t)| ≤ r 1 (x, t) y d 1 + s 1 (x, t); (H 1 ) -there exist 0 ≤ d 2 ≤ p and r 2 , s 2 ∈ C (R d × [a, b], R + ) such that: ∀(x, y, t) ∈ R d × R d × [a, b], ∂L ∂x (x, y, t) ≤ r 2 (x, t) y d 2 + s 2 (x, t); (H 2 ) -there exist 0 ≤ d 3 ≤ p -1 and r 3 , s 3 ∈ C (R d × [a, b], R + ) such that: ∀(x, v, t) ∈ R d × R d × [a, b], ∂L ∂y (x, y, t) ≤ r 3 (x, t) y d 3 + s 3 (x, t); (H 3 ) -coercivity condition: there exist γ > 0, 1 ≤ d 4 < p, c 1 ∈ C (R d × [a, b], [γ, ∞[), c 2 , c 3 ∈ C ([a, b], R) such that: ∀(x, y, t) ∈ R d × R d × [a, b], L(x, y, t) ≥ c 1 (x, t) y p + c 2 (t) x d 4 + c 3 (t); (H 4 ) -convexity condition: ∀t ∈ [a, b], L(•, •, t) is convex, ( H 5 )
then (EL α ) admits a weak solution.

Hypotheses denoted by (H 1 ), (H 2 ), (H 3 ) are usually called regularity hypotheses, see [START_REF] Cesari | Optimization-theory and applications[END_REF][START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF]. In Section 5, we prove that Hypothesis (H 5 ) can be replaced by different convexity assumptions.

1.3. Idea of the proof of Theorem 1. -In the classical case α = 1, D 1 -= -D 1 + = d/dt and consequently (EL α ) is nothing else but the classical Euler-Lagrange equation formulated in the 1750's. In this case, a lot of results of existence of solutions have been already proved. Let us recall that there exist different approaches:

-A first approach is to develop the classical Euler-Lagrange equation in order to obtain an implicit second order differential equation, see [START_REF] Godbillon | Géométrie différentielle et mécanique analytique[END_REF]. Then, under a hyper regularity or non singularity condition on the Lagrangian L, the equation can be written as an explicit second order differential equation and the Cauchy-Lipschitz theorem gives the existence of local or global regular solutions; -A second approach consists in using the variational structure of the equation, see [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF].

Indeed, under some assumptions, the critical points of L correspond to the solutions of the classical Euler-Lagrange equation. The idea is then to prove the existence of critical points of L. In this way, author makes some assumptions (like coercivity and convexity of the Lagrangian L) ensuring the existence of extrema of L. With this second method, author has to use reflexive spaces of functions and consequently, he deals with weak solutions (in a specific sense). In order to prove Theorem 1, we extend the second approach to the strict fractional case (i.e. 0 < α < 1). Indeed, although there exist fractional versions of the Cauchy-Lipschitz theorem (see [START_REF] Kilbas | Theory and applications of fractional differential equations[END_REF][START_REF] Samko | Fractional integrals and derivatives[END_REF]), there is no simple rules for the fractional derivative of a composition and consequently, we can not write (EL α ) in a simpler way. Hence, in the strict fractional case, we can not follow the first method.

Theorem 1 is based on the following preliminaries:

-The introduction in Section 3 of an appropriate reflexive separable Banach space E α,p (see [START_REF] Hilfer | Applications of fractional calculus in physics[END_REF]); -Assuming Hypotheses (H 1 ), (H 2 ) and (H 3 ), Theorem 2 in Section 4 states that if u is a critical point of L, then u is a weak solution of (EL α ); -Assuming additionally Hypotheses (H 4 ) and (H 5 ), Theorem 3 in Section 5 states that L admits a global minimizer. Hence, the proof of Theorem 1 is complete. Let us note that the method developed in this paper is inspired by:

the reflexive separable Banach space introduced in [START_REF] Jiao | Existence of solutions for a class of fractional boundary value problems via critical point theory[END_REF] allowing to prove the existence of a weak solution for a class of fractional boundary value problems;

the suitable hypotheses of regularity, coercivity and convexity given in [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF] proving the existence of a weak solution for classical Euler-Lagrange equations (i.e. in the case α = 1).

1.4. Organisation of the paper. -The paper is organized as follows. In Section 2, some usual notations of spaces of functions are given. We recall the definitions of the fractional operators of Riemann-Liouville and some of their properties. Section 3 is devoted to the introduction and to the study of the appropriate reflexive separable Banach space E α,p . In Section 4, the variational structure of (EL α ) is considered and we prove Theorem 2. In Section 5, we prove Theorem 3. Then, Section 6 is devoted to some examples. Finally, a conclusion ends this paper.

-C ∞ := C ∞ ([a, b]; R d ) the space of infinitely differentiable functions; -C ∞ c := C ∞ c ([a, b]; R d
) the space of infinitely differentiable functions and compactly supported in ]a, b[. We remind that a function f is an element of AC if and only if ḟ ∈ L 1 and the following equality holds:

∀t ∈ [a, b], f (t) = f (a) + t a ḟ (ξ) dξ, (3)
where ḟ denotes the derivative of f . We refer to [START_REF] Kolmogorov | Eléments de la théorie des fonctions et de l'analyse fonctionnelle[END_REF] for more details concerning the absolutely continuous functions.

Finally, we denote by C a (resp.

AC a or C ∞ a ) the space of functions f ∈ C (resp. AC or C ∞ ) such that f (a) = 0. In particular, C ∞ c ⊂ C ∞ a ⊂ AC a .
Convention: in the whole paper, an equality between functions must be understood as an equality holding for almost all t ∈ (a, b). When it is not the case, the interval on which the equality is valid will be specified.

2.2.

Fractional operators of Riemann-Liouville. -Since 1695, numerous notions of fractional operators emerged over the year, see [START_REF] Kilbas | Theory and applications of fractional differential equations[END_REF][START_REF] Podlubny | Fractional differential equations[END_REF][START_REF] Samko | Fractional integrals and derivatives[END_REF]. In this paper, we only deal with the fractional operators of Riemann-Liouville (1847) whose definitions and some basic results are reminded in this section. We refer to [START_REF] Kilbas | Theory and applications of fractional differential equations[END_REF][START_REF] Samko | Fractional integrals and derivatives[END_REF] for the omitted proofs.

Let α > 0 and f be a function defined a.e. on (a, b) with values in R d . The left (resp. right) fractional integral in the sense of Riemann-Liouville with inferior limit a (resp. superior limit b) of order α of f is given by:

∀t ∈]a, b], I α -f (t) := 1 Γ(α) t a (t -ξ) α-1 f (ξ) dξ, (4) respectively
: ∀t ∈ [a, b[, I α + f (t) := 1 Γ(α) b t (ξ -t) α-1 f (ξ) dξ, (5)
where Γ denotes the Euler's Gamma function. If f ∈ L 1 , then I α -f and I α + f are defined a.e. on (a, b). Now, let us consider 0 < α < 1. The left (resp. right) fractional derivative in the sense of Riemann-Liouville with inferior limit a (resp. superior limit b) of order α of f is given by:

∀t ∈]a, b], D α -f (t) := d dt I 1-α -f (t) resp. ∀t ∈ [a, b[, D α + f (t) := - d dt I 1-α + f (t) . (6)
From [17, Corollary 2.2, p.73], if f ∈ AC, then D α -f and D α + f are defined a.e. on (a, b) and satisfy:

D α -f = I 1-α - ḟ + f (a) (t -a) α Γ(1 -α) and D α + f = -I 1-α + ḟ + f (b) (b -t) α Γ(1 -α) . (7)
In particular, if

f ∈ AC a , then D α -f = I 1-α - ḟ .
2.3. Some properties of the fractional operators. -In this section, we provide some properties concerning the left fractional operators of Riemann-Liouville. One can easily derive the analogous versions for the right ones. Properties 1, 2 and 3 are well-known and one can find their proofs in the classical literature on the subject (see [ The first result yields the semi-group property of the left Riemann-Liouville fractional integral:

Property 1.
-For any α, β > 0 and any function f ∈ L 1 , the following equality holds:

I α -• I β -f = I α+β -f. (8)
From Property 1 and Equalities ( 6) and ( 7), one can easily deduce the following results concerning the composition between fractional integral and fractional derivative. For any 0 < α < 1, the following equalities hold:

∀f ∈ L 1 , D α -• I α -f = f and ∀f ∈ AC, I α -• D α -f = f. (9)
Another classical result is the boundedness of the left fractional integral from L p to L p : Property 2. -For any α > 0 and any p ≥ 1, I α -is linear and continuous from L p to L p . Precisely, the following inequality holds:

∀f ∈ L p , I α -f L p ≤ (b -a) α Γ(1 + α) f L p . ( 10
)
The following classical property concerns the integration of fractional integrals. It is occasionally called fractional integration by parts:

Property 3. -Let 0 < α < 1. Let f ∈ L p and g ∈ L q
where (1/p) + (1/q) ≤ 1 + α (and p = 1 = q in the case (1/p) + (1/q) = 1 + α). Then, the following equality holds:

b a I α -f • g dt = b a f • I α + g dt. (11)
This change of side of the fractional integral (from I α -to I α + ) is responsible of the emergence of D α + in (EL α ) although only D α -is involved in the Lagrangian functional L. We refer to Section 4.2 for more details.

The following Property 4 completes Property 2 in the case 0 < (1/p) < α < 1: indeed, in this case, I α -is additionally bounded from L p to C a : Property 4. -Let 0 < (1/p) < α < 1 and q = p/(p -1). Then, for any f ∈ L p , we have:

-I α -f is Holdër continuous on ]a, b] with exponent α -(1/p) > 0; -lim t→a I α -f (t) = 0.
Consequently, I α -f can be continuously extended by 0 in t = a. Finally, for any f ∈ L p , we have I α -f ∈ C a . Moreover, the following inequality holds:

∀f ∈ L p , I α -f ∞ ≤ (b -a) α-(1/p) Γ(α) (α -1)q + 1 1/q f L p . ( 12
)
Proof. -Let us note that this result is mainly proved in [START_REF] Jiao | Existence of solutions for a class of fractional boundary value problems via critical point theory[END_REF]. Let f ∈ L p . We first remind the following inequality:

∀ξ 1 ≥ ξ 2 ≥ 0, (ξ 1 -ξ 2 ) q ≤ ξ q 1 -ξ q 2 . ( 13
) Let us prove that I α -f is Holdër continuous on ]a, b].
For any a < t 1 < t 2 ≤ b, using the Hölder's inequality, we have:

I α -f (t 2 ) -I α -f (t 1 ) = 1 Γ(α) t 2 a (t 2 -ξ) α-1 f (ξ) dξ - t 1 a (t 1 -ξ) α-1 f (ξ) dξ ≤ 1 Γ(α) t 2 t 1 (t 2 -ξ) α-1 f (ξ) dξ + 1 Γ(α) t 1 a (t 2 -ξ) α-1 -(t 1 -ξ) α-1 f (ξ) dξ ≤ f L p Γ(α) t 2 t 1 (t 2 -ξ) (α-1)q dξ 1/q + f L p Γ(α) t 1 a (t 1 -ξ) α-1 -(t 2 -ξ) α-1 q dξ 1/q ≤ f L p Γ(α) t 2 t 1 (t 2 -ξ) (α-1)q dξ 1/q + f L p Γ(α) t 1 a (t 1 -ξ) (α-1)q -(t 2 -ξ) (α-1)q dξ 1/q ≤ 2 f L p Γ(α) (α -1)q + 1 1/q (t 2 -t 1 ) α-(1/p) .
The proof of the first point is complete. Let us consider the second point. For any t ∈]a, b], we can prove in the same manner that:

I α -f (t) ≤ f L p Γ(α) (α -1)q + 1 1/q (t -a) α-(1/p) --→ t→a 0. ( 14
)
The proof is now complete.

Space of functions E α,p

In order to prove the existence of a weak solution of (EL α ) using a variational method, we need the introduction of an appropriate space of functions. This space has to present some properties like reflexivity, see [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF].

For any 0 < α < 1 and any p ≥ 1, we define the following space of functions:

E α,p := {u ∈ L p satisfying D α -u ∈ L p and I α -• D α -u = u a.e.}. (15)
We endow E α,p with the following norm:

• α,p : E α,p -→ R + u -→ u p L p + D α -u p L p 1/p . ( 16
)
Let us note that:

| • | α,p : E α,p -→ R + u -→ D α -u L p (17)
is an equivalent norm to • α,p for E α,p . Indeed, Property 2 leads to:

∀u ∈ E α,p , u L p = I α -• D α -u L p ≤ (b -a) α Γ(1 + α) D α -u L p . (18)
The goal of this section is to prove the following proposition:

Proposition 1. -Assuming 0 < (1/p) < α < 1, E α,p
is a reflexive separable Banach space and the compact embedding E α,p ֒։ C a holds.

Then, in the rest of the paper, we consider: 0 < (1/p) < α < 1 and q = p/(p -1).

Let us detail the different points of Proposition 1 in the following subsections.

3.1. E α,p is a reflexive separable Banach space. -Let us prove this property. Let us consider (L p ) 2 the set L p × L p endowed with the norm (u, v)

(L p ) 2 = ( u p L p + v p L p ) 1/p . Since p > 1, (L p , • L p )
is a reflexive separable Banach space and therefore, (L p ) 2 , • (L p ) 2 is also a reflexive separable Banach space.

We define

Ω := {(u, D α -u), u ∈ E α,p }. Let us prove that Ω is a closed subspace of (L p ) 2 , • (L p ) 2 . Let (u n , v n ) n∈N ⊂ Ω such that: (u n , v n ) (L p ) 2 ---→ (u, v). ( 20
)
Let us prove that (u, v) ∈ Ω. For any n ∈ N, (u n , v n ) ∈ Ω. Thus, u n ∈ E α,p and v n = D α -u n . Consequently, we have:

u n L p -→ u and D α -u n L p -→ v. ( 21
) For any n ∈ N, since u n ∈ E α,p and I α -is continuous from L p to L p , we have:

u n = I α -• D α -u n L p -→ I α -v. (22)
Thus,

u = I α -v, D α -u = D α -• I α -v = v ∈ L p and I α -• D α -u = I α -v = u. Hence, u ∈ E α,p and (u, v) = (u, D α -u) ∈ Ω.
In conclusion, Ω is a closed subspace of (L p ) 2 , • (L p ) 2 and then Ω is a reflexive separable Banach space. Finally, defining the following operator:

A : E α,p -→ Ω u -→ (u, D α -u), (23)
we prove that E α,p is isometric isomorphic to Ω. This completes the proof of Section 3.1.

The continuous embedding

E α,p ֒→ C a . -Let us prove this result. Let u ∈ E α,p and then D α -u ∈ L p . Since 0 < (1/p) < α < 1, Property 4 leads to I α -• D α -u ∈ C a . Furthermore, u = I α -• D α -u
and consequently, u can be identified to its continuous representative. Finally, Property 4 also gives:

∀u ∈ E α,p , u ∞ = I α -• D α -u ∞ ≤ (b -a) α-(1/p) Γ(α) (α -1)q + 1 1/q |u| α,p . (24)
Since • α,p and | • | α,p are equivalent norms, the proof of Section 3.2 is complete.

3.3. The compact embedding E α,p ֒։ C a .
-Let us prove this property. Since E α,p is a reflexive Banach space, we only have to prove that:

∀(u n ) n∈N ⊂ E α,p such that u n Eα,p ---⇀ u, then u n C -→ u. (25)
Let (u n ) n∈N ⊂ E α,p such that:

u n Eα,p ---⇀ u. (26)
Since E α,p ֒→ C a , we have:

u n C -⇀ u. (27) Since (u n ) n∈N converges weakly in E α,p , (u n ) n∈N is bounded in E α,p . Consequently, (D α -u n ) n∈N is bounded in L p by a constant M ≥ 0. Let us prove that (u n ) n∈N ⊂ C a is uniformly lipschitzian on [a, b].
According to the proof of Property 4, we have:

∀n ∈ N, ∀a ≤ t 1 < t 2 ≤ b, u n (t 2 ) -u n (t 1 ) ≤ I α -• D α -u n (t 2 ) -I α -• D α -u n (t 1 ) ≤ 2 D α -u n L p Γ(α) (α -1)q + 1 1/p (t 2 -t 1 ) α-(1/p) ≤ 2M Γ(α) (α -1)q + 1 1/p (t 2 -t 1 ) α-(1/p) .
Hence, from Ascoli's theorem, (u n ) n∈N is relatively compact in C . Consequently, there exists a subsequence of (u n ) n∈N converging strongly in C and the limit is u by uniqueness of the weak limit. Now, let us prove by contradiction that the whole sequence (u n ) n∈N converges strongly to u in C . If not, there exist ε > 0 and a subsequence (u n k ) k∈N such that:

∀k ∈ N, u n k -u ∞ > ε > 0. ( 28
)
Nevertheless, since (u n k ) k∈N is a subsequence of (u n ) n∈N , then it satisfies:

u n k Eα,p ---⇀ u. (29)
In the same way (using Ascoli's theorem), we can construct a subsequence of (u n k ) k∈N converging strongly to u in C which is a contradiction to [START_REF] Riewe | Nonconservative Lagrangian and Hamiltonian mechanics[END_REF]. The proof of Section 3.3 is now complete.

3.4.

Remarks. -Let us remind the following property:

∀ϕ ∈ C ∞ c , I α -ϕ ∈ C ∞ a . (30)
From this result, we get the two following results:

-

C ∞ a is dense in E α,p . Indeed, let us first prove that C ∞ a ⊂ E α,p . Let u ∈ C ∞ a ⊂ L p . Since u ∈ AC a and u ∈ L p , we have D α -u = I 1-α - u ∈ L p . Since u ∈ AC, we also have I α -• D α -u = u. Finally, u ∈ E α,p . Now, let us prove that C ∞ a is dense in E α,p . Let u ∈ E α,p , then D α -u ∈ L p . Consequently, there exists (v n ) n∈N ⊂ C ∞ c such that: v n L p -→ D α -u and then I α -v n L p -→ I α -• D α -u = u, (31)
since

I α -is continuous from L p to L p . Defining u n := I α -v n ∈ C ∞ a for any n ∈ N, we obtain: u n L p -→ u and D α -u n = D α -• I α -v n = v n L p -→ D α -u. (32)
Finally, (u n ) n∈N ⊂ C ∞ a and converges to u in E α,p . The proof of this point is complete;

-In the case (1/p) < min(α, 1 -α), E α,p = {u ∈ L p satisfying D α -u ∈ L p }. Indeed, let u ∈ L p satisfying D α -u ∈ L p and let us prove that I α -• D α -u = u. Let ϕ ∈ C ∞ c ⊂ L 1 . Since D α -u ∈ L p , Property 3 leads to: b a I α -• D α -u • ϕ dt = b a D α -u • I α + ϕ dt = b a d dt (I 1-α -u) • I α + ϕ dt. (33)
Then, an integration by parts gives:

b a I α -• D α -u • ϕ dt = b a I 1-α -u • D 1-α + ϕ dt. (34)
Indeed, I α + ϕ(b) = 0 since ϕ ∈ C ∞ c and I 1-α -u(a) = 0 since u ∈ L p and (1/p) < 1 -α. Finally, using Property 3 again, we obtain:

b a I α -• D α -u • ϕ dt = b a u • I 1-α + • D 1-α + ϕ dt = b a u • ϕ dt, (35)
which concludes the proof of this second point. In this case, let us note that such a definition of E α,p could lead us to name it fractional Sobolev space and to denote it by W α,p . Nevertheless, these notion and notation are already used, see [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF].

4. Variational structure of (EL α )

In the rest of the paper, we assume that Lagrangian L is of class C 1 and we define the Lagrangian functional L on E α,p (with 0 < (1/p) < α < 1). Precisely, we define:

L : E α,p -→ R u -→ b a L(u, D α -u, t) dt. (36)
L is said to be Gâteaux-differentiable in u ∈ E α,p if the map:

DL(u) : E α,p -→ R v -→ DL(u)(v) := lim h→0 L(u + hv) -L(u) h (37)
is well-defined for any v ∈ E α,p and if it is linear and continuous. A critical point u ∈ E α,p of L is defined by DL(u) = 0.

4.1. Gâteaux-differentiability of L. -Let us prove the following lemma:

Lemma 1.
-The following implications hold:

-L satisfies (H 1 ) =⇒ for any u ∈ E α,p , L(u, D α -u, t) ∈ L 1 and then L(u) exists in R; -L satisfies (H 2 ) =⇒ for any u ∈ E α,p , ∂L/∂x(u, D α -u, t) ∈ L 1 ; -L satisfies (H 3 ) =⇒ for any u ∈ E α,p , ∂L/∂y(u, D α -u, t) ∈ L q .
Proof. -Let us assume that L satisfies (H 1 ) and let

u ∈ E α,p ⊂ C a . Then, D α -u d 1 ∈ L p/d 1 ⊂ L 1 and the three maps t -→ r 1 u(t), t , s 1 u(t), t , |L u(t), 0, t | ∈ C ([a, b], R + ) ⊂ L ∞ ⊂ L 1 . Hypothesis (H 1 ) implies for almost all t ∈ [a, b]: |L(u(t), D α -u(t), t)| ≤ r 1 (u(t), t) D α -u(t) d 1 + s 1 (u(t), t) + |L(u(t), 0, t)|. ( 38
)
Hence, L(u, D α -u, t) ∈ L 1 and then L(u) exists in R. We proceed in the same manner in order to prove the second point of Lemma 1. Now, assuming that L satisfies (H 3 ), we have D α -u d 3 ∈ L p/d 3 ⊂ L q for any u ∈ E α,p . An analogous argument gives the third point of Lemma 1.

Let us prove the following result:

Proposition 2. -Assuming that L satisfies Hypotheses (H 1 ), (H 2 ) and (H 3 ), L is Gâteauxdifferentiable in any u ∈ E α,p and:

∀u, v ∈ E α,p , DL(u)(v) = b a ∂L ∂x (u, D α -u, t) • v + ∂L ∂y (u, D α -u, t) • D α -v dt. ( 39
) Proof. -Let u, v ∈ E α,p ⊂ C a . Let ψ u,v defined for any h ∈ [-1, 1] and for almost all t ∈ [a, b] by: ψ u,v (t, h) := L u(t) + hv(t), D α -u(t) + hD α -v(t), t . (40)
Then, we define the following mapping:

φ u,v : [-1, 1] -→ R h -→ b a L(u + hv, D α -u + hD α -v, t) dt = b a ψ u,v (t, h) dt. ( 41
)
Our aim is to prove that the following term:

DL(u)(v) = lim h→0 L(u + hv) -L(u) h = lim h→0 φ u,v (h) -φ u,v (0) h = φ ′ u,v (0) (42)
exists in R. In order to differentiate φ u,v , we use the theorem of differentiation under the integral sign. Indeed, we have for almost all

t ∈ [a, b], ψ u,v (t, •) is differentiable on [-1, 1] with: ∀h ∈ [-1, 1], ∂ψ u,v ∂h (t, h) = ∂L ∂x u(t) + hv(t), D α -u(t) + hD α -v(t), t • v(t) + ∂L ∂y u(t) + hv(t), D α -u(t) + hD α -v(t), t • D α -v(t). (43)
Then, from Hypotheses (H 2 ) and (H 3 ), we have for any h ∈ [-1, 1] and for almost all t ∈ [a, b]:

∂ψ u,v ∂h (t, h) ≤ r 2 u(t) + hv(t), t D α -u(t) + hD α -v(t) d 2 + s 2 u(t) + hv(t), t v(t) + r 3 u(t) + hv(t), t D α -u(t) + hD α -v(t) d 3 + s 3 u(t) + hv(t), t D α -v(t) . (44)
We define:

r 2,0 := max (t,h)∈[a,b]×[-1,1] r 2 u(t) + hv(t), t (45)
and we define similarly s 2,0 , r 3,0 , s 3,0 . Finally, it holds:

∂ψ u,v ∂h (t, h) ≤ 2 d 2 r 2,0 ( D α -u(t) d 2 + D α -v(t) d 2 ∈L p/d 2 ⊂L 1 ) v(t) ∈Ca⊂L ∞ +s 2,0 v(t) ∈Ca⊂L 1 + 2 d 3 r 3,0 D α -u(t) d 3 + D α -v(t) d 3 ∈L p/d 3 ⊂L q ) D α -v(t) ∈L p +s 3,0 D α -v(t) ∈L p ⊂L 1 . (46)
The right term is then a L 1 function independent of h. Consequently, applying the theorem of differentiation under the integral sign, φ u,v is differentiable with:

∀h ∈ [-1, 1], φ ′ u,v (h) = b a ∂ψ u,v ∂h (t, h) dt. (47)
Hence:

DL(u)(v) = φ ′ u,v (0) = b a ∂ψ u,v ∂h (t, 0) dt = b a ∂L ∂x (u, D α -u, t) • v + ∂L ∂y (u, D α -u, t) • D α -v dt. (48) From Lemma 1, it holds: ∂L ∂x (u, D α -u, t) ∈ L 1 and ∂L ∂y (u, D α -u, t) ∈ L q . ( 49
) Since v ∈ C a ⊂ L ∞ and D α -v ∈ L p , DL(u)(v) exists in R.
Moreover, we have:

|DL(u)(v)| ≤ ∂L ∂x (u, D α -u, t) L 1 v ∞ + ∂L ∂y (u, D α -u, t) L q D α -v L p ≤ (b -a) α-(1/p) Γ(α) (α -1)q + 1 1/q ∂L ∂x (u, D α -u, t) L 1 + ∂L ∂y (u, D α -u, t) L q |v| α,p .
Consequently, DL(u) is linear and continuous from E α,p to R. The proof is complete.

4.2. Sufficient condition for a weak solution. -In this section, we prove the following theorem:

Theorem 2. -Let us assume that L satisfies Hypotheses (H 1 ), (H 2 ) and (H 3 ). Then:

u is a critical point of L =⇒ u is a weak solution of (EL α ). (50)
Proof. -Let u be a critical point of L. Then, we have in particular:

∀v ∈ C ∞ c , DL(u)(v) = b a ∂L ∂x (u, D α -u, t) • v + ∂L ∂y (u, D α -u, t) • D α -v dt = 0. ( 51
) For any v ∈ C ∞ c ⊂ AC a , D α -v = I 1-α - v ∈ C ∞ a . Since ∂L/∂y(u, D α -u, t) ∈ L q , Property 3 gives: ∀v ∈ C ∞ c , b a ∂L ∂x (u, D α -u, t) • v + I 1-α + ∂L ∂y (u, D α -u, t) • v dt = 0. (52)
Finally, we define:

∀t ∈ [a, b], w u (t) = t a ∂L ∂x (u, D α -u, t) dt. (53)
Since ∂L/∂x(u, D α -u, t) ∈ L 1 , w u ∈ AC a and ẇu = ∂L/∂x(u, D α -u, t). Then, an integration by parts leads to:

∀v ∈ C ∞ c , b a I 1-α + ∂L ∂y (u, D α -u, t) -w u • v dt = 0. (54)
Consequently, there exists a constant C ∈ R d such that:

I 1-α + ∂L ∂y (u, D α -u, t) = C + w u ∈ AC. (55)
By differentiation, we obtain: Let us note that the use of Property 3 in the previous proof leads to the emergence of D α + in (EL α ) although L is only dependent of D α -. This asymmetry in (EL α ) is a strong drawback in order to solve it explicitly. However, from Theorem 1, the existence of a weak solution for (EL α ) will be guarantee.

-D α + ∂L ∂y (u, D α -u, t) = ∂L ∂x (u, D α -u, t), (56)

Existence of a global minimizer of L

In this section, under assumptions (H 4 ) and (H 5 ), we prove the existence of a global minimizer u of L, see Theorem 3. Then, u is a critical point of L and then, according to Theorem 2, u is a weak solution of (EL α ). This concludes the proof of Theorem 1.

As usual in a variational method, in order to prove the existence of a global minimizer of a functional, coercivity and convexity hypotheses need to be added on the Lagrangian. We have already define Hypotheses (H 4 ) (coercivity) and (H 5 ) (convexity) in Section 1.2. In this section, we introduce two different convexity hypotheses (H ′ 5 ) and (H ′′ 5 ) under which Theorem 1 is still valid:

-Convexity hypothesis denoted by (H ′ 5 ):

∀(x, t) ∈ R d × [a, b], L(x, •, t) is convex and L(•, y, t) (y,t)∈R d ×[a,b] is uniformly equicontinuous on R d . (H ′ 5 )
We remind that the uniform equicontinuity of L(•, y, t) (y,t)∈R d ×[a,b] has to be understood as:

∀ε > 0, ∃δ > 0, ∀(x 1 , x 2 ) ∈ (R d ) 2 , x 2 -x 1 < δ =⇒ ∀(y, t) ∈ R d × [a, b], |L(x 2 , y, t) -L(x 1 , y, t)| < ε. ( 57
)
Let us note that Hypotheses (H 5 ) and (H ′ 5 ) are independent.

-Convexity hypothesis denoted by (H ′′ 5 ):

∀(x, t) ∈ R d × [a, b], L(x, •, t) is convex.
(H ′′ 5 ) Hypothesis (H ′′ 5 ) is the weakest. Nevertheless, in this case, the detailed proof of Theorem 3 is more complicated. Consequently, in the case of Hypothesis (H ′′ 5 ), we do not develop the proof and we use a strong result proved in [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF].

Let us prove the following preliminary result: Lemma 2. -Let us assume that L satisfies Hypothesis (H 4 ). Then, L is coercive in the sense that: lim

u α,p→+∞ L(u) = +∞. ( 58
)
Proof. -Let u ∈ E α,p , we have:

L(u) = b a L(u, D α -u, t) dt ≥ b a c 1 (u, t) D α -u p + c 2 (t) u d 4 + c 3 (t) dt. (59)
Equation [START_REF] Kiryakova | Generalized fractional calculus and applications[END_REF] implies that:

u d 4 L d 4 ≤ (b -a) 1-d 4 p u d 4 L p ≤ (b -a) α+1- d 4 p Γ(α + 1) D α -u d 4 L p = (b -a) α+1- d 4 p Γ(α + 1) |u| d 4 α,p . (60)
Finally, we conclude that:

∀u ∈ E α,p , L(u) ≥ γ D α -u p L p -c 2 ∞ u d 4 L d 4 -(b -a) c 3 ∞ (61) ≥ γ|u| p α,p - c 2 ∞ (b -a) α+1- d 4 p Γ(α + 1) |u| d 4 α,p -(b -a) c 3 ∞ . ( 62
L(u n ) -→ inf v∈Eα,p L(v) =: K. (63)
Since L satisfies Hypothesis (H 1 ), L(u) ∈ R for any u ∈ E α,p . Hence, K < +∞. Let us prove by contradiction that (u n ) n∈N is bounded in E α,p . In the negative case, we can construct a subsequence (u n k ) k∈N satisfying u n k α,p → +∞. Since L satisfies Hypothesis (H 4 ), Lemma 2 gives:

K = lim k∈N L(u n k ) = +∞, (64)
which is a contradiction. Hence, (u n ) n∈N is bounded in E α,p . Since E α,p is reflexive, there exists a subsequence still denoted by (u n ) n∈N converging weakly in E α,p to an element denoted by u ∈ E α,p . Let us prove that u is a global minimizer of L. Since:

u n Eα,p ---⇀ u and E α,p ֒։ C a , (65)
we have:

u n C -→ u and D α -u n L p -⇀ D α -u. (66)
Case L satisfies (H 5 ): by convexity, it holds for any n ∈ N:

L(u n ) = b a L(u n , D α -u n , t) dt ≥ b a L(u, D α -u, t) dt + b a ∂L ∂x (u, D α -u, t) • (u n -u) dt + b a ∂L ∂y (u, D α -u, t) • (D α -u n -D α -u) dt. (67)
Since L satisfies Hypotheses (H 2 ) and (H 3 ), ∂L/∂x(u, D α -u, t) ∈ L 1 and ∂L/∂y(u, D α -u, t) ∈ L q . Consequently, using (66) and making n tend to +∞, we obtain:

K = inf v∈Eα,p L(v) ≥ b a L(u, D α -u, t) dt = L(u). (68)
Consequently, u is a global minimizer of L.

Case L satisfies (H ′ 5 ): let ε > 0. Since (u n ) n∈N converges strongly in C to u, we have:

∃N ∈ N, ∀n ≥ N, u n -u ∞ < δ, ( 69
)
where δ is given in the definition of (H ′ 5 ). In consequence, it holds a.e. on [a, b]:

∀n ≥ N, |L u n (t), D α -u n (t), t -L u(t), D α -u n (t), t | < ε. (70)
Moreover, for any n ≥ N , we have:

L(u n ) = b a L(u, D α -u, t) dt + b a L(u n , D α -u n , t) -L(u, D α -u n , t) dt + b a L(u, D α -u n , t) -L(u, D α -u, t) dt. (71)
Then, for any n ≥ N , it holds by convexity:

L(u n ) ≥ b a L(u, D α -u, t) dt - b a |L(u n , D α -u n , t) -L(u, D α -u n , t)| dt + b a ∂L ∂y (u, D α -u, t) • (D α -u n -D α -u) dt. (72)
And, using Equation (70), we obtain for any n ≥ N :

L(u n ) ≥ b a L(u, D α -u, t) dt -ε(b -a) + b a ∂L ∂y (u, D α -u, t) • (D α -u n -D α -u) dt. (73)
We remind that ∂L/∂y(u, D α -u, t) ∈ L q since L satisfies (H 3 ). Since (D α -u n ) n∈N converges weakly in L p to D α -u, we obtain by making n tend to +∞ and then by making ε tend to 0:

K = inf v∈Eα,p L(v) ≥ b a L(u, D α -u, t) dt = L(u). (74)
Consequently, u is a global minimizer of L.

Case L satisfies (H ′′ 5 ): we refer to Theorem 3.23 in [START_REF] Dacorogna | Direct methods in the calculus of variations[END_REF]. Finally, combining Theorems 2 and 3, the proof of Theorem 1 is now complete.

Examples

Let us consider some examples of Lagrangian L satisfying Hypotheses of Theorem 1. Consequently, the fractional Euler-Lagrange equation (EL α ) associated admits a weak solution u ∈ E α,p .

The most classical example is the Dirichlet integral, i.e. the Lagrangian functional associated to the Lagrangian L given by: L(x, y, t) = 1 2 y 2 .

(75)

In this case, L satisfies Hypotheses (H 1 ), (H 2 ), (H 3 ), (H 4 ) and (H 5 ) for p = 2. Hence, the fractional Euler-Lagrange equation (EL α ) associated admits a weak solution in E α,p for (1/2) < α < 1.

In a more general case, the following Lagrangian L: L(x, y, t) = 1 p y p + a(x, t),

where p > 1 and a ∈ C 1 (R d × [a, b], R + ), satisfies Hypotheses (H 1 ), (H 2 ), (H 3 ), (H 4 ) and (H ′′ 5 ). Consequently, the fractional Euler-Lagrange equation (EL α ) associated to L admits a weak solution in E α,p for any (1/p) < α < 1. Let us note that if for any t ∈ [a, b], a(•, t) is convex, then L satisfies Hypothesis (H 5 ).

In the unidimensional case d = 1, let us take a Lagrangian with a second term linear in its first variable, i.e.:

L(x, y, t) = 1 p |y| p + f (t)x, (77)
where p > 1 and f ∈ C 1 ([a, b], R). Then, L satisfies Hypotheses (H 1 ), (H 2 ), (H 3 ), (H 4 ) and (H 5 ). Then, the fractional Euler-Lagrange equation (EL α ) associated admits a weak solution in E α,p for any (1/p) < α < 1.

Theorem 1 is a result based on strong conditions on Lagrangian L. Consequently, some Lagrangian do not satisfy all hypotheses of Theorem 1. We can cite the Bolza's example in dimension d = 1 given by: L(x, y, t) = (y 2 -1) 2 + x 4 .

(78) L does not satisfy Hypothesis (H 4 ) neither Hypothesis (H ′′ 5 ). Nevertheless, as usual with variational methods, the conditions of regularity, coercivity and/or convexity can often be replaced by weaker assumptions specific to the studied problem. As an example, we can cite [START_REF] Sidi Ammi | Regularity of solutions to higher-order integrals of the calculus of variations[END_REF] and references therein about higher-order integrals of the calculus of variations. Indeed, in this paper, it is proved that calculus of variations is still valid with weaker regularity assumptions.

Conclusion

The method developed in this paper gives a framework in order to study the existence of weak solutions for fractional Euler-Lagrange equations. In this paper, we have studied the special case of a Lagrangian functional involving fractional derivatives of Riemann-Liouville. Nevertheless, such a method can also be developed in the case of fractional derivatives of Caputo or Hadamard. Indeed, these operators satisfy similar properties than Riemann-Liouville's ones, see [START_REF] Kilbas | Theory and applications of fractional differential equations[END_REF][START_REF] Samko | Fractional integrals and derivatives[END_REF]. In fact, the same method can be developed in the case of general linear operators used in [START_REF]Generalized variational problems and Euler-Lagrange equations[END_REF][START_REF] Kiryakova | Generalized fractional calculus and applications[END_REF][START_REF] Odzijewicz | Fractional calculus of variations in terms of a generalized fractional integral with applications to Physics[END_REF][START_REF] Odzijewicz | Generalized fractional calculus with applications to the calculus of variations[END_REF]: this is the aim of a forthcoming paper.

1. 2 .

 2 Main result. -We denote by • the Euclidean norm of R d and C := C ([a, b]; R d ) the space of continuous functions endowed with its usual norm • ∞ . Definition 1. -A function u is said to be a weak solution of (EL α ) if u ∈ C and if u satisfies (EL α ) a.e. on [a, b].

  and then u ∈ E α,p ⊂ C satisfies (EL α ) a.e. on [a, b]. The proof is complete.

)

  Since d 4 < p and since the norms | • | α,p and • α,p are equivalent, the proof is complete. Theorem 3. -Let us assume that L satisfies Hypotheses (H 1 ), (H 2 ), (H 3 ), (H 4 ) and one of Hypotheses (H 5 ), (H ′ 5 ) or (H ′′ 5 ). Then, L admits a global minimizer.Proof. -Let (u n ) n∈N be a sequence in E α,p satisfying:

	Now, we are ready to prove Theorem 3:


G Fau

Numerical Investigations of Chemical Kinetics for Methane Catalysis and Pyrolysis

Numerical Investigations of Chemical Kinetics for Methane Catalysis and Pyrolysis

Extended Abstracts

In the framework of the active and regenerative cooling of hypersonic and space structures, a kinetic mechanism should be proposed to investigate numerically the fuel pyrolysis and the coke formation with a possible surface effect (catalytic reactions). This chemical mechanism should be suitable for Computational Fluid Dynamics approach (limited number of species and reaction). The objective of this study is to obtain a mechanism which permits to compute methane pyrolysis considering both heterogeneous reactions and the coke production. A For this purpose, this work presents some homogeneous and heterogeneous calculations of methane chemistry with metallic catalysts in order to test a kinetic mechanism with the integration of surface reactions. Three catalysts are considered: platinum, nickel and rhodium. A mechanism has been selected. Then, its size (i.e. number of reactions and of species) has been reduced. A parametric study is achieved and several reactors natures are tested. Small differences are noted between the three catalysts for a low value of site density while a strong one appears for a higher one. Both the product distribution and the dynamic are affected. The reactor type used for the computations has been also tested. The differences are limited considering nickel or platinum catalyst while some important discrepancies are observed for the rhodium. These simulations have highlighted some differences and demonstrated the importance of the site density value on the results.

G

  . Fau 1 , N. Gascoin 2 1 Université d'Orléans, Bourges, France; 2 INSA Centre Val de Loire, Bourges, France Primary Technical Area: Theoretical Modeling and Numerical Simulation Techniques for Chemical Propulsion Topics Secondary Technical Area: Environmentally-Friendly "Green" Propellants

Figure 1 .

 1 Figure 1. Effect of catalyst nature on light specie production (mol.%).


Mounir Sahli

Khaled Chetehouna

Francois Faubert

Nadir Bellel

Nicolas Gascoin

Electrochemical preparation process of Mg-La alloys for solid hydrogen Storage

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Hussain Najmi
email: hussain.najmi@insa-cvl.fr

Eddy El Tabach

Khaled Chetehouna

Nicolas Gascoin

François Falempin

François Falempin Flow Con

Flow Configuration Influence On Darcian And Forchheimer Permeabilities Determination

Keywords: Filtration, Darcy's and Forchheimer's permeabilities, Porous stainless steel, Fuel cell application

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

There are three main technologies for hydrogen separation which include: i) pressure swing adsorption (PSA), ii) cryogenic distillation and iii) selective permeation through a membrane. For fuel cell applications, H 2 separation by selective permeation through membrane has several advantages compared to the first two methods. Indeed, several literature works mention that this filtration technique is a simple operation which is characterised by a low cost and a less energy consumption as well as a high purity of hydrogen [START_REF] Spillman | Economics of gas separation membranes[END_REF][START_REF] Lu | Nanoporous materials: science and engineering[END_REF][START_REF] Phair | Materials for separation membranes in hydrogen and oxygen production and future power generation[END_REF][START_REF] Chi | Transient dynamic of hydrogen permeation through a palladium membrane[END_REF][START_REF] Chi | Preparation of thin Pd membrane on porous stainless steel tubes modified by a two-step method[END_REF]. This filtration process related to fuel cell can be found in other fields of application such as geology (Luquot et al. 2009[START_REF] Abdulgader | Hybrid ion exchange -Pressure driven membrane processes in water treatment: A review[END_REF], petrochemical (Tanaka et al. 2005, Shamsabadi et al. 2013), combustion (Franz et al. 2013, Burrerio et al. 2014) or aeronautics. For example, there have been large numbers of literature regarding transpiration cooling in aeronautics [START_REF] Gascoin | High temperature and pressure reactive flows through porous media[END_REF], Wang et al 2011, Zhao et al. 2014). Performance criteria required for the membrane are high flux rate, stability at variable temperature, pressure and high selectivity even in the presence of other gases and contaminants (e.g. CO, CO 2 , CH 4 , H 2 O and H 2 S etc.). Temperature and pressure are the external parameters which can be controlled outside the porous media (i.e. membrane). But the selectivity depends upon the permeability of the material or of the porous medium [START_REF] Pinnaua | Pure-and mixed-gas permeation properties of a microporous spirobisindane-based ladder polymer (PIM-1)[END_REF][START_REF] Baker | Permeability, permeance and selectivity: A preferred way of reporting pervaporation performance data[END_REF].

FLOW CONFIGURATION INFLUENCE ON DARCIAN AND FORCHHEIMER PERMEABILITIES DETERMINATION

Hussain NAJMI It has been recognized in the past decade that the separation factor for gas pairs varies inversely with the permeability of the more permeable gas of the specific pair. Indeed, an analysis of the literature data for binary gas mixtures from the list of He, H 2 , O 2 , N 2 , CH 4 , and CO 2 reveals the above relationship for these mixtures (Robeson 1990, Chenar et al. 2006). Hence the determination of the permeability and studying the factor affecting it is of prime importance.

In previous works, a high pressure test bench with one inlet and one outlet configuration has been developed [START_REF] Gascoin | High temperature and pressure reactive flows through porous media[END_REF][START_REF] Fau | Fuel pyrolysis through porous media: Coke formation and coupled effect on permeability[END_REF]. It has been tested on different type of metal and composite samples to determine the Darcian's and Forchheimer's permeabilities for inert and reactive fluids at high temperature and pressure. Similar experimental setup has been used by several authors to determine the permeability of disk and bar type specimens (Ilias et al. 1997[START_REF] Gao | Electroless plating synthesis, characterization and permeation properties of Pd-Cu membranes supported on ZrO2 modified porous stainless steel[END_REF], Papaioannou et al. 2015). Ilias et al. (1997) studied the permeability and selectivity of hydrogen through a palladium membrane at high temperature and low pressure. [START_REF] Gao | Electroless plating synthesis, characterization and permeation properties of Pd-Cu membranes supported on ZrO2 modified porous stainless steel[END_REF] investigated the hydrogen permeability on porous stainless steel (PSS) disks coated with a mesoporous palladium impregnated zirconia intermediate layer. Papaioannou et al. (2015) evaluated carbon dioxide permeability for sintered supports with molten carbonate dual-phase membranes and its performances were found to be stable for more than 200 h of operation.

Despite of so many studies on the porous media, still we face a lack in evaluating the correct flow through porous media. Permeability of the material is mostly determined through experiments using disc sample in laboratory. The outflow through the porous media is measured to determine the permeability of the material. But it is not exactly similar to a realistic condition when fluid flows through a tube. In such case, a perpendicular flow to the main one occurs through the porous medium (cf. Figure 1). Thus, it makes necessary to study the effect of second outlet on the determination of Darcian's and Forchheimer's permeabilities (despite the fact that permeability is often considered as a characteristic of the material itself [START_REF] Allaby | A Dictionary of Earth Sciences[END_REF]. The aim of this paper is to improve previously used experimental setup in order to determine accurate Darcian's and Forchheimer's flow behaviour inside a porous media in a realistic configuration. The effect of a second outlet on the Darcian's and Forchheimer's permeabilities is investigated. The next section is devoted to the description of the adapted test bench where different opening of second outlet is studied. The Darcian's and Forchheimer's permeabilities of porous stainless steel materials of class 3 (SS3) and class 20 (SS20) are measured and the obtained results are presented and discussed in the third section of this paper.

Materials & Methods

Permeation experimental and test methodology

The bench used here is composed of a cylindrical permeation cell connected to gas injection system (cf Figure 2). Porous material samples are placed in this permeation cell which is composed of two main parts (High Pressure Chamber: HPC for the inlet and Low Pressure Chamber: LPC for the outlet) in order to maintain the porous media in the fluid flow and to avoid leakage. Despite its small size (external diameter of 40 mm), one inlet and two outlet (outlet-1 and outlet-2) enable to measure the pressure and mass flow rate on each side of the porous sample. The outlet-2 is considered as a Primary Outlet (P.O) whereas the outlet-1 is considered as secondary outlet (S.O). The injection system is capable of handling liquid as well as gaseous fuels with a high pressure pump (80 bars, 0.5 g.s -1 ). Nitrogen gas is used as fluid. Four sensors to monitor transient variations of mass flow rate, pressure and temperature are connected to a data acquisition system (about 1 Hz, 16 bits, 48 channels). The mass flowmeter with a range of 0 to 3 g/s is placed at outlet-2 in order to measure mass flow rate of fluid through porous media. Pressures are measured at inlet and outlet-2 to get the pressure in HP chamber (P inlet ) and pressure difference across the sample. Pressure range of a sensor at inlet side is 0 to 60 bar whereas ∆P can measured in a range of 0 to 635 mbar. Flow from the secondary outlet is controlled by manually operated valve where the three opening percentage values (0%, 50% and 100%) are selected. Figures 2 and3 illustrate respectively a photograph and a schematic overview of the developed permeation experimental setup. Permeability measurements are performed on two different classes of stainless steel that is SS3 and SS20 (cf. Figure 4). Specifications of both porous materials used are given in Table 1 [START_REF] Gascoin | High temperature and pressure reactive flows through porous media[END_REF]. Except one consideration that the outlet of permeation cell should be at atmospheric pressure during the experiments, the norm ISO 4022 for the determination of fluid permeability is followed. Indeed, due to the effect of secondary outlet it makes necessary to place the mass flowmeter at primary outlet (result in back pressure) so that the flow through porous media can be accurately measured and permeabilities of the porous media can be determined. All tests are performed at room temperature and repeated at least three times. Table 1. Specifications of the two porous stainless steel materials.

Determination procedure of Darcian's and Forchheimer's permeabilities

Accurate description of fluid flow behavior in the porous media is essential and is described in several ways in various literature studies. Usually, Darcy's law depicts fluid flow behavior in porous media. According to Darcy's law, the pressure gradient is linearly proportional to the fluid velocity in the porous material. This linear relationship is valid only for very small flow velocities or low pressure gradients. As the Reynolds number (Re) approaches to a critical value (i.e. turbulent regime), the relationship becomes nonlinear. The one-dimensional Darcy equation can be written as:

dP V dx K    ( 1
)
where P is the pressure, x is the direction of fluid flow, µ is the fluid viscosity, V is the superficial velocity, and K is the permeability. The transient behavior in porous media is described by Navier-Stokes equations whereas Brinkman's equation below is used to describe the macroscopic fluid flow in large range flow regimes:

2 D F P V V L K K      (2)
where inlet outlet P P P    is the pressure drop through the porous medium, L is the external mean sample thickness, ρ is the inlet density (with respect to inlet pressure), K D and K F are the Darcy's and Forchheimer's terms. The right term of Eq. ( 2) is composed of two parts, one related to the Darcy's law for low velocity regime filtration (Darcian flows) and the quadratic one is related to the Stokes's law (non-Darcian flows) which takes into account the inertial effects related with flow resistance. The Stokes equation is also called as Forchheimer's equation [START_REF] Tully | Fluid Flow and Heat Transfer in a Liquid Rocket Fuel Injector[END_REF][START_REF] Murthy | Thermal Dispersion Effects on Non-Darcy Convection over a Cone[END_REF][START_REF] Murthy | Thermal Dispersion Effects on Non-Darcy Convection over a Cone[END_REF]. Numerous other formulations of the Brinkman's equation can be found [START_REF] Kim | Fully discrete mixed finite element approximations for non-darcy flows in porous media[END_REF][START_REF] Choi | Permeability modeling of fibrous media in composite processing[END_REF][START_REF] Martin | Simulation of pyrolysis gas within a thermal protection system[END_REF][START_REF] Valdes-Parada | On the effective viscosity for the Darcy-Brinkman equation[END_REF]. Large range of flow regimes through porous media can also be described by using power laws [START_REF] Rathish | Free convection in a non-Darcian wavy porous enclosure[END_REF] and cubic laws (Aulisa et al., 2009[START_REF] Nguyen | Mapping of foam mobility in porous media[END_REF][START_REF] Pazos | Compan Calculation of Klinkenberg permeability, slip factor and turbulence factor of core plugs via nonlinear regression[END_REF].

In order to compute Darcian's and Forchheimer's permeabilities from the measured parameters on the experimental bench, a modified form of the above equation is derived:

1 1 D F p V L V K K       (3) Thus, the left term p L V   is plotted as a function of V  
. The origin of the plot is linked to the Darcian's term while the angle of climb or slope of the plot is related to the Forchheimer's term. Due to the implication of density and dynamic viscosity in the pressure drop formula of Eq. ( 3), their estimation is of great importance. The density is computed on the basis of measured pressure and temperature thanks to the modified perfect gases law with the compressibility factor Z which depends notably on the critical coordinates and on the Pitzer acentric factor [START_REF] Gascoin | High temperature and pressure reactive flows through porous media[END_REF]. The dynamic viscosity is computed by the method proposed by Chung [START_REF] Poling | The Properties of Gases and Liquids[END_REF].

Results and discussion

As mentioned in previous sections, the Darcian's and Forchheimer's permeabilities of two kind of stainless steel porous materials are determined and the influence of different openings of secondary outlet is investigated. Before proceeding, some preliminary experiments are performed on SS3 in order to verify that the placement of mass flowmeter at primary outlet instead of inlet, as per ISO 4022 norm, does not affect the permeabilities estimation. Results of Darcian's permeability in the two cases (i.e. mass flowmeter at primary outlet and mass flowmeter at inlet) show clearly that the relative gap do not exceed 8% whereas those of Forchheimer's permeability is inferior to 13% (these values are in the range of previously estimated accuracy, Gascoin 2011). These relative gap values confirm that the placement of mass flowmeter at primary outlet do not affect the determination of permeabilities. When Brinkman's equation is considered (turbulent flow), the Darcy's term for carbon cooled porous structure is found to range from Using the results plotted in above Figure 5, the intercept of the different lines on Y axis will give the K D coefficients and their slopes lead to K F coefficients. The obtained values of Darcian's and Forchheimer's permeabilities are given in Table 2 for different openings of secondary outlet. It can be clearly seen from this Table that in case of both the porous materials (SS3 and SS20) the determination of K D is not affected by the opening of the secondary outlet. For SS3 porous material, a discrepancies of 1.21% and 0.1% are respectively found for 50% and 100% open cases. Similarly in case of SS20 porous media, relative gaps of 5.4% and 10.8% are respectively obtained for 50% and 100% opening of secondary outlet. But when it comes to determination of K F , the opening of secondary outlet seems to have an effect. Indeed, relative changes of 11% and 71% are found in case of SS3 whereas values of 34% and 237% are obtained for SS20 for 50% and 100% open cases respectively. K F is varying a lot for the 100% cases, it is because of the fact that the flow through the porous media is very low that results in decrease of pressure drop which is linearly proportional to square of velocity at higher flow rates (Non Darcian Flow). Since K F coefficient is associated to the turbulent flow inside the porous media [START_REF] Firoozabadi | An analysis of high-velocity gas flow through porous media[END_REF][START_REF] Phair | Materials for separation membranes in hydrogen and oxygen production and future power generation[END_REF][START_REF] Lemos | Turbulent Impinging Jets into Porous Materials[END_REF], hence the velocity of fluid in the material is too low and the flow is not or less turbulent and it makes its determination less accurate. That's is why the value for the K F is quite far from the "real" one.

Type of material Permeability Term

Secondary outlet opening percentage 0% Open 50% Open 100% Open SS3 K D (m 2 ) 7.38×10 -13 7.29×10 -13 7.37×10 -13 K F (m) 3.08×10 -06 3.43×10 -06 5.28×10 -06 SS20 K D (m 2 ) 1.11×10 -10 1.05×10 -10 1.23×10 -10 K F (m) 3.79×10 -05 5.11×10 -05 1.28×10 -04

Table 2. Darcian's and Forchheimer's permeability values at three different secondary outlet openings for SS3 and SS20.

In order to give a deeper explanation to the above statement and study the influence of Darcian's and Forchheimer's contribution on pressure losses, Figure 6 presents the evolutions of these contribution as well as their ratio as functions of pore Reynolds number. The pore Reynolds number has been computed for metallic samples by using the relation / grain diameter deduced from the literature data. It is observed from the set of plotted data that in case of SS3 porous media Darcian's term is always higher than the Forchheimer's term for all the three different opening of secondary outlet (0%, 50% and 100%) showing that the flow is always in a laminar regime. There is no influence of secondary outlet opening on Darcian's term and its contribution remains constant for all the three cases but Forchheimer's contribution decreases as the secondary outlet opening increases. The similar trend is observed in case of SSS20 except that in case of 0% and 50% opening of secondary outlet where the Forchheimer's contribution overcomes the Darcian's contribution as the pore Reynolds number exceeds a critical value of c 6 e R  . It is also observed that the decrease in Forchheimer's term for SS3 porous material is less evident compare to the Forchheimer's decrease in case of SS20.

Conclusion

In fuel cell applications, porous materials is used for the filtration need. Determining physical properties like porosity and permeability are of utmost importance to predict and manage filtration efficiency of the materials. An experimental test bench was design to determine both Darcian's and Forchheimer's permeabilities. Nevertheless, it was found later that using the experimental values of these permeabilities on complete integrated fuel reforming systems led to discrepancies of several orders of magnitude. One of the hypothesis that was pose is that the geometrical configuration may impact the results. In other words, having one flow inlet and one flow outlet or one inlet and two outlets (as it was discussed in this paper) may lead to different apparent permeabilities. Effect of secondary outlet on permeability determination has thus been investigated. It was found that there is no much change is observed in K D value when the mass flowmeter is placed at outlet-2 instead of the inlet. When Darcian's permeability is determined for SS3 and SS20 with different openings for secondary outlet, negligible changes are observed, hence they show that this term is not affected by change in flow configuration in case of stainless steel material. But secondary outlet opening have significant effect on the determination of K F value for both kinds of porous material. Indeed, its criterion of determination is highly governed by turbulent flow regime. This flow regime is found in just two cases out of six cases and that is only when the pore Reynolds number exceed its critical value.

In future works, Darcian's and Forchheimer's permeabilities will be studied in a tube configuration using different composites. Mathematical background will be developed in order to correlate these permeabilities to the selectivity of the porous materials. After that selectivity of several binary mixtures will be investigated on different porous media using an adapted experimental setup.

Figure 1 :

 1 Figure 1: Overview of realistic flows within channel surrounded by porous medium.

Figure 2 :

 2 Figure 2: Photograph of the permeation test bench.

Figure 3 :

 3 Figure 3: Schematic overview of the permeation experimental setup.

Figure 4 :

 4 Figure 4: Stainless steel porous medium: Fe (62 wt.%), Cr (18 wt.%), Ni (11 wt.%) C (7 wt.%), O (1 wt.%) and Si (1 wt.%).

  studied stainless steel porous media which are illustrated in Figure5for different opening of secondary outlet (0%, 50% and 100%).

Figure 5 .

 5 Figure 5. Determination of Darcian's and Forchheimer's permeabilities for different opening of secondary outlet. Left: SS3 and Right: SS20.

  area and g d is grain diameter. Let us notice that the grain diameter of the SS3 sample is measured in a previous work (

Figure 6 .

 6 Figure 6. Permeation contributions plotted as a function of the estimated pore Reynolds number for class 3 and class 20 for different openings of secondary outlet.

  1,* , Eddy EL TABACH 2 , Khaled CHETEHOUNA 1 , Nicolas GASCOIN 1 , François FALEMPIN 3 1 INSA-CVL, Univ. Orléans, PRISME EA 4229, F-18022 Bourges, France 2 Univ. Orléans, INSA-CVL, PRISME EA 4229, F-18000 Bourges, France 3 MBDA-France, 8 rue le Brix, 18000 Bourges, France