Maryam Al Zohbi
email: maryam.al-zohbi@utc.fr

Ahmad El Hajj
email: ahmad.el-hajj@utc.fr

Mustapha Jazar
email: mjazar@laser-lb.org.

Existence and uniqueness results to a system of Hamilton-Jacobi equations

Keywords: AMS Classification: 35F21, 49L25, 35D40, 34A12, 74H20, 74H25, 35B51, 35F55, 35L40, 35L45 Hamilton-Jacobi equations, non-linear eikonal equations, viscosity solution, uniqueness, comprasion principle, dislocations dynamics

We study the existence and uniqueness of a nonlinear system of eikonal equations in one space dimension for any BV initial data. We present two results. In the first one, we prove the existence of a discontinuous viscosity solution without any monotony conditions neither on the velocities nor on the initial data. In the second, we show the continuity of the constructed solution under continuous initial data, and continuous velocities verifying a certain monotony condition. We present an application to a system modeling the dynamics of dislocations densities.

Introduction and main results

Setting of the problem

In this paper, we are interested in a non-linear strongly coupled Hamilton-Jacobi system of the form

   ∂ t u i (t, x) = λ i (t, x, u(t, x))|∂ x u i (t, x)| in (0, T ) × R, u i (0, x) = u i 0 (x) in R, (1.1)
where T > 0 and i = 1, . . . , d, such that d ∈ N * . The functions u i are real valued, ∂ t u i and ∂ x u i represent the time and spatial derivatives of u i respectively. The velocity λ i is assumed to satisfy, for all i = 1, . . . , d, the following assumption λ i ∈ L ∞ ((0, T ) × R × K) for T > 0 and for all compact K ⊂ R d .

(1.2)

1 Our study of system (1.1) is motivated by the consideration of a model describing the dynamics of dislocations densities (see [START_REF] Hajj | Global continuous solutions for diagonal hyperbolic systems with large and monotone data[END_REF]Section 5] for more details about the model), which is given by

∂ t u i =   j=1,...,d A ij u j   |∂ x u i | for i = 1, . . . , d, (1.3)
where (A ij ) i,j=1,...,d is a real matrix. This model can be seen as a special case of system (1.1).

From another point of view, we remark that system (1.1) can be seen as the "level-set approach" system associated to the motion of the front Γ i t := {x : u i (t, x) = 0} with a normal velocity λ j (t, x, u) depending on the solution u and affected by λ j (t, x, u) for i = j (see for instance Barles et al. [START_REF] Barles | Front propagation and phase field theory[END_REF]).

We aim in this work to establish first the existence of a discontinuous viscosity solution assuming we have (1.2) and the following regularity on the initial data

u i 0 ∈ L ∞ (R) ∩ BV (R) for every i = 1, . . . , d, (1.4)
where BV (R) is the space of functions of bounded variations given by

BV (R) = f ∈ L 1 loc (R); T V (f ) < +∞ ,
with T V (f ) being the total variation of f defined as

T V (f ) = sup R f (x)φ (x)dx; φ ∈ C 1 c (R) and φ L ∞ (R) ≤ 1 .
Then, we will show that this discontinuous solution would be continuous if we assume that the system is quasi-monotone in the sense of Ishii, Koike [START_REF] Ishii | Viscosity solutions for monotone systems of second-order elliptic PDEs[END_REF][START_REF]Viscosity solutions of a system of nonlinear second-order elliptic PDEs arising in switching games[END_REF], along with (1.4) and u i 0 ∈ C(R) for every i = 1, . . . , d,

and assuming the velocities verify (1.2) and the following assumptions for every i = 1, . . . , d, λ i ∈ C((0, T ) × R × K) for T > 0 and for all compact K ⊂ R d , there exists M 1 > 0 such that, for all x, y ∈ R and t ∈ (0, T ),

|λ i (t, x, u) -λ i (t, y, u)| ≤ M 1 |x -y|. (1.6)
We refer the reader to [START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF][START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF][START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations[END_REF]] for a complete overview on viscosity solutions. In the following, we take the space BV (R) endowed with the semi-norm |f | BV (R) = T V (f ). Note that BV functions are integrable functions whose distributional derivative is a finite Radon measure.

In order to prove the existence of a discontinuous solution of (1.1), first we will consider, for every i = 1, . . . , d and 0 < ε, η ≤ 1, the following parabolic regularization

   ∂ t u i ε,η (t, x) = η∂ 2 xx u i ε,η (t, x) + λ i ε (t, x, u ε,η (t, x))|∂ x u i ε,η (t, x)| in (0, T ) × R, u i ε,η (0, x) = u i 0,ε (x) in R.
(1.7)

The functions λ i ε and u i 0,ε are the regularizations of λ i and u i 0 by classical convolution, which are defined as

u i 0,ε (x) = u i 0 ρ 1 ε (x) and λ i ε (t, x, w) = λi ρ d+2 ε (t, x, w) ∀ (t, x, w) ∈ R × R × R d , (1.8)
where λi is an extension of λ i by 0 for all i = 1, . . . , d. Moreover, ρ n ε for n = 1 and n = d + 2 are standard mollifiers defined as

ρ n ε (•) = 1 ε n ρ n • ε , such that ρ n ∈ C ∞ c (R n ), supp{ρ n } ⊆ B(0, 1
), ρ n ≥ 0, and

R n ρ n = 1.
We will show that (1.7) admits a unique Lipschitz solution by the means of a Fixed Point argument. Then, by using stability results of viscosity solutions, we will be able to pass to the limit as (ε, η) → (0, 0), and show that the upper and lower relaxed semi-limits of Barles and Perthame [START_REF] Barles | Exit time problems in optimal control and vanishing viscosity method[END_REF][START_REF]Comparison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations[END_REF], which are defined as

u i (t, x) = lim sup u i ε,η (t, x) = lim sup (ε,η)-→(0,0) (s,y)-→(t,x) u i ε,η (s, y), (1.9)
and

u i (t, x) = lim inf u i ε,η (t, x) = lim inf (ε,η)-→(0,0) (s,y)-→(t,x) u i ε,η (s, y), (1.10)
are, respectively, discontinuous viscosity sub-and super-solutions of system (1.1) in the sense of discontinuous viscosity solutions introduced by Ishii in [24, Definition 2.1] for Hamilton-Jacobi systems that is recalled below in Definition 2.1. Then, by applying the comparison principle, we will be able to show the continuity of these discontinuous solutions under (1.5) and (1.6), in the case where system (1.1) is quasi-monotone (defined below in (H)). The uniqueness of the continuous solution is obtained again by a comparison principle.

A brief review of some related literature

Several studies have been made on (1.1), of which we will mention some. In El Hajj et al. [START_REF]BV solution for a non-linear hamilton-jacobi system[END_REF], the global existence of a discontinuous viscosity solution has been obtained, under a certain monotony condition on the velocities. However, without any such monotony conditions, EL Hajj and Oussaily recently proved in [START_REF] Hajj | Existence and Uniqueness of Continuous Solution for a Nonlocal Coupled System Modeling the Dynamics of Dislocation Densities[END_REF] the global existence and uniqueness of a continuous viscosity solution, basing on an entropy and a BV estimate. In this work, we treat the case of discontinuous solutions without any monotony conditions.

We note that in the case of nondecreasing solutions, system (1.1) becomes a diagonal hyperbolic system. For such systems, many existence and uniqueness results have been made. We will recall some of the most significant ones. First, in the case of (2 × 2) strictly hyperbolic system with nondecreasing initial data, Lax proved in [START_REF] Lax | Hyperbolic systems of conservation laws and the mathematical theory of shock waves[END_REF], the existence and uniqueness of nondecreasing smooth solutions. Also, assuming that the initial data is nondecreasing, an existence and uniqueness result of a continuous solution was proved in El Hajj, Monneau [START_REF]Uniqueness results for diagonal hyperbolic systems with large and monotone data[END_REF] for a general (d × d) diagonal strictly hyperbolic system. The proof in [START_REF]Uniqueness results for diagonal hyperbolic systems with large and monotone data[END_REF] is based on a global existence result of a continuous nondecreasing solution established previously in El Hajj, Monneau [START_REF] Hajj | Global continuous solutions for diagonal hyperbolic systems with large and monotone data[END_REF], where the system is assumed to be hyperbolic but not necessarily strictly hyperbolic. Also, in the case of (d × d) strictly hyperbolic systems, Bianchini and Bressan proved in [START_REF] Bianchini | Vanishing viscosity solutions of nonlinear hyperbolic systems[END_REF] a global existence and uniqueness result assuming that the initial data had small total variation. This approach is mainly based on a careful analysis of the vanishing viscosity approximation. We can also mention that an existence result has been also obtained by LeFloch, Liu [START_REF] Lefloch | Existence theory for nonlinear hyperbolic systems in nonconservative form[END_REF] and LeFloch [START_REF] Lefloch | Entropy weak solutions to nonlinear hyperbolic systems under nonconservative form[END_REF][START_REF] Lefloch | Graph solutions of nonlinear hyperbolic systems[END_REF], in the nonconservative case.

Many authors have worked on Hamilton-Jacobi equations in several space dimensions with discontinuous coefficients. We mention the work of Camilli and Siconolfi [START_REF] Camilli | Hamilton-Jacobi equations with measurable dependence on the state variable[END_REF][START_REF]Time-dependent measurable Hamilton-Jacobi equations[END_REF], where comparison principles, existence and uniqueness results, stability properties, and representations formulas of viscosity solutions have been made, under the assumptions that the Hamiltonian H(x, p) is measurable with respect to the space variable x and convex in p. We can also point out the work of Chen and Hu [START_REF] Chen | Viscosity solutions of discontinuous Hamilton-Jacobi equations[END_REF], where they establish the existence and uniqueness of Lipschitz continuous viscosity solutions, assuming the Hamiltonian is nonnegative and Lipschitz continuous. What is genuine about our work is that we will show the existence of viscosity solutions assuming our Hamiltonians (velocities λ i ) are only bounded functions.

We note that the system understudy in this paper is strongly coupled. However, it is important to mention that the case of weakly coupled Hamilton-Jacobi equations have been widely studied in the literature. For instance, Camilli and Loreti proved in [START_REF] Camilli | Comparison results for a class of weakly coupled systems of eikonal equations[END_REF] two comparison theorems on the system

H i (x, Du i ) + M j=1 c ij (u i -u j ) = 0, i = 1, . . . , M,
by imposing convexity and coercivity conditions on the Hamiltonians H i . Many other results have been brightened up under such conditions. Loreti and Vergara Caffarelli proved the existence and uniqueness of variational solutions in [START_REF] Loreti | Variational solutions of coupled Hamilton-Jacobi equations[END_REF]. Also, the authors in Mitake et al. [START_REF] Mitake | A Lagrangian approach to weakly coupled Hamilton-Jacobi systems[END_REF] were able to characterize all sub-solutions of their system and represent explicitly some of which enjoy a certain maximality property.

In this paper, we present two results that remain valid whether the system is strictly hyperbolic or not strictly hyperbolic, and without any monotony conditions on the initial data. First, we show the global existence of a discontinuous viscosity solution to (1.1) without any monotony conditions on the velocities for any BV initial data. Second, we prove the continuity of the obtained solution by using the Comparison Principle, assuming that the initial data are continuous and the velocities are Lipschits functions in space verifying certain monotony conditions. Then, as a consequence, we obtain the uniqueness of the solution.

Main results

In this subsection, we first present, in Theorem 1.1, the global existence of a discontinuous viscosity solution to (1.1). Then, we show, in Theorem 1.2, that this solution is continuous for continuous initial data and under certain monotony conditions on the velocities. Lastly, we present an application to the case of the dynamics of dislocations densities in Theorem 1.3. Assume that (1.2) and (1.4) are satisfied. Then the following points hold i) Existence and uniqueness to the regularized problem There exists a unique Lipschitz solution u ε,η = (u i ε,η ) i=1,...,d of (1.7) belonging to the space (C([0, T ); W 1,∞ (R))) d , and satisfying for all T > 0 and i = 1, . . . , d, the following uniform estimates

u i ε,η L ∞ ((0,T )×R) ≤ u i 0 L ∞ (R) , (1.11)
∂ x u i ε,η L ∞ ((0,T );L 1 (R)) ≤ ∂ x u i 0 L 1 (R) , (1.12
)

∂ t u i ε,η L ∞ ((0,T );W -1,1 (R)) ≤ 1 + λ i L ∞ ((0,T )×R×K 0 ) u i 0 BV (R) , (1.13)
where

W -1,1 (R) is the dual of W 1,∞ (R), and
K 0 = d i=1 -u i 0 L ∞ (R) , u i 0 L ∞ (R) . (1.14)
ii) Sub-and super-solutions of (1.1) Let u ε,η be the unique solution of (1.7) constructed in (i). Then the upper and lower relaxed semi-limits u = u i i=1,...,d and u = u i i=1,...,d , are respectively discontinuous viscosity suband super-solutions of system (1.1).

iii) Convergence

Assume that the solution u i ε,η of (1.7) satisfies (1.11), (1.12) and (1.13) for i = 1, . . . , d. Then, up to the extract of a subsequence, the function u i ε,η converges, as ε and η tend to zero, to a function

u i ∈ L ∞ (0, T ) × R ∩ L ∞ (0, T ); BV (R) ∩ C [0, T ); L 1 loc (R) , (1.15)
strongly in C [0, T ); L 1 loc (R) . Moreover, u i satisfies, for all T > 0 and for i = 1, . . . , d, the following inequalities

u i L ∞ ((0,T )×R) u i 0 L ∞ (R) , (1.16
)

u i L ∞ ((0,T );BV (R)) u i 0 BV (R) , (1.17)
and the following equality

u i (t, •) = u i (t, •) = u i (t, •), except at most on a countable set in R, for all t ∈ [0, T ). (1.18)
Our second result relies on the supposition that system (1.1) is quasi-monotone. This means that the velocities verify the following condition

λ j (t, x, s) -λ j (t, x, r) ≥ 0 for all vectors r = (r i ) i=1,...,d , s = (s i ) i=1,...,d such that r j -s j = max i∈{1,...,d} (r i -s i ) ≥ 0. (H)
Theorem 1.2 (Existence and uniqueness of a continuous solution to (1.1)).

Suppose that (1.2), (1.4), (1.5), (1.6) and (H) hold. Then, there exists a unique continuous viscosity solution of (1.1) satisfying (1.16) and (1.17).

Unique continuous solution for dislocations' dynamics

Now, we present an application of Theorem 1.2 to a nonlinear system that appears in the modeling of the dynamics of dislocations densities in materials.

A dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of its atoms.

Here, we are interested in a particular 1D model initially proposed in 2D dimensions by Groma and Balogh [START_REF] Groma | Link between the individual and continuum approaches of the description of the collective behavior of dislocations[END_REF][START_REF]Investigation of dislocation pattern formation in a two-dimensional self-consistent field approximation[END_REF], in order to describe the dynamics of dislocations densities. This 2D model is written in a specific geometry, where the dislocations are considered as points in the plane (x 1 , x 2 ), propagating to the left and to the right, following two Burger's vectors ±b = ±(1, 0).

In the 1D sub-model, we suppose that the dislocations densities depend only on the variable x = x 1 + x 2 , which transforms the 2D model into a 1D model. We refer the reader to El Hajj and Forcadel [START_REF] Hajj | A convergent scheme for a non-local coupled system modelling dislocations densities dynamics[END_REF]Lemma 3.1] for more details about the modeling.

More precisely, we consider the following system

         ∂ t ρ + (x, t) = -(ρ + -ρ -)(x, t) + α 1 0 (ρ + -ρ -)(y, t)dy + a(t) ∂ x ρ + (x, t) in R × (0, T ), ∂ t ρ -(x, t) = (ρ + -ρ -)(x, t) + α 1 0 (ρ + -ρ -)(y, t)dy + a(t) ∂ x ρ -(x, t) in R × (0, T ), (1.19)
where ρ + , ρ -are the unknown scalars, that we denote for simplicity by ρ ± . These two functions, ρ + and ρ -, are respectively the representations of the left-propagating and rightpropagating dislocations. Their spatial derivatives ∂ x ρ + , ∂ x ρ -represent the dislocations densities of +, -type respectively. The constant α depends on the elastic coefficients and the material size, while the function a(t) represents the exterior strain field. The initial conditions associated to system (1.19) are defined as follows

ρ ± (0, x) = ρ ± 0 (x) = P ± 0 (x) + a 0 x on R, (1.20)
where P ± 0 are 1-periodic functions. In particular, ρ + 0 -ρ - 0 is 1-periodic. The use of the periodic boundary conditions is a way of regarding what is going on in the interior of the material away from its boundary, assuming the material understudy is made up entirely of small and similar subsets.

Remark 1.1. Mathematically speaking, we could have just considered periodic initial conditions, without the linear part in (1.20). However, the use of periodic plus linear conditions is totally physical, as the dislocations densities are considered to be non-decreasing functions in this model.

Applying Theorem 1.2 to the local case (α = 0) of system (1.19) yields the following result, where we note that the set T = R/Z is the [0, 1) periodic interval.

Theorem 1.3 (Existence and uniqueness of a continuous solution).

Assume that α = 0, and suppose that the functions P ± 0 introduced in (1.20) verify

P ± 0 ∈ C(T) ∩ BV (T), (1.21)
and the function a satisfies

a ∈ C[0, T ). (1.22)
Then, there exists a unique continuous viscosity solution of system (1. [START_REF]Uniqueness results for diagonal hyperbolic systems with large and monotone data[END_REF])- (1.20).

The proof of this theorem derives naturally from Theorem 1.2, since system (1.19) verifies condition (H) in the case where α = 0.

Organization of the paper

This paper is organized as follows: in Section 2, we prove the existence of a discontinuous viscosity solution result, that was announced in Theorem 1.1. Section 3 is devoted to the proof of Theorem 1.2, where we show by the means of a comparison principle, the continuity and the uniqueness of the solution constructed in Theorem 1.1, under certain extra conditions. Finally in Section 4, we present an application of Theorem 1.2 to the dynamics of dislocations, presented in Theorem 1.3.

Existence of a discontinuous solution

In this section, we prove the existence of a discontinuous viscosity solution of (1.1), that was announced in Theorem 1.1. First, we recall in the following subsection a well-known compactness lemma, along with the definition of discontinuous viscosity solutions for system (1.1).

Some useful results

We first recall Simon's Compactness Lemma.

f (X) = lim sup Y →X f (Y ) and f (X) = lim inf Y →X f (Y ) for X ∈ R n . (2.1)
For a vector u = (u 1 , . . . , u d ) locally bounded on [0, T ) × R for all T > 0, we write u = ((u 1 ) , . . . , (u d ) ) and u = ((u 1 ) , . . . , (u d ) ). Also we define U : (0, T ) × R → 2 R d , the graph closure of u, by

U (t, x) = {r ∈ R d : there is a sequence {(t n , x n )} ⊂ (0, T ) × R such that (t n , x n ) -→ n→+∞ (t, x) and u(t n , x n ) → r}.
It should be remarked that U (t, x) is closed; i.e., for all sequence {r m } ⊂ R d , if r m ∈ U (t, x) and r m -→ m→+∞ r for some r ∈ R d , then r ∈ U (t, x).

Definition 2.1. (Discontinuous viscosity sub-solution, super-solution and solution) Assume that λ i is locally bounded on (0, T ) × R × R d and u 0 = (u i 0 ) i=1,...,d is locally bounded on R. Let u = (u i ) i=1,...,d be a locally bounded function defined on [0, T ) × R.

(1) (Discontinuous viscosity sub-solution)

We call u a discontinuous viscosity sub-solution of (1.1) if it satisfies

(i) (u i ) (0, x) ≤ (u i 0 ) (x)
, for all i = 1, . . . , d and x ∈ R.

(ii) If whenever φ ∈ C 1 ((0, T ) × R), i = 1, . . . , d and (u i ) -φ attains its local maximum at (t 0 , x 0 ) ∈ (0, T ) × R, then we have

min ∂ t φ(t 0 , x 0 ) -(λ i ) (t 0 , x 0 , r)|∂ x φ(t 0 , x 0 )| : r ∈ U (t 0 , x 0 ), r i = (u i ) (t 0 , x 0 ) ≤ 0. (2.2)
(2) (Discontinuous viscosity super-solution) Similarly, we call u a discontinuous viscosity super-solution of (1.1) if it satisfies

(i) (u i ) (0, x) ≥ (u i 0 ) (x)
, for all i = 1, . . . , d and x ∈ R.

(ii) If whenever φ ∈ C 1 ((0, T ) × R), i = 1, . . . , d and (u i ) -φ attains its local minimum at (t 0 , x 0 ) ∈ (0, T ) × R, then we have max ∂ t φ(t 0 , x 0 ) -(λ i ) (t 0 , x 0 , r)|∂ x φ(t 0 , x 0 )| : r ∈ U (t 0 , x 0 ), r i = (u i ) (t 0 , x 0 ) ≥ 0. (2.3)
(3) (Discontinuous viscosity solution) Finally, we call u a discontinuous viscosity solution of (1.1) if it is both a discontinuous viscosity sub-solution and super-solution of (1.1).

Noting that the minimum and the maximum in (2.2) and (2.3) are attained, since the sets

r ∈ R d : r ∈ U (t 0 , x 0 ), r i = (u i ) (t 0 , x 0 ) and r ∈ R d : r ∈ U (t 0 , x 0 ), r i = (u i ) (t 0 , x 0 )
are non-empty and compact and moreover (λ i ) and (λ i ) are upper and lower semi-continuous, respectively.

Proof of Theorem 1.1

We proceed in three steps.

Step 1. (Proof of (i)):

The proof of Theorem 1.1-(i) is a classic application of the Fixed Point Theorem in Banach spaces. We will explain briefly what we will do. First, we regularize the non-linear term [START_REF] Boudjerada | Global existence results for eikonal equation with BV initial data[END_REF], by replacing it with the smooth function β δ (∂ x u i ε,η ), where β δ is defined as

|∂ x u i ε,η |, in (1.
β δ (x) = x 2 + δ 2 for all 0 < δ ≤ 1. (2.4)
Then, we truncate the function β δ by plugging in the function This brings us to consider for all 0 < δ ≤ 1 , i = 1, . . . , d, and for u ε,η,δ = (u i ε,η,δ ) i=1,...,d , the following problem

ψ δ (•) = φ( √ δ •), (2.5
   ∂ t u i ε,η,δ (t, x) -η∂ 2 xx u i ε,η,δ (t, x) = λ i ε (t, x, u ε,η,δ )ψ δ (x)β δ (∂ x u i ε,η,δ (t, x)) on (0, T ) × R, u i ε,η,δ (0, x) = u i 0,ε (x) x ∈ R, (2.6
) where for all i = 1, . . . , d, we have

u i 0,ε ∈ C ∞ (R) ∩ L ∞ (R), ∂ x u i 0,ε ∈ L p (R) for all 1 ≤ p ≤ +∞, (2.7)
and

λ i ε ∈ W 1,∞ ((0, T ) × R × K) ∩ C ∞ ((0, T ) × R × R d ), for all compact K ⊂ R. (2.8)
Next, we will write (2.6) in its integral form, for all i = 1, . . . , d, as follows

u i ε,η,δ (t, x) = G η (t) u i 0,ε (x) + t 0 G η (t -s) λ i ε (s, •, u ε,η,δ (s, •))ψ δ (•)β δ (∂ x u i ε,η,δ (s, •)) (x)ds, where G η (t, x) = 1 √ 4πηt e -x 2
4ηt is the standard heat kernel. In other words, we consider the following problem

   u ε,η,δ (t, x) = (u i ε,η,δ (t, x)) i=1,...,d , u ε,η,δ (0, x) = u 0,ε (x) = (u i 0,ε (x)) i=1,...,d , u ε,η,δ (t, x) = G η (t) u 0,ε (x) + B(u ε,η,δ )(t, x), (2.9)
where, for r(t, x) = (r i (t, x)) i=1,...,d , we have

B(r)(t, x) = t 0 G η (t-s)    β δ (∂ x r 1 (s, •))ψ δ (•) • • • 0 . . . . . . . . . 0 • • • β δ (∂ x r d (s, •))ψ δ (•)         λ 1 ε (s, •, r(s, •)) λ 2 ε (s, •, r(s, •)) . . . λ d ε (s, •, r(s, •))      (x)ds.
We introduce, for T > 0, the following three Banach spaces

X = r = (r i ) i=1,...,d ∈ (L ∞ (R)) d ; ∂ x r i ∈ L 1 (R) , (2.10) equipped with the norm r X = d i=1 r i L ∞ (R) + d i=1 ∂ x r i L 1 (R)
, and

X T = r = (r i ) i=1,...,d ∈ (L ∞ ((0, T ) × R)) d ; ∂ x r i ∈ L ∞ ((0, T ); L 1 (R)) , equipped with the norm r X T = d i=1 r i L ∞ ((0,T )×R) + d i=1 ∂ x r i L ∞ ((0,T );L 1 (R))
, and lastly

Y T = r ∈ X T ; r X T ≤ v 0 X + 1 .
Then, we define the mapping

T : Y T → Y T r → T (r) = G η (•) v 0 + B(r).
We prove that this mapping is well-defined and is a contraction on Y T * for a certain T * > 0. This enables us to apply the Fixed Point Theorem in order to say that there exists a unique fixed point which is the unique solution of (2.9). Moreover, we can prove that there exists a constant ξ δ p depending on

u i 0,ε L ∞ (R) , ∂ x u i 0,ε L p (R)
, and T * such that

∂ x u i ε,η,δ L ∞ ((0,T * );L p (R)) ≤ ξ δ p for all 1 ≤ p ≤ +∞. (2.11)
Then, by using L p -regularity of parabolic equations and the fact that β δ and λ i ε are regular, we can show by classical Bootstrap arguments that

u ε,η,δ ∈ (C ∞ ((0, T * ) × R)) d ∩(W 1,∞ ((0, T * ) × R)) d ∩ Y T * .
After that, we can prove via the maximum principle for parabolic equations (see Lieberman [31, Theorem 2.10]) that the smooth solution u i ε,η,δ verifies

u i ε,η,δ L ∞ ((0,T * )×R) ≤ T * δ Λ i + u i 0 L ∞ (R) for all i = 1, . . . , d, (2.12)
where

Λ i = λ i L ∞ ((0,T )×R× K 0 ) , (2.13)
with

K 0 = d i=1 -1 -u i 0 L ∞ (R) , u i 0 L ∞ (R) + 1 .
Then, using the compactness lemma by Simon, inherited from (2.11) and (2.12), we will be able to pass to the limit as δ → 0, and prove the existence of (1.7). Estimate (1.11) comes from passing to the limit as δ → 0 is (2.12). The proof of the BV estimate (1.12) and the time derivative estimate (1.13) can be found in [START_REF] Hajj | Continuous solution for a non-linear eikonal system[END_REF]. Finally, using (1.11) and (1.12) we will be able to prove the global in time existence of a solution u ε,η to (1.7) belonging to the space

(C([0, T )); W 1,∞ (R)) d for T > 0.
We refer to [20, Theorem 5.1] for the complete proof of this step.

Step 2. (Proof of (ii)):

We have to prove that the upper and lower relaxed semi-limits u and u of u ε,η verify the conditions of discontinuous viscosity sub-and super-solutions, given in Definition 2.1, respectively. We introduce the finite speed propagation property, valid on the smooth solutions of (1.7). Indeed, under assumptions (1.2) and (1.4), if u ε,η = (u i ε,η ) i=1,...,d is the unique solution of (1.7), given by Theorem 1.1-(i), then u i ε,η satisfies, for all h ≥ 0, the following estimate

R G η (t, y) min |z-(x-y)|≤Λt u i ε,η (h, z)dy ≤ u i ε,η (t + h, x) ≤ R G η (t,
u i 0,ε (x) ≤ max |x-x0|≤c+ε u i 0 (x)
, where c > 0, allows us to give sense to the initial data, as it is precised in Definition 2.1 ( 1)-(i) and ( 2)-(i). See [1, Section 5] for more details. However, to give meaning to the system, in other words, to prove Definition 2.1 ( 1)-(ii) and ( 2)-(ii), we use the stability results of discontinuous viscosity solutions, along with the following properties

λ i (t, x, r) ≤ (λ i ) (t, x, r) and (λ i ) (t, x, r) ≤ λ i (t, x, r) for all (t, x, r) ∈ [0, T ) × R × R d ,
where

λ i (t, x, r) = lim sup ε-→0 (s,y,w)-→(t,x,r) λ i ε (s, y, w), and λ i (t, x, r) = lim inf ε-→0 (s,y,w)-→(t,x,r) λ i ε (s, y, w).
We refer the reader also to [1, Section 5] for a complete elaboration of the preceding information.

Step 3. (Proof of (iii)):

We proceed as in [1, Section 6]. Indeed, for u ε,η = u i ε,η i=1,...,d being the solution of (1.7), constructed in Theorem 1.1-(i), and from estimates (1.11), (1.12), (1.13) and Simon's Lemma, we can extract a subsequence, denoted by (u i (εn,ηn) K 0 ) (εn,ηn) K 0 , that converges strongly in L ∞ ((0, T ); L 1 (K 0 )) to some limit u i , as n → +∞. By a standard diagonalization procedure, we can extract a subsequence (u i εn,ηn ) εn,ηn (independent of i and K) that converges to the limit u i strongly in C([0, T ); L 1 (K)) for all compact K ⊂ R. Now, thanks to estimates (1.11) and (1.12) we can extract a subsequence, still denoted by (u i εn,ηn ) εn,ηn , satisfying the following convergences

u i εn,ηn -→ u i , strongly in C([0, T ); L 1 (K)), for all compact K ⊂ R, u i εn,ηn -→ u i , weakly- in L ∞ ((0, T ) × R), u i εn,ηn -→ u i , weakly- in L ∞ ((0, T ); BV (R)).
Taking the lim inf in estimates (1.11), (1.12) and using the lower semi-continuity of

• L ∞ (R) and | • | BV (R)
, we can prove that u i satisfies (1.15), (1.16) and (1.17). The proof of (1.18) is based on the Finite Speed Propagation Property (2.14) and the BV estimate (1.12), in particular, the discontinuity of BV functions in dimension one. See [1, Section 7] for more details.

Unique continuous solution

In this section, we give the proof of Theorem 1.2. We will first show that the discontinuous viscosity sub-solution u and super-solution u, given by Theorem 1.1, are continuous under assumptions (1.2), (1.4), (1.5), (1.6) and (H). We already have u i ≤ u i for every i = 1, . . . , d, from the definition of upper and lower relaxed semi-limits (see (1.9) and (1.10)). Then, by the use of the comparison principle, we will be able to show that we also have u i ≥ u i for every i = 1, . . . , d, which proves that the solution u is indeed continuous. Thus, we present the following proposition.

Proposition 3.1. Assume (1.2), (1.4), (1.5), (1.6) and (H) hold. Let u = (u i ) i=1,...,d and u = (u i ) i=1,...,d be respectively discontinuous viscosity sub-and super solutions of (1.1), in the sense of Definition 2.1, where the functions u i satisfy max i∈{1,...,d}

u i L ∞ (T×(0,T )) ≤ M 0 . (3.1)
Then, if u i (•, 0) ≤ u i (•, 0) in R we get u i ≤ u i in R × [0, T ) for every i = 1, . . . , d.
Proof of Proposition 3.1: Let us denote by

M sup = max i∈{1,...,d} sup R×[0,T ] V i -V i ,
where V i (x, t) = u i (x, t)e -γt and V i (x, t) = u i (x, t)e -γt . We remark that V i and V i are respectively discontinuous viscosity sub-and super-solutions of the equation

∂ t V i = -γV i + λ i (t, x, u)|∂ x V i |. (3.2)
It is sufficient to prove that M sup ≤ 0. Let us suppose by contradiction that M sup > 0. We proceed in two steps.

Step 1. (Doubling the variables):

We duplicate the variables by considering, for all , β, η and α positives ψ(

= V i (x, t) -V i (y, s) - |x -y| 2 2 - |t -s| 2 2β - η T -t -α(|x| 2 + |y| 2 ). x, y, t, s, i)
We note that ψ(•,

•, •, •, •) is a locally bounded function in R 2 × [0, T ) 2 × {1, . . . , d}.
We can think that the maximum of ψ noted M ( , β, α, η, γ) = sup (x,y)∈R 2 ,(t,s)∈[0,T ] 2 i∈{1,...,d} ψ(x, y, t, s, i), is similar with M sup . The idea is justified by the following lemma.

Lemma 3.1. Let (x, y, t, s, k) be a maximum of ψ. If we define M = lim Proof of Lemma 3.1: Using the fact that V i and V i are bounded for every i ∈ {1, . . . , d}, we deduce that

(h,k)→(0,0) M k h , where M k h = sup |x-y|,|t-s|≤h |x|,|y|≤ 1 √ k V i (x, t) -V i (y, s) , then the following properties hold 1. α|x| 2 , α|y| 2 ≤ 2M 0 . 2. lim β→0 |t -s| = lim →0 |x -y| = 0. 3. t < T -η 2M0 < T . 4. lim inf α→0 lim inf →0 lim inf β→0 lim inf η→0 V i (x, t) -V i (y, s) = lim sup α→0 lim sup →0 lim sup β→0 lim sup η→0 V i (x, t) -V i (y, s) = M .

lim inf

lim |x|,|y|→∞ ψ(x, y, t, s, i) = -∞,
and so the function ψ reaches a maximum at a point (x, y, t, s, i) ∈ R 2 × (0, T ) 2 × {1, . . . , d}.

We know that M sup = max k∈{1,...,d}

sup

R×[0,T ] V i -V i > 0, then there exists (x * , t * , i * ) ∈ (0, 1) × (0, T ) × {1, . . . , d}, such that V i * (x * , t * ) -V i * (x * , t * ) > 0.
Then, we have ψ(x * , x * , t * , t * , i * ) ≤ M ( , β, α, η),

which implies 0 < V i * (x * , t * ) -V i * (x * , t * ) ≤ M ( , β, α, η) + η T -t * + 2α|x * | 2 , 0 < M ( , β, α, η, γ) + η T -t * + 2α|x * | 2 .
Then, for η and α small enough, we get M ( , β, α, η, γ) > 0.

(3.3)

We then deduce α(|x|

2 + |y| 2 ) < V i (x, t) -V i (y, s) ≤ 2M 0 ,
where we have used (3.1) for the second inequality. Multiplying the previous inequality by α yields (1) and lim α→0 α|x| = lim α→0 α|y| = 0.

In the same way, we have |x -y|

2 2 + |t -s| 2 2β < 2M 0 (3.4)
and so lim

→0 |x -y| 2 = lim β→0 |t -s| = 0.
We also deduce that

η T -t < V i (x, t) -V i (y, s) ≤ 2M 0 , Which leads to t < T - η 2M 0 < T.

Now we prove (4). We recall that M

k h = sup |x-y|,|t-s|≤h |x|,|y|≤ 1 √ k V i (x, t) -V i (y, s) . Let (x h,k n , y h,k n , t h,k n , s h,k n ) be such that V i (x h,k n , t h,k n ) -V i (y h,k n , s h,k n ) ≥ M k h - 1 n , with |x h,k n -y h,k n |, |t h,k n -s h,k n | ≤ h and |x h,k n |, |y h,k n | ≤ 1 √ k .
We then have

M k h - 1 n - h 2 2 - h 2 2β - η T -t h n - 2α k ≤ V i (x h,k n , t h,k n ) -V i (y h,k n , s h,k n ) - |x h,k n -y h,k n | 2 2 - |t h,k n -s h,k n | 2 2β - η T -t h,k n -α |x h,k n | 2 + |y h,k n | 2 ≤ M ( , β, η, α, γ) ≤ V i (x, t) -V i (y, s).
We pass to the limit infimum and limit supremum in the following order: η → 0, h → 0, β → 0, → 0, α → 0 and lastly k → 0. Thus, using the fact that the function V i (x, t) -V i (y, s) is upper semi-continuous and so M k h has a limit, we get

M - 1 n ≤ lim inf α→0 lim inf →0 lim inf β→0 lim inf η→0 V i (x, t) -V i (y, s) ≤ lim sup α→0 lim sup →0 lim sup β→0 lim sup η→0 V i (x, t) -V i (y, s) ≤ lim sup α→0 lim sup →0 lim sup β→0            sup |x-y|≤ √ 4 M0 |t-s|≤ √ 4βM0 |x|,|y|≤ 2M 0 α V i (x, t) -V i (y, s)            ≤ lim h →0 k →0     sup |x-y|,|t-s|≤h |x|,|y|≤ 1 √ k V i (x, t) -V i (y, s)     ≤ M , for some h , k such that √ 4 M 0 , √ 4βM 0 ≤ h and k ≤ α 2M 0
, where we have used (3.4) and Lemma 3.1-(1) in the third line. Then, by passing to the limit as n → +∞, we deduce (4).

In the same way we can prove ( 5) and [START_REF] Bianchini | Vanishing viscosity solutions of nonlinear hyperbolic systems[END_REF].

Finally, to prove (7), we suppose for example t = 0. From (1), we can see that x, y, t, s are uniformly bounded with respect to η, β, and . Then we have, up to the extract of a subsequence lim (x, y, t, s) = (ẑ, ẑ, τ , τ ), lim ( ,β,η)→(0,0,0)

i = i = î. (3.5)
Passing to the limit as η → 0, β → 0 then → 0, and using the fact that V i and V i are upper and lower semi-continuous functions respectively, we get

lim sup →0 lim sup β→0 lim sup η→0 V i (x, 0) -V i (y, s) ≤ V î(ẑ, 0) -V î(ẑ, 0) = u î(ẑ, 0) -u î(ẑ, 0) = 0. (3.6)
The last equality is valid since we have

u i 0 (•) = (u i 0 ) (•) ≤ u i (•, 0) ≤ u i (•, 0) ≤ u i (•, 0) ≤ (u i 0 ) (•) = u i 0 (•)
for every i = 1, . . . , d, using the fact that u i and u i are discontinuous viscosity sub-and super-solutions of (1.1) respectively. This means that u i (•, 0) = u i (•, 0) = u i 0 . Then, from (3.6) we deduce that M ≤ 0. However, M > 0 (see Lemma 3.1-( 5)) and for , β, η, α small, we get a contradiction. A similar proof can be made if we consider the case where s = 0.

Step 2. (Obtaining a contradiction): We take , β, α and η small enough such that t > 0 and s > 0 (see Lemma 3.1-( 7)). We can remark that the function

(x, t) → V i (x, t) -V i (y, s) + |x -y| 2 2 + |t -s| 2 2β + η T -t + α(|x| 2 + |y| 2 )
reaches a maximum at (t, x). By using the test-function

φ 1 (x, t) = V i (y, s) + |x -y| 2 2 + |t -s| 2 2β + η T -t + α(|x| 2 + |y| 2 ),
and the fact that u = (u i ) i=1,...,d is a sub-solution of (1.1) and that λ i is continuous for every i = 1, . . . , d, we deduce that

min γV i (t, x) + (t -s) β + η (T -t) 2 -λ i (t, x, r(t, x)) (x -y) + 2αx : r ∈ U (t, x), r i = u i (t, x) ≤ 0.
Then we get

γV i (t, x) + (t -s) β + η (T -t) 2 -λ i (t, x, r 1 (t, x)) (x -y) + 2αx ≤ 0, (3.7)
for some r 1 = (r j 1 ) j=1,...,d such that r i 1 (t, x) = u i (t, x).

On the other hand, we deduce that the function

(s, y) → V i (s, y) -V i (t, x) - |x -y| 2 2 - |t -s| 2 2β - η T -t -α(|x| 2 + |y| 2 )
reaches a minimum at (s, y). By using the test-function

φ 2 (s, y) = V i (t, x) - |x -y| 2 2 - |t -s| 2 2β - η T -t -α(|x| 2 + |y| 2 ),
and the fact that u = (u i ) i=1,...,d is a super-solution of (1.1), we deduce that max γV i (s, y) + (t -s) β -λ i (s, y, r(s, y)) (x -y) -2αy : r ∈ U (s, y), r i = u i (s, y) ≥ 0.

Then we get

γV i (s, y) + (t -s) β -λ i (s, y, r 2 (s, y)) (x -y) -2αy ≥ 0, (3.8)
for some r 2 = (r j 2 ) j=1,...,d such that r i 2 (s, y) = u i (s, y).

By subtracting (3.8) from (3.7), we deduce that

γ(V i (t, x) -V i (s, y)) -λ i (t, x, r 1 (t, x)) (x -y) + 2αx + λ i (s, y, r 2 (s, y)) (x -y) -2αy ≤ 0. (3.9)
As we have done in (3.5), we can extract a subsequence such that lim (β,η)→(0,0) (x, y, t, s) = ( x, y, τ , τ ), lim

(β,η)→(0,0) i = i = i
Sending η → 0 then β → 0 in (3.9), we obtain

γ lim inf β→0 lim inf η→0 V i (t, x) -V i (s, y) -λ i ( τ , x, r 1 ( τ , x)) x -y + 2α x + λ i ( τ , y, r 2 ( τ , y)) x -y -2α y ≤ 0. (3.10)
By adding and subtracting the terms λ i ( τ , x, r 2 ( τ , y)) x-y + 2α x and λ i ( τ , y, r 2 ( τ , y)) x-y + 2α x to inequality (3.10), we get

γ lim inf β→0 lim inf η→0 V i (t, x) -V i (s, y) + λ i ( τ , x, r 2 ( τ , y)) -λ i ( τ , x, r 1 ( τ , x))
x -y + 2α x + λ i ( τ , y, r 2 ( τ , y))

x -y -2α y -x -y + 2α x + λ i ( τ , y, r 2 ( τ , y)) -λ i ( τ , x, r 2 ( τ , y))

x -y + 2α x ≤ 0.

(3.11)

From (3.3), we can see that

V i (t, x) -V i (s, y) ≥ 0.
Thus we have

0 ≤ V i (t, x) -V i (s, y) ≤ lim sup β→0 lim sup η→0 V i (t, x) -V i (s, y) = V i ( τ , x) -V i ( τ , y) = e -γ τ u i ( τ , x) -u i ( τ , y) ≤ u i ( τ , x) -u i ( τ , y).
Then by applying (H) in the first line of (3.11), we get γ lim inf β→0 lim inf η→0 V i (t, x) -V i (s, y) + λ i ( τ , y, r 2 ( τ , y))

x -y -2α y -x -y + 2α x + λ i ( τ , y, r 2 ( τ , y)) -λ i ( τ , x, r 2 ( τ , y))

x -y + 2α x ≤ 0.

Finally, using (1.6), we obtain

γ lim inf β→0 lim inf η→0 V i (t, x) -V i (s, y) -2αΛ i (| x| + | y|) -M 1 | x -y| 2 -2αM 1 | x|| x -y| ≤ 0.
Passing to the limit as → 0 then α → 0, we get according to (1), ( 2), ( 4) and ( 6) in Lemma 3.1 γM ≤ 0, which contradicts our supposition. Hence, we obtain the continuity of the solution.

Application to dislocations dynamics

In this section, we present an application of the results proven in the previous sections to a system modeling the dynamics of dislocations densities. In other words, we will prove Theorem 1.3.

Let us mention some previous results for system (1.19). In El Hajj [START_REF] Hajj | Well-posedness theory for a nonconservative Burgers-type system arising in dislocation dynamics[END_REF], basing on an energy estimate, the global existence and uniqueness of a non-decreasing solution in W 1,2 loc (R×(0, T )) has been obtained. However, without any monotony assumptions on the initial data, El Hajj and Forcadel proved in [START_REF] Hajj | A convergent scheme for a non-local coupled system modelling dislocations densities dynamics[END_REF] the existence and uniqueness of a Lipschitz solution, using the notion of viscosity solutions. They also proposed a convergent numerical scheme and proved a Crandall-Lions error type estimate between the continuous solution and its numerical approximation. Moreover, we can point out the result done by El Hajj and Boudjerada in [START_REF] Boudjerada | Global existence results for eikonal equation with BV initial data[END_REF], who were able to prove the global existence of discontinuous viscosity BV solutions for scalar one dimensional nonlinear and nonlocal eikonal equations, including in particular the case d = 1 in system (1.1), where the velocity does not contain the solution. This result has been extended to a more general nonlinear (2 × 2) system in El Hajj et al. [START_REF] Hajj | Global BV solution for a non-local coupled system modeling the dynamics of dislocation densities[END_REF]. Further more, in El Hajj, Oussaily [START_REF] Hajj | Existence and Uniqueness of Continuous Solution for a Nonlocal Coupled System Modeling the Dynamics of Dislocation Densities[END_REF], the global existence of a continuous viscosity solution has been presented, which was based on an entropy estimate and under a control on the gradient of the solution.

We omit the proof of the existence to system (1. [START_REF]Uniqueness results for diagonal hyperbolic systems with large and monotone data[END_REF]), and we refer the reader to El Hajj et al. [START_REF] Hajj | Global BV solution for a non-local coupled system modeling the dynamics of dislocation densities[END_REF], where the global existence of a discontinuous viscosity solution has been obtained under (1.20) and assuming we have the following conditions

P ± 0 ∈ L ∞ (T) ∩ BV (T), a ∈ L ∞ (0, T ). (4.1)
Proof of Theorem 1.3: It suffices to prove that system (1.19) satisfies the quasi-monotony condition (H) when α = 0. We denote, for ρ = (ρ + , ρ -), by λ + (t, x, ρ) = ρ + (t, x) -ρ -(t, x) + a(t), and λ -(t, x, ρ) = -(ρ + (t, x) -ρ -(t, x)) + a(t).

Let r = (r + , r -) and s = (s + , s -) be two vectors such that r j -s j = max k∈{+,-} (r k -s k ) ≥ 0. We have λ j (t, x, r) -λ j (t, x, s) = j r + -r -+ a(t) -j s + -s -+ a(t)

= j (r + -s + ) -(r --s -)

= sign (r + -s + ) -(r --s -) (r + -s + ) -(r --s -) = (r + -s + ) -(r --s -) ≥ 0.

This ends the proof.

Theorem 1 . 1 (

 11 Existence of a discontinuous viscosity solution to (1.1)).

Lemma 2 . 1 .

 21 (Simon's Lemma[START_REF] Simon | Compact sets in the space L p (0, T ; B)[END_REF] Corollary 4]) Let X, B and Y be three Banach spaces, where X → B with compact embedding and B → Y with continuous embedding. If (θ n ) n is a sequence uniformly bounded in L ∞ ((0, T ); X) and(∂ t θ n ) n is uniformly bounded in L r ((0, T ); Y ) where r > 1, then, (θ n ) n is relatively compact in C((0, T ); B).Next, we are going to recall the definition of discontinuous viscosity solutions for system (1.1) introduced by Ishii in [24, Definition 2.1]. We denote by f and f the respective upper and lower semi-continuous envelopes of a locally bounded function f defined on an open domain in R n and given by

  ) where φ ∈ C ∞ (R) is a cut-off function taking values in [0, 1], supported by the interval [-2, 2] and φ(x) ≡ 1 on [-1, 1].

M 7 .

 7 ( , β, α, η, γ) = lim sup α→0 lim sup →0 lim sup β→0 lim sup η→0 M ( , β, α, η, γ) = M . t, s are positive if , β, α and η are sufficiently small.

Remark 3 . 1 .

 31 Under the same conditions of Proposition 3.1, we can obtain the uniqueness of the solution by proving the following Comparison Principleif u i (•, 0) ≤ v i (•, 0) in R we get that u i ≤ v i in R × [0, T ) for every i ∈ {1, . . . , d},where u = (u i ) i=1,...,d and v = (v i ) i=1,...,d are two bounded continuous viscosity solutions of (1.1).

( ,β,η)→(0,0,0)


Nicolas Guérin

Stefan Enoch

Gérard Tayeb

Pierre Sabouroux

Patrick Vincent

Hervé Legay

A Metallic Fabry-Perot Directive Antenna

Keywords: Directive antennas, Fabry-Perot resonators, leaky wave antennas

We report the design of a directive antenna using the electromagnetic resonances of a Fabry-Perot cavity. The Fabry-Perot cavity is made of a ground plane and a single metallic grid. The resonance is excited by a patch antenna placed in the cavity at the vicinity of the ground plane. The two remarkable features of Fabry-Perot cavity antennas are, first, that they are very thin and second that only one excitation point is needed. A directivity of about 600 is measured at = 14 80 GHz which is to our knowledge one of the highest directivities reported for an antenna using Fabry-Perot resonances.

I. INTRODUCTION

I N the past few years new solutions have been proposed to design directive sources that combine both compactness and a single feeding point. The first solution proposed is to use resonant cavities in dielectric photonic crystals. The simplest proposed design is based on one-dimensional (1-D) photonic crystals [START_REF] Thèvenot | Directive photonicbandgap antennas[END_REF], [START_REF] Cheype | An electromagnetic bandgap resonator antenna[END_REF].

B. Temelkuran et al. [START_REF] Temelkuran | Photonic crystal-based resonant antenna with a very high directivity[END_REF], [START_REF] Ozbay | Microwave applications of photonic crystals[END_REF] have proposed to use threedimensional (3-D) dielectric photonic crystals rather than 1-D leading to a structure slightly more intricate but still easily realizable in the microwave domain. This group has shown that the 3-D structures lead to better performances than 1-D and justify the higher complexity.

Another solution proposed by our group was to use a source embedded in a metamaterial that simulates a low optical index homogeneous medium [START_REF] Enoch | A metamaterial for directive emission[END_REF]. The metamaterial was made of a stack of copper grids and slices of foam (mainly used as a mechanical spacer). A simple way to understand the physical principle is to consider grazing rays radiated by the source on the interface metamaterial-air. Because of the low value of the optical index, they will be refracted in a direction close to the normal of the interface as required by the Snell-Descartes laws. Consequently, the radiated energy will be concentrated in a lobe around the normal. Performances similar to the 3-D photonic crystal based antenna have been obtained.

The Fabry-Perot cavity antenna are obviously cavity antennas that have been studied for a long time in the microwave community. Several applications have been proposed for similar structures such as antennas with the required phase-front and amplitude linearity for helicopter stabilization [START_REF] Klass | Helicopter stabilized by microwave beam[END_REF] systems for missiles [START_REF] Belee | Image element antenna array for a monopulse tracking system for a missile[END_REF]. An exhaustive source about the theory, design and practical applications of the cavity-type antennas can be found in [START_REF] Kumar | Microwave Cavity Antennas[END_REF]. More recently, it has also been proposed to design cavity antennas with nonuniform mirrors to obtain very low sidelobes [START_REF] Sauleau | A new concept of focusing antennas using plane-parallel Fabry-Perot cavities with nonuniform mirrors[END_REF].

In this paper, we report the design of a source that uses a Fabry-Perot resonance with a cavity made of a ground plane and we use a single metallic grid as reflector. The structure is designed for the Ku-band [START_REF] Feresidis | High gain planar antenna using optimized partially reflective surfaces[END_REF][START_REF] Brigham | The Fast Fourier Transform[END_REF](14)(15)(16)(17)(18).

First, we describe the actual prototype of the antenna. Then, we present the 3-D modeling which allowed us to optimize the parameters of the structure in order to obtain a high directivity at the suitable frequency. Finally, experimental results are shown, and a good agreement with the modeling is obtained. The realized antenna has a directivity of about 600 and presents a very low cross-polarization level.

II. DESCRIPTION OF THE STRUCTURE

The antenna is a parallelepipedal flat structure with a square base of 23 cm side (11 side at 14.3 GHz) and a thickness of about 1 cm. Note that the lateral size of the actual antenna has not been optimized since our goal was not to obtain large surface efficiency, thus it can probably be reduced without important change in the obtained directivity. The cavity is bounded by a copper ground plane and a copper grid (Fig. 1): the ground plane acts as a perfect mirror, and the grid acts as a partially reflecting mirror.

The feeding device is a square patch with 8.6 mm width and placed 1.84 mm above the ground plane. It is fed by a coaxial cable coming from the ground plane. The feeding point of the patch is off-center along the -axis in order to have the currents predominantly along this axis (arrows in Fig. 1) and the emitted electric field mainly polarized linearly in the plane (E plane). The grid is a bi-periodic array with 5.8 mm periods of crossed copper strips, which are 0.14 mm thick (along the -axis) and 3.06 mm large. This array is made of 40 40 elementary cells. The cavity is filled with a foam whose permittivity is close to 1 ( at 14 GHz) that ensures the spacing between the grid and the ground plane. The thickness of the foam is .

III. MODELING OF THE STRUCTURE

A 3-D simulation code was developed to calculate the electrical field emitted by such a metallic structure. This code is based on the method of moments using a thin wire approximation [START_REF] Guida | Electromagnetic modeling of three-dimensional metallic photonic crystals[END_REF]. The ground plane is taken into account using image theory. Thus we study a symmetric cavity made of two grids with the thickness . The mirrors are modeled as a bi-periodic array of crossed wires, the development of a fast bi-periodic Green's function [START_REF] Guérin | Combined method for the computation of the doubly periodic Green's function[END_REF] reduces the model discretization to only a single cell of the grating. Note that the actual grids are made of thin copper plates, so that the cross section of the strips is rectangular. In order to model the strips with a sufficient accuracy with our code, we replace each strip by a mesh made of 15 parallel perfectly conducting wires. The diameter of the wires is equal to the thickness of the copper strips (equal to 0.14 mm due to available copper sheets thickness).

The parameters of the meshes are optimized with the 3-D code to obtain a high directivity at a frequency around 14 GHz. The optimization is done in two steps: first, the optimum propagation properties of the cavity are determined by considering only the transmission of two grids without the patch. The reason is that the beam width of the antenna is directly linked with the angular selectivity of the structure illuminated by a plane wave under the incidence [START_REF] Feresidis | High gain planar antenna using optimized partially reflective surfaces[END_REF]. It means that the choice of the parameters of the antenna (first section) can be achieved thanks to the modeling of the structure when the cavity is excited by a plane wave. This simplifies the computations. Secondly, the antenna is simulated and the directivity and cross polarization are estimated.

Fig. 2 shows the transmission as a function of the wavelength under normal incidence. The wavelength giving the maximum Fig. 3. Transmission of the cavity with respect to the incidence angle for different wavelengths. The incident electric field is perpendicular to the plane of incidence.

of transmission is

, and the transmission-bandwidth is equal to 0.6%. However, only wavelengths slightly larger than must be considered. This is shown in Fig. 3: for , the transmission maximum is not along the direction (normal of the structure), but along the direction defined by . As a consequence the expected emission diagram for the antenna is not a lobe centred around the normal but a conical radiation pattern. Hence, for the antenna will not have the required properties in terms of directivity. For , the maximum of transmission corresponds to , thus, it is expected that the antenna will emit a main lobe focussed around the normal of the structure. Moreover, the best angular selectivity is obtained for and not for , consequently the maximum directivity of the antenna will be expected for and not for . In order to model the antenna excited by the patch, we assume that the incident field on the bi-periodic grid is the field radiated by the patch in vacuum. This field is expressed as a superposition of plane waves whose amplitudes are obtained using a bidimensional fast Fourier transform (FFT) [START_REF] Brigham | The Fast Fourier Transform[END_REF]. The drawback of this procedure is that the number of grating problems to solve is equal to the number of propagating plane waves obtained from the FFT. Nevertheless, we are able to run our code on a personal computer. Note that due to the image theory, the source is composed by the actual one (patch) and its image.

Fig. 4 shows the far-field radiation patterns for the patch without the grid (but with the ground plane). Obviously, we get a broad pattern as usual for this kind of antennas. The asymmetry in the E-plane (i.e., plane

) is due to the position of the feeding point on the patch. When the patch is covered with the bi-periodic grid, we observe (Fig. 5) that the emission is concentrated in a narrow lobe. The emitted field is quasilinearly polarized and the maximum level of cross-polarization is about 30 dB. Fig. 6 shows the behavior of the antenna when the wavelength is varying. As previously, we observe a typical behavior for such resonators: putting , when the maximum of emission is not along the normal to the antenna, and when we can get a narrower lobe than and the H plane. In the E plane, the E component is not visible since it is much less than 030 dB.

for . The narrowest lobe is obtained for . The highest theoretical value of the antenna directivity is 715 at (see Fig. 7).

IV. EXPERIMENTAL STUDY

The prototype of the antenna has already been depicted in Section II. The measurements have been made in an anechoic chamber whose dimensions are 14.5 m long, 6.20 m wide The radiation patterns for three different wavelengths are shown in Fig. 8 for both the E-plane and the H-plane. A qualitative agreement is obtained in the behavior of the antenna. The resonance wavelength is slightly below the wavelength corresponding to the maximal directivity . For lower values of the wavelength, several lobes are observed. Moreover we have also a good quantitative agreement: the experimental value of the wavelength is measured to be 20.26 mm while the theoretical value is 19.25 mm (5% of difference). These slight discrepancies are attributed to two factors. First, we do not know exactly the permittivity of the foam. Second, the foam thickness is not perfectly constant and equal to 9 mm. The obtained discrepancies are consistent with the tolerance given by the manufacturer of the foam (thickness tolerance equal to 0.5 mm).

The E-plane radiation pattern in Fig. 8 exhibits an asymmetry that is not present in the theoretical model. We attribute this to the excitation point of the feeding patch and to the tolerance in the realization of the antenna. This asymmetry induces a lower value for the directivity than expected, but it is still about 600. The very low level of the side lobes should also be noted. Fig. 9 shows the measured directivity as a function of the wavelength. As predicted by our theory we can find a passband for the directivity of about 0.3%. Note that, although the design of the structure has not be optimized to obtain impedance matching, we have measured a value of about 5 dB for the S11 coefficient for the frequency corresponding to the maximum of directivity.

V. CONCLUSION

We have presented a numerical and experimental study of a Fabry-Perot cavity antenna and a good agreement has been obtained. High directivities have been obtained both theoretically and experimentally (resp. 715 at and about 600 at

). Beside the small thickness and the single feeding device, the weight of our prototype of the antenna is low, thus it could be interesting for spatial applications.

The small value of bandwidth should be increased by considering aperiodic grids. This could also open new opportunities for high surface efficiency antennas. Finally, the use of a high impedance surface as ground plane should be considered. This might improve the performances, specially it might reduce the thickness of the antenna.

  or tracking N. Guérin is with the Laboratory for Electromagnetic Fields and Microwave Electronics, Swiss Federal Institute of Technology, 8092 Zürich, Switzerland (e-mail: nguerin@ifh.ee.ethz.ch). S. Enoch, G. Tayeb, P. Sabouroux, and P. Vincent are with the Fresnel Institute, CNRS 6133, Université d'Aix-Marseille III, Faculté des Sciences et Techniques, 13397 Marseille cedex 20, France. H. Legay is with Alcatel Space, 31037 Toulouse cedex, France.

Fig. 1 .

 1 Fig. 1. Schematic representation of the antenna based on a Fabry-Perot cavity, the black lines delimit the volume occupied by the foam. For the prototype L = 3:06 mm, t = 0:14 mm, d = d = 5:8 mm, and h = 9 mm.

Fig. 2 .

 2 Fig. 2. Transmission of the cavity with respect to the wavelength, for normal incidence. The electric field is linearly polarized along the x-axis.

Fig. 4 .

 4 Fig. 4. Radiation patterns (normalized dB scale) of the patch without the grid in the E plane and the H plane. In the E plane, the E component is not visible since it is much less than 030 dB.

Fig. 5 .

 5 Fig. 5. Radiation patterns (normalized dB scale) of the antenna in the E plane

Fig. 6 .

 6 Fig. 6. Radiation patterns (normalized dB scale) of the antenna for different values of the wavelength.

Fig. 7 .

 7 Fig. 7. Directivity deduced from the computed radiation patterns with respect to the wavelength: the bandwidth is about 0.3%.

Fig. 8 .

 8 Fig. 8. Experimental radiation patterns (normalized decibels scale) of the antenna in the E plane (top) and the H plane (bottom). The radiation patterns have been normalized with respect to the maximum of emission for the three wavelengths.

Fig. 9 .

 9 Fig. 9. Experimental directivity of the antenna.


M Rodriguez Pascual

F Ravelet

R Delfos

G J Witkamp

MEASUREMENT OF FLOW FIELD AND WALL TEMPERATURE DISTRIBUTION IN A SCRAPED HEAT EXCHANGER CRYSTALLIZER

During crystallization the control and effective distribution of heat transfer from the solution to the heat exchanger (HE) plays an important role. Inhomogeneity in temperature on the HEsurface limits the production capacity and increases the tendency of an isolating scale layer formation. To avoid scaling, scraper blades on the HE are commonly used.

In a typical axi-symmetric geometry the scale layer formation was investigated. We studied the influence of the flow field on the actual heat transfer distribution 1) by directly measuring the surface temperature field of the HE using LCT and 2) by measuring the flow field inside the crystallizer using 3C-PIV.

From Liquid Crystal Thermometry, we can clearly see that, except for the centre region, near the inside the heat transfer is considerable better than near the outside. Stereoscopic PIV velocity measurements reveal that the central region of the crystallizer is in solid body rotation; in the outer parts there is a strong secondary flow, where crystallizer solution descends onto the HE-surface, causing a high heat transfer; near the outer wall the solution rises again having a small heat transfer.

We conclude that the present design of the stirrer / scraper system in the crystallizer provides sufficient secondary flow to keep the system mixed. However, near the outer wall and near the shaft the heat transfer is considerably lower than average. This is also the zone where during Eutectic Freeze Crystallization experiments the first not-removed ice-scaling layer is observed. This is thus not due to a limited scraping function of the scraper, but due to the HE getting the lowest temperature, hence the highest sub-cooling.

Introduction

Several crystallization processes consist of liquid solutions that by the action of heat exchangers are brought into supersaturated regions where crystals will form. In industrial continuous processes, where solution is constantly fed into the crystallizer, control and stability of the bulk solution temperature is mandatory. For this reason a degree of turbulent flow in the crystallizers is necessary to achieve good mixing of the whole solution. The heat transfer rates are directly responsible for the production rates. The residence times of the solution in the crystallizer will determine size and quality of the crystals.

To obtain the desired supersaturation the solution is cooled down (as in Eutectic Freeze crystallization) or heated up (as in Evaporative Crystallization). This action is done by Heat Exchangers (HE) in direct contact with the solution. Near the HE surface exists a thermal boundary layer that depends on the crystallizer flow characteristics. Here the supersaturation is higher that in the bulk solution. Because of the higher supersaturation on the HE surface the nucleation and growth rates of the crystals are faster than in the bulk solution. This situation is responsible for the formation of a scaling layer of crystals on the HE surface. As a consequence produces a decrease of heat transfer, affecting the stability of the crystallization process. To avoid this scale layer formation, it is common to use mechanical actions, such as scraping.

The scraping efficiency is directly proportional to the temperature difference between the bulk solution and the heat exchanger surface temperature [START_REF] Vaessen | Scale formation of ice from electrolyte solutions on a scraped surface heat exchanger plate[END_REF] and Pronk (2005)]. Depending on our solution and the characteristics of our mechanical scraping action (velocity, shapes and applied forces), we will have a maximum temperature difference between the solution and the heat exchanger surface where scaling will appear. In order to achieve the desired production rate and avoid the scaling layer problem we will have to apply the consequent scraping characteristics. For these reasons the heat exchanger surface temperature has to be as uniform as possible to control the unwanted scaling layer.

Ice scaling Experiments

Setup

The experimental set-up for ice-scaling research (Ice Scaling Crystallizer, ISC) consists of a 10-liter scraped surface crystallizer as shown in Figure 1. The crystallizer has a 1 mm stainless steel bottom plate with a heat transfer area of 0.031 m 2 , which is scraped by four rotating Teflon scraper blades that are driven by a vertical shaft. Halfway this shaft, a turbine mixer is installed to keep the slurry homogenously mixed. The bottom plate is the topside of a Heat Exchanger (HE), which is cooled by a 50 wt% potassium formate solution ('Freezium'). The coolant follows a spiral channel through the HE. Height & width of the channel measure 5 & 17 mm, respectively. The coolant flow rate is 1000 l/h, resulting in a channel velocity of over 3 m/s, guaranteeing turbulent channel flow. The HE inlet temperature is controlled within 0.1 K by a cooling machine.

The crystallizer overflows to an ice-melting vessel were the produced ice crystals are molten and from which aqueous solution is pumped back to the crystallizer, such facilitating stationary flow conditions. To minimise heat leaks to the ambient, the experimental system was thermally insulated wherever appropriate besides some selected parts required for (visual) observation.

Temperature of the coolant (at inlet and outlet), the crystallizer liquid (near top and bottom of the vessel) and the feed flow were measured with ASL F250 precision thermometer connected to PT-100 temperature sensors with an accuracy of ±0.01 °C. The pressure of the coolant in the HE channel was measured at inlet and outlet by two analogue pressure sensors with an accuracy of ±0.001 bar. The flow rate of feed solution and coolant were measured with two electro-magnetic flowmeters with an accuracy of ±1l/h. The five temperatures, the two pressures and the two flow rates were collected every 1 second in a computer using the LabView data acquisition program. This same behaviour was observed in the CDCC ending with a damaged outer scraper because of the impossibility to remove the thick ice layer formed in this area.

Observations

During all experiments in the ISC, the temperature of the crystallizer liquid constant, i.e. at the freezing temperature of the water, or at the eutectic temperature for a salt water system. With this system, we have run experiments in which we gradually increase the heat flux through the HE by gradually lowering the HE temperature. Doing so, we find that ice scaling (i.e. the formation of ice that is not removed by the scraper blades) starts in two distinct areas of the HE surface: 1) near the centre of the crystallizer and 2) close to the outside wall. The net effect is the formation of two rings of ice. When one keeps the process going anyway, the scrapers gradually get lifted up, and the function of the scrapers quickly deteriorates. In experiments in the 'CDCC' crystallizer (a scaled-up version of the 'ISC', [START_REF] Genceli | Eutectic freeze crystallization of a 2nd generation cooled disk column crystallizer for MgSO4-H2O system[END_REF]), exactly the same behaviour is observed. As we can see in Figure 2, continuation of the process ended with damage to the outer scraper.

Possible explanations for this reproducible behaviour were as follows:

1) The scrapers function better in the middle section. This argument was discarded, because also when we scraped with arms of three independent scraper sections (each with their own spring loading) we observed exactly the same behaviour. 2) Influence of the feed flow inlet. The feed flow inlet is always slightly warmer than the crystallizer fluid. We have visualized the flow pattern by injecting a blue dye into the feed flow.

There we observed that, when entering the crystallizer, the dye is first transported upwards, then mixed in the top section, before it finally descends into the lower section where it contacts the HE surface. By that time, it has mixed well, and cooled down sufficiently. 3) Possibly the local heat transfer varies over the plate. Since we could cross out the first two mechanisms, we have done a more elaborate study on the latter process.

We have done in-situ measurements of the heat exchanger surface temperature distribution, and consequently the local heat transfer distribution, which is described in section 3. Also, we have done quantitative flow measurements, which are described in section 4. In Section 5 we will combine and discuss the results, and consider the consequences.

3 Heat Exchanger Surface Temperature Measurements The set-up for measuring the HE surface temperature is shown in Figure 3. This temperature was measured using an adhesive thermochromatic liquid crystal sheet (TLC, Hallcrest R34C1W) glued directly onto the HE-surface. The TLC changes colour with temperature, with a working range between 34-35 °C (provider-specified). The TLC-sheet is illuminated using a slide projector with 'white light' (320 W halogen lamp, colour temperature 3400 K). Its light is passed through a linear polarising filter, then projected via a mirror onto the bottom of the tank under an angle of incidence of about 15° with the vertical ('lighting angle'). The TLC-sheet surface was observed from above under an angle of 15° as well ('viewing angle') through a second polarizer using a digital photo camera (Canon Powershot S2 IS). The two polarizers suppress unwanted reflections of the illuminating light, resulting in more saturated TLC-reflected colours. Colour TLC surface photographic images (jpeg-compressed, 2592*1944 px) were recorded and transferred to a PC every 2 seconds, on which further image processing was done using MatLab, v. 6.0. Calibration of the TLC-colours was done in situ, in the same manner as described in [START_REF] Delfos | Influence of large scale flow structures on convection at moderate Rayleigh numbers[END_REF]. In short: during a gradual heat-up and cool-down procedure when the temperature on top the TLC and the temperature of the coolant readings were stable at the same value within 0.01 o C, an image was selected for the calibration curve. The Red-Green-Blue (RGB) data was mapped to Hue-Saturation-Intensity (HSI) data using MatLab's rgb2hsv function. Based on the Hue values H, we obtain a one-to-one relationship for T vs. H. The useful part of the resulting calibration curve runs is in the range 33.5 °C < T < 36 °C (Figure 4), as in [START_REF] Delfos | Influence of large scale flow structures on convection at moderate Rayleigh numbers[END_REF] and was shown to depend only little on illumination intensity with an accuracy of ±0.03 °C.

Experimental Setup

To avoid hysteresis effects of the TLC due to the working range we decide to study the HE surface temperature in an inverted experiment, where instead of cooling the solution we heated up to avoid exposing the TLC sheet to high temperatures. This means: the solution is maintained at a temperature of some 32 o C; it is heated through the HE with 'coolant' of 42 o C; the feed flow is about 28 o C. After a stationary situation was obtained (monitored using thermometers), imaging was started. In Figure 4 we see a typical picture of (half of) the TLC sheet on the heat exchanger as measured halfway during this time interval. The shiny part is one of the steel arms that hold the scraper. This result immediately shows that, even though at first glance the process as a whole is stationary, that there is a strongly non-uniform temperature distribution on the HE surface.

Results -integral heat transfer

To get a view on the energy balance of the system, we applied the following steps. The total heat flux through the heat exchanger surface into the crystallizer, HE Q & , is directly found from the integral energy balance:

( ) ( ) HE cool cool V cool loss Q c T p Q φ ρ = ⋅ ⋅ ⋅ ∆ -∆ - & & , (1)
where φ cool , ρ cool and c V are the flow rate, density and its specific heat of the coolant; ∆T cool and ∆p its temperature and pressure drop in the coolant channel, and loss Q & the heat flux to the surroundings. The latter was determined experimentally earlier for this HE [START_REF] Pronk | Fluidized Bed Heat Exchangers to Prevent Fouling in Ice Slurry Systems and Industrial Crystallizers[END_REF]), and is here estimated as 1.5 W/K. For the experimental conditions as above, we have an estimated heating of the crystallizer solution of 220 W. For instationary heating as in our case, this net heat flux is absorbed by the solution and the feed liquid:

HE feed sol V cryst Q Q Vol c dT dt ρ = + ⋅ ⋅ ⋅ & & , (2)
where feed Q & is calculated the same as for the HE flux, and the solution temperature rise, dT cryst /dt is obtained from the measurements as 0.0017 o C/s. This give a feed Q & of 120 W average during the measurement time, which means a total heat taken by the solution of 100 W and an increase of 0.57 o C, whereas the experiment has a measured increase of 0.4 o C. Taking into account the limited accuracies, we consider this a sufficient approximation.

Results -local heat transfer

In the previous we have considered the total heat transfer from coolant to crystallizer solution. Since we have observed that the HE surface temperature is much dependent on position, we need to search for the local surface heat flux, q & and local heat transfer coefficient h. Since we observe that the TLC colour pattern on the HE surface (so presumably also the temperature) is more or less axis symmetric, we assume further that all profiles are functions of the distance to the centre, r, only. Then we can write in general the local relation between q & and h (Mills 1999):

( ) ( ) (
)

sol cool h r q r T T = - & (3)
where cool T is the average temperature of the coolant, and the solution temperature T sol already is assumed to be constant throughout the crystallizer. The heat transfer coefficient for the system, h is found by adding up the individual heat resistances:

) ( / 1 / / / 1 ) ( / 1 r h d d h r h cryst TLC TLC plate plate cool + + + = λ λ (4)
Here the conductive heat transfer coefficients for the stainless steel plate and the TLC layer are 15*10 3 W/m 2 .K and 1.2*10 3 W/m 2 .K, respectively. To find the convective heat transfer coefficients at the coolant channel side, h cool , is more complicated: For a flowing liquid, it is given by .

cool cool cool h Nu d λ =
, with d the hydraulic diameter of the duct, which is 7.7 mm. The Nusselt number Nu is of the form as for turbulent duct flow (Mills, 1999):

0.8 0.3 Nu 0.024 Re Pr = ⋅ ⋅
(5) A result found previously for turbulent flow in our cooling channel [START_REF] Pronk | Fluidized Bed Heat Exchangers to Prevent Fouling in Ice Slurry Systems and Industrial Crystallizers[END_REF]) is about twice as large due to the curvature of the channel improving the heat transport in the channel: 0.699 0.33 Nu 0.0507 Re Pr = ⋅ ⋅ (6) For our typical process conditions, we find Nu = 42 hence h cool = 2800 W/m 2 .K. Now with the three known heat resistances (h known = h steel+cool+TLC ), we can calculate the local heat transfer coefficient between crystallizer and HE surface based on the temperature difference between solution (T sol ) and coolant (T cool ), and the calculated surface temperature (T surf ) from the direct TLC Temperature values under the polyester sheet (Temp[oC] in Figure 6) as follows:

) /( ) (

surf sol cool surf known cryst T T T T h h - - ⋅ =
(7) To get the numerical values, we process the TLC images as described. Visually it looks as below: From the photographic image of the TLC-surface we cut out the area in between scrapers (Figure 5a), we convert the RGB values into Hue values (false colours, Figure 5b), and with the calibration curve into temperature (Figure 5c). As expected, the heat exchanger surface temperature is far from uniform: with a temperature difference between coolant and solution of just 10 o C, the HE surface temperature varies with differences of more than 4 o C, and in fact even outside the working range of the TLC! We converted these data into a radial profile of the temperature distribution T(r), as is shown in Figure 6, together with the related local heat transfer coefficient, h cryst (r). Figure 6. Radial distribution of temperature T(r) and convective heat transfer coefficient h cryst (r) on HE surface (between scrapers). Note: temperatures above 36 o C are approximate (being outside the proper calibration range), as commented before.

We now indeed clearly see near the crystallizer centre and close to the outside that the heat transfer between HE and solution, h cryst (r), is lower than that somewhat halfway the radius by a factor of at least five. This observation of temperature and heat transfer distribution in our opinion completely explains the observation of ice scaling in the ISC experiments: in regions where the heat transfer is low, the HE surface temperature falls to lower values; thereby it increases the supersaturation of the solution, hence in these regions the nucleation will be much stronger and so the tendency of scaling. The remaining question is however the reason why h cryst varies so much over the HE plate. For this, we will investigate the flow pattern and the turbulence present above the HE plate.

Stereoscopic PIV Flow Measurements

To investigate the origin of the so much varying local heat transfer coefficient, we decided to try to get more insight into the flow field and the turbulence distribution inside the crystallizer. To do so, part of the field was investigated using Stereoscopic Particle Image Velocimetry, (3C-PIV). A general description of 3C-PIV, and methods to set it up can be found in [START_REF] Raffel | Particle Image Velocimetry[END_REF]; some specific details of our 3C-PIV set-up in [START_REF] Ravelet | Experimental studies of liquid liquid dispersion in a turbulent shear flow[END_REF]. The set-up is summarised below and sketched in Figure 7.

Experimental Setup

The shaft with scraper and stirrer was taken from the original 200 mm ID geometry to a slightly larger glass cylinder of 240 mm ID, which was well aligned in the two-camera stereoscopic PIV set-up of [START_REF] Ravelet | Experimental studies of liquid liquid dispersion in a turbulent shear flow[END_REF]. With the verticalradial light laser light sheet we were able Figure 7. Sketch of set-up for PIV measurements.

with a PIV calibration grid-image inside/ to capture double-frame PIV images from two different projections of the particle motion. The 3D calibration, the laser and camera timing and the PIV image processing were all done using Davis 7.1-7.2 (www.Lavision.de).

The measuring area measures height 45 mm and width 80 mm, starting just above the scraper and at the outer wall, all as indicated in figure 9. From 500 instantaneous flow fields obtained the average velocities and all six Reynolds stresses. Here we only consider the average velocity field and its role in the heat transfer.

Results

The average flow field is shown in figure 8a. Also, since the flow varies only little with height, we can obtain the axially averaged radial profiles, as shown in figure 8b. The biggest velocity component is the tangential velocity, while it is forced directly by the rotating aerodynamically bluff scraper arms, and friction with the smooth outer wall and the free surface top is small. We can observe, besides a thin underresolved boundary layer at the outer wall, three distinct regions in the profile: 1) Up to a radius of roughly 50 mm, the tangential liquid velocity (blue) matches that of the scraper-stirrer shaft (dotted red). Even though data for most of this inner part is failing, it only seems realistic that this whole region, the 'core', is in solid-body rotation with the scraper-stirrer shaft, and there is no centrifugal or axial forcing at all; the latter is reflected as well in the small axial velocity.

2) In the annulus around the core (beyond 60 mm), the velocity decreases again, and in this region the interesting transport processes take place. The axial profile shows a strong secondary flow; near the outer wall the fluid is flowing upwards, with a magnitude as much as 15% of the tangential velocity. This strong secondary flow component driven by the rotating scraper arms, which near the bottom transport fluid outward by centrifugal forcing. Arriving at the outer wall, this flow is deflected upward as we measure it. By continuity, the fluid has to flow somewhere radially inward and then downward; here we see a maximum downward flow occurring at 80 mm from the centre. When moving radially inward or outward, the flow tends to conserve its angular momentum [START_REF] Batchelor | An Introduction to Fluid Dynamics[END_REF]; in a frictionless flow the result would be a potential vortex. As a simplified physical model to the flow, we fitted the measured tangential velocity profile with the Oseen vortex solution [START_REF] Batchelor | An Introduction to Fluid Dynamics[END_REF], which gradually transits from a solid-body rotation inside region to a free vortex flow outside region, as shown in figure 10b. Here we can see that in our case the picture matches qualitatively, but that the transition between the two regions is more abrupt than in the model.

The measured flow field completely supports the picture obtained from the heat transfer distribution on the HE-surface: in the core, the heat transfer is low because of the absence of a mean flow. In the annulus, there is the secondary flow, which causes much better heat transport. Locally, the heat transfer is much increased where the solution descends onto the HE-surface; the boundary layer is thinned here by the stagnation flow. In contrast, the local heat transfer is reduced where the fluid temperature has adapted to the boundary temperature and ascends. This picture of heat transfer is much similar to that measured in turbulent Benard convection, where underneath stgantion regions the heat transfer is enhanced and underneath rising plumes reduced (Delfos &[START_REF] Delfos | Influence of large scale flow structures on convection at moderate Rayleigh numbers[END_REF][START_REF] Petracci | Combined PIV/LIF measurements in a Rayleigh-Bénard convection cel[END_REF]).

Conclusions

During experiments of Eutectic Freeze Crystallization we observed that ice scaling was formed in specific areas on the heat exchanger. Because the driving force for ice scaling is the local supersaturation, and this is related to the solution temperature, we measured the temperature and heat transfer on the HE-surface.

The temperature results showed that indeed the temperature on the HE-surface was not uniform having temperature differences bigger than 4 o C, a difference more than sufficient to explain the local variation in scaling tendency. The temperature difference could directly be related to a variation in the local heat transfer coefficient over the HE-surface.

Stereoscopic PIV measurements were carried out, which showed that the secondary fluid flow in this crystallizer geometry is responsible for the inhomogeneous heat transfer distribution.

When designing cooling crystallizers and heat exchanger geometries, it is important to take flow-induced inhomogeneity in heat trasnfer into account, while this forms a strong limitation in the crystallizer performance.

Figure 1 .

 1 Figure 1. Scheme for performing continuous crystallising experiments.Figure 2. The bad local heat transfer distribution on the HE surface translates into high temperature differences through the HE surface. As observed during ice scaling experiments in the ISC lower temperatures where found in the centre of the HE and in the outer area, forming two rings with difference on temperature compare with the middle ring higher than 4 o C. Due to this difference on temperature the nucleation and crystal growth are higher in the centre and in the outer ring and is this zones where ice scaling starts.

Figure 2 .

 2 Figure 1. Scheme for performing continuous crystallising experiments.Figure 2. The bad local heat transfer distribution on the HE surface translates into high temperature differences through the HE surface. As observed during ice scaling experiments in the ISC lower temperatures where found in the centre of the HE and in the outer area, forming two rings with difference on temperature compare with the middle ring higher than 4 o C. Due to this difference on temperature the nucleation and crystal growth are higher in the centre and in the outer ring and is this zones where ice scaling starts.

Figure 3 .

 3 Figure 3. Sketch for the set-up for surface temperature measurement.Figure 4. Calibration curve for the TLC material in the useful working range.

Figure 4 .

 4 Figure 3. Sketch for the set-up for surface temperature measurement.Figure 4. Calibration curve for the TLC material in the useful working range.

Figure 5a .

 5a Figure 5a. Cut area in between scrapers of the HE surface.Figure 5b. Converted into Hue values.Figure 5c. Via the calibration curve converted into temperatures.

  Figure 5a. Cut area in between scrapers of the HE surface.Figure 5b. Converted into Hue values.Figure 5c. Via the calibration curve converted into temperatures.

Figure 5c .

 5c Figure 5a. Cut area in between scrapers of the HE surface.Figure 5b. Converted into Hue values.Figure 5c. Via the calibration curve converted into temperatures.

Figure 8a

 8a Figure 8a Average velocity field Note: the radial and tangential components are depicted as vectors, the tangential component in false colours. Figure 8b. Radial profiles of average velocity.

5th European Thermal-Sciences Conference, TheNetherlands, 2008