B ÉR ÉNICE GREC, MILANA PAVI Ć Laurent Boudin
email: laurent.boudin@upmc.fr

Bérénice Grec
email: berenice.grec@parisdescartes.fr

Milana Pavic
email: milana.pavic@cmla.ens-cachan.fr

AND Francesco Salvarani
email: francesco.salvarani@unipv.it

B G : Map5

Diffusion asymptotics of a kinetic model for gaseous mixtures

In this work, we consider the non-reactive fully elastic Boltzmann equations for mixtures in the diffusive scaling. We mainly use a Hilbert expansion of the distribution functions. After briefly recalling the H-theorem, the lower-order non trivial equality obtained from the Boltzmann equations leads to a linear functional equation in the velocity variable. This equation is solved thanks to the Fredholm alternative. Since we consider multicomponent mixtures, the classical techniques introduced by Grad cannot be applied, and we propose a new method to treat the terms involving particles with different masses.

Introduction

The study of the asymptotic behaviour of the Boltzmann equation for small mean free path is known as Hilbert's sixth problem, after its formulation by Hilbert himself during the International Congress of Mathematicians held in Paris in 1900 [START_REF] Hilbert | Mathematical problems[END_REF]. Since then, it has been a very active field of research, and many results have been obtained, both at a formal level and in the context of rigorous limits.

The main tools are based on asymptotic (Hilbert, Chapman-Enskog) expansions with respect to the mean free path, see for instance [START_REF] Chapman | The mathematical theory of nonuniform gases[END_REF]. The translation in a rigorous mathematical language has been performed in a series of pioneering papers by Bardos, Golse and Levermore [START_REF] Bardos | Sur les limites asymptotiques de la théorie cinétique conduisant à la dynamique des fluides incompressibles[END_REF][START_REF] Bardos | Fluid dynamic limits of kinetic equations. I. Formal derivations[END_REF][START_REF] Bardos | Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation[END_REF], where the authors established the ground of the subsequent results concerning the rigorous asymptotic limits. They stated a program that led to many interesting results, such as [START_REF] Lions | From the Boltzmann equations to the equations of incompressible fluid mechanics. I[END_REF]. This trail culminated in [START_REF] Golse | The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels[END_REF], where Golse and Saint-Raymond established a Navier-Stokes limit for the Boltzmann equation considered over the infinite spatial domain R 3 : appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations whose limit points are governed by Leray solutions of the limiting Navier-Stokes equations. We can also refer to [START_REF] Golse | The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials[END_REF], which extends the results of [START_REF] Golse | The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels[END_REF] for hard cutoff potentials in Grad's sense.

Apart from the research concerning with the classical Boltzmann equation (see [START_REF] Degond | Macroscopic limits of the Boltzmann equation: a review[END_REF] as a review article), which can be seen as a model describing a mono-species, monoatomic and ideal gas, one can focus on the study, at a kinetic level, of gaseous mixtures, without excluding the possibility of chemical reactions. In such a framework, the models are much more intricate. It is indeed necessary to treat systems of Boltzmann-like equations, rather than one single equation, with multi-species kernels and cross interactions between the different distribution functions describing each component of the mixture [START_REF] Sirovich | Kinetic modeling of gas mixtures[END_REF][START_REF] Greene | Mathematical aspects of kinetic model equations for binary gas mixtures[END_REF][START_REF] Morse | Kinetic model equations for a gas mixture[END_REF]. The complexity of the models grows dramatically if exchanges of internal energy and chemical reactions are allowed [START_REF] Ross | Some deductions from a formal statistical mechanical theory of chemical kinetics[END_REF][START_REF] Sirovich | Kinetic modeling of gas mixtures[END_REF][START_REF] Bourgat | Microreversible collisions for polyatomic gases and Boltzmann's theorem[END_REF][START_REF] Rossani | A note on the kinetic theory of chemically reacting gases[END_REF].

The derivation of macroscopic equations from kinetic models remains crucial for mixtures, both at a mathematical level and for deducing relevant macroscopic equations based on the modelling of microscopic binary interactions. In this spirit, in [START_REF] Desvillettes | A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions[END_REF], the authors propose a model describing a reacting mixture of polyatomic gases and recover in the limit, via the appropriate scaling (t, x) → (t/ε, x/ε) for ε > 0, the reactive Euler equations.

The diffusive scaling we investigate here, i.e. (t, x) → (t/ε 2 , x/ε), is more complicated, and even the formal structure of the asymptotic hierarchy is not trivial at all. After the first attempts of Chapman and Cowling [START_REF] Chapman | The mathematical theory of nonuniform gases[END_REF], in [START_REF] Bastea | Binary fluids with long range segregating interaction. I. Derivation of kinetic and hydrodynamic equations[END_REF], the authors consider a binary mixture of red and blue particles which interact via strong short range (hard core) and weak long range pair potentials, and study the small free path limit in various situations. Ref. [START_REF] Dogbe | Fluid dynamic limits for gas mixture. I. Formal derivations[END_REF] handles the same kind of model of binary mixture with a heavy species and a lighter one.

In this paper, we consider a mixture of several different non reactive gases, which evolves in time via the classical Boltzmann system for non-reacting multicomponent mixtures, and study the formal Hilbert expansion leading to the Navier-Stokes system for mixtures. Even if this strategy dates back to Grad [START_REF] Grad | Asymptotic theory of the Boltzmann equation[END_REF][START_REF] Grad | Asymptotic theory of the Boltzmann equation. II[END_REF], who studied the formal small free path limit for the monatomic and monospecies Boltzmann equation, the development leading to write the Chapman-Enskog hierarchy owns some particular properties, which are inherited from the multispecies feature of the mixture.

First of all, we can observe that the structure of the small free-path asymptotics for mixtures, composed by three or more species, differs from the case of a mono-species gas or a binary mixture. The most apparent feature which is peculiar to a (at least) ternary mixture is the phenomenon of uphill diffusion, independently foreseen by Maxwell [START_REF] Maxwell | On the dynamical theory of gases[END_REF] and Stefan [START_REF] Stefan | Ueber das Gleichgewicht und die Bewegung insbesondere die Diffusion von Gasgemengen[END_REF], and experimentally recovered by Duncan and Toor [14]. This kind of diffusion is described by second-order coupled terms, which make the asymptotics quite different from the limit related to a binary mixture [START_REF] Boudin | The Maxwell-Stefan diffusion limit for a kinetic model of mixtures[END_REF].

A second important aspect is that we cannot apply Grad's methodology when we have to deal with different species, with different masses and different cross sections. We hence propose a new approach to the problem which only works when the binary collisions involve particles with different masses, and we then use Grad's procedure [START_REF] Grad | Asymptotic theory of the Boltzmann equation. II[END_REF] when considering collisions between same mass particles. Note that both approaches are needed to give a complete answer to the question of resolvability of the first-order (in ε) equations of the hierarchy, obtained by means of the Fredholm alternative. We point out that our approach holds for generic cut-off cross sections, as those described in Section 2.

Note that this type of asymptotics is also numerically investigated, see, for instance, [START_REF] Bennoune | An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit[END_REF][START_REF] Jin | A BGK-penalization asymptotic-preserving scheme for the multispecies Boltzmann equation[END_REF], using BGK approaches.

The article is structured as follows. In the next section, we describe the model, its basic properties and the main results which are proved in the rest of the paper. Then Sections 3 and 4 are devoted to the proof of the mathematical results stated in Section 2. More precisely, Fredholm's alternative is applied in Section 3 (provided the compactness of some operator is known), and the proof of the compactness is treated in Section 4. As we already mentioned, the case of particles of same masses is treated following Grad's method [START_REF] Grad | Asymptotic theory of the Boltzmann equation. II[END_REF] (see Section 4.1), whereas the case of different masses needs a new approach (see Section 4.2, in particular Lemma 3).

Model

We consider an ideal gas mixture constituted with I ≥ 2 species. Each species A i of the mixture, 1 ≤ i ≤ I, is described by a microscopic density function f i . It depends on time t ∈ R + , space position x ∈ R 3 and molecular velocity v ∈ R 3 , and is nonnegative. More precisely, f i (t, x, v) dxdv allows to quantify the number of molecules of species A i at time t in an elementary volume of size dx, and whose velocities equal v up to dv. We can also define the macroscopic density n i of each species A i by

n i (t, x) = R 3 f i (t, x, v) dv.
We assume that the mixture only involves molecular elastic collisions. Let us consider two colliding molecules of species A i and A j , 1 ≤ i, j ≤ I. Their masses are m i and m j , and their pre-collisional velocities v ′ and v ′ * . After a collision, the particles belong to the same species (no chemical reactions), so their masses remain as they were, and their velocities become v and v * . Since the collisions are elastic, both momentum and kinetic energy are conserved, i.e.

(1)

m i v ′ + m j v ′
where ω ∈ S 2 is arbitrary, and T ω is the symmetry with respect to the plane {ω} ⊥ , i.e.

T ω z = z -2(ω • z)ω, ∀z ∈ R 3 .
2.1. Collision operators. Let 1 ≤ i, j ≤ I. The collision operator associated to species A i and A j is defined by

(3) Q ij (f, g)(v) = R 3 ×S 2 f (v ′ )g(v ′ * ) -f (v)g(v * ) Bij (v, v * , ω) dω dv * ,
where v ′ and v ′ * are defined by [START_REF] Bardos | Fluid dynamic limits of kinetic equations. I. Formal derivations[END_REF], and f and g are two functions of the velocity variable. The crosssection Bij (v, v * , ω) only depends on v, v * and ω. In fact, Bij is only a function of |v -v * | and the angle θ between ω and V := v -v * , by Galilean invariance. Let us set

B ij (ω, v -v * ) = Bij (v, v * , ω), ∀ω ∈ S 2 , ∀v, v * ∈ R 3 .
The collisions are also supposed microreversible. That ensures that (4)

B ij (ω, v -v * ) = B ji (ω, v -v * ), B ij (ω, v -v * ) = B ij (ω, v ′ -v ′ * ), ∀ω ∈ S 2 , ∀v, v * ∈ R 3 .
Moreover, we assume that B ij satisfies a general condition

(5) B ij (ω, V ) ≤ a | sin θ| | cos θ| |V | + 1 |V | 1-δ , ∀ ω ∈ S 2 , ∀ V ∈ R 3 ,
where a > 0, 0 < δ < 1. As emphasized in [START_REF] Grad | Asymptotic theory of the Boltzmann equation. II[END_REF], this corresponds to intermolecular potentials with finite range and it means that B ij linearly approaches 0 near θ = 0 and θ = π/2, and is of restricted growth for both small and large |V |.

Condition ( 5) is, for instance, satisfied by hard spheres of diameter σ ij > 0:

B ij (ω, V ) = σ 2 ij
|V | sin θ cos θ and, by all cutoff power-law potentials:

B ij (ω, V ) = |V | γ ij β ij (θ), γ ij = s ij -5 s ij -1 ,
where β ij (θ) is a bounded function and linearly approaches 0 when θ tends to π/2, and s ij > 3.

The collision operators can be also written under weak forms, obtained from (3) using the changes of variables (v, v * ) → (v * , v) and (v, v * ) → (v ′ , v ′ * ) for a fixed ω ∈ S 2 . Weak forms in the cases i = j and i = j are intrinsically different, and are explained in detail in [START_REF] Desvillettes | A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions[END_REF][START_REF] Boudin | The Maxwell-Stefan diffusion limit for a kinetic model of mixtures[END_REF]. Mention that, if we choose suitable test-functions, the weak forms of (3) allow to formally write, for any i and j, and any functions f and g for which the following equations make sense:

R 3 Q ij (f, g)(v) dv = 0, (6) R 3 Q ij (f, g)(v) m i v m i v 2 /2 dv + R 3 Q ji (g, f )(v) m j v m j v 2 /2 dv = 0. (7)
2.2. H-theorem. Let us now write down the H-theorem corresponding to the collisional operators we defined in the previous subsection, and discuss the mechanical equilibrium. The main tool is the entropy production functional, whose explicit form is given by

D(f 1 , . . . , f I ) := I i=1 I j=1 R 3 Q ij (f i , f j )(v) log f i (v) m 3 i dv.
It is well defined when all f i are non-negative and satisfy some suitable assumptions of regularity and decay to infinity, which we will not discuss here since our purpose consists in finding formal results.

The following properties hold [START_REF] Boudin | The Maxwell-Stefan diffusion limit for a kinetic model of mixtures[END_REF] (see also [START_REF] Desvillettes | A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions[END_REF]).

Proposition 1. Let us assume that the cross sections ( Bij ) 1≤i,j≤I are positive almost everywhere and that all f i ≡ f i (v) ≥ 0, 1 ≤ i ≤ I, are such that both the collisional integrals Q ij and the entropy production D are well defined. Then (a) The entropy production is non-positive, i.e.

D(f 1 , . . . , f I ) = I i=1 I j=1 R 3 Q ij (f i , f j )(v) log f i (v) m 3 i dv ≤ 0.
(b) Moreover, the three following properties are equivalent. i. For any

1 ≤ i, j ≤ I and v ∈ R 3 (8) Q ij (f i , f j )(v) = 0.
ii. The entropy production vanishes, that is

D(f 1 , . . . , f I ) = I i=1 I j=1 R 3 Q ij (f i , f j )(v) log f i (v) m 3 i dv = 0.
iii. There exist T > 0 and u ∈ R 3 such that, for any i, there exists n i ≥ 0 such that

(9) f i (v) = n i m i 2 π k T 3/2 e -m i 2kT |v-u| 2 .

Statement of the problem.

In this work, we focus on the diffusion limit of the Boltzmann equations for mixtures. That limit is obtained from the framework of the classical diffusive scaling, where the scaling parameter is the mean free path. Let us choose ε > 0 as the mean free path. Hence, for any i, each distribution function f ε i must solve the following scaled Boltzmann equation, that is

(10) ε ∂ t f ε i + v • ∇ x f ε i = 1 ε I j=1 Q ij (f ε i , f ε j ), t > 0, x ∈ R 3 , v ∈ R 3 .
We look for f ε i as a formal power series in ε, replace f ε i in [START_REF] Chapman | The mathematical theory of nonuniform gases[END_REF] and identify the same order terms. We only focus on the first two orders. Thanks to Proposition 1, the order -1 of (10) allows to find the zero-th order term of the series: Maxwell functions (9). Therefore, each distribution function f ε i , 1 ≤ i ≤ I, can be seen as a perturbation of the equilibrium [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF]. Without any loss of generality, we set u = 0 (diffusion limit) and kT = 1. Thus we can write f ε i as ( 11)

f ε i (t, x, v) = M i (v) n i (t, x) + ε M i (v) 1/2 g i (t, x, v) + ..., ∀t ≥ 0, ∀x, v ∈ R 3 , where M i (v) is the normalized, centred Maxwell function M i (v) = m i 2π 3/2 e -m i 2 v 2 , ∀v ∈ R 3 .
In [START_REF] Degond | Macroscopic limits of the Boltzmann equation: a review[END_REF], we choose to put M i (v) 1/2 within the first-order term of f ε i . As a matter of fact, it allows us to work in a plain L 2 framework in the variable v for g i .

We then focus on the zero-th order in ε coming from [START_REF] Chapman | The mathematical theory of nonuniform gases[END_REF]. Taking ( 8) into account, we obtain the following equation, holding for any 1 ≤ i ≤ I,

(12) M i -1/2 I j=1 n i Q ij (M i , M 1/2 j g j ) + n j Q ij (M 1/2 i g i , M j ) = M 1/2 i (v • ∇ x n i ) .
In this work, we investigate the existence of g = (g 1 , . . . , g I ) with respect to a given (n 1 , . . . , n I ) satisfying [START_REF] Desvillettes | A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions[END_REF]. Note that the dependence of (12) on t and x is not crucial, in the sense that t, x, (n i ) and (∇ x n i ) can be seen as parameters.

The role of the first-order in ε coming from [START_REF] Chapman | The mathematical theory of nonuniform gases[END_REF] is briefly discussed at the end of this article, in Section 5.

For any function g ∈ L 2 (R 3 ) I of v, we shall write the L 2 norm of g:

g 2 L 2 = I j=1 g j 2 L 2 = I j=1 R 3 g j (t, x, v) 2 dv.
We can write the left-hand side of ( 12) in a more suitable form if we introduce the operator K, where the i-th component of Kg is given by

(13) [Kg] i (v) = I j=1 m j 2π 3/4 R 3 ×S 2 B ij (ω, v -v * ) e -1 4 m i v 2 e -1 2 m j v 2 * n i m i 2π 3/4 e 1 4 m j v ′2 * g j (v ′ * ) -e 1 4 m j v 2 * g j (v * ) + n j m j 2π 3/4 e 1 4 m i v ′2 g i (v ′ ) dω dv * ,
for any i, and the positive function

ν = ν(v), whose i-th component is (14) ν i (v) = I j=1 n j m j 2π 3/2 R 3 ×S 2 e -1 2 m j v 2 * B ij (ω, v -v * ) dω dv * .
Consequently, ( 12) can be written as a functional equation in the variable v:

(15) (K -ν Id) g = M 1/2 i (v • ∇ x n i ) 1≤i≤I .
Let us now state the main result of this work.

Theorem 1. Suppose that the cross sections (B ij ) 1≤i,j≤I are positive functions satisfying [START_REF] Bennoune | An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit[END_REF]. If we assume that (16)

I i=1 n i (t, x) does not depend on x,
then, for any t, x, there exists g(t, x, [START_REF] Golse | The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels[END_REF], where K and ν are given by ( 13)- [START_REF] Duncan | An experimental study of three component gas diffusion[END_REF].

•) ∈ L 2 (R 3 v ) I satisfying
Condition [START_REF] Golse | The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials[END_REF] means that we consider a situation where the total number density of gaseous particles is uniform in space, see Section 5 for further discussions. Let us briefly draw the sketch of the proof of this theorem. We shall first need the following proposition.

Proposition 2. The operator K, defined by [START_REF] Dogbe | Fluid dynamic limits for gas mixture. I. Formal derivations[END_REF],

is compact from L 2 (R 3 v ) I to L 2 (R 3 v ) I .
The proof of Proposition 2 is given in Section 4. Let us emphasize again that, in Proposition 2, t, x and (n i ) are considered as parameters and the compactness is only related to the variable v. Since K is compact, we can apply the Fredholm alternative to the operator K -ν Id. This is explained in the next section.

Proof of Theorem 1

This section is devoted to the proof of the main result of our article. We here assume that the compactness of the operator K is known. Let us denote L = K -ν Id, and study the null space of L, as required by the Fredholm alternative.

Step 1 -Study of ker L. Writing down the i-th component of Lg and performing the change of variable (v, v * ) → (v ′ , v ′ * ) while ω remains fixed, we obtain

[Lg] i (v) = I j=1 R 3 ×S 2 M i (v) 1/2 M j (v * ) n j M i (v ′ ) -1/2 g i (v ′ ) + n i M j (v ′ * ) -1/2 g j (v ′ * ) -n j M i (v) -1/2 g i (v) -n i M j (v * ) -1/2 g j (v * ) B ij (ω, v -v * ) dω dv * .
Thanks to the H-theorem, g ∈ ker L if and only if there exist α ∈ R I , β ∈ R 3 , γ ∈ R such that, for any i,

g i (t, x, v) = n i (t, x) M i (v) 1/2 α i + m i β • v + γ m i 2 v 2 , ∀ t > 0, x, v ∈ R 3 .
Consequently, ker L = {0}, and the Fredholm alternative allows to state that (15) has a solution if and only if

(17) M 1/2 i (v • ∇ x n i ) i=1,...,I ∈ (ker L * ) ⊥ , ∀ t > 0, x ∈ R 3 .
Step 2 -Computation of L * . Let us compute the adjoint operator L * by studying the inner product between Lg and a vector h ∈ L 2 (R 3 v ) I . We successively write, using the change of variables

(v, v * ) → (v ′ , v ′ * ) and (v, v * ) → (v * , v) I i=1 R 3 [Lg] i (v) h i (v)dv = I i,j=1 R 3 ×R 3 ×S 2 h i (v) M i (v) -1/2 n i M i (v ′ )M 1/2 j (v ′ * )g j (v ′ * ) -n i M i (v)M j (v * ) 1/2 g j (v * ) + n j M j (v ′ * )M i (v ′ ) 1/2 g i (v ′ ) -n j M j (v * )M i (v) 1/2 g i (v) B ij (ω, v -v * ) dω dv * dv = I i,j=1 R 3 ×R 3 ×S 2 g i (v) n j M i (v) -1/2 M i (v ′ )M 1/2 j (v ′ * )h j (v ′ * ) -M i (v)M j (v * ) 1/2 h j (v * ) + M j (v ′ * )M i (v ′ ) 1/2 h i (v ′ ) -M j (v * )M i (v) 1/2 h i (v) B ij (ω, v -v * ) dω dv * dv.
Consequently, we have

[L * h] i = M -1/2 i I j=1 n j Q ij (M i , M 1/2 j h j ) + Q ij (M 1/2 i h i , M j ) .
Thanks to the H-theorem, h ∈ ker L * if and only if there exist

a ∈ R I , b ∈ R 3 , c ∈ R such that, for any i, (18)
h i (v) = M i (v) 1/2 a i + m i b • v + c m i 2 v 2 , ∀ v ∈ R 3 .
Step 3 -Conclusion. Now, taking (18) into account, condition (17) can be rewritten as

I i=1 3
k=1 ∂n i ∂x k R 3   v k m i v k v j m i v k v 2 /2   M i (v) dv = 0, 1 ≤ j ≤ 3.
Using parity arguments, the first and third integrals are immediately satisfied, as well as the second ones if k = j. In the case when k = j, the condition ∇ x n i = 0, which is assumed in [START_REF] Golse | The incompressible Navier-Stokes limit of the Boltzmann equation for hard cutoff potentials[END_REF], allows to complete the proof.

Proof of Proposition 2

We still have to prove that K is compact. In this section, (n i ) 1≤i≤I are assumed to be nonnegative constants. Let g ∈ L 2 (R 3 ) I . Note that we do not need g to be the function defined in [START_REF] Degond | Macroscopic limits of the Boltzmann equation: a review[END_REF]. First, we write K as the sum of four operators K 1 ,..., K 4 . For any i, the i-th component of each K ℓ g, 1 ≤ ℓ ≤ 4, is given by

[K 1 g] i (v) = -n i I j=1 m i m j 4π 2 3/4 R 3 ×S 2 e -1 4 m i v 2 e -1 4 m j v 2 * g j (v * ) B ij (ω, v -v * ) dω dv * , [K 2 g] i (v) = n i j ∈M i m i m j 4π 2 3/4 R 3 ×S 2 e -1 4 m i v 2 e -1 2 m j v 2 * e 1 4 m j v ′2 * g j (v ′ * ) B ij (ω, v -v * ) dω dv * , [K 3 g] i (v) = j∈M i m i 2π 3/2 R 3 ×S 2 e -1 4 m i v 2 e -1 2 m i v 2 * n i e 1 4 m i v ′2 * g j (v ′ * ) + n j e 1 4 m i v ′2 g i (v ′ ) B ij (ω, v -v * ) dω dv * , [K 4 g] i (v) = j ∈M i n j m j 2π 3/2 R 3 ×S 2 e -1 4 m i v 2 e -1 2 m j v 2 * e 1 4 m i v ′2 g i (v ′ ) B ij (ω, v -v * ) dω dv * .
We denoted, for any i,

M i := {1 ≤ j ≤ I | m j = m i } ,
which is non empty since i ∈ M i . It is crucial to dissociate the cases when m i = m j or m i = m j , because the proofs are quite different. We successively prove that K ℓ , 1 ≤ ℓ ≤ 4, is compact. To this end, we shall use the following property. An operator J is compact if it satisfies • a uniform decay at infinity

(19) J g L 2 (B(0,R) c ) ≤ σ(R) g L 2 (R 3 ) , ∀R > 0,
where B(0, R) denotes the open ball of R 3 v centred at 0 and of radius R, and σ(R) goes to 0 when R goes to +∞; • an equicontinuity property, i.e., for any ε > 0, there exists ̺ > 0 such that, for all w ∈ B(0, ̺),

(20) (τ w -I)J g L 2 (R 3 ) ≤ ε g L 2 (R 3 ) ,
where τ w denotes the translation operator, i.e.

τ w J g (v) = J g (v + w), ∀v, w ∈ R 3 . 4.1. Compactness of K 1 .
Let us denote, for any i, j,

k ij 1 (v, v * ) = S 2 e -1 4 m i v 2 e -1 4 m j v 2 * B ij (ω, v -v * ) dω, ∀ v, v * ∈ R 3 .
We immediately have, for any i,

[K 1 g] i (v) = -n i I j=1 m i m j 4π 2 3/4 R 3 g j (v * )k ij 1 (v, v * ) dv * , ∀ v ∈ R 3 . 4.1.1. Properties of k ij 1 . First of all, note that k ij 1 (v, v * ) = k ji 1 (v * , v
), for any i, j and v, v * , thanks to (4). In order to prove the uniform decay to infinity and the equicontinuity for K 1 (i.e. the inequalities [START_REF] Greene | Mathematical aspects of kinetic model equations for binary gas mixtures[END_REF] and ( 20)), we need to establish some preliminary properties of k ij 1 . Lemma 1. There exists C > 0 such that, for any i, j,

R 3 k ij 1 (v, v * )dv * ≤ C e -1 4 m i v 2 (1 + |v|) , ∀ v ∈ R 3 . (21)
Proof. Thanks to (5) and using the change of variables v * → V * = v * -v, we can write

R 3 k ij 1 (v, v * )dv * ≤ C R 3 e -1 4 m i v 2 e -1 4 m j (V * +v) 2 |V * | + |V * | δ-1 dV * ≤ C e -1 4 m i v 2 |V * |≤1 |V * | + |V * | δ-1 dV * + |V * |≥1 (1 + |V * |) e -1 4 m j (V * +v) 2 dV * .
This inequality allows us to get [START_REF] Jin | A BGK-penalization asymptotic-preserving scheme for the multispecies Boltzmann equation[END_REF].

Lemma 2. For any i, j, k ij 1 belongs to L 2 (R 3 × R 3 ). Proof. The proof follows the same strategy as the previous one, using [START_REF] Bennoune | An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit[END_REF] and the same change of variables. We can write

R 3 ×R 3 k ij 1 (v, v * ) 2 dv * dv ≤ C R 3 ×R 3 e -1 2 m i v 2 e -1 2 m j v 2 * |v -v * | 2 + |v -v * | 2δ-2 dv * ≤ C R 3 e -1 2 m i v 2 1 + v 2 dv,
which is clearly finite.

4.1.2. Uniform decay. The L 2 norm of K 1 g decreases at infinity. More precisely, the following proposition holds.

Proposition 3. Let g ∈ L 2 (R 3 ) I . For any R > 0 and any i, we have

(22) [K 1 g] i L 2 (B(0,R) c ) ≤ C n i R g L 2 (R 3 ) ,
where C > 0 is a constant.

Proof. Let 1 ≤ i ≤ I and write

R 3 v 2 [K 1 g] i (v) 2 dv ≤ Cn 2 i I j=1 R 3 v 2 R 3 k ij 1 (v, v * ) g j (v * ) dv * 2 dv.
Thanks to the Cauchy-Schwarz inequality, the previous inequality becomes

R 3 v 2 [K 1 g] i (v) 2 dv ≤ Cn 2 i I j=1 R 3 v 2 v * ∈R 3 g j (v * ) 2 k ij 1 (v, v * )dv * R 3 k ij 1 (v, v * )dv * dv.
Using Lemma 1 and Fubini's theorem, we get

R 3 v 2 [K 1 g] i (v) 2 dv ≤ Cn 2 i I j=1 R 3 g j (v * ) 2 R 3 k ij 1 (v, v * )φ i (v)dv dv * ,
where

φ i (v) = v 2 (1 + |v|) e -1 4 m i v 2 is clearly bounded. Consequently, since k ij 1 (v, v * ) = k ji 1 (v * , v), we have R 3 v 2 [K 1 g] i (v) 2 dv ≤ Cn 2 i I j=1 R 3 g j (v * ) 2 R 3 k ji 1 (v * , v)dv dv * .
Eventually, using Lemma 1 again, we obtain

R 3 v 2 [K 1 g] i (v) 2 dv ≤ Cn 2 i g 2 L 2 .
Besides, we can deduce, for any R > 0,

R 3 v 2 [K 1 g] i (v) 2 dv ≥ |v|≥R v 2 [K 1 g] i (v) 2 dv ≥ R 2 |v|≥R [K 1 g] i (v) 2 dv.
It is then easy to recover [START_REF] Krishna | The Maxwell-Stefan approach to mass transfer[END_REF].

4.1.3. Equicontinuity. The following property of equicontinuity of K 1 holds.

Proposition 4. For any w ∈ R 3 , set

̺ 1 (w) = C max i,j n i R 3 ×R 3 k ij 1 (v + w, v * ) -k ij 1 (v, v * ) 2 dv * dv 1/2
, where C is a suitable nonnegative constant. Then, for any i, we have

(23) [(τ w -Id)K 1 g] i L 2 (R 3 v ) ≤ ̺ 1 (w) g L 2 (R 3 v ) , ∀ w ∈ R 3
, and ̺ 1 (w) tends to 0 when w tends to 0.

Proof. First, thanks to Lemma 2, it is clear that ̺ 1 is a continuous function of w, and goes to 0 when w goes to zero. Let us now focus on [START_REF] Lions | From the Boltzmann equations to the equations of incompressible fluid mechanics. I[END_REF]. For any i, using the Cauchy-Schwarz inequality, we have

[(τ w -Id)K 1 g] i 2 L 2 (R 3 ) ≤ Cn 2 i I j=1 g j 2 L 2 R 3 ×R 3 k ij 1 (v + w, v * ) -k ij 1 (v, v * ) 2 dv * dv.
Estimate ( 23) is an immediate consequence of the previous inequality.

4.2.

Compactness of K 2 . As in section 4.1, we first write K 2 in a more convenient form. Indeed, thanks to (1), we have

- 1 4 m i v 2 - 1 2 m j v 2 * + 1 4 m j v ′2 * = - 1 4 m j v 2 * - 1 4 m i v ′2 . Hence, [K 2 ] i becomes (24) [K 2 g] i (v) = j ∈M i n i m i m j 4π 2 3/4 R 3 ×S 2 e -1 4 m j v 2 * e -1 4 m i v ′2 g j (v ′ * ) B ij (ω, v -v * ) dω dv * .
The next step consists in writing [START_REF] Maxwell | On the dynamical theory of gases[END_REF] in a form that allows to apply the same strategy used for K 1 .

In order to get this particular form, we need the following lemma.

Lemma 3. There exists b > 0 such that, for any i, j satisfying

m i = m j , (25)
m i v ′2 + m j v 2 * ≥ b m i v 2 + m j v ′2 *
for any v, v * ∈ R 3 and v ′ , v ′ * given by (2).

Remark.

The assumption on the masses is here crucial, as we shall see in the proof. Indeed, [START_REF] Morse | Kinetic model equations for a gas mixture[END_REF] somehow gives a property of norm equivalence in R 3 × R 3 , linking (v, v ′ * ) and (v ′ , v * ). Such a property does not hold when we deal with molecules of the same mass.

Proof. Choose j ∈ M i . Equation ( 2) can be rewritten as

v ′ = I 3 -2 m j m i + m j ω ω T v + 2 m j m i + m j ω ω T v * , (26) v ′ * = I 3 -2 m i m i + m j ω ω T v * + 2 m i m i + m j ω ω T v, ( 27
)
where I 3 is the identity matrix of R 3 . Then, from [START_REF] Rossani | A note on the kinetic theory of chemically reacting gases[END_REF], we get

I 3 -2 m i m i + m j ω ω T v * = v ′ * - 2
m i m i + m j ω ω T v.
Let us now set

A = I 3 -2 m i m i + m j ω ω T .
This matrix A is invertible, since det A = (m j -m i ) / (m i + m j ) and j ∈ M i . Consequently, we can write

(28) v * = I 3 -A -1 v + A -1 v ′ * ,
where we used the equality -2

m i m i + m j A -1 ω ω T = I 3 -A -1 .
Then we put ( 28) in ( 26) to obtain

v ′ = m i + m j m i I 3 - m j m i A -1 v - m j m i I 3 -A -1 v ′ * .
Consider now the following block matrix

A =   m i +m j m i I 3 - m j m i A -1 - m j m i I 3 -A -1 m j m i I 3 -A -1 A -1   ,
which is invertible: det A = -1 and A -1 = A. The following vector equality holds:

√ m i v ′ √ m j v * = A √ m i v √ m j v ′ * .
In fact, ( 25) is obtained by finding a lower bound of

A √ m i v √ m j v ′ * T 2 √ m i v √ m j v ′ * T 2 , which is A -1 -2 2 = A -2 2 .
Since A is clearly a continuous function of ω, we may conclude that the matrix norm A 2 is a positive continuous function of ω on S 2 , which is compact. Therefore, it reaches its minimum, which of course remains positive. Choosing b = min

ω∈S 2 A -2
2 > 0 leads to the required estimate [START_REF] Morse | Kinetic model equations for a gas mixture[END_REF].

Using [START_REF] Bennoune | An asymptotic preserving scheme for the Kac model of the Boltzmann equation in the diffusion limit[END_REF] and Lemma 3, we obtain the upper bound

[K 2 g] i (v) ≤ C n i j ∈M i e -1 4 m j v 2 R 3 e -1 4 m i v ′2 * g j (v ′ * ) |v -v * | + |v -v * | δ-1 dv * .
Let us then perform the change of variable v * → v ′ * , whose Jacobian is 1/det A. Since

v -v * = A -1 v -v ′ * and A -1 2 ≤ A -1 (v -v ′ * ) |(v -v ′ * )| ≤ A -1 2 , we can write |v -v * | + |v -v * | δ-1 ≤ A -1 2 v -v ′ * + A 1-δ 2 v -v ′ * δ-1 .
Eventually, we obtain

[K 2 g] i (v) ≤ C n i
The upper bound in the previous equality has exactly a kernel form, which allows us to conclude on the compactness of K 2 in the same way as in section 4.1.

4.3.

Compactness of K 3 . This operator describes interactions between molecules with the same mass. Note that it does not mean that species A i and A j are the same, since B ii , B ij and B jj can be different. We only have to adapt ideas from [START_REF] Grad | Asymptotic theory of the Boltzmann equation. II[END_REF] and [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF] used in the monospecies case.

4.3.1. Obtaining a kernel form. Note that if m i = m j , (2) becomes (29)
v ′ = v -(ω • (v -v * )) ω, v ′ * = v * + (ω • (v -v * )) ω.
Symmetry properties allow us to write [K 3 g] i in terms of v, v * and v ′ , and not v ′ * anymore. More precisely, we have the following lemma. Lemma 4. For any i, there exist nonnegative functions Bij 1≤j≤I satisfying (5), such that

(30) [K 3 g] i (v) = j∈M i R 3 ×S 2 e -1 4 m i v 2 -1 2 m j v 2 * + 1 4 m i v ′2 g j (v ′ ) Bij (ω, v -v * ) dω dv * , ∀v ∈ R 3 .
Proof. The key idea of the proof lies in [START_REF] Stefan | Ueber das Gleichgewicht und die Bewegung insbesondere die Diffusion von Gasgemengen[END_REF]. Indeed, if we consider the relative velocity V = v -v * , we can choose one unit vector ω ⊥ ∈ Span(V, ω) orthogonal to ω (the choice of either ω ⊥ or -ω ⊥ is relevant, but must be performed in a continuous way with respect to ω, and not randomly). Consequently, we can write

V = ω(ω • V ) + ω ⊥ (ω ⊥ • V ), from which we immediately get (31) v -(ω • V )ω = v * + (ω ⊥ • V )ω ⊥ , v * + (ω • V )ω = v -(ω ⊥ • V )ω ⊥ .
We can see that, if we look for the post-collision relative velocity for the same pre-collisional V , but with respect to ω ⊥ , we just exchange the velocities v ′ and v ′ * : for instance, the new v ′ * , depending on ω ⊥ , will be the old v ′ , depending on ω. Hence, it is clear that ω

→ ω ⊥ implies v ′ → v ′ * and v ′ * → v ′ . Consequently, if we replace ω by ω ⊥ in the integral R 3 ×S 2 e -1 2 m i v 2 * e 1 4 m i (v * +(ω•V )ω) 2 g j (v * + (ω • V )ω) B ij (ω, V ) dω dv * , it becomes R 3 ×S 2 e -1 2 m i v 2 * e 1 4 m i (v * +(ω ⊥ •V )ω ⊥ ) 2 g j (v * + (ω ⊥ • V )ω ⊥ ) B ij (ω ⊥ , V ) dω ⊥ dv * .
Changing the variable ω into ω ⊥ is obtained thanks to a rotation, so dω ⊥ = dω. Hence, using (31), the previous integral becomes

R 3 ×S 2 e -1 2 m i v 2 * e 1 4 m i v ′2 g j (v ′ ) B ij (ω ⊥ , V ) dω ⊥ dv * = R 3 ×S 2 e -1 2 m i v 2 * e 1 4 m i v ′2 g j (v ′ ) B ij (ω ⊥ , V ) dω dv * . Let us set Bij (ω, V ) = m i 2π 3/2        n i B ij (ω ⊥ , V ) if i = j, k∈M i n k B ik (ω, V ) + n i B ii (ω ⊥ , V ) if i = j.
Assumption (5) on both B ij (ω, V ) and B ij (ω ⊥ , V ) ensures that, for any i, j

(32) Bij (ω, V ) ≤ 2 a m i 2π 3/2 max k∈M i n k | sin θ| | cos θ| |V | + |V | δ-1 ,
as well as (30).

Lemma 4 allows to obtain the kernel form of K 3 . More precisely, we have Proposition 5. Denote, for any i, j,

k ij 3 (η, v) = e -1 8 m i (η-v) 2 -1 8 m i (η 2 -v 2 ) 2 (η-v) 2 |η -v| -1 ϕ ij 3 (η -v), ∀ η, v ∈ R 3 , where (34) ϕ ij 3 (p) = 2 |p| {p} ⊥ e -1 2 m i (q+z 2 ) 2 Bij (p, q) | sin(p, p + q)| -1 dq, ∀ p ∈ R 3 . (33)
Then we have

[K 3 g] i (v) = j∈M i R 3 g j (η) k ij 3 (η, v)dη, ∀ v ∈ R 3 .
Proof. We perform the change of variable v * → V * = v * -v in (30), whose Jacobian equals 1, and get

(35) [K 3 g] i (v) = j∈M i R 3 ×S 2 e -1 4 m i v 2 e -1 2 m i (V * +v) 2 e 1 4 m i v ′2 g j (v ′ ) Bij (ω, V * ) dω dV * .
Next, we consider the components of V * respectively parallel and orthogonal to ω, i.e. we write We now perform the change of variables (see Figure 1)

V * = p + q, where p = ω(ω • V * ), q = V * -ω(ω • V * ). The component q which is orthogonal to ω belongs to the plane Π = {ω} ⊥ = {p} ⊥ . Π ∈ Π x 2 x 3 x 1 ω V = v -v * θ ω ⊥ V * = v * -v p q θ
(36) (V * , ω) → (p, q), R 3 × S 2 → R 3 × Π.
Let us compute its Jacobian. For ω fixed, the replacement of V * by p and q has unit Jacobian. Note that V * and ω are independent of each other, which is not the case with p and q. Therefore, the integration order is crucial. Hence we first integrate with respect to q since Π = {p} ⊥ . Then we combine the one-dimensional integration in the direction ω with the integral of ω over the unit sphere to give a threedimensional integration over the three rectangular components of |p|ω. We have to introduce the factor 2, since p = ±|p|ω. The Jacobian from p to (|p|, ω) (Cartesian to spherical coordinates) is p 2 sin(p, p + q). Consequently, we can write dV * dω = 2 p 2 sin(p, p + q) dp dq.

Eventually, it is clear that

v ′ = v -ω(ω • (v -v * )) = v + ω(ω • V * ) = v + p.
Hence, (35) becomes

(37) [K 3 g] i (v) = 2 j∈M i R 3 Π e -1 4 m i v 2 -1 2 m i (p+q+v) 2 + 1 4 m i (v+p) 2 g j (v + p) Bij (p, q) |p| -2 | sin(p, p + q)| -1 dq dp.
Since p • q = 0, we can deduce

- 1 4 v 2 + 1 4 (v + p) 2 - 1 2 (p + q + v) 2 = - 1 8 p 2 - 1 2 q + 1 2 (2v + p) 2 .
Consequently, we obtain

[K 3 g] i (v) = 2 j∈M i R 3 Π e -1 8 m i p 2 -1 2 m i[ q+ 1 2 (2v+p)] 2 g j (v + p) Bij (p, q) |p| -2 | sin(p, p + q)| -1 dq dp.
Furthermore, let us set

z = 1 2 (2v + p),
and denote by z 1 the component of z which is parallel to ω and z 2 = z -z 1 ∈ Π. Then we can write

q + 1 2 (2v + p) 2 = (q + z 1 + z 2 ) 2 = z 1 2 + (q + z 2 ) 2 ,
and

[K 3 g] i becomes [K 3 g] i = 2 j∈M i R 3 e -1 8 m i p 2 -1 2 m i z 1 2 g j (v + p) |p| -2 Π e -1 2 m i (q+z 2 ) 2 Bij (p, q) | sin(p, p + q)| -1 dq dp.
Finally, we perform the change of variable p → η = p + v, and write

z 1 2 = z • η -v |η -v| 2 = 1 2 (η + v) • (η -v) |η -v| 2 = 1 4 (η 2 -v 2 ) 2 |η -v| 2 .
This completes the proof.

Properties of k ij

3 . Let us first prove the following lemma, and then investigate some properties of k ij 3 .

Lemma 5. The function ϕ ij 3 : R 3 → R, defined by (34) for any i, j, belongs to L ∞ (R 3 ). Proof. Let 1 ≤ i, j ≤ I, and choose p ∈ R 3 and q ∈ {p} ⊥ . From (32), we obtain

0 ≤ Bij (p, q) | sin(p, p + q)| ≤ 2 a m i 2π 3/2 max k∈M i n k | cos(p, p + q)| |p + q| + |p + q| δ-1 .
Since | tan(p, p + q)| = |q|/|p|, we can write

0 ≤ Bij (p, q) | sin(p, p + q)| ≤ C i max k∈M i n k 1 + q 2 p 2 -1 2 (p 2 + q 2 ) 1 2 + (p 2 + q 2 ) δ-1 2
, where C i = 2 a (m i /2π) 3/2 > 0. In what follows, C i will denote any nonnegative constant only depending on m i . This implies

0 ≤ Bij (p, q) |p| | sin(p, p + q)| ≤ C i max k∈M i n k 1 + p 2 + q 2 δ 2 -1 ≤ C i max k∈M i n k 1 + |q| δ-2 ,
using the fact that δ < 1. Now, we split the range of integration in (34) into |q| ≤ 1 and |q| ≥ 1, and finally get

0 ≤ ϕ ij 3 (p) ≤ C i max k∈M i n k |q|≤1 (1 + |q| δ-2 )dq + |q|≥1 e -1 2 (q+z 2 ) 2 dq ≤ C i max k∈M i n k .
This ends the proof of Lemma 5.

Let us now investigate two properties of k ij 3 , which are related to Lemmas 1 and 2 for k ij 1 . Lemma 6. There exists C > 0 such that, for any i, j,

R 3 k ij 3 (η, v) dη ≤ C |v| , ∀ v ∈ R 3 \ {0}, R 3 k ij 3 (η, v) dη ≤ C, ∀ v ∈ R 3 .
Proof. Let 1 ≤ i, j ≤ I. We integrate (33) with respect to η and perform the change of variable η → p = η -v. Using Lemma 5, we get

R 3 k ij 3 (η, v) dη ≤ C R 3 e -1 8 m i p 2 -1 8 m i (p 2 +2p•v) 2 p 2 1 |p| dp, ∀ v ∈ R 3 .
We split the right-hand integral into I 1 + I 2 , where I 1 refers to |p| ≥ |v| and I 2 to |p| ≤ |v|. On the one hand, we have

I 1 ≤ |p|≥|v| e -1 8 m i p 2 1 |p| dp ≤ C e -1 8 m i v 2 .
On the other hand, for the second integral, with spherical coordinates, we have

I 2 = |v| 0 2π 0 π 0 e -1 8 m i r 2 -1 8 m i (r+2 |v| cos ψ 1 ) 2 r sin ψ 1 dψ 1 dψ 2 dr = 2π |v| 0 re -1 8 m i r 2 π 0 e -1 8 m i (r+2 |v| cos ψ 1 ) 2 sin ψ 1 dψ 1 dr,
where ψ 1 corresponds to the angle between p and v. We then have to consider two situations for

I 2 : |v| ≥ 1 and |v| ≤ 1. Simple changes of variables give, if v = 0, π 0 e -1 8 m i (r+2 |v| cos ψ 1 ) 2 sin ψ 1 dψ 1 = 1 2|v| r+2|v| r-2|v| e -1 8 m i s 2 ds. If |v| ≥ 1, I 2 ≤ C |v| +∞ 0 r e -1 8 m i r 2 dr +∞ -∞ e -1 8 m i s 2 ds = C i |v| ≤ C i .
On the contrary, if |v| ≤ 1,

I 2 ≤ C i |v| r+2|v| r-2|v| ds = C i , if v = 0.
Consequently, we get the required estimates.

Using the same strategy as above, the following lemma can also be proved.

Lemma 7. For any i, j, k ij 3 belongs to L 2 loc R 3 v ; L 2 R 3 η . 4.3.3. Uniform decay. Let us now prove the uniform decay property at infinity. Proposition 6. Let g ∈ L 2 (R 3 ) I . For any R > 0 and any i, we have

[K 3 g] i L 2 (B(0,R) c ) ≤ C √ R g L 2 (R 3 ) ,
where C > 0 is a constant.

Proof. Using the Cauchy-Schwarz inequality and Lemma 6, we can write

[K 3 g] i 2 L 2 (B(0,R) c ) ≤ C |v|≥R j∈M i R 3 g j (η) 2 k ij 3 (η, v) dη R 3 k ij 3 (η, v) dη dv ≤ C R j∈M i R 3 R 3 k ij 3 (v, η) dv g j (η) 2 dη = C R g 2 L 2 (R 3 ) ,
where we also used the fact that k ij 3 (η, v) = k ij 3 (v, η). This ends the proof. 4.3.4. Equicontinuity. This property is described in the following proposition.

Proposition 7. For all ε > 0, there exists α > 0 (not depending on g or i) such that

(38) [(τ w -Id)K 3 g] i L 2 (R 3 ) ≤ ε g L 2 (R 3 ) , ∀ w ∈ B(0, α).
Proof. Let R > 0. We obviously have, for any w ∈ B(0, R),

[(τ w -Id) K 3 g] i 2 L 2 (R 3 ) ≤ B(0,2R) ([K 3 g] i (v + w) -[K 3 g] i (v)) 2 dv + B(0,R) c [K 3 g] i (v) 2 dv. (39)
Proposition 6 ensures that the second integral is upper-bounded by g 2 L 2 /R. The first one can also be upper-estimated by

j∈M i g j 2 L 2 B(0,2R) R 3 k ij 3 (η, v + w) -k ij 3 (η, v) 2 dη dv.
We can now choose R ≥ 2/ε 2 > 0. Thanks to Lemma 7, there exists α > 0 such that

|w| < α ⇒ B(0,2R) R 3 k ij 3 (η, v + w) -k ij 3 (η, v) 2 dη dv ≤ ε 2 2 .
It is then immediate to recover (38) from (39). e -1 2 m j (q+z 2 ) 2 B ij m i + m j 2m j (η -v) , q sin m i + m j 2m j (η -v) , m i + m j 2m j (η -v) + q -1 dq dη.

To write [K 4 g] i into the convenient kernel form, we introduce the function

ϕ ij 4 (p) = m j 2π
3/2 m i + m j 2m j |p| q∈Π e -m j 1 2 (q+z 2 ) 2 B ij (p, q) |sin (p, p + q)| -1 dq.

It is easy to prove, in the same way as in Lemma 5, that there exists C > 0 such that ϕ ij 4 L ∞ (R 3 ) ≤ C for any i, j. The i-th component of K 4 g can be written in the kernel form

[K 4 g] i (v) = j ∈M i n j R 3 k ij 4 (η, v)g i (η) dη,
where k ij 4 (η, v) is given by

k ij 4 (η, v) = 1 4 e -1 8
m 2 i m j (η-v) 2 -1 8 m j (η 2 -v 2 ) 2 (η-v) 2 |η -v| -1 ϕ ij 4 m i + m j 2m j (η -v) .
The form of each k ij 4 is exactly the same as in (33). Consequently, k ij 4 inherits the same properties as k ij 3 (see Lemmas 5 and 6), which allows to obtain ( 19)-( 20) as in Propositions 6 and 7, and the compactness of K 4 .

Diffusion limit

To conclude this work, let us add a few words about the diffusion limit itself. Theorem 1 ensures the existence of g = (g i ) 1≤i≤I , in the expansion [START_REF] Degond | Macroscopic limits of the Boltzmann equation: a review[END_REF], with respect to (n i ). Let us then write the first order in ε coming from [START_REF] Chapman | The mathematical theory of nonuniform gases[END_REF], that is

M i ∂ t n i + M 1/2 i v • ∇ x g i = I j=1 Q ij (M 1/2 i g i , M
1/2 j g j ).

Integrating the previous equality with respect to v, and using ( 6)-( 7), we can write the usual continuity equation (40)

∂ t n i + ∇ x • N i = 0,
where the flux N i of species A i is given by (41)

N i (t, x) = R 3 v g i (t, x, v) M i (v) 1/2 dv, ∀x, v 1 ≤ i ≤ I.
Since g i only depends on (n j ), the flux N i shares the same property thanks to (41). Consequently, the continuity equations give a system, which only involves (n j ). The existence of (n j ) was not investigated here. The reader can refer to [START_REF] Bisi | Incompressible Navier-Stokes equations from kinetic models for a mixture of rarefied gases[END_REF][START_REF] Boudin | The Maxwell-Stefan diffusion limit for a kinetic model of mixtures[END_REF] for more details about the related macroscopic equations. The expression of the flux (41) allows to clarify Assumption ( 16) of Theorem 1: the equimolar diffusion situation, which means N i = 0, is common in closed experimental settings [START_REF] Duncan | An experimental study of three component gas diffusion[END_REF][START_REF] Krishna | The Maxwell-Stefan approach to mass transfer[END_REF]. This property clearly implies that n i equals its initial value (sum (40) over i). Hence, if we suppose that the initial value does not depend on x, assumption ( 16) is fulfilled.

Figure 1 .

 1 Figure 1. Geometrical situation for the change of variables (36)

4. 4 . 2 j ∈M i n j m j 2π 3 / 2 2 j ∈M i n j m j 2π 3 / 2 R 3 e- m j m 2 i 2 (m i +m j ) 2 p 2 -1 2 m j z 1 2 g i v + 2 m j m i + m j p |p| - 2 Π e - 1 2 3 ,

 4232232322213 Compactness of K 4 . The proof of the compactness of K 4 is very similar to the final part of the proof for K 3 . The main difficulty is to obtain a kernel form of K 4 . Once it is done, (19)-(20) can easily be proven as in section 4.3. Using the same change of variables as to obtain (37), we can write[K 4 g] i (v) = If we denote z = v + m j m i + m j p,and decompose it into the component z 1 parallel to ω and the component z 2 orthogonal to ω (z 2 ∈ Π), we obtain the new form of [K 4 g] i :[K 4 g] i (v) = m j (q+z 2 ) 2 B ij (p, q) | sin(p, p + q)| -1 dq dp.Next, we perform the change of variablesp → η = v + 2 m j m i + m jp, whose Jacobian equals 2 m j m i + m j v) 2 g i (η) |η -v| -2 Π

Acknowledgment. The authors would like to thank Laurent Desvillettes for stimulating and helpful discussions.


Integration and religiosity among the Turkish second generation in Europe: A comparative analysis across four capital cities

Keywords: Integration, Religion, Islam, Second Generation, Institutions, comparative approach integration, religion, Islam, second generation, institutions, comparative approach

Integration and religiosity among the Turkish second generation in Europe: A comparative analysis across four capital cities

Fenella Fleischmann, Karen

Introduction

Large-scale immigration of Muslims to North-Western Europe's highly secularized societies has raised the question of how the local-born children of Muslim immigrants relate to, and practice their religion. Being socialized with Islam within their families and local co-ethnic communities (Güngör et al., forthcoming), the second generation grows up in societies where the majority is historically Christian, highly secularized and increasingly anti-Islamic. This study examines religiosity among the Turkish second generation and asks how it relates to their religious socialization in immigrant families and communities and to their structural integration into European societies. Moreover, we investigate contextual variation in the relationship between structural integration and religiosity across four European capital cities -Amsterdam, Berlin, Brussels and Stockholm -which represent different national traditions of church-state relations and distinct patterns of institutional incorporation of Islam.

Due to limited data availability, we currently lack comparative empirical evidence on second-generation Muslims' religiosity in Europe. Previous studies were often restricted to the foreign-born first generation; they used narrow measures of religious affiliation and worship; and/or they were single-country studies [START_REF] Diehl | Religiosität türkischer Migranten im Generationenverlauf: ein Befund und einige Erklärungsversuche[END_REF], Phalet et al., 2008[START_REF] Van Tubergen | Religious affiliation and participation among immigrants in a secular society: A study of immigrants in the Netherlands[END_REF]. Comparative studies of religiosity among Muslim minorities in Europe remain an exception (but see [START_REF] Connor | Contexts of immigrant receptivity and immigrant religious outcomes: the case of Muslims in Western Europe[END_REF]. Recent cross-national surveys from the TIESproject (The Integration of the European Second generation, see e.g. [START_REF] Crul | Comparative integration context theory: participation and belonging in new diverse European cities[END_REF] allow comparative investigations of multiple dimensions of religiosity among the Turkish second generation. Due to their different national histories of church-state relations, the four countries in our study -Belgium, Germany, the Netherlands and Sweden -are characterized by different opportunity structures for the practice of Islam by Muslim minorities (Bader, 2007, Fetzer and[START_REF] Fetzer | Muslims and the state in Britain[END_REF]. Against this comparative background, we assess theoretical expectations of religious decline and vitality in the second generation.

In the following, we first present our theoretical framework and derive hypotheses about individual-level determinants of religious decline and religious vitality in the second generation. Next, we derive hypotheses about contextual differences in the relation between second-generation integration and religiosity from an institutionalist approach to the incorporation of Islam in European societies. Before putting our hypotheses to a test, we briefly describe the groups and contexts under study and the data and methods applied.

The hypothesis of religious decline: structural integration and secularization

The secularization paradigm posits the erosion of religion through modernization processes, as evident from the progressive separation of church and state and from declining religious attendance [START_REF] Dobbelaere | Secularization: A Multi-Dimensional Concept[END_REF]. Religious decline has been firmly established among European majority populations (Gorski and Altinordu, 2008) and secularization has become an ideology -in stark contrast with the US. As Casanova (2003, p.19) puts it: 'Americans think that they are supposed to be religious, while Europeans think that they are supposed to be irreligious'. Contrary to the US, in European societies religion is commonly portrayed as hindering immigrant integration [START_REF] Foner | Immigrant religion in the U.S. and Western Europe: Bridge or barrier to inclusion?[END_REF].

Consequently, the classic assimilation hypothesis, which predicts that immigrants' life chances and life styles will gradually converge with those of natives over time and generations [START_REF] Gordon | Assimilation in American Life: The role of race, religion and national origins[END_REF][START_REF] Gordon | Assimilation in American Life: The role of race, religion and national origins[END_REF]), when applied to the religious domain, amounts to a secularization hypothesis in European societies. Accordingly, our first hypothesis expects that second-generation Muslims will be less religious the more they are integrated into the mainstream (Hypothesis 1) in terms of social mixing and socio-economic inclusion relative to a non-immigrant reference population [START_REF] Esser | Integration und ethnische Schichtung[END_REF]. Specifically, we predict that higher education and labour market participation as well as intermarriage with a non-Turkish partner will go together with lower levels of religiosity. Similar predictions can also be derived from the sociology of religion. Here, the expected inverse relationship of religiosity with education and labour market attainment is explained through increased exposure to a secular worldview [START_REF] Berger | The sacred canopy. Elements of a sociological theory of religion[END_REF] or through decreased economic uncertainty (Norris and Inglehart, 2004).

Research among Turkish and Moroccan minorities in the Netherlands [START_REF] Van Tubergen | Religious affiliation and participation among immigrants in a secular society: A study of immigrants in the Netherlands[END_REF], Phalet et al., 2008) and in Belgium [START_REF] Lesthaeghe | RON 2000 'Communities and generations. Turkish and Moroccan populations in Belgium[END_REF]Neels, 2000, Smits et al., 2010) largely confirms secularization in the sense of an inverse relationship between socio- [START_REF] Maliepaard | Generational differences in ethnic and religious attachment and their interrelation. A study among Muslim minorities in the Netherlands[END_REF]. Similarly in Germany, [START_REF] Diehl | Religiosität türkischer Migranten im Generationenverlauf: ein Befund und einige Erklärungsversuche[END_REF] found remarkable stability instead of religious decline between migration generations. To our knowledge, there is no comparable evidence from large-scale surveys on religion and integration in Sweden.

The hypothesis of religious vitality: religious socialization and reactive religiosity

An alternative hypothesis expects religious vitality, defined as the continuity of a minority religion through the religious socialization of the next generation within the family and the co-ethnic community, in parallel with a notion of ethno-linguistic vitality in socio-linguistic studies of minority languages [START_REF] Harwood | The genesis of vitality theory: historical patterns and dissoursal dimensions[END_REF]. Turkish immigrants come from a majority Muslim society where levels of religiosity are much higher than in Germany, Belgium, the Netherlands or Sweden (Norris and Inglehart, 2004). This raises the question to what extent Turkish immigrants pass on Islamic traditions and practices to the next generation who are growing up in historically Christian and highly secularized European societies.

Religious studies document the importance of a religious upbringing so that the children of more religious parents will themselves be more religious as adults [START_REF] Kelley | National context, parental socialization, and religious belief: Results from 15 nations[END_REF]. Accordingly, we predict that the Turkish second generation will be more religious when their parents visited the mosque more often during their youth and/or when they attended Koran lessons as children (Hypothesis 2).

A complementary hypothesis of religious reactivity expects increased religiosity in the second generation in response to societal exclusion and experiences of discrimination. This hypothesis extends a notion of reactive ethnicity among most deprived secondgeneration groups in the US [START_REF] Portes | Legacies. The story of the immigrant second generation[END_REF] to the religious domain. The reactive religiosity hypothesis mirrors the inverse relation between socio-economic attainment and religiosity that is also predicted by the secularization hypothesis. In addition, however,

The institutionalization of Islam: contextual differences

European nation-states differ substantially in their accommodation of Islam, because the institutionalization of this minority religion builds on pre-existing national patterns of churchstate relations [START_REF] Bader | The Governance of Islam in Europe: The Perils of Modelling[END_REF][START_REF] Koenig | Europeanising the governance of religious diversity: An institutionalist account of Muslim struggles for public recognition[END_REF]. Distinct institutional pathways thus provide different opportunity structures for the institutionalization of Islam [START_REF] Statham | Resilient or adaptable Islam? Multiculturalism, religion and migrants' claims-making for group demands in Britain, the Netherlands and France[END_REF].

Among our four comparison countries, the formal accommodation of Islam is most advanced in the Netherlands. Due to the history of 'pillarization' (Lijphart, 1968) and despite increasing secularization among the majority population, pluralist church-state relations created opportunities for Muslims to develop their own institutions [START_REF] Rath | Nederland en zijn islam. Een ontzuilende samenleving reageert op het ontstaan van een geloofsgemeenschap Amsterdam: Het Spinhuis RENSTRAND, NILS & LUNDSTRÖM, SIXTEN 2008 'Personliga ervarenheter av integration i Stockholms län. En undersökning på uppdrag av Centrum för forskning om internationell migration och etniska relationer (CEIFO)[END_REF]. Thus, Dutch Muslims have established numerous local mosque associations, as well as state-funded Islamic broadcasting networks and Islamic schools [START_REF] Doomernik | The institutionalization of Turkish Islam in Germany and the Netherlands: a comparison[END_REF]. From an institutional perspective, Dutch Muslims were granted formal equality with Christian and other religious groups from the early 1980s onwards; and they made the most of the opportunities offered by the Dutch system.

In Germany, the institutional position of Islam is the most underprivileged by comparison. Due to their formal status as corporations of public law, Christian churches in Germany profit from taxes collected by the state. In the absence of a centralized organizational structure of German Muslims similar to that of the Christian churches, German authorities have refrained from granting Islam the same legal status [START_REF] Fetzer | Muslims and the state in Britain[END_REF].

Consequently, Islamic organisations in Germany, lacking legal recognition and financial support, remain disadvantaged relative to established churches.

In Belgium, the status of Islam is formally equal to that of the historically dominant Catholic Church since 1974. Yet, in order to receive the state funding for religious services to which they are legally entitled, Muslim communities were required to set up a nationally representative Islamic council as a partner for the Belgian state (Foblets and [START_REF] Manço | Voix et voies musulmanes de Belgique[END_REF][START_REF] Hu | Cutoff criteria for fit indexes in covariance structure analysis: conventional criteria versus new alternatives[END_REF]. Because of the delayed implementation of the recognition of Islam, Islamic organizational structures are less fully developed in Belgium than they are in the Netherlands.

Finally, in Sweden Islam enjoys the same legal status as other religions and Islamic organisations, which are entitled to state funding proportionally to the size of their membership. However, Sweden has historically known a state church, which still counts more than 80 per cent of the Swedish population as its members [START_REF] Alwall | Religious liberty in Sweden: an overview[END_REF]. Although the privileged position of the Swedish Lutheran Church has been dismantled in the second half of the twentieth century, the legacy of the state church system implies that Islam occupies a relatively marginal position. In spite of recent reforms, the Swedish religious landscape continues to be marked by a quasi-monopoly of the Swedish Lutheran Church.

From an institutionalist perspective, the position of Islam as a minority religion is most equal in the Netherlands and most unequal in Germany, with Belgium and Sweden granting some degree of recognition. Regarding our main research question about the relation between integration and religiosity, we reason that institutional opportunity structures can make the difference between complementary expectations of second-generation religious vitality or decline. Thus, we expect that upward integration will go together with decreased religiosity mainly in national contexts which do not recognize Islam nor facilitate its practice.

When Islamic religion is excluded by mainstream institutions, the continued religiosity of Muslims stands in the way of successful integration. Conversely, in national contexts where the state recognizes and accommodates Islam, there will be more room for religiosity also among more integrated Muslims. We therefore predict an inverse relationship between integration and religiosity in contexts where the institutional accommodation of Islam is most restricted; and zero or even positive relationships where Islam has been more fully institutionalized (Hypothesis 4).

Institutionalist approaches in migration studies have been criticized for their formalistic and overly static conception of social processes, which would by-pass alternate resources and forms of agency 'from below' as well as ongoing socio-political struggles about institutional inequalities [START_REF] Bousetta | Institutional theories of immigrant ethnic mobilisation: relevance and limitations[END_REF]. Moreover, institutional arrangements may be far removed from public opinion in European societies. For instance, while the institutional accommodation of Islam is most advanced in the Netherlands, the position of Islam and Muslims is highly contested in Dutch media and politics and, despite constitutionally enshrined religious pluralism, Muslims are increasingly and overtly rejected by the Dutch majority population (Sniderman and Hagendoorn, 2007). Moreover, the capital cities each add their own local flavour to the distinct national approaches of church-state relations. Yet, they are strategic sites for comparative research as the localities where national and local policies intersect and where most minorities are concentrated. Due to the path-dependency of institutional arrangements, distinct national histories of church-state relations impose differential constraints on the development of religious organizations and institutions by immigrant communities. From a comparative perspective, therefore, we expect a long-term impact of institutional arrangements on patterns of integration and religiosity, over and above variation resulting from short-term changes in public opinion and policy. Moreover, we argue that our comparison of capital cities is well-suited to bring out such national differences.

The Turkish second generation in North-Western Europe

Turkish immigrants first arrived in the Netherlands, Germany and Belgium as so-called 'guest workers' and later through family-based migration. [START_REF] Lesthaeghe | Islamic communities in Belgium: religious orientations and secularization[END_REF], Vermeulen and Penninx, 2000[START_REF] Münz | Zuwanderung nach Deutschland. Strukturen, Wirkungen, Perspektiven[END_REF]. In the Swedish context labour migration has been less important as the country received relatively more refugees [START_REF] Corman | Sveriges invandring och utvandring[END_REF]. Therefore, the Turkish-Swedish population is comparatively small and includes substantial shares of Kurdish and Assyrian refugees. Assyrians, who are mainly Orthodox Christians (Sander, 1990), are excluded from our analysis for reasons of comparability. Turks are the largest non-European minority group in the Netherlands; including naturalized Turks and the second generation, they numbered 0.38 million, which is 2.3 per cent of the total population in 2009 (Statline, 2010). Similarly, in Germany, Turkish immigrants and their offspring are the largest immigrant minority with an estimated 2.5 million members, which constitutes 3.1 per cent of the total German population in 2009 [START_REF] Statistisches Bundesamt | Bevölkerung mit Migrationshintergrund -Ergebnisse des Mikrozensus 2009[END_REF]. In Belgium, Turks are the third largest ethnic minority group, which numbered approximately 150,000 or 1.5 per cent of the total Belgian population in the 2001 Census [START_REF] Phalet | Old and new inequalities in educational attainment. Ethnic minorities in the Belgian Census 1991-2001[END_REF]. Finally, in Sweden, the size of the Turkish minority amounted to approximately 60,000 persons in 2000, or about 0.7 per cent of the Swedish population, including the Swedish-born second generation (Westin, 2003).

Data and Method

The TIES-surveys Our analysis draws on new comparative survey data among the Turkish second generation in Amsterdam (NIDI-IMES, 2007, TIES06-07-Netherlands), Berlin (IMIS, 2008, TIES08-Germany), Brussels (CESO-CSCP, 2008, TIES07-08-Belgium) and Stockholm (CEIFO, 2008, TIES08-Sweden). The Turkish second generation is narrowly defined as the local-born children of immigrants from Turkey. Only young adults between 18 and 35 years old were sampled. Different sampling frames and response rates challenge the representativeness and comparability of minority samples in comparative surveys. In Amsterdam and Stockholm, random samples of native-born children of Turkish-born parents could be drawn from population registers. In Brussels, screening had to be applied to addresses in street segments which were sampled proportionally to the numbers of Turkish inhabitants. In Berlin, a random sample was drawn from a list of names generated by onomastic analysis identifying persons of Turkish origin within the population of 18-35 year old German-born residents of Berlin. For further information regarding sampling and fieldwork, readers are referred to the technical reports (Renstrand andLundström, 2008 for Sweden, Groenewold, 2008 for the Netherlands, [START_REF] Swyngedouw | Technical Report TIES 2007-2008 Belgium -Extended. Trajectories and Experiences of Turkish, Moroccan and Native Belgians in Antwerp and Brussels[END_REF] for Belgium; no technical report is available so far for Germany). Reported response rates were 29 per cent in Stockholm, 30 per cent in Amsterdam, and 31 per cent in Brussels and Berlin. Low response rates are due in part to the quality of available address lists in cities and neighbourhoods with high mobility rates and in part to generally low cooperation rates in inner cities, in young age groups, and in ethnic minority populations (Stoop et al., 2010). These data constraints should be kept in mind when drawing conclusions from our comparative analyses.

To ensure that our comparative results are not driven by differential selectivity of Turkish immigrants across the cities, a detailed comparison was undertaken of the characteristics of the parents of TIES-participants, i.e., the Turkish first generation. 1 The comparison revealed very few differences in parental characteristics between the four cities.

Most parents matched the typical profile of Turkish guest worker migration to North-Western Europe. Because the parents of Turkish Muslims in Brussels and Stockholm had somewhat higher levels of education, we include parental education as a control variable. Furthermore, we conducted logistic regression analyses to examine differential selectivity of selfcategorized Muslims across cities (see Note 4).

Measures

Religious identification was measured by four items reflecting participants' attachment to a common Muslim identity, e.g., "Being a Muslim is an important part of my self" (see Table 1 for the other items). They rated their agreement on five-point Likert-scales ranging from one (totally disagree) to five (totally agree).

Four indicators of religious practices were used to form two composite indices, labelled dietary practices and worship. Dietary practices refer to fasting during the last Ramadan and eating halal food; worship refers to saying daily prayers and attending religious services (i.e., visiting a mosque). 2 While interrelated, it is useful to distinguish between worship and dietary practices as they form distinct clusters in K-means cluster analyses across the same Turkish Muslim populations (Phalet et al., in preparation).

An additional political dimension of religiosity, labelled political religion, is assessed by four statements expressing support for the role of Islam in politics and its representation in the public sphere, e.g., "Islam should be represented in politics and society" (see Table 1 for the remaining items).

Explanatory variables

To assess the structural integration of the Turkish second generation, measures of educational attainment and employment status were used, along with marital status and ethnicity of the partner as a measure of social mixing in primary ties.

Four categories of educational attainment were distinguished based on participants' highest level of education completed or currently attended: lower secondary or less, vocational and academic upper secondary, and tertiary qualifications. 3 Regarding employment status, we distinguish between working (employed or selfemployed), unemployed, full-time student, or otherwise inactive participants. Furthermore, we looked at the ethnicity of the partner, with a non-Turkish partner as a proxy for intermarriage outside the ethnic community of co-religionists. We distinguish between participants who had no partner, those who had a partner of Turkish origin, or a non-Turkish partner. Partners were categorized as Turkish when they, or one of their parents, were born in Turkey, thus including the second generation.

Two indicators were used to assess religious childhood socialization. A dummy variable indicates whether participants have attended Koran lessons outside school hours as a child. In addition, participants reported the frequency of parents' mosque visits during their youth on a five-point scale ranging from one (never) to five (once a week or more).

To test the reactive religiosity hypothesis, we used two questions on perceived discrimination. Participants were asked to rate the frequency with which they had personally experienced unfair treatment or hostility due to their background (perceived personal discrimination); and how often they thought that Muslims as a group experienced unfair treatment or hostility (perceived group discrimination). Although personal and group level perceptions of discrimination are inter-related, moderate correlations (< .40) confirm that they are distinct phenomena, with group discrimination being reported more frequently than personal discrimination. Gender (female dummy) and age (in years) were added as control variables, along with parental education (dummy which is one if one or both parents have maximally completed primary education). Table 1 displays descriptive information about all variables used in the analysis.

TABLE 1 ABOUT HERE

As Table 1 shows, Muslim participants in Stockholm are the least religious of the four comparison groups. Conversely, overall levels of religious identification and practices and political religion are highest among Muslim participants in Berlin. Keeping in mind previously described data constraints, however, mean differences across cities should be interpreted with caution due to the fact that mean levels are more susceptible to sampling bias than the associations between variables (Stoop et al., 2010, Van de Vijver and[START_REF] Van De Vijver | Methods and data analysis for cross-cultural research[END_REF].

Method

Multi-group confirmatory factor analysis was applied to establish measurement equivalence of latent dimensions of religiosity across the four cities for subsamples of self-categorized Muslims. 4 Measurement equivalence is confirmed when the difference in χ 2 between a fourgroup model with equality constraints on factor loadings and an unconstrained model is not significant [START_REF] Schmitt | Measurement invariance: review of practice and implications[END_REF]. Throughout our analysis, we report the values of χ 2 in relation to the degrees of freedom (df), CFI and RMSEA as indicators of model fit, which is considered to be good if the χ 2 /df ratio is approaching 1, CFI is above .90, and RMSEA is below .05 [START_REF] Kline | Principles and practice of structural equation modelling[END_REF]. Subsequently, we estimated a second-order factor model of religiosity with religious identification, dietary practices, worship and political religion as first-order factors. Using multi-group structural equation models, we replicated the hypothesized relationships of religiosity with religious socialization, structural integration, and perceived discrimination in the four cities, while controlling for gender, age and parental education. After establishing the measurement equivalence of religious identification, worship, dietary practices, and political religion across cities, a second-order factor model was estimated specifying one common factor, religiosity, which explains inter-factor correlations between the four dimensions of religiosity. This model fitted the data reasonably well: χ 2 (271, 615) = 512.010, CFI = 0.880, RMSEA = 0.038. However, the loadings of the four religiosity dimensions on the second-order factor differ significantly between the four cities, as the model fit is significantly worse when loadings are set equal: ∆ χ 2 (9) = 39.667, p < .001.

In comparative analyses, the loadings on the second-order factor religiosity are estimated separately in each city while the loadings on the four dimensions of religiosity are constrained to be equal across cities. Figures 1a, b, c andd show the standardized factor loadings of religious identification, religious practices, and political religion in each city, revealing different patterns of religiosity among second-generation Turkish Muslims in the four cities.

FIGURES 1A-1D ABOUT HERE

In Berlin and Stockholm, very high loadings (>.80) on all four dimensions of religiosity suggest rather undifferentiated patterns of religiosity. Turkish Muslims in these cities who are strongly committed to their Muslim identity also engage more frequently in religious practices and more strongly support political religion. In Amsterdam and Brussels, in contrast, patterns of religiosity are more selective; and political religion in particular is distinctive of only highly involved Muslims. To test our hypotheses, we predict religiosity from indicators of structural integration, religious socialization, and perceived discrimination while controlling for age, gender and parental education.

Structural integration and secularization

With regard to degrees of religiosity, including religious identification, worship, dietary practices, and political religion, the secularization hypothesis, which predicts religious decline with the structural integration of the second generation, is not supported in Amsterdam, Brussels and Stockholm; yet educational attainment is associated with lower religiosity in Berlin. Logistic regression of self-categorization as a Muslim yields parallel results (see Note 4), which shows that our findings apply not only to the degree of religiosity of the second generation, but also to religious self-identification. In support of differential associations, constraining the effects of educational and labour market attainment on religiosity to zero yields a significantly worse model fit only in Berlin, but not in the other three cities: ∆ χ 2 (18) = 21.859, p > .05. Similarly, having a Turkish partner is associated with higher religiosity only in Berlin (although a non-Turkish partner has no additional effect on religiosity). In the other cities, religiosity is dissociated from the ethnicity of one's partner so that constraining the effects to zero does not significantly affect the model fit. This is found in all cities except Berlin: ∆ χ 2 (6) = 7.762, p > .05). These findings are in line with hypothesized contextual differences as a function of the institutional accommodation of Islam. As expected, structural integration and religion are inversely related in the German capital, where Islam is least accommodated, yet they are decoupled in the other cities.

Religious vitality and reactive religiosity

In line with religious vitality, religious socialization has a strong impact on second-generation religiosity in all cities: Turkish Muslims who attended Koran lessons as a child or whose parents visited a mosque regularly are more religious. 6 Finally, reactive religiosity is not supported, as associations with perceived discrimination disappear after taking into account religious socialization, structural integration, and controls. Accordingly, constraining the effects of personal and group-level discrimination to zero in all cities does not significantly affect model fit: ∆ χ 2 (8) = 9.323, p > .05. Again, these effects are paralleled in logistic regressions predicting Muslim self-categorization. Table 2 shows hypothesis, which expects religious decline with increasing levels of structural integration in mainstream societies, is tested together with a complementary hypothesis of religious vitality.

The latter expects continued religiosity due to the religious socialization of the second generation within immigrant families and communities. In parallel, second-generation religion has been conceived as 'reactive' in response to perceived discrimination. Recently gathered cross-national data from the TIES-surveys among the Turkish second generation in major European cities allowed us to examine multiple dimensions of religiosity and to compare patterns of second-generation integration and religiosity between four European capital cities: Amsterdam, Berlin, Brussels, and Stockholm. Across cities, a majority of the Turkish second generation self-categorizes as Muslim and reports high overall levels of religious identification and practices, in line with earlier findings of sustained religiosity across generations of Turkish Muslims [START_REF] Maliepaard | Generational differences in ethnic and religious attachment and their interrelation. A study among Muslim minorities in the Netherlands[END_REF][START_REF] Diehl | Religiosität türkischer Migranten im Generationenverlauf: ein Befund und einige Erklärungsversuche[END_REF], Phalet et al., 2008).

Comparative analyses revealed distinct components of religiosity, including religious identification, worship, dietary practices and political religion. Across cities, the same components constituted different patterns of religiosity, however, with less differentiated religious involvement among the second generation in Berlin and Stockholm and more selective patterns of religiosity in Amsterdam and Brussels. In the latter cities, political religion in particular was distinctive of only a subgroup of highly involved Muslims, while it was more closely associated with religious identity and practices in Berlin and Stockholm.

Turning to theoretical expectations of religious decline versus vitality in the second generation, we find support for the religious vitality hypothesis across the four cities whereas support for the secularization hypothesis is restricted to Berlin. Only here, educational attainment was inversely related to religiosity; in all other cities, second-generation religiosity was decoupled from structural integration. Differential associations of religiosity with integration point to the hypothesized moderating role of institutional accommodation. As expected, integration and religiosity were inversely related in the German context, where the state does not accommodate Islam as a minority religion. In the other contexts, where there is In line with our religious vitality hypothesis, we find that religious childhood socialization within Turkish immigrant families and communities strongly predicts continued religiosity in young adulthood. Second-generation Muslims whose parents visited a mosque more regularly or who attended Koran lessons as children were more religious later on in terms of their religious identification, practices and political religion in all four cities. This finding implies that the regulatory impact of different institutional arrangements is mostly restricted to that part of the social life of the second generation which is situated outside the co-ethnic community. Finally, and after taking into account religious socialization, we found no evidence of reactive religiosity.

Our findings should be interpreted with caution in light of data constraints. Firstly, the cross-sectional nature of the data does not permit us to empirically establish the direction of causality. Thus, we cannot be sure whether in Berlin higher education decreases religiosity, or high levels of religiosity hinder educational attainment, or whether the negative association of religion and integration works both ways. The issue of causal direction was not the main focus of our study, however, and our conclusions stand regardless of whether the negative association between religiosity and integration in Berlin works in either or both ways. The strength of the study is the comparative design, which allows us to replicate associations between integration and religion across four European capital cities which represent different models of church-state relations. Our finding of differential associations, in particular the lack of support for an inverse relationship of integration and religion in Amsterdam, Brussels and Stockholm, strongly suggests that there is nothing inherent in Islamic faith which would stand in the way of the integration of second-generation Muslims in European societies. The orthogonality of religion and structural integration is not an artefact due to a lack of variation in the structural integration of the Turkish second generation outside Berlin, since the marginal distributions of the four samples in Table 1 show that participants were similarly or Moreover, our analyses take into account competing explanations of religiosity from the differential selection of the first generation across the four cities; from childhood religious socialization; and from experiences of discrimination against Muslims. Taking into consideration these precautions, it seems plausible that different patterns of religiosity and integration between Berlin and the other capitals can be attributed at least in part to the cumulative impact of (the absence of) institutional recognition and accommodation in the receiving societies.

Secondly, our findings may be affected by low response rates to the TIES-surveys. In combination with different sampling frames, selective non-response implies that we cannot assume that the TIES-data are representative of the Turkish second generation, nor that participants are similarly selected in the four cities. Yet we found few differences in the selectivity of the first generation of Turkish immigrants across the four samples. Furthermore, we could establish measurement equivalence of latent dimensions of religiosity across cities. Finally, our comparative analyses are concerned with differential patterns and associations of religiosity rather than differences in absolute levels, which are most vulnerable to sampling and measurement bias (Van de Vijver and [START_REF] Van De Vijver | Methods and data analysis for cross-cultural research[END_REF], Stoop et al., 2010). For these reasons, the present analysis is a first step towards more comprehensive comparative analyses of religion among the European second generation, which should cross-validate our main finding of the conditional decoupling of religiosity and integration with other data sources.

Finally, our comparison is restricted to capital cities. To the extent that the differential institutionalization of Islam affects the relation between integration and religiosity, these effects of national institutions should show up in our comparative analysis. In support of our approach, previous research on public claims-making related to immigrant integration, revealed more variation between countries than between cities within the same country [START_REF] Koopmans | Migrant mobilisation and political opportunities: variation among German cities and a comparison with the United Kingdom and the Netherlands[END_REF]. Our findings should not be generalized to the national level, however, because, with only one city and one immigrant group per country, it is not possible to disentangle contextual variation at the level of cities or ethnic communities from countrylevel differences.

Keeping these limitations in mind, our results cast serious doubt on common assumptions of an inherent conflict between Islamic religiosity and the integration of the Muslim second generation in Europe. The fact that Berlin was the exception to the general rule of sustained religiosity combined with structural integration among the second generation highlights the distinctiveness of the German receiving context. Particularly in the religious domain, the lack of recognition and accommodation of the Muslim minority implies that Islam is treated as a second-class religion, leaving little room for a positive attachment to and active practice of this religion among most integrated Turkish Germans. The absence of a consistent trend towards lower religiosity with social mobility and social mixing in Amsterdam, Brussels and Stockholm implies that Islamic religiosity does not signify failed integration. Neither can it be interpreted in any straightforward way as a reaction of the second generation against perceived discrimination -despite widely shared perceptions that Muslims as a group are treated unfairly in European societies. Most importantly, in the presence of some degree of public recognition of Islam as a minority religion, structural integration is not significantly associated with religiosity. We conclude that religiosity is compatible with structural integration in more welcoming receiving contexts which accommodate religious diversity -albeit to varying degrees. fifteen years old, while mothers were mostly homemakers; in Brussels and Stockholm, however, mothers' levels of labour market participation were higher. 2 Because mosque visiting is gendered with higher attendance among men (cf. [START_REF] Diehl | Religiosität türkischer Migranten im Generationenverlauf: ein Befund und einige Erklärungsversuche[END_REF], a direct effect of gender on mosque attendance is included in all models. 3 The levels of education distinguished correspond to the International Standard Classification of Education (ISCED; UNESCO 1997), with low education representing ISCED-2 and below, vocational referring to ISCED-3B, upper secondary to ISCED-3A and tertiary to ISCED-5 and higher. 4 Self-categorized Muslims are participants who indicated that they currently had a religion and that this religion was Islam. Religious affiliation was used as a routing question in computer-assisted personal interviews such that participants who indicated that they had no religion did not answer questions on religious identification and practice. The wording of the routing question deviated in Berlin where participants were asked "Are you currently religious?" This wording may have resulted in a greater selectivity of participants who completed the religiosity measures in Berlin. We analysed selfcategorization as Muslims using all independent variables as predictors in logistic regression. These results (not shown, available upon request) replicate similar context-dependent associations of religious membership with structural integration, religious socialization and perceived discrimination. 5 Moreover, model fit statistics suggest that the distinction between the two religious practice scales fits better to the data, compared to a model where all four indicators load on one common practice scale.

Akaike's Information Criterion (AIC) is used to compare these competing models, with smaller values indicating better fit (Kline, 2005, Hu and[START_REF] Hu | Cutoff criteria for fit indexes in covariance structure analysis: conventional criteria versus new alternatives[END_REF]

  religiosity. A more recent analysis, however, indicates an emerging trend towards religious revival particularly among more highly educated second-generation Dutch Muslims

  Integration and religiosity among the Turkish 2 nd generation 10

  Confirmatory factor analyses largely support the equivalence of the four latent dimensions of religiosity across groups. Loadings of the four indicators on the latent variable religious identification could be set equal across the four cities, as evident from a non-significant χ 2difference between models with and without equality constraints (∆ χ 2 (9) = 14.806, p > .05) and a good model fit of the factorially invariant model: χ 2 (17, 615) = 27.272, CFI = 0.979, RMSEA = 0.031. Concerning religious practices, full factorial invariance of the four indicators of worship and dietary practices is rejected: due to a lower loading of the halal item in Amsterdam, the χ 2 -difference test is significant at p = 0.042 (∆ χ 2 (6) = 13.081). Because all loadings are significant and positive in the four cities and because the factorially invariant model fits the data well (χ 2 (22, 615) = 39.780, CFI = 0.971, RMSEA = 0.036), loadings of religious practice items are set equal in comparative analyses across cities for the sake of parsimony.5 Regarding political religion, full factorial invariance was again rejected. The χ 2difference test is significant at p = 0.038 (∆ χ 2 (9) = 18.041) because the loading of the private religion item is non-significant in Stockholm. Again, we set loadings on political religion equal in comparative analysis since loading patterns are similar across cities and because the factorially invariant model fits the data well (χ 2 (17, 615) = 30.104, CFI = 0.959, RMSEA = 0.046).

  within-group standardized parameter estimates for the final most restrictive structural equation models in the four cities. Only effects that are different from zero are included in this model. Squared multiple correlations indicate substantial explained variances in religiosity in all four cities, with most explained variance in Berlin and least in Stockholm. 7TABLE 2 ABOUT HEREDiscussionAgainst the background of heated public debates, yet scant empirical evidence on the integration of Muslim minorities in Europe, this study analysed the relationship between structural integration and religiosity among the Turkish second generation. The secularization

  accommodation, religiosity was compatible with the structural integration of the second generation, in the sense of socio-economic inclusion and social mixing in mainstream society.

  different integration outcomes in Amsterdam, Brussels or Stockholm.

Notes1

  Results not shown, available upon request. We compared rural or urban origin, year of migration, age at migration, migration motives and educational qualifications, as well as parents' employment status during participants' youth. Most parents originate from rural areas in Turkey and migrated to Western Europe in the late 1960s and early 1970s, with mean age at migration between 18 and 23 years. Most fathers migrated for economic reasons, while mothers generally migrated for family reunification or formation. Most parents had little or no formal education, with somewhat higher levels in Brussels and Stockholm. Most fathers were working in unskilled or skilled manual jobs when participants were

Figure 1 Figure 1 Figure 1 Figure 1

 1111 Figure 1(a). Second-order factor model of religiosity (standardized loadings): Amsterdam

Table 1 .

 1 . AIC is 186.316 for a model with only one religious practice scale, while a model with two correlated practice scales has an AIC of 155.780, VERMEULEN, HANS & PENNINX, RINUS 2000 Immigrant integration: the Dutch case, Descriptive statistics

	ceteris paribus.

Table 2 .

 2 Multi-group structural equation models of religiosity (standardized coefficients)

		Amsterdam	Berlin	Brussels	Stockholm
	N	166	156	177	116
	Predictors
	Age	-0.184 *	-0.108	-0.023	-0.196
	Female (0/1)	-0.017	0.103	-0.041	-0.206
	Parental education	0.013	-0.019	-0.053	0.097
	Koran school (0/1)	0.270 *	0.441 ***	0.174 *	0.159
	Parental mosque visit	0.373 **	0.434 ***	0.364 ***	0.170

The effects of Koran lessons and parental religiosity are not significant in Stockholm, yet constraining them to zero results in significantly worse model fit (∆ χ 2 (2) = 8.144, p = 0.017).

Table 2 also shows the loadings of the religiosity dimensions on the second-order factor after including all explanatory and control variables. Despite some changes in magnitude, the loadings still differ significantly between the four cities; constraining them to be equal results in significantly worse model fit: ∆ χ 2 (9) = 38.733, p < .001.