Thibaut Allemand

THE INCOMPRESSIBLE EULER LIMIT OF THE BOLTZMANN EQUATION FOR A GAS OF FERMIONS

. Our result is analogous to the classical case result, but the treatment is slightly complicated by the cubic nonlinearity of the collision operator. * ) and (v, v ′ ) → (v * , v ′ * ) have unit jacobian, imply that, at least formally,

(1.5) and then R 3

(1, v, |v| 2 )Q(f )dv = 0.

Introduction

The study of quantum gases has been given increasing interest in the literature over the last decade. In particular, quantum kinetic theory is an expanding field of research. By quantum gases we mean gases made of quantum particles, that are, bosons or fermions. The first ones aim at aggregating together to form the so-called "Bose-Einstein condensates". Conversely, fermions obey Pauli's exclusion principle, which prevents any pair of fermions from being in the same state.

Among the possible models for quantum gases, the Boltzmann-type models first proposed by Nordheim in 1928 [START_REF] Nordheim | On the kinetic method in the new statistics and its application in the electron theory of conductivity[END_REF] then Uehling and Uhlenbeck in 1933 [START_REF] Uehling | Transport phenomena in Einstein-Bose and Fermi-Dirac gases[END_REF] are very popular. Although their range of validity is not clear, they seem to capture some aspects of the behaviour of bosonic or fermionic particles. Indeed, it has been proved, in the bosonic (and space homogeneous) case, that under a threshold temperature, a condensate occurs in infinite time [START_REF] Escobedo | Entropy maximisation problem for quantum relativistic particles[END_REF][START_REF] Escobedo | Homogeneous Boltzmann equation in quantum relativistic kinetic theory[END_REF]. On the contrary, solutions of the fermionic Boltzmann equation satisfy a natural L ∞ bound which reflects Pauli's exclusion principle and makes the Cauchy problem easier. There is no need for renormalized solutions in this case since one can prove the existence of global weak solutions [START_REF] Lions | Compactness in Boltzmann's equation via Fourier integral operators and applications[END_REF][START_REF] Dolbeault | Kinetic models and quantum effects: a modified Boltzmann equation for Fermi-Dirac particles[END_REF][START_REF] Allemand | Existence and conservation laws for the Boltzmann-Fermi equation in a general domain[END_REF].

In the present work, we investigate the incompressible Euler limit of the Boltzmann equation for a gas of fermions (or Boltzmann-Fermi equation).

1.1. The Boltzmann equation for fermions. It reads in nondimensional form

(1.1) Ma∂ t f + v.∇ x f = 1 Kn Q(f )
where f (t, x, v) is the density of particles which at time t ∈ R + are at the point x ∈ R 3 of space with velocity v ∈ R 3 . The left-hand term expresses the free transport of particles in absence of interactions, while the right-hand term takes into account the effects of collisions between particles. The Knudsen and Mach numbers are positive constants determined by the physical situation of the gas and defined by Kn = mean free path observation length scale Ma = bulk velocity speed of sound .

The Knudsen number is a measure of the rarefaction of the gas whereas the Mach number measures its compressibility.

The collision integral Q(f ) is given by (1.2)

Q(f ) = S 2 R 3 B(v -v * , ω) (f ′ f ′ * (1 -f )(1 -f * ) -f f * (1 -f ′ )(1 -f ′ * )) dv * dω
with the usual notations

f * = f (t, x, v * ), f ′ = f (t, x, v ′ ), f ′ * = f (t, x, v ′ * )
and where the precollisional velocities (v ′ , v ′ * ) are deduced from the postcollisional ones by the relations

v ′ = v -(v -v * ).ωω, v ′ * = v * + (v -v * )
.ωω, ω being a unit norm vector. These relations express the conservation of momentum and kinetic energy at each collision.

The collision integral (1.2) differs from the classical one by the presence of cubic terms, due to Pauli's exclusion principle, and thus takes into account the quantum nature of the gas. Indeed the probability for a particle with velocity v ′ of taking velocity v after a collision is all the more penalized as other particles at the same point already have velocity v. This prevents any pair of two fermions from being in the same quantum state. As a consequence the solution f satisfy a natural L ∞ bound : if the initial datum f 0 is chosen such that 0 ≤ f 0 ≤ 1, this bound is preserved by the solution f for all times. This property is of great help in the study of this equation and implies very different features from the classical or the bosonic Boltzmann equation even if they look very similar.

The function B(z, ω), known as the collision kernel, is measurable, a.e. positive, and depends only on |z| and on the scalar product z • ω. It is often assumed to satisfy Grad's cutoff assumption:

(1.3) 0 < B(z, ω) ≤ C B (1 + |z|) β a.e. on S 2 × R 3

(1.4)

S 2 B(z, ω)dω ≥ 1 C B |z| 1 + |z|
a.e. on R 3

for some constants C B > 0 and β ∈ [0, 1]. These assumptions guarantee the existence of a solution to equation (1.1) in the whole domain R 3 or in the torus T 3 [START_REF] Dolbeault | Kinetic models and quantum effects: a modified Boltzmann equation for Fermi-Dirac particles[END_REF][START_REF] Lions | Compactness in Boltzmann's equation via Fourier integral operators and applications[END_REF]. However, in order to have existence of solutions in more general domains and to ensure that they satisfy the local conservation of mass, momentum, and energy, as well as the entropy inequality, we will have to make more restrictive assumptions [START_REF] Allemand | Existence and conservation laws for the Boltzmann-Fermi equation in a general domain[END_REF], as will be explained later.

The symmetry properties of the collision operator, coming from the fact that the transformations (v, v * ) → (v ′ , v ′ As a consequence, the solution f of equation (1.1) formally satisfies the conservations of mass, momentum and kinetic energy:

               ∂ t f dv + 1 Ma ∇ x . vf dv = 0 ∂ t vf dv + 1 Ma ∇ x . v ⊗ vf dv = 0 ∂ t |v| 2 f dv + 1 Ma ∇ x . v|v| 2 f dv = 0.
Moreover, taking ϕ = log f 1-f in (1.5) leads to the so-called H-theorem, which expresses the second principle of thermodynamics:

∂ t R 3 s(f )(v)dv + 1 Ma ∇ x . R 3 vs(f )(v)dv = - 1 MaKn D(f ) with s(f ) = f log f + (1 -f ) log(1 -f )
and

D(f ) = 1 4 R 3 ×S 2 B(v -v * , ω) (f ′ f ′ * (1 -f )(1 -f * ) -f f * (1 -f ′ )(1 -f ′ * )) • log f ′ f ′ * (1 -f )(1 -f * f f * (1 -f ′ )(1 -f ′ * ) dv * dω.
Notice that D(f ) is non-negative. The minimizers of the entropy, which are also the functions that cancel the collision operator, are given by : Proposition 1.1. Assume that g ∈ L 1 (R 3 ) is such that Q(g) and D(g) are well defined and satisfy the bounds 0 ≤ g ≤ 1.

Then

Q(g) = 0 ⇐⇒ D(g) = 0 ⇐⇒ g = M 1 + M or g = ½ v∈Λ
where M is a maxwellian distribution, that is,

M = ae -|v-u| 2 2b
with a ≥ 0, b > 0, and for some subset Λ ⊂ R 3 . In the first case the distributions are called Fermi-Dirac or Planckian distributions, whereas they are called degenerate Fermi-Dirac distributions in the second case.

Proofs and details are to be found in [START_REF] Dolbeault | Kinetic models and quantum effects: a modified Boltzmann equation for Fermi-Dirac particles[END_REF][START_REF] Escobedo | Entropy maximisation problem for quantum relativistic particles[END_REF]. The coefficiens of the Planckian distribution a, b, u are fully determined by macroscopic parameters of the fluid with distribution g, namely the mass ρ g , bulk velocity u g and pressure p g , defined by

ρ g (t, x) = R 3 g(t, x, v)dv, u g (t, x) = 1 ρ g (t, x) R 3 vg(t, x, v)dv and p g (t, x, v) = R 3 |v -u g (t, x)| 2 g(t, x, v)dv.
This relation will be detailed in the next Section.

1.2. The formal hydrodynamic limit. In the fast relaxation regime, that is, when the Knudsen number goes to 0, the collision process becomes predominant and we expect in view of (1.1) that a local thermodynamic equilibrium is reached almost instantaneously. The distribution f describing the gas is then close to a Planckian distribution fully determined by some coefficients a, b, u which in turn are caracterized by the hydrodynamic fields ρ f (t, x), u f (t, x) and p f (t, x). The conservation laws then become up to order O(Kn)

Ma∂ t ρ f + ∇ x • (ρ f u f ) = 0 Ma∂ t (ρ f u f ) + ∇ x • ρ f u f ⊗ u f + 1 3 p f I 3 = 0 Ma∂ t (ρ f u 2 f + p f ) + ∇ x • ρ f |u f | 2 u f + 5 3 p f u f = 0
which are the compressible Euler equations for perfect gases (I 3 stands for the three dimensional identity matrix). If now the Mach number also goes to 0, that is, in incompressible regime, the first equation becomes ∇ x • (ρ f u f ) = 0, which is nothing but the incompressibility constraint. The other equations of motions are obtained by a systematic multiscale expansion, depending on another parameter: the Reynolds number, defined by Re = Ma Kn for perfect gases, which measures the viscosity of the gas. In all the sequel, we are interested in the inviscid incompressible regime, so that we consider Ma = ε, Kn = ε q , q > 1 and investigate the asymptotic ε → 0. In this scaling, Kn = ε 1-q → ∞. The Boltzmann equation for fermions now writes

(1.6) ε∂ t f + v.∇ x f = 1 ε q Q(f ).
Moreover, the hydrodynamic fields will be assumed to be fluctuations around a global constant equilibrium (ρ 0 , 0, p 0 ), so that, denoting the fluctuations around the mass, momentum and pressure by ρ, ũ and p,

ρ f = ρ 0 + ε ρ, u f = εũ, p f = p 0 + εp.
We also define the temperature T f by

p f = ρ f T f
and its fluctuation T by T f = T 0 + ε T where T 0 is a constant equilibrium value. Plugging these expressions into the hydrodynamic equations (1.2) we get at leading order ∇ x ũ = 0, ∇ x (T 0 ρ + ρ 0 T ) = 0 which are known as the incompressibility and the Boussinesq relations; at next order it comes

∂ t ρ + ũ • ∇ x ρ = 0 ∂ t ũ + ũ • ∇ x ũ + ∇ x (ρ T ) = 0.
The challenge is now to make rigorous this formal limit. The rest of the paper is organized as follows: in Section 2, we recall some facts about equation (1.6). In Section 3 we explain our strategy which is based on the modulated entropy and we give our main result (Theorem 3.1). In Section 4 we compute the time derivative of the modulated entropy. In Section 5 we construct an approximate solution of (1.6) the parameters of which satisfy the incompressible Euler equation. We give some useful intermediate results and estimates in Section 6. Finally in Section 7 we end the proof of the main result by controlling some flux terms.

Some details on the Boltzmann equation for fermions

In this Section we explain the relation between the hydrodynamic fields and the coefficients a, u, b of the Planckian distribution defined in Proposition (1.1). We then state the existence result to be used in the sequel.

Relations between the coefficients and the macroscopic quantities.

We recall here the definition of the hydrodynamic fields: for a given distribution function f the mass ρ f , the bulk velocity u f and the pressure p f are defined by

ρ f (t, x) = R 3 f (t, x, v)dv, u f (t, x) = 1 ρ f (t, x) R 3 vf (t, x, v)dv and p f (t, x, v) = R 3 |v -u f (t, x)| 2 f (t, x, v)dv.
When the thermodynamic equilibrium is reached, the density function is an equilibrium solution

f = M 1 + M , M = ae -|v-u| 2 2b
, with a, b ≥ 0. It is then clear that u = u f . Let us define for comodity the functions

F p (a) = R 3 |v| p ae -|v| 2 2 1 + ae -|v| 2 2 .
They are well defined for a ∈ [0, +∞). Then we write

ρ f = b 3/2 F 0 (a), p f = b 5/2 F 2 (a).
If we define now the internal energy

e f = 1 2ρ f (p f -4ρ f |u f | 2 ),
we can quote the following result from [3]:

Proposition 2.1. There exists a positive constant l such that the mapping

(a, b) ∈ (0, +∞) 2 → (ρ f , e f ) ∈ E, where E = {(ρ f , e f ) s.t. e f > lρ 2/3
f }, is one-to-one.

From now on, we will denote by P 0 a global equilibrium defined by (2.1)

P 0 = M 0 1 + M 0 , M 0 = e a0 e -|v| 2 2b 0
where a 0 , b 0 > 0 and a 0 is such that F 2 (e a0 ) > 2lF 0 (e a0 ) 5 3 .

This ensures that perturbating a 0 and b 0 will leave the corresponding perturbed mass and internal energy inside E.

Existence theory.

Let Ω be a subset of R 3 regular enough such that the normal is well defined on the boundary. The equation

(2.2) ∂ t f + v.∇ x f = Q(f )
must be supplemented with an initial condition:

(2.3) f (0, x, v) = f 0 (x, v) ∀(x, v) ∈ Ω × R 3 ,
and a boundary condition we choose to be specular reflection for simplicity:

(2.4)

f (t, x, v) = f (t, x, R x (v)) ∀(x, v) ∈ ∂Ω × R 3 such that n(x).v < 0
where n(x) is the outer unit vector normal to the boundary and R x (v) is the specular reflection law

(2.5) R x (v) = v -2(v.n(x))n(x).
The choice of the specular reflection as a boundary condition makes all the boundary terms vanish in the weak formulation of the equation. It also cancels the Prandtl layer along the boundary.

The following existence result was proved in [START_REF] Dolbeault | Kinetic models and quantum effects: a modified Boltzmann equation for Fermi-Dirac particles[END_REF][START_REF] Allemand | Existence and conservation laws for the Boltzmann-Fermi equation in a general domain[END_REF]:

Theorem 2.2.
Let Ω be either R 3 or a regular subset of R 3 . Let the collision kernel B be such that

(2.6) 0 ≤ B ∈ L 1 (R 3 × S 2 ),
and let

(2.7) f 0 ∈ L ∞ (Ω × R 3 ), 0 ≤ f 0 ≤ 1.
Then, the problem (2.2)-(2.4) has a unique solution f satisfying

f ∈ L ∞ (R + × Ω × R 3 ), 0 ≤ f ≤ 1 a.e.
Moreover, f is absolutely continuous with respect to t.

The assumption made on the collision kernel (2.6) is very strong, much more than Grad's cut-off assumption (1.3)-(1.4) which would be enough to ensure the existence of a solution [START_REF] Lions | Compactness in Boltzmann's equation via Fourier integral operators and applications[END_REF]. However, it is necessary (at the time of the writing) in order to prove that the local conservation laws are satisfied by the solution. It is the case if we assume that the collision kernel has the symmetry

(2.8) B(w, ω) = q(|w|, |w.ω|),
which is physically relevant. The following proposition was proved in [START_REF] Allemand | Existence and conservation laws for the Boltzmann-Fermi equation in a general domain[END_REF] :

Proposition 2.3. Assume that the collision kernel B is as in (2.6) and (2.8).

Assume moreover that the initial datum is as in (2.7), and

Ω×R 3 (1 + |v| 3 )f 0 (x, v)dxdv < +∞.
Then, the solution f to (2.2)-(2.4) satisfies, in the distributional sense, the local conservation laws

(2.9)

               ∂ t R 3 f dv + ∇ x . R 3 vf dv = 0 ∂ t R 3 vf dv + ∇ x . R 3 v ⊗ vf dv = 0 ∂ t R 3 |v| 2 f dv + ∇ x . R 3 v|v| 2 f dv = 0
Finally, under the same assumptions on B, Boltzmann's H-theorem is true:

Proposition 2.4. Let the collision kernel B satisfy (2.6) and (2.8). Assume that the initial datum satisfies (2.7), and

Ω×R 3 (|x| 2 + |v| 2 )f 0 (x, v)dxdv < +∞,
and let f be the solution to (2.2)-(2.4). Then,

Ω×R 3 s(f )(t)dvdx + t 0 Ω D(f )dxds = Ω×R 3
s(f 0 ).

2.3.

The linearized collision operator and the fluxes. It is sometimes interesting to linearize the Fermi-Dirac collision operator Q(f ) around a global equilibrium. Let P ε = Mε 1+Mε be a global equilibrium depending on the parameter ε, with

M ε = a ε e -|v-uε| 2 2bε
. Then the linearized collision operator writes

L ε g = S 2 R 3 B(v -v * , ω) M ε, * 1 + M ε, * 1 1 + M ′ ε 1 1 + M ′ ε, * × g(1 + M ε ) + g * (1 + M ε, * ) -g ′ (1 + M ′ ε ) -g ′ * (1 + M ′ ε, * ) dv * dω.
The properties of this operator will be detailed and proved in Section 6. Let us just mention that if the collision kernel B satisfies Grad's cutoff assumptions

(1.3)-(1.4) then L ε is a Hilbert-Schmidt operator with kernel KerL ε = span 1 1 + M ε , v 1 + M ε , |v| 2 1 + M ε .
This property will be crucial in the proof of our main result.

Remark 2.5. Let us make here an important remark. The collision kernel B cannot satisfy both assumptions (2.6) (required to prove the existence of solutions) and (1.3)-(1.4). We will work with two different collision kernels :

B ε (z, ω) = ½ |z|≤ 1 ε 2 satisfies (2.
6) and will be used in the definitions of the collision operator Q(f ε ) and the entropy dissipation D(f ε ).

However, B = ½ will be considered when studying the properties of the linear operator.

In proving the hydrodynamic limit, the collision kernel B ε will naturally appear in expressions to control. When the linear operator L ε will be needed, we will write

B ε = B + (B ε -B)
and use the linear operator properties on the term containing B, while the other one will be controlled by hands using the fact that B ε and B are equal on a very large ball.

Let us define the energy and heat fluxes

Φ ε = 1 b ε (v -εu) ⊗2 -|v -εu| 2 I Ψ ε = v -εu b 2 ε |v -εu| 2 -τ ε ,
where τ ε is the unique number such that Ψε 1+Mε ∈ (KerL ε ) ⊥ and I is the 2 × 2 identity matrix. These quantities naturally appear in Section 4. Taking ε = 0, we can prove that Φ, Ψ ∈ (KerL 0 ) ⊥ where Φ = Φ 0 and Ψ = Ψ 0 . Since L 0 is a Hilbert-Schmidt operator, we can define (2.10) Φ = L -1 0 Φ and Ψ = L -1 0 Ψ. Φ and Ψ will appear in the control of the flux terms.

Strategy and main result

In investigating the incompressible Euler limit of the Boltzmann-Fermi equation, we use the modulated entropy method as in the classical case [START_REF] Saint-Raymond | Hydrodynamic limits: some improvements of the relative entropy method[END_REF]. Let us first introduce the relative entropy of the solution f ε to (1.6) with respect to the global thermodynamic equilibrium P 0 (defined by (2.1)):

H(f ε |P 0 ) = Ω×R 3 f ε log f ε P 0 -f ε + P 0 + (1 -f ε ) log 1 -f ε 1 -P 0 -(1 -f ε ) + 1 -P 0 dvdx.
It is a non-negative Lyapunov functional for the Boltzmann-Fermi equation thanks to the H theorem. In the fast relaxation limit, f ε is supposed to be close to a thermodynamic equilibrium, we therefore define the modulated entropy by

H(f ε |P ε ) = Ω×R 3 f ε log f ε P ε -f ε + P ε + (1 -f ε ) log 1 -f ε 1 -P ε -(1 -f ε ) + 1 -P ε dvdx.
where

P ε = Mε 1+Mε is some local equilibrium which approximates the solution f ε , with M ε = a ε e -|v-uε | 2 2bε
. The relative entropy functional measures this approximation since

H(f ε |P ε ) ≥ Ω×R 3 f ε -P ε 2 dvdx, consequence of the pointwise inequality ( √ x - √ y) 2 ≤ x log x y -x + y ∀x, y > 0.
Our strategy consists in studying the time evolution of 1 ε 2 H(f ε |P ε ). This has the good scaling since we want to observe the fluctuations of f ε around an equilibrium. To do that we

• determine the equation satisfied by 1 ε 2 H(f ε |P ε ), using the local conservation laws satisfied by f ε ; it contains an acoustic (fast oscillating) term and flux terms • control the acoustic terms by specifying a ε , u ε and b ε • control the flux terms thanks to a priori bounds on the linear collision operator. Our main result is the following: Theorem 3.1. Let Ω be some regular bounded domain of R 3 . Let (f ε,in ) be a family of measurable nonnegative functions over Ω × R 3 satisfying the bounds

Ω R 3 (1 + |x| 2 + |v| 3 )f ε,in dvdx ≤ C ε , 0 ≤ f ε,in ≤ 1,
and the scaling condition

(3.1) 1 ε 2 H(f ε,in |P 0 ) ≤ C.
Without loss of generality, assume that the fluctuation g ε,in defined by f ε,in = P 0 (1 + εg ε,in ) converges

g ε,in → g in = 1 1 + M 0 a in a 0 + u in b 0 • v + b in 2b 0 |v| 2 .
We assume moreover that the initial data is well-prepared:

(3.2) 1 ε 2 H(f ε,in |P ε,in ) → 0.
Let f ε be some family of solutions to the scaled Boltzmann-Fermi equation

(3.3) ε∂ t f ε + v.∇ x f ε = 1 ε q Q(f ε ) f ε (0, x, v) = f ε,in (x, v) on Ω × R 3
with q > 1, and with a maxwellian truncated collision kernel B ε (z, ω) = ½ |z|≤ 1 ε 2 , endowed with the boundary condition

(3.4) f ε (t, x, v) = f ε (t, x, R x (v)) ∀(x, v) ∈ ∂Ω × R 3 such that n(x).v < 0
where R x (v) is the specular reflection law (2.5). Assume that we have the following control on the tails :

(3.5) R 3 P 0 f ε -P 0 P 0 2 dv ≤ C a.e.
and, finally, assume that

(3.6) Φ = L -1 0 Φ and Ψ = L -1 0 Ψ are at most polynomial as |v| → ∞. Then the fluctuation (g ε ) defined by f ε = P 0 (1 + εg ε ) converges in L 1 ([0, T ]; (1 + |v| 2 )P 0 dvdx) weak g ε -→ ε→0 1 1 + M 0 ā a 0 + ū b 0 • v + b 2b 0 |v| 2
where (ā, ū, b) is the unique Lipschitz solution to the incompressible Euler equations

∂ t ū + ū.∇ x ū + ∇ x p = 0, ∇ x .ū = 0 supplemented with ∂ t ā + ū.∇ x ā = 0, ∂ t b + ū.∇ x b = 0, ∇ x (b 0 ā + τ 0 b) = 0 on [0, T ],
and T is the maximal lifespan of the solution.

The result is not very surprising since it is very similar to the classical case [START_REF] Saint-Raymond | Hydrodynamic limits: some improvements of the relative entropy method[END_REF]. Assumption (3.6) was proved in the classical case [START_REF] Golse | A remark about the asymptotic theory of the Boltzmann equation[END_REF], and it should be possible (although very technical) to prove it also in the fermion case.

Assumption (3.2) allows us to consider solutions which are almost instantaneously at thermodynamic equilibrium, thus avoiding the Knudsen layer. To get rid of it we need to get a better understanding of the relaxation mecanism.

As to assumption (3.5), it is a purely technical control of large velocities which allows us to treat flux terms. The exact same assumption is done in the classical case and it seems to be a great challenge to avoid it. It has been proved that some regular solutions to the classical Boltzmann equation satisfying (3.5) exist, but such solutions in the fermion case are still missing to the author's knowledge.

The cubic nonlinearity of the collision operator adds a lot of technical difficulties in the proof: a lot more terms, respect to the classical case, appear and must be controlled. The L ∞ bound is of great help to treat these difficulties, but does not bring more, so that at the end we do not recover a better result than in the classical case.

The modulated entropy

The main idea to prove the convergence of the solutions to the scaled Boltzmann-Fermi equation toward a solution of the incompressible Euler equation is to study the time evolution of the modulated entropy. Let

P ε = M ε 1 + M ε , M ε = a ε e -|v-εu| 2 2bε
, be a local equilibrium which approximates the solution f ε , and take (4.1)

a ε = e a0+εa1 , b ε = b 0 e εb1 .
Recall that a 0 and b 0 are positive constants which were defined in Subsection 2.1; a 1 , u and b 1 are regular functions of (t, x).

Proposition 4.1. Any solution to the scaled Boltzmann-Fermi equation (1.6) satisfies the following identity

1 ε 2 H(f ε |P ε )(t) = 1 ε 2 H(f ε,in |P ε,in ) - 1 ε q+3 t 0 Ω D(f ε )dxds - 1 ε t 0 Ω×R 3 f ε 1, v -εu b ε , |v -εu| 2 2b ε .A ε (a 1 , u, b 1 )dxdvds - 1 2ε 2 t 0 Ω×R 3 f ε b ε Ψ ε .∇ x b 1 dxdvds - 1 2ε 2 t 0 Ω×R 3 f ε (Φ ε • ∇ x u)dxdvds + 1 3ε 2 t 0   d dt Ω×R 3 e -εb1 b 0 |v| 2 e a0+εa1-e -εb 1 |v| 2 2b 0 1 + e a0+εa1-e -εb 1 |v| 2 2b 0 dvdx   ds.
where the acceleration operator A ε (a 1 , u, b 1 ) is defined by (4.2)

A ε (a, u, b) =   ∂ t a + u.∇ x a ∂ t u + u.∇ x u + b 0 e εb -1 ε ∇ x a + 1 ε (τ ε -τ 0 )∇ x b + 1 ε ∇ x (b 0 a + τ 0 b) ∂ t b + u.∇ x b + 2 3ε ∇ x .u  
and τ 0 is the limit of τ ε as ε goes to 0.

Proof. The modulated entropy can be written as

H(f ε |P ε ) = H(f ε |P 0 ) + Ω×R 3 f ε log M 0 M ε + log 1 + M ε 1 + M 0 dvdx.
The H-theorem allows to write

H(f ε |P ε )(t) = H(f ε,in |P ε,in ) - 1 ε q+1 t 0 Ω D(f ε )dxds + t 0 d dt Ω×R 3 f ε -log a ε + |v -εu| 2 2b ε - |v| 2 2 dxdv ds + t 0 d dt Ω×R 3 log (1 + M ε ) dxdv ds. (4.3)
An integration per part changes the last term:

Ω×R 3 log (1 + M ε ) dxdv = 1 3 Ω×R 3 1 b ε |v| 2 a ε e -|v| 2 2bε 1 + a ε e -|v| 2 2bε dvdx.
Computing the time derivative and using the conservation laws (2.9) leads to:

d dt Ω×R 3 f ε -log a ε + |v -εu| 2 2b ε - |v| 2 2 dxdv = - Ω×R 3 f ε ∂ t log a ε + u • ∇ x log a ε + ε b ε (v -εu) ∂ t u + u • ∇ x u + 1 ε ∇ x u : (v -εu) + 1 ε 2 b ε ∇ x log a ε + |v -εu| 2 2b ε ∂ t b ε b ε + u • ∇ x b ε b ε + 1 ε (v -εu) • ∇ x b ε b ε dvdx.
Introducing the rescaled translated versions of the momentum and energy fluxes

Φ ε = 1 b ε (v -εu) ⊗2 -|v -εu| 2 I and Ψ ε = v -εu b 2 ε |v -εu| 2 -τ ε
and replacing a ε and b ε by (4.1) we obtain:

d dt Ω×R 3 f ε -log a ε + |v -εu| 2 2b ε - |v| 2 2 dxdv = - Ω×R 3 f ε ε (∂ t a 1 + u • ∇ x a 1 ) + ε b ε (v -εu) ∂ t u + u • ∇ x u + 1 ε b ε ∇ x a 1 + 1 ε τ ε ∇ x b 1 + ε |v -εu| 2 2b ε ∂ t b 1 + u • ∇ x b 1 + 2 3ε ∇ x • u dvdx - 1 2ε Ω×R 3 f ε Ψ ε • ∇ x b ε dxdv - 1 2ε Ω×R 3 f ε (Φ ε • ∇ x u)dxdv.
Summarizing, the modulated entropy satisfies

1 ε 2 H(f ε |P ε )(t) = 1 ε 2 H(f ε,in |P ε,in ) - 1 ε q+3 t 0 Ω D(f ε )dxds - 1 ε t 0 Ω×R 3 f ε 1, v -εu b ε , |v -εu| 2 2b ε • A ε (a 1 , u, b 1 )dxdvds - 1 2ε 2 t 0 Ω×R 3 f ε (b ε Ψ ε • ∇ x b 1 + Φ ε • ∇ x u) dxdvds + 1 3ε 2 t 0   d dt Ω×R 3 e -εb1 b 0 |v| 2 e a0+εa1-e -εb 1 |v| 2 2b 0 1 + e a0+εa1-e -εb 1 |v| 2 2b 0 dvdx   ds.
where the acceleration operator is defined by (4.2).

Construction of the approximate solutions

A global equilibrium solution is not expected to be a good approximation of f ε in the fast relaxation limit since fast oscillations can take place, such as acoustic waves. Hence we have to find correctors in order to obtain a refined approximation which will lead to the convenient asymptotics.

We want to find

V ε = (a ε 1 , u ε , b ε 1 ) such that A ε (a ε 1 , u ε , b ε 1 ) → 0 in L 2
, that is, such that V ε is an approximate solution of the system (5.1) Such a constrution is done by filtering methods, and we refer to [START_REF] Saint-Raymond | Hydrodynamic limits: some improvements of the relative entropy method[END_REF] for all the details. We will just here mention the main points of the proof of

∂ t V + 1 ε W V + B(V, V ) = 0, where V = (a 1 , u, b 1 ), W V = 0, ∇ x (b 0 a 1 + τ 0 b 1 ) , 2 3 ∇ x .u and
B(V, V ) =   u.∇ x a 1 u.∇ x u + b 0 b 1 ∇ x a 1 + τ 1 ∇ x b 1 u.∇ x b 1   with moreover the constraint (5.2) A ε = 1 ε 2 d dt
Theorem 5.2. Let (a in 1 , u in , b in 1 ) belong to H s (Ω) for some s > 5 2 .
Then there exists some T > 0 and some family

(a ε,N 1 , u ε,N , b ε,N 1 ) such that sup N ∈N lim ε→0 (a ε,N 1 , u ε,N , b ε,N 1 ) L ∞ ([0,T ];H s (Ω)) ≤ C T , (a ε,N,in 1 , u ε,N,in , b ε,N,in 1 ) → (a in 1 , u in , b in 1 ) in H s (Ω) as ε → 0 and N → +∞, A ε (a ε,N 1 , u ε,N , b ε,N 1 ) → 0 in L 2 ([0, T ] × Ω)
as ε → 0 and N → +∞, and

A N ε = 1 ε 2 d dt Ω×R 3 e -εb ε,N 1 b 0 |v| 2 e a0+εa ε,N 1 -e -εb ε,N 1 |v| 2 2b 0 1 + e a0+εa ε,N 1 -e -εb ε,N 1 |v| 2 2b 0 dvdx -→ 0 in L 1 ([0, T ]) as ε → 0 and N → +∞.
In L 2 equipped with the norm

V 2 L 2 = a 1 2 L 2 + u 2 L 2 + 3 2τ 0 b 0 a 1 + τ 0 b 1 2 L 2 ,
the operator W is skew-symmetric. We can thus define the associated semigroup W. If we conjugate (5.1) with W( t ε ), it comes

∂ t (W t ε V ) + W t ε B(V, V ) = 0. Defining Ṽ = W t ε V, it becomes (5.3) ∂ t Ṽ + W t ε B W - t ε Ṽ , W - t ε Ṽ = 0.
We obtain the first order approximation by taking strong limits in the filtered systems. However, the error is not expected to converge strongly to 0 due to high frequency oscillations. We therefore need to construct a second order approximation. We then need a third order approximation in order to have mass conservation at sufficient order.

5.1. Study of W . We expect the solutions of (5.3) to have a very different behaviour depending on the spectrum of W . Since it is skew-symmetric, its eigenvalues are purely imaginary, and its eigenvectors satisfy

W V λ = iλV λ , which implies if λ = 0      a λ = 0 ∆ x b λ = -3 2τ0 λ 2 b λ ∇ x b λ = i λ τ0 u λ .
Hence, the operator W has the same spectral structure as the laplacian on Ω, which means that is is diagonalizable on the orthogonal of its kernel.An explicit computation shows that the orthogonal projection on ker W is

Π 0 (a, u, b) = a, P u, |Ω| -1 (b 0 a + τ 0 b)dx -b 0 a τ 0
where P is the Leray projection, that is, the orthogonal projection onto the divergencefree vectors. In the sequel, Π λ is the orthogonal projection onto ker(W -λI).

Construction.

We just sketch here the construction of approximate solutions to (5.3), and we refer to [START_REF] Saint-Raymond | Hydrodynamic limits: some improvements of the relative entropy method[END_REF] for all the details. The method is as follows:

• Decompose the operator W as W = λ∈σp λΠ λ and plug this into (5.3). It leads to

∂ t Ṽ + k1,k2,k3∈σp e i t ε (λ k 1 -λ k 2 -λ k 3 ) Π λ k 1 B(Π λ k 2 Ṽ , Π λ k 3 Ṽ ) = 0
• The first order approximation Ṽ0 of Ṽ is defined as the solution of equation ( 5.3) when we take into account only the resonant modes, that is,

∂ t Ṽ0 + λ k 1 =λ k 2 +λ k 3 Π λ k 1 B(Π λ k 2 Ṽ0 , Π λ k 3 Ṽ0 ) = 0
This equation is known to have solutions in L ∞ loc ([0, T * ), H s (Ω)) provided that V in ∈ H s (Ω) and s > 5 2 . However, Ṽ0 is not an approximation of Ṽ in the sense that A ε ( Ṽ0 ) converges weakly but not strongly in L 2 to 0. We therefore have to add correctors. • Hence, we construct the second order approximation plugging Ṽ = Ṽ0 + ε Ṽ1 + o(ε) into (5.3). To avoid the problem of small divisors, we introduce the projection onto a finite dimensional subset of C ∞ ( Ω) J N , which is the orthogonal projection onto the N first harmonics of W and the N first harmonics in ker W . We then denote Ṽ N 0 = J N Ṽ0 , and we check that Ṽ0 dx = Ṽ N 0 dx. We define Ṽ N 1 by

Ṽ N 1 = J N λ k 1 +λ k 2 =λ k exp it ε (λ k -λ k1 -λ k2 ) i(λ k1 + λ k2 -λ k ) Π λ k B(Π λ k 1 Ṽ N 0 , Π λ k 2 Ṽ N 0 ).
Then, Ṽ N 0 + ε Ṽ N 1 is an approximate solution of (5.3) strongly in L 2 . However, it does not satisfy 1 ε 2 ( Ṽ N 0 + ε Ṽ N 1 )dx → 0 when ε → 0.

• We therefore define the third order approximation Ṽ N 2 as in [START_REF] Saint-Raymond | Hydrodynamic limits: some improvements of the relative entropy method[END_REF] by plugging

Ṽ N ε = Ṽ N 0 + ε Ṽ N 1 + ε 2 Ṽ N 2 + o(ε 2 ) into (5.
3), and we check that

-Ṽ N ε = Ṽ N 0 + ε Ṽ N 1 + ε 2 Ṽ N 2 is an approximate solution of (5.3), -1 ε 2 Ṽ N ε dx → 0 when ε → 0.
The key ingredient for the above computations to work (see [START_REF] Saint-Raymond | Hydrodynamic limits: some improvements of the relative entropy method[END_REF]) is that, for λ, µ = 0 with λ = µ, we have

Π 0 B(Π λ V, Π µ V ) = 0. Indeed, writing Π λ V = (a λ , u λ , b λ ), we get Π 0 B(Π λ V, Π µ V ) = 1 2 Π 0     u λ .∇ x a µ + u µ .∇ x a λ u λ .∇ x u µ + u µ .∇ x u λ + b 0 b λ ∇ x a µ + b 0 b µ ∇ x a λ +(c 1 a λ + c 2 b λ )∇ x b µ + (c 1 a µ + c 2 b µ )∇ x b λ u λ .∇ x b µ + u µ .∇ x b λ     ;
but a λ = a µ = 0 since λ, µ = 0, so that

Π 0 B(Π λ V, Π µ V ) = 1 2 Π 0   0 u λ .∇ x u µ + u µ .∇ x u λ + c 2 b λ ∇ x b µ + c 2 b µ ∇ x b λ u λ .∇ x b µ + u µ .∇ x b λ   =   0 P (u λ .∇ x u µ + u µ .∇ x u λ ) (u λ .∇ x b µ + u µ .∇ x b λ )dx)   . Note that u λ .∇ x b µ dx = -3µ 2 2iλ b λ .b µ dx = 0 since λ = µ. Moreover, (u λ .∇ x u µ + u µ .∇ x u λ ) i = (∇ x (u λ .u µ )) + u j λ ∂ j u i µ -∂ i u j µ + u j µ ∂ j u i λ -∂ i u j λ , so that Π 0 B(Π λ V, Π µ V ) = 0.
In addition, if λ = 0,

Ω b λ dx = -i 2 3λ Ω ∇ x .u λ dx = i 2 3λ ∂Ω u.ndσ x = 0.
At the end, we get that

d dt Ω bdx = d dt Ω b 0 dx = 0 and d dt Ω ãdx = 0,
which is the key to see that 1

ε 2 Ṽ N ε dx → 0 when ε → 0.
The equation for the non-oscillating part can be decoupled from the rest of the system, and writes

∂ t Π 0 Ṽ + Π 0 B(Π 0 Ṽ , Π 0 Ṽ ) = 0,
which can be rewritten, with the notation Π 0 Ṽ = (ā, ū, b),

∂ t ā + ū.∇ x ā = 0, ∂ t b + ū.∇ x b = 0, ∇ x (b 0 ā + τ 0 b) = 0, ∂ t ū + ū.∇ x ū + ∇ x p = 0, ∇ x .ū = 0.
This is the incompressible Euler system. Now it remains to show that the approximate solution we constructed satisfies the constraint (5.2). This is the object of the following result:

Proposition 5.3. With the above contruction of Ṽ N ε = (a N 1 , u N , b N 1 )
, the quantity (5.4)

A N ε = 1 ε 2 d dt Ω×R 3 e -εb N 1 b 0 |v| 2 e a0+εa N 1 -e -εb N 1 |v| 2 2b 0 1 + e a0+εa N 1 -e -εb N 1 |v| 2 2b 0
dvdx goes to 0 as ε → 0 for all N ≥ 1.

Proof. A N ε can be rewritten

A N ε = 1 ε 2 d dt (β N ε ) 3/2 F 2 (α N ε )dx
where

β N ε = b 0 e εb N 1 , α N ε = e a0+εa N 1
and

F p (α) = |v| p αe -|v| 2 2 1 + αe -|v| 2 2 dv.
Using the equality

αF ′ 2 (α) = 3 2 F 0 (α),
we compute

A N ε = 3b 3/2 0 2ε e 3 2 εb N 1 F 2 (α N ε )∂ t b N 1 + e 3 2 εb N 1 F 0 (α N ε )∂ t a N 1 dx.
We easily prove the two identities

e 3 2 εb N 1 (u.∇ x b N 1 + 2 3ε ∇ x .u N ) = ∇ x . 2 3ε e 3 2 εb N 1 u N and F 2 (α N ε )∇ x . 2 3ε e 3 2 εb N 1 u N dx = -e 3 2 εb N 1 F 0 (α N ε )u N .∇ x a N 1 dx, which imply that A N ε = 3b 3/2 0 2ε F 2 (α N ε )e 3 2 εb N 1 ∂ t b N 1 + u N .∇ x b N 1 + 2 3ε ∇ x .u N + F 0 (α N ε )e 3 2 εb N 1 ∂ t a N 1 + u N .∇ x a N 1 dx.
Using the fact that A(a N 1 , u N , b N 1 ) → 0 in L 2 and the conservation of the mass, we deduce that

A N ε -→ ε→0 0.

Useful intermediate results

In this section we define and study the linearized collision operator, and then give some bounds that will be very useful in the proof of Theorem 3.1.

The linearized collision operator.

As we look at solutions of (1.6) which are fluctuations around an equilibrium state, that is,

f ε = P ε (1 + εḡ ε ),
it makes sense to use the linearized collision operator, which is defined as

L ε g = S 2 R 3 B(v -v * , ω) M ε, * 1 + M ε, * 1 1 + M ′ ε 1 1 + M ′ ε, * × g(1 + M ε ) + g * (1 + M ε, * ) -g ′ (1 + M ′ ε ) -g ′ * (1 + M ′ ε, * ) dv * dω. Let us note (6.1) ν ε (v) = S 2 R 3 B(v -v * )M ε, * (1 + M ε ) (1 + M ε, * )(1 + M ′ ε )(1 + M ′ ε, * ) dv * dω,
and recall that

P ε = Mε 1+Mε with M ε = a ε e -|v-uε | 2 2bε
. Keep in mind that we will use L ε with the collision kernel B = ½.

D(L ε ) = {g ∈ L 2 (M ε dv) | ν ε g ∈ L 2 (M ε dv)} = L 2 (ν ε M ε dv).
and kernel

KerL ε = span 1 1 + M ε , v 1 + M ε , |v| 2 1 + M ε .
It can be decomposed as

L ε g(v) = ν ε (v)g(v) + K ε g(v)
where K ε is a compact integral operator on L 2 (M ε dv) and

0 < ν -≤ ν ε (v) ≤ ν + (1 + |v| β ) provided that (6.2) 1 C 0 ≤ a ε ≤ C 0 and 1 C 0 ≤ b ε ≤ C 0
with C 0 a positive constant independent of ε, and ν + , ν -depend only on C 0 and

C B from (1.4) (for B = ½ we have ν -= 8 √ 2π 5/2
(1+C0) 9/2 and ν + = 4πC 9/2 0 ). Remark 6.2. In the framework of Theorem 3.1, assumption (6.2) is satisfied, since we consider for a ε and b ε small perturbations around the constant values e a0 and b 0 .

Proof. Let g, h ∈ L 2 (M ε dv). Then hL ε gM ε dv = h(1 + M ε )L ε g M ε 1 + M ε dv = 1 4 S 2 B(v -v * , ω) M ε 1 + M ε M ε, * 1 + M ε, * 1 1 + M ′ ε 1 1 + M ′ ε, * × g(1 + M ε ) + g * (1 + M ε, * ) -g ′ (1 + M ′ ε ) -g ′ * (1 + M ′ ε, * ) × h(1 + M ε ) + h * (1 + M ε, * ) -h ′ (1 + M ′ ε ) -h ′ * (1 + M ′ ε, * ) dvdv * dω which easily shows that L ε is a non-negative self-adjoint operator. Moreover, letting h = g implies that g is in the nullspace of L ε if and only if, for almost all (v, v * , ω) ∈ R 3 × R 3 × S 2 , g(1 + M ε ) + g * (1 + M ε, * ) = g ′ (1 + M ′ ε ) + g ′ * (1 + M ′ ε, * ).
In other words, g(1 + M ε ) must be a collision invariant, and therefore the nullspace of L ε is made of all linear combinations of 1 1+Mε , vi 1+Mε , |v| 2 1+Mε (see [START_REF] Cercignani | The Mathematical Theory of Dilute Gases[END_REF]). The next step is to split L ε into two parts:

L ε g = ν ε (v)g + K ε h(v)
with ν ε defined by (6.1) and

K ε g(v) = S 2 R 3 B(v -v * , ω) M ε, * 1 + M ε, * 1 1 + M ′ ε 1 1 + M ′ ε, * × g * (1 + M ε, * ) -g ′ (1 + M ′ ε ) -g ′ * (1 + M ′ ε, *
) dv * dω. The bounds on ν ε can be proved using Grad's cutoff assumptions (1.3)-(1.4). They are uniform in ε thanks to (6.2). The operator K ε can be split into two operators in the following way:

K ε g(v) = K 1 ε g(v) -K 2 ε g(v) with K 1 ε g(v) = R 3 S 2 B(v -v * , ω)M ε, * 1 1 + M ′ ε 1 1 + M ′ ε, * g * dv * dω and K 2 ε g(v) = S 2 R 3 B(v -v * , ω) M ε, * 1 + M ε, * 1 1 + M ′ ε 1 1 + M ′ ε, * × g ′ (1 + M ′ ε ) + g ′ * (1 + M ′ ε, *
) dv * dω. In the classical case, the same decomposition holds, and a clever change of variables known as "Carleman's parametrization" allows to show that the following two operators are compact on L 2 (M ε dv) (see [START_REF] Saint-Raymond | Hydrodynamic limits of the Boltzmann equation[END_REF] for example):

K1 g(v) = R 3 S 2 B(v -v * , ω)M ε, * g * dv * dω K2 g(v) = R 3 S 2 B(v -v * , ω)M ε, * (g ′ + g ′ * )dv * dω.
It is easy to see that

K 1 ε g L 2 (Mεdv) ≤ K1 g L 2 (Mεdv) and K 2 ε g L 2 (Mεdv) ≤ K2 g L 2 (Mεdv) + K2 (M ε g) L 2 (Mεdv) from which we deduce that K ε is a compact operator on L 2 (M ε dv).
With these results, we can assert that L ε is coercive, and therefore is a Fredholm operator:

Corollary 6.3. There exists C L > 0 such that, for each g ∈ D(L ε ) ∩ (Ker(L ε )) ⊥ , gL ε gM ε dv ≥ C L g 2 L 2 (νεMεdv) .
If (6.2) holds, then C L depends only on C 0 and C B , but not on ε.

Proof. This result is a consequence of Proposition 6.1. Indeed, the multiplication operator g → ν ε g is self-adjoint on L 2 (ν ε M ε dv) and has continuous spectrum, namely the numerical range of

ν ε [ν - ε , +∞[ where ν -≤ ν - ε = inf v∈R 3 ν ε (v).
Then, since K ε is self-adjoint and compact on L 2 (M ε dv), Weyl's theorem ensures that the spectrum of L ε is made of [ν - ε , +∞[ and of a sequence of eigenvalues on the interval [0, ν - ε ] with ν - ε as unique possible accumulation point. Consequently, there exists a smallest eigenvalue λ 1 ε which is bounded by below by λ 1 if (6.2) holds, and the following spectral gap inequality is satisfied for each g ∈ D(L ε ) ∩ (Ker(L ε ))

⊥ :

gL ε gM ε dv ≥ C L g 2 L 2 (Mεdv) .
Recalling that

gL ε gM ε dv = ν ε g 2 M ε dv -gK ε gM ε dv
and using the continuity of K ε , we get the inequality with the weighted norm as stated above.

6.2. Useful bounds. This subsection lists some a priori bounds which are needed for the proof of convergence in the hydrodynamic limit. The first estimate comes from Young's inequality: Proposition 6.4. For z > -1 we define the function

h(z) = (1 + z) log(1 + z) -z. It satisfies h(z) ≥ 0 for z > -1 and p|z| ≤ λh * (p) + 1 λ h(z)
where h * is the Legendre transform of h h * (p) = max z>-1 (pz -h(z)) = e p -p -1.
The study of the function h is motivated by the relation

H(f ε |P ε ) = Ω×R 3 P ε h f ε -P ε P ε dvdx + Ω×R 3 (1 -P ε )h P ε -f ε 1 -P ε dvdx,
Proof. The first property is immediate from the definitions of h. The second one comes from Young's inequality

pz ≤ h * (p) + h(z),
supplemented with the two inequalities

h(|z|) ≤ h(z) z > -1 h * (λp) ≤ λ 2 h * (p) p ≥ 0, λ ∈ [0, 1].
It will be useful to work in L 2 since it is the natural space for the study of L ε , and then we will use renormalized fluctuations instead of the natural one defined as f ε = P ε (1 + εḡ ε ): Proposition 6.5. Let us define the renormalized fluctuations

ĝε = 1 ε f ε P ε -1 and ĥε = 1 ε 1 -f ε 1 -P ε -1 .
Then (6.3)

Ω×R 3 P ε ĝ2 ε dvdx + Ω×R 3 (1 -P ε ) ĥ2 ε dvdx ≤ 1 ε 2 H(f ε |P ε ).
Moreover, if (6.2) holds holds, (6.4)

ε P ε ĝε L ∞ (dvdx) + ε ĥε L ∞ (dvdx) ≤ 4 + C 0 .
Proof. Estimate (6.3) is a direct consequence of the definition of ĝε and ĥε and the following inequality:

x log x y -x + y ≥ ( √ x - √ y) 2 , ∀x, y > 0.
Recalling that 0 ≤ f ε ≤ 1 and that P ε is bounded indipendently of ε the second point comes by direct inspection.

In the L 2 setting, the collision kernel needs also to be renormalized. We repeat here the définition of the entropy dissipation:

D(f ε ) = 1 4 R 3 ×S 2 B ε (v -v * , ω) f ′ ε f ′ ε, * (1 -f ε )(1 -f ε, * ) -f ε f ε, * (1 -f ′ ε )(1 -f ′ ε, * ) • log f ′ ε f ′ ε, * (1 -f ε )(1 -f ε, * f ε f ε, * (1 -f ′ ε )(1 -f ′ ε, * ) dv * dω.
Proposition 6.6. Define the renormalized collision kernel by

qε = 1 P ε B ε Λ 1/2 ε f ′ ε f ′ ε, * (1 -f ε )(1 -f ε, * ) -f ε f ε, * (1 -f ′ ε )(1 -f ′ ε, * ) dv * dω, with Λ ε = M ε M ε, * (1 + M ε )(1 + M ε, * )(1 + M ′ ε )(1 + M ′ ε, * ) . Then qε 2 L 2 (ν -1
ε Pεdv) ≤ D(f ε ) where ν ε is the collision frequency defined by (6.1).

Note that the scaling condition (3.1) and the H theorem imply the following bound on the entropy dissipation

1 ε q+3 T 0 Ω D(f ε )dxds ≤ C.
Proof. This estimate is obtained by the Cauchy-Schwarz inequality:

q2 ε ≤ 1 P 2 ε B ε G 2 ε dv * dω B ε Λ ε dv * dω with G ε = f ′ ε f ′ ε, * (1 -f ε )(1 -f ε, * ) -f ε f ε, * (1 -f ′ ε )(1 -f ′ ε, *
). We easily see that

B ε Λ ε dv * dω = ν ε P ε 1 + M ε .
Next, using the classical inequality

(x -y) log x y ≥ 4( √ x - √ y) 2 , ∀x, y > 0,
we get the result.

Mixing together the previous estimates, we can prove a relaxation result. Define Π ε as the orthogonal projection in L 2 (P ε dv) on Ker L ε . The following proposition will be useful in the control of the flux terms: coupled with the control on the tails (3.5) and an interpolation argument, it will allow the control of the third moment of f ε . Proposition 6.7. Define ĝε and ĥε as in Proposition 6.5 and assume (6.2). Then

ĝε -ĥε 1 + M ε -Π ε ĝε -ĥε 1 + M ε L 2 (Pεdv) = O(ε) ĝε 2 L 2 (Pε) + ĥε 2 L 2 + ε 1 ε 2 P ε h f ε -P ε P ε dv + o(ε).
Proof. Plugging the identities

f ε = P ε (1 + εĝ ε ) 2 , 1 -f ε = (1 -P ε )(1 + ε ĥε ) 2 into f ′ ε f ′ ε, * (1 -f ε )(1 -f ε, * ) gives f ′ ε f ′ ε, * (1 -f ε )(1 -f ε, * ) = P ′ ε P ′ ε, * (1 -P ε )(1 -P ε, * )(1+εĝ ′ ε )(1+εĝ ′ ε, * )(1+ε ĥ)(1+ε ĥε, * )
which leads to the key following identity: Raising it to the square and dividing by ε 2 , we get, thanks to (6.4):

f ′ ε f ′ ε, * (1 -f ε )(1 -f ε, * ) -f ε f ε, * (1 -f ′ ε )(1 -f ′ ε, * ) = εΛ 1/2 ε ĝ′ ε -ĥ′ ε + ĝ′ ε, * -ĥ′ ε, * -(ĝ ε -ĥε ) -(ĝ ε, * -ĥε, * ) + ε 2 Λ 1/
1 C R 3 ĝε -ĥε 1 + M ε L ε ĝε -ĥε 1 + M ε M ε dv ≤ 1 ε 2 f ′ ε f ′ ε, * (1 -f ε )(1 -f ε, * ) -f ε f ε, * (1 -f ′ ε )(1 -f ′ ε, * ) 2 dvdv * dω + ε 2 ( ĝε 4 L 2 (Pεdv) + ĥε 4 L 2 (dv) ) (6.6)
where C depends on C 0 . Note that here we used the fact that, for a given integrable function u, we have (see [START_REF] Alexandre | Entropy dissipation and longrange interactions[END_REF] for more details)

R 3 ×S 2 u(v ′ )dvdω ≤ C R 3 u(v)dv. The next step is to decompose B = B ε + (B -B ε ), so that 1 ε 2 B f ′ ε f ′ ε, * (1 -f ε )(1 -f ε, * ) -f ε f ε, * (1 -f ′ ε )(1 -f ′ ε, * ) 2 dωdvdv * ≤ C ε 2 D(f ε ) + C ε 2 (B -B ε )f ε f ε, * dωdvdv * .
Recalling that f ε can be written

f ε = P ε (1 + 2εĝ ε + ε 2 ĝ2 ε ) and M ε = a ε e -|v-uε| 2 2bε
, and thanks to the following inequality (6.7)

½ |v-v * |> 1 ε 2 M ε M ε, * ≤ a 1/2 ε M 3/4 ε M 3/4 ε, * e - 1 16C 0 ε 4 , we get C ε 2 ½ |v-v * |> 1 ε 2 f ε f ε, * dωdvdv * ≤ C ε 2 e -1 Cε 4 + 2C ½ |v-v * |> 1 ε 2 P ε P ε, * ĝ2 ε dvdv * + 4εC ½ |v-v * |> 1 ε 2 P ε P ε, * ĝ2 ε ĝε, * dvdv * + Cε 2 ĝε 4 L 2 (Pεdv) . (6.8)
Using Proposition 6.4 we obtain for δ > 0

½ |v-v * |> 1 ε 2 P ε P ε, * ĝ2 ε dvdv * ≤ δ P ε, * P ε h f ε -P ε P ε dvdv * + ½ |v-v * |> 1 ε 2 P ε P ε, * h * 1 δε 2 dvdv * . We choose δ = 20C 0 ε 2 : ½ |v-v * |> 1 ε 2 P ε P ε, * ĝ2 ε dvdv * ≤ Cε 4 1 ε 2 P ε, * P ε h f ε -P ε P ε dvdv * 2 1/2 + o (1)
.

We now apply the following variant of Young's inequality

Y α ≤ 1 β α 1-α + βY, Y ≥ 0, β > 0, α ∈ (0, 1) with β = 1, α = 1 2 and Y = 1 ε 2 P ε, * P ε h fε-Pε Pε dvdv * 2 : ½ |v-v * |> 1 ε 2 P ε P ε, * ĝ2 ε dvdv * ≤ Cε 4 1 ε 2 P ε, * P ε h f ε -P ε P ε dvdv * 2 + o(1).
Since ε P ε, * ĝε, * is bounded, the same computation works for the second term in the right hand side of (6.8), so that we finally get

1 ε 2 B f ′ ε f ′ ε, * (1 -f ε )(1 -f ε, * ) -f ε f ε, * (1 -f ′ ε )(1 -f ′ ε, * ) 2 dωdvdv * ≤ Cε 2 ĝε 4 L 2 (Pεdv) + Cε 4 1 ε 2 P ε h f ε -P ε P ε dv 2 + o(1) since 1 ε 2 D(f ε ) = o(1)
. Pluging this inequality into (6.6) and using the coercivity of the linearized operator (Corollary 6.3) gives the result.

End of the proof

We now have all the ingredients to prove Theorem 3.1. We begin by controlling the flux terms, and then end the proof of the theorem. 7.1. Control of the flux terms. Our goal is to estimate the terms

1 ε 2 t 0 Ω×R 3 f ε b ε Ψ ε • ∇ x b 1 dxdvds and 1 ε 2 t 0 Ω×R 3 f ε (Φ ε • ∇ x u)dxdvds
with respect to the modulated entropy and the entropy dissipation. The main difficulty is that Ψ ε = O(|v| 3 ) whereas the modulated entropy allows us to control the powers of v only up to 2, via Young's inequality. We therefore try to gain as much integrability as possible. Here the relaxation estimate (6.

3) plays an central role. Since it gives a control on ĝε-ĥε 1+Mε , we use a new decomposition of f ε (7.1)

f ε = P ε + ε 2 P ε ĝε -ĥε 1 + M ε ĝε + ĥε + 2 ε
and we obtain the following estimate:

Proposition 7.1. Assume that Φε and Ψε have at most polynomial growth and that (6.2) holds. Then the flux terms are bounded by the following quantities:

1 ε 2 t 0 Ω×R 3 f ε b ε Ψ ε .∇ x b 1 dxdvds + 1 ε 2 t 0 Ω×R 3 f ε (Φ ε • ∇ x u)dxdvds ≤ C ε 2 t 0 D x (u, b 1 )(s) L 2 ∩L ∞ (Ω) H(f ε |P ε )(s) + Ω D(f ε )(s)dx ds + o(1).
Proof. The proof for the momentum flux (involving Φ ε ) is identical to the one with the energy flux (involving Ψ ε ), hence we will focus on the first one. Writing f ε as in (7.1), we use a first decomposition of the flux terms:

(7.2) 1 ε 2 R 3 Φ ε f ε dv = 2 ε Φ ε P ε ĝε -ĥε 1 + M ε dv + Φ ε P ε ĝε -ĥε 1 + M ε (ĝ ε + ĥε )dv since R 3 P ε Φ ε dv = 0.
To treat the first term in the right hand side, we use the identity (6.5), wich implies The first term can be estimated by the entropy dissipation as showed by the next lemma, while the other terms are controlled straightforwardly by the L 2 (M ε dv) norm of ĝε , and hence by the modulated entropy.

1 ε R 3 P ε ĝε -ĥε 1 + M ε Φ ε dv = R 3 M ε Φε 1 ε L ε ĝε -ĥε 1 + M ε dv = 1 ε 2 B Φε Λ 1/2 ε f ′ ε f ′ ε, * (1 -f ε )(1 -f ε, * ) -f ε f ε, * (1 -f ′ ε )(1 -f ′ ε, * ) dvdv * dω - B Φε Λ ε ĝ′ ε ĝ′ ε, * -ĝε
Lemma 7.2. Assume that Φε has at most a polynomial growth and that (6.2) holds.

Then the first term in the decomposition (7.3) is estimated by:

Ω B Φε Λ 1/2 ε f ′ ε f ′ ε, * (1 -f ε )(1 -f ε, * ) -f ε f ε, * (1 -f ′ ε )(1 -f ′ ε, * ) dvdv * dω 2 dx ≤C Ω D(f ε )dx + e - 1 16C 0 ε 4 Proof. Let G ε = f ′ ε f ′ ε, * (1 -f ε )(1 -f ε, * ) -f ε f ε, * (1 -f ′ ε )(1 -f ′ ε, * ); We decompose B Φε Λ 1/2 ε G ε dvdv * dω = B ε Φε Λ 1/2 ε G ε dvdv * dω + (B -B ε ) Φε Λ 1/2 ε G ε dvdv * dω. (7.4)
The first term is dealt with thanks to the Cauchy-Schwarz inequality and Proposition (6.6):

B ε Φε Λ 1/2 ε G ε dvdv * dω ≤ C B( Φε ) 2 Λ ε dv * dvdω 1/2 D(f ε ) 1/2 .
By Corollary 6.3, the coercivity inequality implies, since Φε ∈ (KerL ε ) ⊥ , that

B( Φε ) 2 Λ ε dv * dvdω ≤ C M ε (1 + M ε ) 2 (Φ ε ) 2 dv,
with C > 0 independent of ε.

The second term in the right hand side of (7.4), containing the high velocities, is handled thanks to the inequality The next terms in (7.3) are dealt with using Cauchy-Schwarz inequality and the a priori bound (6.4), and therefore are easily bounded by C( ĝε

2 L 2 [Pεdv + ĥε 2 L 2 ).
Coming back to (7.2), the second term on the right hand side is splitted as follows, in order to use the relaxation estimate:

Φ ε P ε ĝε -ĥε 1 + M ε (ĝ ε + ĥε )dv = Φ ε P ε ĝε -ĥε 1 + M ε -Π ε ĝε -ĥε 1 + M ε (ĝ ε + ĥε )dv + Φ ε P ε Π ε ĝε -ĥε 1 + M ε (ĝ ε + ĥε )dv
The last term is easily handled:

Φ ε P ε Π ε ĝε -ĥε 1 + M ε (ĝ ε + ĥε )dv ≤ C( ĝε L 2 (Pεdv) + ĥε L 2 )
while we need some control on the high velocities for the first one. From the hypothesis (3.5), we deduce that

R 3 P ε f ε P ε 2p dv ≤ C p a.e.
for all p < 1 and uniformly in ε (which depends on p), since the moments of P ε differ from that of P 0 only by terms of order ε. This estimate leads to some control on the rescaled fluctuations:

ε|ĝ ε | + ε| ĥε | = O(1) L ∞ t,x L 4p (Pεdv)
so that (7.5)

ε ĝε -ĥε 1 + M ε -Π ε ĝε -ĥε 1 + M ε = O(1) L ∞ t,x L 4p (Pεdv) .
We easily find the following interpolation inequality for all functions φ ∈ L 2 ∩ L 4p

φ L 8p 2p+1 ≤ φ 1/2 L 2 φ 1/2 L 4p ,
which implies, together with (7.5) and Proposition 6.7:

ĝε -ĥε 1 + M ε -Π ε ĝε -ĥε 1 + M ε L 8p 2p+1 (Pεdv) = O ĝε L 2 (Pεdv) + ĥε L 2 + 1 ε 2 P ε h f ε -P ε P ε dv 1/2
+ o(1).

Choosing p > 1 2 implies 8p 2p+1 > 2, and Hölder's inequality gives such that taking φ 1 = ĝε + ĥε , φ 2 = ĝε-ĥε 1+Mε -Π ε ĝε-ĥε 1+Mε and φ 3 = Φ ε leads to the final estimate:

Φ ε P ε ĝε -ĥε 1 + M ε -Π ε ĝε -ĥε 1 + M ε (ĝ ε + ĥε )dv = O ĝε 2 L 2 (Pεdv) + ĥε 2 L 2 + 1 ε 2 P ε h f ε -P ε P ε dv + o(1)
7.2. Proof of convergence. In view of the preceding results, we are now able to prove Theorem 3.1. Proposition 4.1 together with Proposition 7.1 imply the following:

1 ε 2 H(f ε |P ε )(t) + 1 ε q+3 t 0 Ω D(f ε )dxds ≤ 1 ε 2 H(f ε,in |P ε,in ) - 1 ε t 0 Ω×R 3 f ε 1, v -εu b ε , |v -εu| 2 2b ε .A ε (a 1 ,

Ω×R 3 e

 3 -εb1 b 0 |v| 2 e a0+εa1-e -εb 1 |v| 2 2b 0 1 + e a0+εa1-e -εb 1 |v| 2 2b 0 dvdx → 0. Remark 5.1. B is a bilinear operator since we can decompose τ ε as τ ε = τ 0 + ετ 1 + o(ε) and τ 1 is a linear combination of a 1 and b 1 .

½ 3 1 16bε ε 2 , 1 32bε ε 4 .

 31214 |v-v * |> 1 ε M ε M ε, * ≤ a 1/2 ε M using that Φε (1 + M ε ) is at most polynomial: (B -B ε ) Φε (1 + M ε )Λ 1/2 ε Gdvdv * dω ≤ Ce -

φ 1 φ 2 φ 3 P

 3 ε dv ≤ φ 1 L 2 (Pε) φ 2

1 + 0 D

 10 e a0+εa1-e -εb 1 |v| 2 x (u, b 1 )(s) L 2 ∩L ∞ (Ω) H(f ε |P ε )(s) + Ω D(f ε )(s)dx ds + o(1).

  ĝε, * + ĥε ĥε, * -ĥ′

	(7.3)
							ε	ĥ′ ε, * + ĝ′ ε ĥε -ĝε	ĥ′ ε
		+ ĝ′ ε ĥε, * -ĝε	ĥ′ ε, * + ĝ′ ε, * ĥε -ĝε, *	ĥ′ ε + ĝ′ ε, * ĥε, * -ĝε, *	ĥ′ ε, * dvdv * dω
	-ε	B Φε Λ ε	ĝ′ ε	ĝ′ ε, * ( ĥε + ĥε, * ) -ĝε ĝε, * ( ĥ′ ε + ĥ′ ε, * )
		+ ĥε ĥε, * (ĝ ′ ε + ĝ′ ε, * ) -ĥ′ ε	ĥ′ ε, * (ĝ ε + ĝε, * ) dvdv * dω
	-ε 2	B Φε Λ ε	ĝ′ ε	ĝ′ ε, * ĥε ĥε, * -ĝε ĝε, *	ĥ′ ε	ĥ′ ε, * dvdv * dω.

Acknowledgment

This work was partialy supported by the GDRE 224 CNRS-INdAM GREFI-MEFI and by Projet CBDif-Fr ANR-08-BLAN-0333-01. I thank the anonymous referee for his comments that helped a lot improving this manuscript.

We plug the functions (a ε,N 1 , u ε,N , b ε,N 1 ) constructed in Theorem 5.2 in the approximate solution P ε . It follows by Gronwall's lemma that

The last term go to 0 as ε → 0 and N → 0 thanks to Theorem 5.2, using the fact that

The fist term can be decomposed as

P N ε,in

1 -P N which is a stronger result respect to the one stated above.


Marc Milesi

Yvan Chastel

Elie Hachem

Roland E Logé

Pierre-Olivier O Bouchard

M Bernacki

Digital microstructures matching statistical distributions of features in real materials -Example of forgings

Keywords: Statistical similar microstructures, Digital Material, Forging

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

Introduction

Over the last decades, micromechanics models [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF] have been developed in order to approximate the material behaviour by taking into account explicitly the heterogeneous nature of materials. Today, explicit representations of microstructures are commonly constructed for allowing calculations meant to determine local variables and fields such as strains or stresses. Such representations require an important investment in terms of computing power to the quantity of elements to obtain a Representative Elementary Volumes (R.E.V.) which would be statistically representative of the real material organization. Several methods have been reported in the literature to generate microstructures by means of topological mappings [START_REF] Lilliu | 3D lattice type fracture model for concrete[END_REF][3] [START_REF] Yang | Simulation and quantitative assessment of homogeneous and inhomogeneous particle distributions in particulate metal matrix composites[END_REF]. Rigorous statistical approaches are needed to perform such realistic microstructures. Data needs to be determined from fairly intensive experimental analyses so as to obtain the needed mechanical characteristics of each constituent and local geometrical features. The first part of this work describes the panel of different experimental methods which were performed. The second part provides an overview of numerical tools used to perform numerical computations on R.E.Vs. The third part presents the link between values and fields obtained at different microstructure scales to simulate fatigue properties of a forging component.

Microstructure characterization

In a first approach, two main categories of microscopic constituents are often defined and used to describe a metallic microstructure : grains and inclusions (or voids). Scanning Electron Microscopy (S.E.M.) together with Electron Back-Scattered Diffraction (E.B.S.D.) analyses were performed to determine the statistics of morphological characteristics of the microstructure and of all constituents which are known to be the key parameters governing the fatigue behaviour of the material. Note that some of these features are known to have a bigger impact on fatigue behaviour than others. E.B.S.D. analyses of the grains of the matrix for the material of interest in this study display a fairly random crystallographic texture. Therefore, the anisotropic mechanical response of the R.E.V. and of the component is assumed to be induced by the presence and the organisation of inclusions in a matrix, and the crystallographic nature of the grains is not taken into account in the R.E.V. representation.

Forging processes are known to induce orientations of particles which cause the final material anisotropy. And indeed, several analyses and tests on the material of interest for this work have showed that the anisotropy of the final component was essentially due to the orientation of inclusions clusters, mainly MnS (manganese sulfide) inclusions. Statistical studies were performed in order to determine all parameters meant to generate a realistic R.E.V. The shapes of the inclusions were approximated as ellipsoids which match the statistics of the real morphology of MnS inclusions. Note that these inclusions are gathered in cluster (Figure 1). The method of the "Localized cluster" [START_REF] Yang | Simulation and quantitative assessment of homogeneous and inhomogeneous particle distributions in particulate metal matrix composites[END_REF] was chosen to generate the cluster by controlling the number of inclusions and their widths. Each parameter follows a Gaussian (Euler orientation) or a log-normal distribution (length, width, thickness, number …) [START_REF] Milesi | Impact des caractéristiques microstructurales des pièces forgées sur leur tenue en fatigue à grand nombre de cycles : Modélisation multi-échelle et validation expérimentale[END_REF]. An example of stochastic data about the length and the orientation of the inclusions is represented in Figure 2 : statistical distributions are in good agreement with experimental results. Precise statistics were determined for components at each of the forging stages for an industrial sequence. To apply such distributions and to take into account all stochastic parameters, a specific algorithm has been developed [START_REF] Milesi | Impact des caractéristiques microstructurales des pièces forgées sur leur tenue en fatigue à grand nombre de cycles : Modélisation multi-échelle et validation expérimentale[END_REF] [START_REF] Milesi | Explicit microstructure fatigue analysis of forged components[END_REF]. Numerical statistical selections were performed using a Box-Müller method [START_REF] Box | A note on the generation of random normal deviates[END_REF] with random sampling carried out with the "mersenne twister" algorithm [START_REF] Matsumoto | Mersenne twister: A 623-dimensionnally equidistributed uniform pseudorandom number generator[END_REF]. Figure 3 shows an example of such a digital cluster. Each inclusion is separated using its specific topology in order to define it independently from the other inclusions. A robust and automatic 3D mesher (MTC) was used in order to refine the mesh in the vicinity of each inclusion [START_REF] Coupez | Ducloux: Parallel meshing and remeshing[END_REF]. The meshing operation (global meshing and the size of all elements) allows to control the local refinement of the topology in the vicinity of each interface. The interface of each inclusion can then be refined and, overall, an 'anisotropic' (in the sense of an anisotropic metrics of the finite elements) mesh is produced. This anisotropic mesh prevents numerical stress discrepancy which could be due to finite element methods with linear approximation of the constitutive law near the interfaces.

Computation of the mechanical response of a R.E.V.

An elastic-plastic or viscoplastic finite element computation can then be performed in order to study the influence of particles distribution and morphology on the stress field throughout the R.E.V. Inclusions are assumed -and this is checked later on-to remain in the elastic regime. A lognormal statistical distribution of the Young's moduli for the MnS inclusions was determined by nano-indentation. Then, a value of Young's modulus is associated to each inclusion. The matrix is considered to follow a standard plastic Hollomon law. Interestingly, the role of MnS inclusions on the global response was found not to be significant on the global intensity during a forming operation. And the parameters of the Hollomon law could therefore be determined from standard tensile tests in the longitudinal direction.

In order to assess the role of each morphological parameter of inclusions in a standard high cycle fatigue simulation, a total of 150 R.E.Vs were constructed and studied under different loading modes. The analyses of the calculations concentrated on different characteristics. Figure 5 describes an example of such simulations. The fatigue criterion used was Papadopoulos criterion [START_REF] Milesi | Impact des caractéristiques microstructurales des pièces forgées sur leur tenue en fatigue à grand nombre de cycles : Modélisation multi-échelle et validation expérimentale[END_REF][6] [START_REF] Papadopoulos | Fatigue limit of metals under multiaxial stress conditions: the microscopic approach[END_REF] which gives a critical domain under which the structure is found to be safe. Each parameter had to be carefully assessed to provide, statistically, this critical domain at the mesoscopic scale.

Linking lengthscales in a forming process analysis

The morphology of the particles depends essentially on the strains undergone by the material during the forging process. Two macroscopic parameters are commonly used to assess the impact of such forming history : -the kneading rate, defining the deformation undergone by the part during forming [START_REF] Milesi | Impact des caractéristiques microstructurales des pièces forgées sur leur tenue en fatigue à grand nombre de cycles : Modélisation multi-échelle et validation expérimentale[END_REF] ; -the direction of microstructural flow orientation given by a fiber vector [5][6].

In our approach, morphological characteristics at the microscopic scale are described with these two parameters. For instance, the orientation of inclusions can be associated to the fiber vector, and the shape and organisation of a cluster is connected to the kneading rate.

The industrial case chosen in this study is a suspension component described in Figure 6. Various virtual fatigue tests at the microscopic scale were performed on R.E.Vs with different kneading rate and anisotropy orientation (fiber vector) distributions. These fatigue simulations at the microscopic scale were needed to determine two material parameters in the anisotropic Papadopoulos criterion [START_REF] Papadopoulos | Fatigue limit of metals under multiaxial stress conditions: the microscopic approach[END_REF]. Indeed, the only mean to obtain these coefficients without experimental tests (expensive and difficult to perform) is to simulate several digital cases. Figure 7 shows the endurance limits, which were used to determine these Papadopoulos coefficients, for different configurations of R.E.V.s and relying on macroscopic results of the forging component. The values of the two Papadopoulos coefficients are necessary to perform high cycle fatigue tests on the real components. Moreover, this multi-scale approach gives not only one value, but a distribution of these coefficients.

A numerical testing campaign for various Papadopoulos coefficients provided standard deviation prediction of the endurance limit for real forgings (Table 1.) Critical areas in terms of fatigue resistance can then be determined in the thickest area of the suspension arm.

Critical values of the Papadopoulos criterion

Conclusion

A multi-scale pragmatic approach for high cycle fatigue computation was presented and applied to forging applications. Microstructural flow orientation and kneading rate have been taken into account in order to perform fatigue analyses. This approach consists in describing the microstructure morphology and rheological characteristics of the main microstructure constituents in order to create numerically adapted elementary volumes. Fatigue simulations were then performed on different statistically representative morphologies to obtain material coefficients of the Papadopoulos criterion which allows to describe the anistropy of the material. Results seem to be more conservative than experimental endurance limits but they remain within the standard deviations. It is also possible to obtain, at the end of the forging process, anisotropic characteristics which can be linked to material coefficients obtained statistically from calculations on elementary volumes. Endurance limits obtained for the real forged components are quantitatively and qualitatively close to the experimental results.

Perspectives

This method is being confronted to R.E.Vs obtained from tomography imaging. The reconstruction from different cross sections gives a more realistic description of the material. As an example of such representation, Figure 9 shows a R.E.V. reconstructed from experimental data. And it is possible to perform simulations on such R.E.V.s following the same approach as previously described.

Figure 1 .

 1 Figure 1. Example of a MnS cluster (S.E.M.)

Figure 2 .Figure 3 .

 23 Figure 2. Examples of statistic representation of MnS inclusions

Figure 4 .

 4 Figure 4. Mesh adaptation near inclusions using MTC Mesher.

Figure 5 .

 5 Figure 5. Numerical simulation of a tensile fatigue test for a specific loading direction

Figure 6 .

 6 Figure 6. Experimental fatigue test bench on the real component

Figure 7 .

 7 Figure 7. Determination of the fatigue limit evolution from different calculations of various digital R.E.Vs with MnS clusters.

Figure 8 .

 8 Figure 8. Most critical area predicted by the modified Papadopoulos criterion Numerical predictions are in good agreement with experimental results, quantitatively and qualitatively.

Figure 9 .

 9 Figure 9. Numerical reconstruction of a R.E.V. from imaging obtained by tomography.

Table 1 .

 1 Example of predictions of endurance limit forces

Acknowledgement

This work has been performed within the ANR (French National Research Agency) Optiforge project, in collaboration several industrial partners (Ascoforge, Ascometal, Cetim, PSA, Setforge, Transvalor) and academic institutions (MINES ParisTech -CEMEF, Arts et Métiers Paris-Tech -Angers, INSA Lyon).