FRANC ¸OIS GOLSE Francesco Salvarani

THE ROSSELAND LIMIT FOR RADIATIVE TRANSFER IN GRAY MATTER

Keywords: 85A25 35K55 (82A70, 35B25, 45K05) Radiative transfer, diffusion approximation, Rosseland equation, Relative entropy, Compensated compactness

This paper establishes the Rosseland approximation of the radiative transfer equations in a gray atmosphere -i.e. assuming that the opacity is independent of the radiation frequency. The problem is set in a smooth bounded domain Ω of the Euclidian space R 3 , assuming that the incoming radiation intensity at the boundary of Ω is Lipschitz continuous and isotropic -including for instance the case of a blackbody radiation with nontrivial temperature gradient.

Presentation of the model

Radiative transfer is an important phenomenon in many branches of physics and, in many situations, the most significant mode of transmission of energy.

As established in all textbooks on radiation hydrodynamics (see, for example, [START_REF] Mihalas | Foundations of Radiation Hydrodynamics[END_REF]), radiative transfer in a host medium can be described by the following set of partial differential equations, with unknowns the radiative intensity I = I(t, x, ω, ν) and the temperature T = T (t, x). The independent variables in these equations are the time t ∈ [0, τ ], τ > 0, the position x ∈ Ω ⊆ R 3 , the spherical angle with respect to a suitable reference frame ω ∈ S 2 and the frequency ν ∈ R + . With these notations, the radiative transfer equations are (1)

         1 c ∂I ∂t + ω • ∇ x I + Σ(T, ν)(I -B ν (T )) = 0 ∂E ∂t (T ) = +∞ 0 Σ(T, ν)( Ī -B ν (T )) dν,
where B ν = αν 3 e hν/kT -1 Ī = 1 4π S 2 I(t, x, ω, ν) dω.

1

The parameter c > 0 is the speed of light and B ν (T ) is Planck's formula for the black-body radiation at temperature T , the parameters h and k being the Planck and the Boltzmann constants respectively. We have introduced the normalization constant α = 15 h kπ 4 , so that Stefan's law assumes the simple form:

+∞ 0 B ν (T ) dν = T 4 .
The function Σ = Σ(T, ν), is the opacity -or absorption cross-section -of the host medium. In general, the opacity takes into account how energy is transferred from the incoming radiation to the electrons in the background medium. Its expression is complicated in general, as it must take into account the contributions of bound electrons, of free electrons as well as transitions from bound to free states. The respective importance of each such transition in the opacity strongly depends upon the nature of the background and the temperature. (For instance, bound-bound transitions are more important for heavy elements; however, their importance decreases at high temperatures as most electrons are in free states). We refer to chapter VII of [START_REF] Pomraning | The Equations of Radiation Hydrodynamics[END_REF] and to [START_REF] Alexander | Low temperature Rosseland opacities[END_REF] for more information on this subject.

Thus we seek to establish the validity of the Rosseland approximation under the mildest possible assumptions on the opacity, since so little is known about it in general.

There are however, two particular features of the opacity that complicate the Rosseland approximation, and that we wish to take into account in the present paper.

Specifically, our proof of the Rosseland approximation allows considering opacities that may tend to infinity as T → 0 + . As we shall see, such opacities lead to a limit equation that is parabolic degenerate, and hence has singular solutions.

This kind of behavior of the opacity can be seen on the example of the Kramers opacity (an explicit formula for the opacity in the case of free-free transitions and for a hydrogen-like atom). The Kramers opacity is of the form

Σ K (T, ν) = C
1 -e -hν kT (hν) 3 (kT ) 1/2 1 + O kT hν -see fla (7.72) on p. 174 in [START_REF] Pomraning | The Equations of Radiation Hydrodynamics[END_REF]. In this example Σ(T, ν) ∼ T -1/2 . Otherwise, the opacity can oscillate strongly because of bound-free and bound-bound transitions.

In view of the above considerations, we only assume in this paper that the opacity satisfies Σ(T, ν) ∼ T -λ as T → 0, with λ ∈ (0, 1].

The other important nonlinear term in system (1) is the internal energy E = E(T ). This quantity depends on the temperature, but the experimental curves suggest that its dependence upon T is simpler than that of the cross section.

Henceforth, we assume that E is increasing and such that E (T ) ∼ T γ as T → 0, where γ ∈ [0, 2]. This assumption includes the perfect gas law, E(T ) = c V T . Without loss of generality we further assume that E(0) = 0.

In some relevant physical situations, for example in the external layers of a stellar atmosphere [START_REF] Mihalas | Foundations of Radiation Hydrodynamics[END_REF], the opacity is independent of the photons frequency. It is therefore convenient to integrate system (1) over frequencies and obtain the following system of equations, called the gray model:

(2)          1 c ∂u ∂t + ω • ∇ x u + Σ(T )(u -T 4 ) = 0 ∂E ∂t (T ) = Σ(T )(ū -T 4 ), ū = 1 4π S 2 u(t, x, ω) dω.
Here, the unknowns are the photon density

u = u(t, x, ω) = +∞ 0 I(t, x, ω, ν) dν
and, again, the temperature T = T (t, x).

In this paper, we restrict our attention to the gray system [START_REF] Bardos | The Rosseland approximation for the radiative transfer equations[END_REF]. The mathematical study of system (1) will be left for future research.

We consider situations where the interaction between radiation and matter is the dominant phenomenon.

Through multiple interactions, the photon experiences a sort of random walk in the host medium. Since the equilibrium (Planck's function) has mean zero velocity, it is therefore natural to investigate a diffusive regime for the photons gas.

From a mathematical point of view, this would mean that it is possible to obtain diffusion-type equations through the usual parabolic scaling:

(3)

         ε c ∂u ε ∂t + ω • ∇ x u ε + 1 ε Σ(T ε )(u ε -T 4 ε ) = 0 ε 2 ∂E ∂t (T ε ) = Σ(T ε )(ū ε -T 4 ε ), ūε = 1 4π S 2 u ε (t, x, ω)dω.
Here ε > 0 is a non-dimensional quantity which represents the ratio between the mean free path of the photons and some characteristic length of the host medium (for example, in the case of the Sun, the mean free path is typically of the order of magnitude of a few centimeters, which is negligeable if compared to the size of the Sun).

It is therefore obvious to look at the behavior of the angular average of the photon density, governed by system (3), when ε → 0.

The first step in obtaining the formal limit is the definition of the macroscopic density

ρ ε (t, x) = ūε (t, x, ω) = 1 4π S 2 u ε (t, x, ω) dω
and the flux vector j ε (t, x) = (j

ε (t, x), j (1)
ε (t, x), j (2)
ε (t, x)):

j (i) ε (t, x) = 1 4πε S 2 ω (i) u ε (t, x, ω) dω,
where ω (i) denotes the i-th component of the vector ω.

Then, by integrating the first equation of System (3), it is easy to deduce that ∂ ∂t

ρ ε c + E(T ε ) + ∇ x • j ε = 0.
Moreover, integrating the first equation of System (3) with respect to ω in S 2 after multiplying each side of this equation by ω (i) gives

ε 2 c ∂j (i) ε ∂t + 1 4π S 2 ω (i) ∇ x • (ωu ε ) dω + Σ(T ε ) j (i) ε = 0.
Assume the existence of the two limits:

ρ = lim ε→0 ρ ε , j = lim ε→0 j ε .
If j ε is bounded in some sense and that lim ε→0

ε 2 E t (T ε ) = 0 lim ε→0 ε 2 (j ε ) t = 0,
the second equation of (3) implies that lim ε→0

T 4 ε = ρ.
At leading order in ε, System (3) forces u to be independent of ω, so that ρ = u. Moreover, ρ solves the following diffusion-type equation, which is known in the literature under the name of Rosseland equation:

ρ c + E(ρ 1/4 ) t -∇ x • ∇ x ρ 3Σ(ρ 1/4 ) = 0.
One major difficulty with the Rosseland approximation is about boundary conditions. As is well known, natural boundary conditions for transport equations and for diffusion equations are of a very different nature -for instance, Dirichlet boundary conditions are admissible for diffusion equations, not for transport equations. A typical instance of natural boundary condition for a transport equation consists of prescribing the density of particles entering the spatial domain where the equation is to be solved.

So far, the Rosseland approximation has been established in the case of matter radiating in vacuum -see [START_REF] Bardos | The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation[END_REF].

In the more complicated case of a prescribed incident radiation intensity, the Rosseland approximation is based on asymptotic analysis involving nontrivial boundary layer analysis: see [START_REF] Larsen | Asymptotic analysis of radiative transfer problems[END_REF], and [START_REF] Bardos | The Rosseland approximation for the radiative transfer equations[END_REF] for a mathematical justification of these asymptotic ansatz, based on somewhat unrealistic monotonicity assumptions on the opacity.

Here, we discuss the case of an incident radiation field that is a Planck distribution (thereby avoiding boundary layers), however with a possible temperature gradient at the boundary. This type of boundary condition cannot be handled with the ideas of either [START_REF] Bardos | The Rosseland approximation for the radiative transfer equations[END_REF][START_REF] Bardos | The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation[END_REF] for realistic opacities.

Mathematical assumptions and results

We start this section by declaring some notation which we will use in the sequel.

We will work in a bounded domain Ω ⊂ R 3 with boundary Γ = ∂Ω of class C 1 .

If we denote with n x the outer normal to the boundary of Γ in x, we can define the sets Γ -, Γ + and Γ 0 , subsets of Γ, as:

Γ -= {x ∈ Γ : ω • n x < 0}, Γ + = {x ∈ Γ : ω • n x > 0} and Γ 0 = {x ∈ Γ : ω • n x = 0}
. Moreover, since in the paper we will heavily use indices, in order to avoid confusion, any vector v ∈ R n will be written in the following way, when it will be convenient to give the components in explicit form: v = (v (1) , . . . , v (n) ).

We are now ready to give a brief presentation of our results. Goal of the paper is the mathematical study of system (3): we prove existence of a weak solution and we deduce rigorously the Rosseland approximation as a diffusive limit of such system.

We assume that the opacity and the internal energy satisfy the following properties: Definition 2.1. Let Σ = Σ(T ) and E = E(T ) be the opacity and the internal energy of system (3). We say that Σ and E are admissible if and only if

(1) Σ(y) > 0 for any y > 0 and is of class C 1 ((0, +∞));

(2) lim y→+∞ Σ(y) = 0;

(3) lim y→0 Σ(y) = +∞ and Σ(y) ∼ y -β when y → 0, with β ∈ (0, 1].

(4) E is continuous in [0, +∞) and belongs to the class C 1 ((0, +∞)).

Moreover E(0) = 0 and E (y) ≥ 0 for all y > 0.

(5) E (y) ∼ y α for y → 0, where α

∈ [0, 2].
In what follows, we will consider the initial-boundary value problem for System (3), with (t, x, ω) ∈ (0, τ ) × Ω × S 2 , supplemented with appropriate initial and boundary conditions.

Our first theorem concerns an existence result:

Theorem 2.2. Let us consider System (3) for (t, x, ω) ∈ (0, τ ) × Ω × S 2 , with initial conditions

u ε (0, x, ω) = u in (x, ω) ∈ L ∞ (Ω × S 2 ) T ε (0, x) = T in (x) ∈ L ∞ (Ω)
and boundary data

u ε (t, x, ω)| Γ -= u b (t, x, ω) ∈ L ∞ ((0, τ ) × ∂Ω × S 2 ),
where Ω ⊂ R 3 is a bounded domain with boundary of class C 1 and ε > 0.

Then System (3) admits a weak solution

(u ε (t, x, ω), T ε (t, x)) ∈ L ∞ ((0, τ ) × Ω × S 2 ) × L ∞ ((0, τ ) × Ω).
This existence result improves on the earlier references [START_REF] Mercier | Application of accretive operators theory to the radiative transfer equations[END_REF][START_REF] Golse | Generalized solutions of the radiative transfer equations in a singular case[END_REF] which used unrealistic monotonicity assumptions on the opacity σ, even in the gray case. It also improves on the existence result in [START_REF] Bardos | The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation[END_REF], which further reduces the system of two equations (2) to a scalar equation on the single unknown u by neglecting the time-derivative of the internal energy in the second equation of ( 2). An earlier attempt at treating the system (2) can be found in [START_REF] Dogbe | The radiative transfer equations: diffusion approximation under accretiveness and compactness assumptions[END_REF], with however a lesser degree of generality in the opacity and internal energy admissible.

Moreover, we give a rigorous justification of the Rosseland limit equation of the radiative transfer system. Here, in order to avoid additional difficulties dues to the presence of a boundary layer, we do not allow that the boundary data depend on ω.

The result is summarized in the following theorem:

Theorem 2.3. Let us consider System (3), posed for (t, x, ω) ∈ (0, τ ) × Ω × S 2 , with initial conditions

u ε (0, x, ω) = u in (x, ω) ∈ L ∞ (Ω × S 2 ) T ε (0, x) = T in (x) ∈ L ∞ (Ω)
and boundary data

u ε (t, x, ω)| Γ -= u b (t, x) ∈ W 1,∞ ((0, τ ) × ∂Ω),
where Ω ⊂ R 3 is a bounded domain with boundary of class C 1 and ε > 0.

Let {u ε } be a family of solutions of System (3) which satisfy the same initial and boundary conditions.

Then, there exists a function ρ ∈ L ∞ ((0, τ ) × Ω) such that ρ = lim ε→0 ūε strongly in L 2 ((0, τ )×Ω). The limit function ρ moreover solves the following initial-boundary value problem in (0, τ ) × Ω, for τ > 0:

ρ c + E(ρ 1/4 ) t -∇ x • ∇ x ρ 3Σ(ρ 1/4 ) = 0.
with initial conditions ρ(0, x) = ūin (x, ω) and boundary data ρ(t, x)| ∂Ω = u b (t, x).

For technical reasons, the regularity condition on the boundary data for the family {u ε } in Theorem 2.3 is slightly more stringent than the requirements on the boundary conditions in Theorem 2.2.

It is interesting to note that, even if only the incoming density on the boundary is prescribed for the family {u ε }, we obtain that the boundary condition for the limit is fully defined.

The Rosseland approximation has been established in [START_REF] Bardos | The Rosseland approximation for the radiative transfer equations[END_REF] by a method based on a multiscale expansion in powers of ε, assuming that the opacity Σ is decreasing, while T → Σ(T )T 4 is increasing.

In [START_REF] Bardos | The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation[END_REF], the same Rosseland approximation for a reduced version of the gray model ( 2) neglecting the time derivative of E(T ) in the energy balance equation. The method in [START_REF] Bardos | The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation[END_REF] used compactness arguments based on velocity averaging, and hence could handle the most general class of opacities imaginable. However, it assumed an absorbing boundary conditioncorresponding to radiation expanding in the vacuum. The same method could also handle the case of a incoming black-body radiation at a constant temperature. The case of a temperature gradient at the boundary, treated in the present paper, requires a new idea for controling the radiation flux, which is the key to the Rosseland approximation.

Finally, we would like to mention the reference [START_REF] Goudon | Approximation by homogenization and diffusion of kinetic equations[END_REF], where a nonlinear diffusion approximation is established by compensated compactness -at variance with the argument in [START_REF] Bardos | The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation[END_REF] based on velocity averaging instead.

Because of the conditions on the opacity given in Definition 2.1, the Rosseland equation is degenerate parabolic, so that the temperature may display some singularity in the form of a propagating front when radiation penetrates into cold matter. Hence result above proves that the Rosseland approximation is robust in the sense that it holds even in situations in which such singularities may occur -something that is not at all clear with classical asymptotic expansions.

The method for proving existence of a weak solution (Theorem 2.2) uses a compactness argument based on a maximum principle and velocity averaging. The main difficulty is that the absorption-emission nonlinear term is not a Lipschitz continuous perturbation of the transport operator, since the opacity tends to infinity as the temperature vanishes.

The method for proving the Rosseland approximation is based on using the relative entropy to control the radiation flux. Usually, the relative entropy method is used when solutions of the target equation are known to be smooth. This is not the case here, as temperature may fail to be differentiable at interfaces with cold matter. Here, as in the case of Carleman's kinetic model [START_REF] Golse | The nonlinear diffusion limit for generalized Carleman models: the initial-boundary value problem[END_REF], we use both relative entropy and a compactness argument that allows for possibly singular solutions of the target equations. Relative entropy is not used all the way through the proof to control the distance between the radiative intensity and the (fourth power of) the temperature that solves the Rosseland limiting equation, but only to control the radiation flux. This control is in turn one essential step in the following compactness argument.

That it is possible to use relative entropy within a compactness argument to establish the validity of such a macroscopic (or hydrodynamic) limit seems to be special to the case of limits leading to a pure diffusion equation (degenerate or not), but without streaming term. In particular, the strategy presented here does not seem to apply to the derivation of the incompressible Navier-Stokes equations from the Boltzmann equation.

From a mathematical point of view, it will be more convenient to work with two new unknowns: the photon density u = u(t, x, ω) and the normalized temperature θ = θ(t, x), the last one being defined as θ = T 4 . Moreover, we will set c = 1.

We write now the radiative transfer equations (3) with respect to the two new unknowns.

We first define the normalized cross section σ and the internal energy

E as σ(θ) = Σ(θ 1/4 ), E(θ) = E(θ 1/4
). The pair (σ, E), must satisfy the same constraints we have imposed on the pair (Σ, E). The precise meaning of admissible cross section and energy is given in the following definition: (1) σ(y) is strictly positive for any y > 0 and is of class C 1 ((0, +∞));

(2) lim y→+∞ σ(y) = 0;

(3) lim y→0 σ(y) = +∞ and σ(y) ∼ y -β when y → 0. We will always suppose that β ∈ (0, 1/4].

(4) E is continuous in [0, +∞) and is of class C 1 ((0, +∞)). Moreover E(0) = 0 and E (y) ≥ 0 for all y > 0.

(5) E (y) ∼ y α for y → 0, where α ∈ [-3/4, -β], where β is the same exponent that gives the behavior of σ when y → 0.

Then, System (3) becomes, in the new unknowns,

         ε 2 ∂u ε ∂t + εω • ∇ x u ε + σ(θ ε )(u ε -θ ε ) = 0 ε 2 E t (θ ε ) = σ(θ ε )(ū ε -θ ε ), ūε = 1 4π S 2 u ε (t, x, ω)dω. (4)
Our results, already given in Theorems 2.2 and 2.3, will hence be proved under the following form:

Theorem 2.5. Let us consider System (4), posed for (t, x, ω) ∈ (0, τ ) × Ω × S 2 , with initial conditions u ε (0, x, ω) = u in (x, ω) ∈ L ∞ (Ω × S 2 ) θ ε (0, x) = θ in (x) ∈ L ∞ (Ω) and boundary data u ε (t, x, ω)| Γ -= u b (t, x, ω) ∈ L ∞ ((0, τ ) × ∂Ω × S 2 ),
where Ω ⊂ R 3 is a bounded domain with boundary of class C 1 and ε > 0.

Then System (4) admits a weak solution

(u ε (t, x, ω), θ ε (t, x)) ∈ L ∞ ((0, τ ) × Ω × S 2 ) × L ∞ ((0, τ ) × Ω). Theorem 2.6. Let us consider System (4), posed for (t, x, ω) ∈ (0, τ ) × Ω × S 2 , with initial conditions u ε (0, x, ω) = u in (x, ω) ∈ L ∞ (Ω × S 2 ) θ ε (0, x) = θ in (x) ∈ L ∞ (Ω)
and boundary data

u ε (t, x, ω)| Γ -= u b (t, x) ∈ W 1,∞ ((0, τ ) × ∂Ω),
where Ω ⊂ R 3 is a bounded domain with boundary of class C 1 and ε > 0. Let {u ε } be a family of solutions of System (4) which satisfy the same initial and boundary conditions.

Then, there exists a function ρ ∈ L ∞ ((0, τ ) × Ω) such that ρ = lim ε→0 ūε strongly in L 2 ((0, τ )×Ω). The limit function ρ moreover solves the following initial-boundary value problem in (0, τ ) × Ω, for τ > 0:

(5) [ρ + E(ρ)] t -∇ x • ∇ x ρ 3σ(ρ) = 0.
with initial conditions ρ(0, x) = ūin (x, ω) and boundary data ρ(t, x)| ∂Ω = u b (t, x).

Existence proof

This section is devoted to the proof of Theorem 2.5. In order to simplify the notation, in this section we will specialize system (4) by considering ε = 1, and by eliminating all the subscripts in the unknowns. Obviously, the results are valid for all parameters ε > 0.

3.1. The maximum principle. As a first step, we prove that (4) preserves the non-negativity of the solutions, which are uniformly bounded in time. This property is proved in the following proposition: Proposition 3.1. Let us suppose that there exists a solution (u, θ) for the radiative transfer system (4) in (t, x, ω) ∈ (0, τ ) × Ω × S 2 , with bounded initial conditions

u ε (0, x, ω) = u in (x, ω) ∈ L ∞ (Ω × S 2 ) θ ε (0, x) = θ in (x) ∈ L ∞ (Ω)
and bounded boundary data

u ε (t, x, ω)| Γ -= u b (t, x, ω) ∈ L ∞ ((0, τ ) × ∂Ω × S 2 ).
Then there exists a constant M , which depends only on the initial and boundary data, such that 0 ≤ u, θ ≤ M a.e. (uniformly in x, ω and t).

Proof. We develop first the part of the proof which establishes the bound from above. We define

M = max u in L ∞ (Ω×S 2 ) , θ in L ∞ (Ω) , u b L ∞ (Γ -×S 2 ×(0,τ ]) .
We consider now the pair of constant functions (v = M, S = M ), solutions of the system (6)

         ∂v ∂t + ω • ∇ x v + σ(θ)(v -S) = 0 ∂ ∂t E(S) = σ(θ)(v -S),
where θ satisfies the radiative transfer system (4), with initial and boundary conditions

v(x, ω, t = 0) = M v(x, ω, t)| Γ -= M.
S(x, t = 0) = M. We subtract the first equation of system [START_REF] Dogbe | The radiative transfer equations: diffusion approximation under accretiveness and compactness assumptions[END_REF] from the first equation of system (4) and subtract the second equation of system (6) from the second equation of system [START_REF] Cessenat | Théorèmes de trace L p pour des espaces de fonctions de la neutronique[END_REF].

We then multiply the two obtained equations by sign + (u-v) and sign + (θ-S) respectively, where sign + (y) = 1 when y ≥ 0 and sign + (y) = 0 when y < 0.

Finally we add the two equations and integrate with respect to x and ω in the domain Ω × S 2 . We note that the term

Ω×S 2 ω • ∇ x (u -v) dωdx = Γ×S 2 (u -v) + ω • n x dsdω,
where (u -v) + = (u -v) sign + (u -v), is always non-negative: indeed, in the region Γ 0 ∪ Γ + , the factor (ω • n x ) is non-negative, as well as (u -v) + ; on the other hand, in the region Γ -, the term (ω • n x ) is negative, but there u < v because of the boundary conditions imposed to v, and therefore (u-v) + = 0 and the integral vanishes.

Since E is non-negative, we can moreover use the identity

sign + (θ -S) = sign + (E(θ) -E(S)).
We hence obtain

d dt Ω×S 2 (u -v) + dωdx + Ω×S 2 (E(θ) -E(S)) + dωdx ≤ 0, since Ω×S 2 σ(θ)(θ -S)( sign + (u -v) -sign + (θ -S)) dωdx- Ω×S 2 σ(θ)(u -v)( sign + (u -v) -sign + (θ -S)) dωdx ≤ 0.
But in t = 0 we have that

Ω×S 2 (u(0) -v(0)) + dωdx + Ω×S 2 (E(θ(0)) -E(S(0))) + dωdx = 0,
and therefore the bound from above is proved.

In order to prove that System (4) preserves the sign of the solution, we apply a similar strategy: we consider the function sign -(y), defined as sign -(y) = 0 when y ≥ 0 and sign -(y) = -1 when y < 0; then we multiply the first equation of ( 4) by sign -(u) and the second one by sign -(θ). We integrate the equations in Ω × S 2 with respect to the space variable x and ω and deduce (notice that, by hypothesis on E, the identity sign -(E(θ)) = sign -(θ) is always satisfied):

d dt Ω×S 2 u -dxdω+ Γ×S 2 u -(ω•n x ) dsdω+ Ω×S 2 σ(θ)(u-θ) sign -(u) dxdω = 0 and d dt Ω×S 2 (E(θ)) -dxdω = Ω×S 2 σ(θ)(u -θ) sign -(θ) dxdω = 0,
where we have defined the function y -as y -= 0 when y ≥ 0 and y -= -y when y < 0. By summing the two equation, with the same strategy used to prove the maximum principle, we deduce that, separately,

Γ×S 2 u -(ω • n x ) dsdω ≥ 0 and Ω×S 2 σ(θ)(u -θ)( sign -(u) -sign -(θ)) dxdω ≥ 0. We can hence conclude that d dt Ω×S 2 [u -+ (E(θ)) -] dxdω ≤ 0.
Since the initial and boundary conditions are non-negative, E is monotone and E(0) = 0, this implies that u ≥ 0 and E(θ) ≥ 0 a.e. in Ω for all t ∈ [0, τ ].

3.2. The truncated problem. In the proof on the existence of a weak solution of System (4), we use a trick introduced by Mercier in [START_REF] Mercier | Application of accretive operators theory to the radiative transfer equations[END_REF], by working with the solutions of a truncated approximation of (4). Then, thanks to the maximum principle of the truncated problem, it will be possible to construct sequences of solutions of such approximated problems which will be relatively compact in L ∞ -weak * . Once passed to the limit, the existence proof will be achieved if one proves that the limit of the solutions of the approximated problems solves System (4).

We first consider the following truncated approximation of System ( 4):

(7)          ∂u n ∂t + ω • ∇ x u n + σ n (θ n )(U n -θ n ) = 0 E t (θ n ) = σ n (θ n )( Ūn -θ n ), Ūn = 1 4π S 2 U n (t, x, ω)dω,
where

σ n (y) = min{σ(y), n}, U n (t, x, ω) = max{0 , min{u n , M }}, n ∈ N.
System [START_REF] Golse | The Milne problem for the radiative transfer equations (with frequency dependence)[END_REF] has the form of a Lipschitzian perturbation of the free transport equations. Therefore, thanks to Theorems 6.1.2 and 6.1.6 in [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF], System [START_REF] Golse | The Milne problem for the radiative transfer equations (with frequency dependence)[END_REF]

admits a strong solution (u n , θ n ) ∈ C 0 ([0, τ ]; L 2 (Ω × S 2 )) × C 0 ([0, τ ]; L 2 (Ω)).
By applying the same strategy of Lemma 3.1, we can prove that, if we suppose that there exists a solution (u n , θ n ) for the truncated approximation (7) in (t, x, ω) ∈ (0, τ )×Ω×S 2 , with the same initial and boundary conditions of Theorem 2.5, then there exists a constant M , which depends only on the initial and boundary data, such that 0 ≤ u n , θ n ≤ M a.e. in Ω, uniformly in x, ω and t.

Thanks to the maximum principle for System [START_REF] Golse | The Milne problem for the radiative transfer equations (with frequency dependence)[END_REF], the solution of such set of equations coincides therefore with the solution of the following system:

(8)          ∂u n ∂t + ω • ∇ x u n + σ n (θ n )(u n -θ n ) = 0 E t (θ n ) = σ n (θ n )(ū n -θ n ), ūn = 1 4π S 2 u n (t, x, ω)dω,
where σ n (y) = min{σ(y), n}, n ∈ N. We hence deduce the following Lemma:

Lemma 3.2. Consider System (8) for (t, x, ω) ∈ (0, τ ) × Ω × S 2 , with initial conditions u n (0, x, ω) = u in (x, ω) ∈ L ∞ (Ω × S 2 ) θ n (0, x) = θ in (x) ∈ L ∞ (Ω) and boundary data u n (t, x, ω)| Γ -= u b (t, x, ω) ∈ L ∞ ((0, τ ) × ∂Ω × S 2 ).
Then, there exists a strong solution (u n (t, x, ω), θ n (t, x)) of System (8) which belongs to the space C 0 ([0, τ ]; L 2 (Ω×S 2 ))×C 0 ([0, τ ]; L 2 (Ω)). Moreover, there exists a constant M > 0 such that

u n L ∞ ((0,τ )×Ω×S 2 ) , θ n L ∞ ((0,τ )×Ω×S 2 ) ≤ M
uniformly with respect to n.

Thanks to Lemma 3.2, the sequences {u n } and {θ n } are relatively compacts in L ∞ -weak * . Nevertheless, this property is not enough to pass to the limit because of the nonlinearities of the problem. We need strong compactness for the sequences {ū n } (obtained through an averaging lemma) and {θ n }.

We prove therefore the following result:

Lemma 3.3. Let {u n } and {θ n } be sequences of solutions of the truncated approximation (8). Then ūn is relatively compact in L p loc (t, x) for all p ∈ [1, +∞).

Proof. Since θ n is uniformly bounded in L ∞ t,x and the internal energy E is of class C 0 ([0, +∞)), the second equation of System [START_REF] Golse | Generalized solutions of the radiative transfer equations in a singular case[END_REF] shows that we deduce that

Ω E(θ n )(τ, x) dx - Ω E(θ n )(0, x) dx+ τ 0 Ω σ n (θ n ) θ n dtdx = τ 0 Ω σ n (θ n ) ūn dtdx.
Therefore, there exists a constant C > 0 such that

(0,τ )×Ω×S 2 σ n (θ n )u n dtdxdω ≤ C.
By the first equation of System (8), we hence deduce that (∂ t + ω • ∇ x )u n is bounded in L 1 loc (t, x, ω). Since u n bounded in L ∞ (t, x, ω), we then use simply the averaging lemma proved in [START_REF] Golse | The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels[END_REF] to conclude.

As a next step we prove that the unknown θ n of System (8) depends only on ūn in a very special way: Lemma 3.4. Consider System (8). Then there exists a . . . functional F :

L p loc ((0, τ ) × Ω) → L p loc ((0, τ ) × Ω), p ∈ [1, +∞) such that θ n = F(ū n ). Proof.
Let us consider the second equation of [START_REF] Golse | Generalized solutions of the radiative transfer equations in a singular case[END_REF]. Since E is of class C 1 , we can introduce the function

H n = H n (θ n ) defined in such a way that (9) H n (θ n ) = E (θ n ) σ n (θ n ) .
This allows us to rewrite the second equation of ( 8) in the form

(10) ∂H n ∂t (θ n ) + θ n = ūn .
Thanks to the hypotheses on E and σ n , we can deduce that H n is one-toone and of class C 1 . Indeed, by definition of H n , we have that H n is always non-negative. The only point which can give troubles is

θ n = 0. But lim θ→0 H n (θ n ) = lim θn→0 E (θ n ) σ n (θ n ) ∼ θ α+β n ,
because E and σ are admissible internal energy and opacity respectively. Since β ∈ (0, 1/4] and α ∈ [-3/4, -β], we can deduce that

lim y→0 (H -1 n ) (y)
exists and is always finite.

The quantity H n being defined by integration of H n , we can choose the arbitrary constant; the best choice is to put H n (0) = 0:

H n (θ n ) = θn 0 E (ζ) σ n (ζ)
dζ.

Thanks to the maximum principle, the behavior of H n when θ n → +∞ plays no role. We have then proved that

H n ∈ C 1 ([0, H n (M )]), H -1 n ∈ C 1 ([0, M ]
) and H n is invertible. By setting 10) becomes [START_REF] Goudon | Approximation by homogenization and diffusion of kinetic equations[END_REF] ∂φ n ∂t + H -1 n (φ n ) = ūn . We build the difference between the previous equation and a shifted version of the same equation. We can deduce that 1 2

φ n = H n (θ n ), Equation (
d dt [φ n (t, x + h) -φ n (t, x)] 2 + [H -1 n (φ n (t, x + h)) -H -1 n (φ n (t, x))][φ n (t, x + h) -φ n (t, x)] = [ū n (t, x + h) -ūn (t, x)][φ n (t, x + h) -φ n (t, x)].
We integrate respect to x in Ω and obtain, thanks to the monotonicity of H n (which is a consequence of the monotonicity of E):

1 2 φ n (t, x + h) -φ n (t, x) 2 L 2 x ≤ 1 2 φ n (0, x + h) -φ n (0, x) 2 L 2 x + t 0 Ω [ū n (s, x + h) -ūn (s, x)][φ n (s, x + h) -φ n (s, x)] dsdx ≤ 1 2 φ n (0, x + h) -φ n (0, x) 2 L 2 x + t 0 ūn (s, • + h) -ūn (s, •) L 2 x φ n (s, • + h) -φ n (•, x) L 2 x ds.
Thanks to Gronwall's Lemma, in the limit |h| → 0 we obtain that

φ n (t, • + h) -φ n (t, •) 2 L 2 x → 0, Moreover, since ūn is relatively compact in L p loc , p ∈ [1, +∞), t 0 ūn (s, • + h) -ūn (s, •) L 2 x ds → 0 when |h| → 0. Finally, since ūn is bounded in L ∞ t,x and H -1 n (φ n ) is also bounded in L ∞ t,
x , we are guaranteed that φn is bounded in L ∞ t,x . Hence, we can deduce that the family of functions {φ n } is relatively compact in L p loc ((0, τ ) × Ω), and therefore there exists a subsequence, which converges to a limit point φ in L p loc ((0, τ ) × Ω) and a.e.. 3.3. Proof of theorem 2.5. Thanks to Lemma 3.2 and the Banach-Alaouglu theorem, u n and θ n are weakly* compact. This means that, by subsequences, there exists two limits u and θ such that

u n u ∈ L ∞ ([0, τ ] × Ω × S 2 ) θ n θ ∈ L ∞ ([0, τ ] × Ω)
weakly-*. The next step consists in proving that, if {u n } and {θ n } are two sequences of solutions of System ( 8) with initial and boundary data as in Theorem 2.5, then the two weak* limits u and θ solve the non-truncated version of Equations ( 4) with initial and boundary conditions

u(x, ω, t = 0) = u in (x, ω) u(x, ω, t)| Γ -= u b (t, x, ω)
and θ(x, t = 0) = θ in (x). The previous equations are highly nonlinear; hence the passage to the limit is not straightforward. We analyze therefore each term of the two equations.

i) We have easily that

σ n (θ n ) θ n = O(1) in L ∞ ([0, τ ] × Ω): indeed,
in any interval of type [a, +∞) with a > 0, σ is uniformly bounded and, when y → 0, we have that

σ n (y) y ∼ y 1-β with β ∈ [0, 1/4).
ii) We look now at the term

σ n (θ n ) u n .
Thanks to the second equation of system (8), we deduce, by integrating with respect to (t,

x, ω) in [0, τ ] × Ω × S 2 ) that 1 4π τ 0 Ω S 2 σ n (θ n ) u n dtdxdω = τ 0 Ω σ n (θ n ) ūn dtdx = τ 0 Ω [E t (θ n ) + σ n (θ n ) θ n ] dtdx.
We already know that σ n (θ n ) θ n belongs to L ∞ ([0, τ ] × Ω). We now consider the other term of the last integral: we have that

τ 0 Ω E t (θ n ) dtdx = Ω E(θ n )(x, τ ) dx - Ω E(θ in )(x) dx.
Thanks to the hypotheses on E, we have proved that also

σ n (θ n ) u n belongs to L ∞ ([0, τ ] × Ω) since Ω is bounded.
In order to pass to the limit, we use the compactness properties of ūn and θ n . By Lemma 3.3, we have that ūn is relatively compact in L p loc ((0, τ ) × Ω) for all p ∈ [1, +∞) and, by Lemma 3.4, that θ n is relatively compact in

L p loc ((0, τ ) × Ω) for all p ∈ [1, +∞). We know that (1) u n u in L ∞ t,x,ω -weak * (2) θ n θ in L ∞ t,
x,ω -weak * and, by Lemma 3.4, that θ n → θ in L p loc and a.e.. Consequently, σ n (θ n ) θ n σ(θ) θ in in L p loc and a.e.. We can now apply the dominated convergence theorem to control the nonlinear terms. We recall moreover that the only term which depends on the angular variable ω is u n . Therefore we can deduce that, modulo extraction of a subsequence, there exists a measure µ such that

σ n (θ n ) u n * µ.
We only need to identify µ with the limit product σ(θ) u.

For any m < n, we have that

σ m (θ n ) u n ≤ σ n (θ n ) u n ,
and therefore, in the limit m, n → +∞ σ(θ) u ≤ µ.

On the other hand, the second equation of ( 8), says us that

Ω E(θ n )(τ, x) dx - Ω E(θ in )(x) dx + τ 0 Ω σ n (θ n ) θ n dtdx = 1 4π τ 0 Ω S 2 σ n (θ n ) u n dtdxdω.
The l.h.s converges to

Ω E(θ)(τ, x) dx - Ω E(θ in )(x) dx + τ 0 Ω σ(θ) θ dtdx,
whereas the r.h.s converges to τ 0 Ω µ dtdx.

But in the limit, by strong convergence

σ(θ) ū = E t (θ) + σ(θ) θ,
and therefore

τ 0 Ω σ(θ)θ u dtdx = τ 0 Ω µ dtdx.
We have hence proved that the term σ n (θ n ) u n converges weakly * to σ(θ) u.

Hence, the limit (u, θ) of the sequence (u n , θ n ) solves System (4).

iii) The proof will be complete once we show that the initial and boundary conditions for the limit problem are satisfied (in the sense of traces). We know that the sequence u n is bounded in L p ((0, τ ) × Ω × S 2 ) for any p ∈ [1, +∞). If we can find an exponent q > 1 such that (∂ t + ω • ∇ x )u n is bounded in L q ((0, τ ) × Ω × S 2 ), we can deduce that, thanks to Cessenat's results [START_REF] Cessenat | Théorèmes de trace L p pour des espaces de fonctions de la neutronique[END_REF][START_REF] Cessenat | Théorèmes de trace pour des espaces de fonctions de la neutronique[END_REF], u n has a trace in L q -sense and that u n u weakly* in

L q ({t = 0} × Ω × S 2 ∪ ∂Ω × S 2 × (0, τ ]).
Let us therefore consider the first equation of System [START_REF] Golse | Generalized solutions of the radiative transfer equations in a singular case[END_REF]. Thanks to the behavior in zero of σ n , we deduce that

σ n (θ n ) θ n is bounded in L ∞ ((0, τ )×Ω). Therefore we deduce that (∂ t + ω • ∇ x )u n is bounded in L q ((0, τ ) × Ω × S 2 ) if we can prove that σ n (θ n ) u n ∈ L q ((0, τ ) × Ω × S 2 ); we have (0,τ )×Ω×S 2 σ q n (θ n ) u q n dtdxdω ≤ u n q-1 L ∞ (0,τ )×Ω σ q n (θ n ) ūn dtdx.
By the second equation of System (8), we have

(0,τ )×Ω σ q-1 n (θ n )E (θ n ) ∂θ n ∂t dtdx + (0,τ )×Ω σ q n (θ n ) θ dtdx = (0,τ )×Ω σ q n (θ n ) ūn dtdx.
But the first term in the l.h.s. of the previous equation is bounded thanks to fundamental theorem of integral calculus and the second term in the l.h.s. is bounded thanks to the behavior in zero of σ n , provided that q ∈ (1, 4]. Hence, we deduce that

(0,τ )×Ω×S 2 σ q n (θ n ) u q n dtdxdω is bounded and therefore u n u weakly* in L q ({t = 0} × Ω × S 2 ∪ ∂Ω × S 2 × (0, τ ]) for any q ∈ (1, 4].
In order to prove, finally, that the limit θ satisfies the initial condition, we consider the second equation of System [START_REF] Cessenat | Théorèmes de trace L p pour des espaces de fonctions de la neutronique[END_REF]. Since E is of class C 1 and strictly increasing and the second member of the equation is bounded in L p loc and a.e., we deduce that θ(x, t) → θ in (x) strongly in L p (Ω) for all 1 ≤ p < ∞.

Hence the proof of Theorem 2.5 is complete.

Relative entropy estimates

This section is devoted to give the tools which will permit to handle the limiting behavior of system [START_REF] Cessenat | Théorèmes de trace L p pour des espaces de fonctions de la neutronique[END_REF].

From now on, we set

(12) ν = u b W 1,∞ (∂Ω×[0,τ ]
) and m = min 0≤y≤M σ(y).

The main mathematical tool we will need is a functional, which is nothing but the relative entropy with respect to a gauge profile.

There are many ways to define this gauge profile. Its most important property concerns the behavior on the boundary. For example, the gauge function f = f (t, x) could be the solution of the Laplace equation with boundary data given by u b (t, x):

(13)    ∆ x f (t, x) = 0 x ∈ Ω f (t, x)| ∂Ω = u b (t, x) ∈ L ∞ ((0, τ ] × ∂Ω).
Classical theory on the Laplace equation ensures that f exists, is unique and satisfies the strong maximum principle. In particular, 0

≤ f ≤ M = u b ∞
in Ω and f ∈ W 2,p (Ω) for any p ∈ [1, +∞). Obviously, here the time t acts only as a parameter.

4.1. An elementary inequality. The natural entropy generating function to control the L 2 -norm of radiation flux should be y 2 /2. However, one needs to control boundary terms, and the natural choice stated before does not seems the best one. To do so, it is more convenient to use the following entropy generating function

Φ(y) = 1 2 y 2 + 1 2 (ν + 1) 2 ,
where ν has been introduced in [START_REF] Larsen | Asymptotic analysis of radiative transfer problems[END_REF]. With this definition Φ(y) ≥ (ν + 1)y = Φ (ν)y + y, i.e. ( 14) Φ(y) -Φ (ν)y ≥ y and this last inequality is helpful in handling boundary terms.

Entropy inequalities.

In this subsection we show in which way the control on radiation flux come from a entropy production control. We multiply the first equation of system (4) by Φ (u ε ) and integrate with respect to x and ω in Ω × S 2 . We obtain:

ε 2 d dt Ω×S 2 Φ(u ε ) dω dx + ε ∂Ω×S 2 ω • n x Φ(u ε ) dω ds(x)+ (15)
Ω×S 2 σ(θ ε )(u ε -θ ε )Φ (u ε ) dω dx = 0.
We write now Equation [START_REF] Mihalas | Foundations of Radiation Hydrodynamics[END_REF] in the form

d dt Ω×S 2 Φ(u ε ) dω dx + 1 ε ∂Ω×S 2 ω • n x [Φ(u ε ) -Φ (f )u ε ] dω ds(x) = (16) - 1 ε 2 Ω×S 2 σ(θ ε )(u ε -θ ε )Φ (u ε ) dω dx - 1 ε Ω×S 2 ∇ • (ωΦ (f )u ε ) dω dx.
Let us study now the various terms of the previous equation.

(i) We have that

∂Ω×S 2 ω•n x [Φ(u ε )-Φ (f )u ε ] dω ds = ∂Ω ω•nx>0 ω•n x [Φ(u ε )-Φ (f )u ε ] dω ds + ∂Ω ω•nx<0 ω • n x [Φ(u ε ) -Φ (f )u ε ] dω ds ≥ ∂Ω ω•nx>0 ω•n x [Φ(f )-Φ (f )f ] dω ds+ ∂Ω ω•nx<0 ω•n x [Φ(u ε )-Φ (f )u ε ] dω ds = ∂Ω (Φ(f ) -Φ (f )f ) S 2 ω • n x dω ds = 0.
Here we have used the convexity of Φ, inequality [START_REF] Mercier | Application of accretive operators theory to the radiative transfer equations[END_REF] and the fact that [Φ(f ) -Φ (f )f ] does not depend on ω. This term is therefore non-negative.

(ii) We perform now some manipulations of the last term in Equation ( 16):

1 ε Ω×S 2 ∇•(ωΦ (f )u ε ) dω dx = 1 ε Ω×S 2 ω•[u ε Φ (f )∇ x f +Φ (f )∇ x u ε ] dω dx = 1 ε Ω×S 2 [u ε Φ (f )ω • ∇ x f ] dω dx + Ω Φ (f )(∇ x • j ε ) dx.
We notice that

Ω Φ (f )(∇ x • j ε ) dx = Ω Φ (f ) ∂ ∂t E(θ ε ) - ∂ ∂t ρ ε dx = Ω Φ (f )E(θ ε ) t dx - Ω Φ (f )E(θ ε )f t dx- Ω Φ (f )ρ ε t dx + Ω Φ (f )ρ ε f t dx.
Also the other term can be controlled: we have that 1

ε Ω×S 2 [u ε Φ (f )ω • ∇ x f ] dω dx = Ω×S 2 [Φ (f )j ε • ∇ x f ] dω dx ≤ 1 2γ Ω |∇ x Φ (f )| 2 dx + γ Ω |j ε | 2 dx.
(iii) Finally, we deduce that

1 ε 2 Ω×S 2 σ(θ ε )(u ε -θ ε )Φ (u ε ) dω dx = 1 ε 2 Ω×S 2 σ(θ ε )(u ε -θ ε )[Φ (u ε ) -Φ (θ ε )] dω dx + Ω×S 2 E t (θ ε )Φ (θ ε ) dx.
If we remember the form of Φ, we see immediately that the first member in the right-hand side of the previous equation controls the L 2 -norm of the flux:

1 ε 2 Ω×S 2 σ(θ ε )(u ε -θ ε ) 2 dω dx ≥ 1 ε 2 Ω σ(θ ε ) S 2 (u ε -θ ε ) 2 dω S 2 (ω (i) ) 2 dω dx ≥ 1 ε 2 Ω σ(θ ε ) S 2 (u ε -θ ε )ω (i) dω 2 dx = 1 ε 2 Ω σ(θ ε ) S 2 u ε ω (i) dω 2 dx = Ω σ(θ ε )(j (i) ε ) 2 dx. Hence 1 ε 2 Ω×S 2 σ(θ ε )(u ε -θ ε ) 2 dω dx ≥ 1 3 Ω σ(θ ε )|j ε | 2 dx ≥ m 3 Ω |j ε | 2 dx.
We therefore deduce that Equation ( 16) implies

d dt Ω×S 2 1 2 u 2 ε + f (E(θ ε ) -ρ ε ) dxdω ≤ γ - m 3 Ω |j ε | 2 dx- Ω E t (θ ε )θ ε dx - Ω f t [ρ ε -E(θ ε )] dx + 1 2γ Ω |∇ x f | 2 dx.
If we choose γ < m/3 in the previous inequality and denote by

Q(y) = y 0 E(s)s ds, we deduce that d dt Ω×S 2 1 2 u 2 ε + f (E(θ ε ) -ρ ε ) + Q(θ ε ) dxdω + m 3 -γ Ω |j ε | 2 dx ≤ C,
where C is a positive constant which can be computed explicitly. We have then deduced the following result:

Proposition 4.1. Let (u ε , θ ε ) be a solution of the scaled system (4), in (0, τ ) × Ω × S 2 , with initial conditions

u ε (0, x, ω) = u in (x, ω) ∈ L ∞ (Ω × S 2 ) θ ε (0, x) = θ in (x) ∈ L ∞ (Ω) and boundary data u ε (t, x, ω)| Γ -= u b (t, x, ω) ∈ W 1,∞ ((0, τ ) × ∂Ω × S 2 ).
Then there exists a positive constant J = J(τ, u b , u in , θ in , m) such that the current j ε satisfies

τ 0 Ω |j ε (t, x)| 2 dt dx ≤ J and 1 ε 2 τ 0 Ω×S 2 σ(θ ε )(u ε -θ ε ) 2 dt dx dω ≤ J
uniformly for each ε > 0.

Thanks to Proposition 4.1, we can deduce the following corollary:

Corollary 4.2. Let (u ε , θ ε ) as in Proposition 4.1. Then, if one of the sequences u , or θ , or ū admits a limit, then all the three sequences go to the same limit in strong L 2 -sense.

Proof. Since σ is bounded below by a strictly positive constant, we obtain from Proposition 4.1 that

1 ε u ε -θ ε L 2 t,x,ω = O(1) and therefore lim →0 u -θ L 2 t,x,ω = 0. Since ūε -θ ε L 2 t,x ≤ u ε -θ ε L 2 t,x,ω and u ε -ūε L 2 t,x,ω ≤ u ε -θ ε L 2 t,x,ω + ūε -θ ε L 2 t,x , we obtain that also lim →0 u -θ L 2 t,x,ω = lim →0 θ -ū L 2 t,x = 0.

Convergence to the Rosseland equation

This section is devoted to conclude the proof on the convergence of the diffusive limit stated in Theorem 2.6. Since we will use various relatively compact sequences, when we will indicate that a sequence converges to a limit, we will mean that there exists a subsequences which converges to the limit.

Let (u in , θ in ) and u b be initial and boundary data which satisfy the hypotheses of Theorem 2.6; then, for each > 0, let (u , θ ) be the solution to the scaled system (4).

It follows from Propositions 3.1 and 4.1 that, for each τ > 0, one has

ρ L ∞ ((0,τ )×Ω) ≤ K and j L 2 ([0,τ ]×Ω) ≤ J 1/2
for each > 0.

By the Banach-Alaoglu theorem, for each τ > 0 (17) the family ρ is relatively compact in L ∞ ((0, T ) × Ω) weak-*, and therefore there exists ρ ∈ L ∞ ((0, τ ) × Ω) such that ρ * ρ, while (18) the family j is relatively compact in L 2 ([0, τ ] × Ω) weak, which means that there exists j ∈ L 2 ((0, τ ) × Ω) such that j j. Because of the nonlinearities, that are present both in the scaled system (4) and in the limiting nonlinear diffusion equation, weak compactness results as above are not enough to pass to the limit as → 0. Strong L 2 compactness of the family ρ is required, which will be obtained by using Minty's trick [START_REF] Marcati | The one-dimensional Darcy's law as the limit of a compressible Euler flow[END_REF] and compensated compactness.

Consider the vector fields p = (ρ + E(θ ), j ) and q = (ρ , 0, 0, 0) .

By the arguments as above, both vector fields satisfy p and q are bounded in (L 2 ([0, τ ] × Ω)) 4 .

Moreover, if we integrate the first equation of system (4) with respect to ω in S 2 , we obtain

(19) ∂ ∂t ρ ε c + E(θ ε ) + ∇ x • j ε = 0,
which means: div t,x p = 0. In order to prove that curl t,x q is relatively compact in H -1 t,x , we simply use the averaging lemma [START_REF] Golse | The Navier-Stokes limit of the Boltzmann equation for bounded collision kernels[END_REF], which give us compactness in x, the behavior of the time derivative being irrelevant, since the only non-zero component of q ε is the first one.

Let E * the weak- * limit of E(θ ε ). Since (ū ε -θ ε ) tends strongly to zero in L 2 , we can deduce that also E(ρ ε ) converges weakly- * to E * . By compensated compactness (the div-curl lemma in [START_REF] Murat | Compacité par compensation[END_REF]) we find that, by subsequences, (20)

p n • q n = ρ 2 n + ρ n E(θ n ) p • q = ρ 2 + ρE *
in the sense of Radon measures on (0, T ) × (0, 1) as n → 0. The family (21) ρ is relatively compact in L 2 ([0, T ] × [0, 1]) strong: indeed, (ρ ε -ρ) 2 + (ρ ε -ρ)(E(ρ ε ) -E(ρ)) ≥ 0 since E is monotone non-decreasing. Moreover, we deduce that, by (20), (ρ ε -ρ) 2 + (ρ ε -ρ)(E(ρ ε ) -E(ρ)) → 0 in D . A non-negative sequence which converges in D to zero converges also strongly to zero. Since also the term

(ρ ε -ρ)(E(ρ ε ) -E(ρ))
is non-negative, we deduce finally that ρ ε → ρ (strongly) in L 2 ((0, τ ) × Ω).

Because of the regularity on E (see Definition 2.4) and because (θ -ū ) → 0 strongly in L 2 ((0, τ ) × Ω), then E(θ ε ) → E(ρ) (strongly) in L 2 ((0, τ ) × Ω).

Let us now take into account the flux equation

(22) ε 2 ∂j (i) ε ∂t + 1 4π S 2 ω (i) ∇ x • (ωu ε ) dω + σ(θ ε ) j (i) ε = 0.
Since σ is strictly positive, we can state that

σ(θ ε ) 1/2 j ε = -σ(θ ε ) -1/2 ε 2 ∂j ε ∂t + 1 4π ∇ x • S 2
(ω ⊗ ω)u ε dω . Therefore, there exists η ∈ L 2 ((0, τ ) × Ω) such that σ(θ ε ) 1/2 j ε η in L 2 ((0, τ ) × Ω). Since σ is strictly positive and θ ε → ρ strongly in L 2 ((0, τ ) × Ω), we deduce that j ε σ(θ ε ) -1/2 η = j. On the other hand, if we multiply the first equation of the system (4) by u ε , we deduce:

(23) 1 2 ε ∂u 2 ε ∂t + ω • ∇ x u 2 ε = [u ε σ(θ ε ) 1/2 ] 1 σ(θ ε ) 1/2 (u ε -θ ε ) .
(ρ + E(ρ)) t -∇ x • ∇ x ρ 3σ(ρ) = 0,
which is equation [START_REF] Cessenat | Théorèmes de trace pour des espaces de fonctions de la neutronique[END_REF].

In order to verify that the initial and boundary conditions are the correct limits of the conditions imposed on the radiative transfer system, we will use a regularity argument.

Let us now consider the continuity equation ( 19): since the flux j ε is bounded in L 2 ((0, τ ) × Ω), we deduce that (ρ ε + E(θ ε )) t is bounded in L 2 (0, T ; H -1 (Ω)) ; which, joint to the bound on ρ and E, implies that (24)

ρ + E(θ ) is relatively compact in C([0, τ ]; H -1 (Ω))

by Arzela-Ascoli's theorem and therefore the target equation recovers the correct initial data.

Because of the initial condition of the radiative transfer system (4), one has (25) ρ ∈ C([0, τ ]; H -1 (Ω)) and ρ t=0 = 1 4π S 2 u in (x, ω) dω.

Let us now care about the boundary conditions: we consider equation (23) and perform the limit as → 0. We note immediately, thanks to the identity

σ(θ ε )(u ε -θ ε )u ε = σ(θ ε )(u ε -θ ε ) 2 + σ(θ ε )u ε θ ε -σ(θ ε )θ 2
ε , that ω•∇ x u ε is bounded in L 2 ((0, τ )×Ω×S 2 ), we can deduce that, thanks to Cessenat's results [START_REF] Cessenat | Théorèmes de trace L p pour des espaces de fonctions de la neutronique[END_REF][START_REF] Cessenat | Théorèmes de trace pour des espaces de fonctions de la neutronique[END_REF], u ε has a trace in L 2 -sense and that u ε u weakly in L 2 ((0, τ ] × ∂Ω × S 2 ), and therefore the trace of the limit exists. Thus the proof of the nonlinear diffusion limit is competed.

Definition 2 . 4 .

 24 The normalized opacity σ : R + → R + and the internal energy E : [0, +∞) → [0, +∞) are admissible if and only if

Now, by Proposition ( 4 . 1 ), we have that σ(θ ε ) 1 /2 j ε 2 L 2 ( 2 L 2 (

 4112222 (0,τ )×Ω) ≤ σ(θ ε ) 1/2 (u ε -θ ε ) 2 L 2 ((0,τ )×Ω×S 2 ) + σ(θ ε ) 1/2 θ ε (0,τ )×Ω) ≤ ε 2 J + Cτ,where C is the maximum of the function σ(y)y 2 for y ∈ [0, M ].

Thanks to Proposition 4 . 1 ,

 41 there exists a functionζ ∈ L 2 ((0, τ ) × Ω) such that 1 σ(θ ε ) 1/2 (u ε -θ ε ) ζ in L 2 ((0, τ ) × Ω) and therefore we can pass to the limit and obtain thatζ = 1 σ(ρ) ω • ∇u,since σ is bounded below by a positive constant by hypothesis. But it is easy to see that η = -

Acknowledgment: This paper has been written when the second author was visiting the Centre de Mathématiques Laurent Schwartz of the Ecole Polytechnique, France. FS is very grateful to the Ecole Polytechnique for the invitation and the facilities provided to him.

Financial support of the Italian National Group for Mathematical Physics (GNFM) is acknowledged.


Henri Gouin

R Fosdick

E Dunn

M Slemrod

DYNAMIC EFFECTS IN GRADIENT THEORY FOR FLUID MIXTURES

Keywords: Fluid mixtures, Interfaces, Gradient theory PACS: 47.55.N-, 47.61.Jd, 68.03.Kn, 68.18.Jk 1991 MSC: 37N10, 74A50, 76T30

. Maxwell rule is extended to interfaces of fluid mixtures out of equilibrium. Formulae like Clapeyron relation are obtained for isothermal layers.

Introduction

In this paper we propose a new attempt to study motions in interfaces of fluid mixtures. First, we recall the study of interfaces between mixture bulks in equilibrium. We remind problems for an interface moving in a single fluid and extend them to cases of mixtures.

How to calculate the proportions of constituents in each bulk of a n-components mixture? This problem is solved in the case of isothermal equilibrium for mixtures whose free energy is known [START_REF] Dunn | Interstitial working and a Nonclassical Continuum Thermodynamics[END_REF]. A simple calculus consists in writing the total free energy of the mixture is minimum.

Practically, calculus are available only if:

Email address: henri.gouin@univ-cezanne.fr (Henri Gouin).

Preprint submitted to : IMA Volumes in Mathematics and its Applications, 52, pp. 111-121 (1993).

-the container has a simple shape allowing one dimension calculus (cylinder or sphere)

-the mixture has a few number of components -the mixture free energy is not a too complex function of component densities and temperature.

Let us denote by ρ i the density of constituent i ; ρ = n i=1 ρ i is the density of the mixture, v = 1 ρ the specific volume and c i = ρ i ρ the concentration of constituent i. The volume free energy at temperature θ is in the form

G o = G o (v, c i , θ).
The total free energy is the sum of the thermodynamic free energy and the interfacial energy.

The determination of the function G o is certainly of first importance, but it is not the topic of this paper. This is both a theoretical and an experimental problem and an equation of state with coefficients given by experiments is used for calculations. The free energy must be independent of the proportion of constituents when the density is vanishing [START_REF] Casal | Non-isothermal liquid-vapour interfaces[END_REF][START_REF] Müller | Theory of mixtures of fluids[END_REF][START_REF] Sampaio | On the viscosities of liquid mixtures[END_REF].

In the case of flat interface, the knowledge of interfacial energy is not necessary for the calculation of balance equations between the bulks. The knowledge of interfacial energy is necessary to study spherical interfaces (bulks, drops, aerosols...). A Landau-Ginzburg model consisting in a quadratic form of density gradients representing the interfacial energy [12]. Coefficients of the quadratic form are assumed to be constant (in the case of a single fluid, this corresponds to a mean field theory [START_REF] Redlich | On the thermodynamics of solutions[END_REF]). A quadratic form of the gradients of specific volumes can be also used; nevertheless, assuming that the coefficients of the quadratic form are constant, it has a different physical meaning. In fact, the choice of a model is of no consequence outside of the interfacial layer. The Landau-Ginzburg model represents an interface and the bulks and builds a complete theory of the mixtures in dynamics [START_REF] Emschwiller | Chimie-Physique[END_REF][START_REF] Galdi | Mathematical problems for miscible, incompressible fluids with Korteweg stresses[END_REF][START_REF] Hohenberg | Theory of Dynamic Critical Phenomena[END_REF].

In this paper, we apply the model to motions through interfaces for both single fluids and mixtures. Because of physic scales, motions can be often considered as permanent [START_REF] Casal | Connection between the energy equation and the motion equation in Korteweg's theory of capillarity[END_REF]. Interface moving in a single fluid was investigated by Slemrod [START_REF] Serrin | Mathematical principles of classical fluid mechanics[END_REF][START_REF] Slemrod | Admissibility criteria for propagating phase boundaries in a van der Waals fluid[END_REF] as a traveling wave whose image in a reference space is propagating with the constant velocity C. It is equivalent to say the fluid crosses a stationary interface with the flow q [4] (q is the product of velocity and density). When dissipative phenomena are neglected, the problem boils down to equilibrium with a specific free energy increased with -

1 2 q 2 v 2 .
The general study of mixtures in dynamics can be applied in the case of interfaces [START_REF] Galdi | Mathematical problems for miscible, incompressible fluids with Korteweg stresses[END_REF]. The constituent i crosses the interface with the flow q i . We are back to a static problem with the same specific free energy and an additional term in the form

- 1 2 v 2 n i=1 q i 2 c i .
Each constituent of the mixture has its own reference space [1, [START_REF] Galdi | Mathematical problems for miscible, incompressible fluids with Korteweg stresses[END_REF][START_REF] Rocard | Thermodynamique[END_REF]. Due to the fact that the interface has different velocities in each reference space, the generalization of traveling wave is less intuitive.

For a n-component mixture, a permanent motion through an interface yields n + 1 equations of dynamical equilibrium. The equations represent a linear system of the n + 1 variables q 2 i and θ. When the specific volume v and the concentrations c i are given in each bulk, a simple calculus yields the temperature of the mixture and the flow for each constituent. Obviously, solutions are acceptable only if values q 2 i and θ are positive.

Surprisingly, the problem of an interface in equilibrium is more difficult to study than the dynamical problem. It is more difficult to choose q 2 i and θ (q i is null in equilibrium case) and to deduce concentrations and specific volume than to choose concentrations and specific volume in each bulk and to deduce q 2 i and θ from a system of linear equations.

2 Isothermal motion of a single fluid in the "Korteweg -van der Waals" theory of capillarity.

Let us consider the one dimensional motion of van der Waals like fluid. We notice that the same equations describe a traveling wave of dynamic phase transition and a flow through an interface in the case of permanent motion.

M. Slemrod [START_REF] Slemrod | Admissibility criteria for propagating phase boundaries in a van der Waals fluid[END_REF] consider the one-dimensional motion of fluid possessing a specific free energy of the form 1 :

G(v, θ) = G o (v, θ) + e 2 ( ∂v ∂X ) 2
where X is the lagrangian mass variable, e is a small positive parameter. In

1
The notations are different from the ones given in [START_REF] Serrin | Mathematical principles of classical fluid mechanics[END_REF][START_REF] Slemrod | Admissibility criteria for propagating phase boundaries in a van der Waals fluid[END_REF] 3 the referential space, the balance laws of motion may be written in the form:

                               ∂v ∂t = ∂V ∂X ∂V ∂t = ∂ ∂X {-p(v, θ) + µ ∂V ∂X -e 2 ∂ 2 v ∂X 2 } ∂E ∂t = ∂ ∂X {v (-p + µ ∂V ∂X -e 2 ∂ 2 v ∂t 2 ) + e 2 ( ∂V ∂X ∂v ∂X ) + k ∂θ ∂ξ } (1)
where V denotes the velocity of the fluid, µ the viscosity, p the pressure, E the specific total energy, k the coefficient proportionality between the heat flux and the gradient of temperature in the Fourier Law and ξ = X -Ct.

The traveling wave theory of phase transitions attempts to determine when two homogeneous phases may be joined by a traveling wave solution {v = v(X -Ct), V = V (X -Ct), q = q(X -Ct)} of equations (1). Slemrod finds that the volume mass is solution of the equation:

e 2 d 2 v dξ 2 = -C 2 (v -v o ) -(p -p o ) -µ C v (2)
p o and v o denotes the values of p and v in the bulks. Because of the physical scales far from critical conditions, we established in [START_REF] Casal | Connection between the energy equation and the motion equation in Korteweg's theory of capillarity[END_REF] the motions of van der Waals fluids crossing interlayers can be considered as permanent. In the case of permanent motion of inviscid fluid, Eq. (1) 2 yields

dV dt = d dx {-p(v, θ) -e 2 ∂ 2 v ∂X 2 } v
where x denotes eulerian variable and ∂x ∂X = v [START_REF] Serrin | Mathematical principles of classical fluid mechanics[END_REF]. From V = q v where q denotes the constant flow of the fluid for an unidimensional permanent motion, we deduce:

e 2 ∂ 2 v ∂X 2 = -q 2 (v -v o ) -(p -p o ) At t
given, the form of the equation is similar to Eq. (2), which allows to obtain interpretations about dynamical Maxwell rule as in [START_REF] Serrin | Mathematical principles of classical fluid mechanics[END_REF][START_REF] Slemrod | Admissibility criteria for propagating phase boundaries in a van der Waals fluid[END_REF].

3 Isothermal motion of fluid mixture in the "Korteweg -van der Waals" theory of capillarity [START_REF] Galdi | Mathematical problems for miscible, incompressible fluids with Korteweg stresses[END_REF] For the sake of simplicity, we study a mixture of two fluids. The method can be immediately extended to any number of constituents. No assumption has to be done about composition or miscibility.

The motion of a two-fluid continuum can be represented with two surjective differentiable mappings:

z → X 1 = M 1 (z) and z → X 2 = M 2 (z)
(Subscripts 1 and 2 are associated with each constituent of the mixture)

X 1 and X 2 denote the positions of each constituent in reference spaces D 01 and D 02 . The lagrangian of the mixture is:

L = 1 2 ρ 1 V 2 1 + 1 2 ρ 2 V 2 2 -ε -ρ 1 Ω 1 -ρ 2 Ω 2
where V 1 and V 2 denote the velocity vectors of each constituent, ρ 1 and ρ 2 are the densities, Ω 1 and Ω 2 are the extraneous force potentials depending only on z = (t, x) and ε is the volume internal energy.

The expansion of the Lagrangian is in a general form. In fact dissipative phenomena imply that V 1 is almost equal to V 2 and we do not take into account some kinetic energy associated with the relative velocity of the components.

Because of the interaction between the constituents, ε does not divide into energies related to each constituent of the mixture, like for simple mixtures of fluids [START_REF] Joseph | Fluid Dynamics of two miscible liquids with slow diffusion and Korteweg stresses[END_REF]. The mixture is supposed to be no chemically reacting. Conservations of masses require:

ρ i det F i = ρ oi (X i ) (3)
where subscript i belongs to {1,2}. At t fixed, the Jacobian associated with M i is denoted by F i and ρ oi is the reference specific mass in D oi .

In differentiable cases, Eq. ( 3) is equivalent to:

∂ρ i ∂t + div ρ i V i = 0
The volumic internal energy ε is given by the thermodynamic behavior of the mixture. Each constituent has a specific mass; in the same way, two specific entropies s 1 and s 2 are supposed to be associated with constituents 1 and 2.

For an internal energy depending on gradients of densities, the volume internal energy is:

ε = ε(s 1 , s 2 , ρ 1 , ρ 2 , gradρ 1 , gradρ 2 ) (4)
The quantity

h i = ε, ρ i -ε, ρ i,γ,γ
defines the specific enthalpy of the constituent i of the mixture. Subscript γ corresponds to the spatial derivatives associated with gradient terms; as usually, summation is made on repeated subscript γ.

θ i = ε, s i ρ i
defines the temperature of the constituent i of the mixture.

In practise we use the expression

ε = α(s 1 , s 2 , ρ 1 , ρ 2 ) + 1 2 Q
where Q is a quadratic form with constant coefficients:

Q = C 1 (grad ρ 1 ) 2 + 2D gradρ 1 gradρ 2 + C 2 (grad ρ 2 ) 2
and α(s 1 , s 2 , ρ 1 , ρ 2 ) is the value of internal energy in the homogeneous bulks.

To obtain the equations of motions, we used variational principle whose original feature is to choice variations in reference spaces (Fig. 1) [START_REF] Galdi | Mathematical problems for miscible, incompressible fluids with Korteweg stresses[END_REF].

Figure 1 Variations δ 1 and δ 2 of motions of particles are deduced from X 1 = ψ 1 (x, t, β 1 ) and X 2 = ψ 2 (x, t, β 2 ). They are associated with a two-parameter family of virtual motions of the mixture. The real motion corresponds to β 1 = 0 and β 2 = 0. Obviously, a classical variation of function in space W (as Serrin's p. 145, [START_REF] Soave | Equilibrium constants from a modified Redlich-Kwong equation of state[END_REF]) with only one parameter cannot give as many informations as the ones given by any two-parameters family of virtual displacements.

The equation of the motion of each constituent of the mixture is:

Γ i = θ i grad s i -grad (h i + Ω i ) (5)
Let us note that the motion of each constituent is described by the same equation that for a single fluid [START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF][START_REF] Soave | Equilibrium constants from a modified Redlich-Kwong equation of state[END_REF]. In applications, the motions are supposed to be isothermal and θ i = θ (θ common temperature value for all the components). This case corresponds to strong heat exchange between components.

Eq. ( 5) yields:

Γ i + grad (ϕ i + Ω i ) = 0
where ϕ i = h iθs i is the free specific enthalpy (or chemical potential) of constituent i. In the bulks, the volumic free energy associated with α is denoted by g o = g o (ρ 1 , ρ 2 , θ). A complete study of the barycentric motion of the mixture and of the equation of energy is given in [START_REF] Galdi | Mathematical problems for miscible, incompressible fluids with Korteweg stresses[END_REF]. This generalizes results for single fluid [START_REF] Cahn | Free energy of a non-uniform system[END_REF][START_REF] Peng | A new two-constant equation of state[END_REF][START_REF] Gouin | Thermodynamic form of the equation of motion for perfect fluids of grade n[END_REF].

Motion of an isothermal fluid mixture through a plane interface

An interface in a two-phase mixture is generally schematized by a surface without thickness. Far from critical conditions, this layer is of molecular size and density and entropy gradients are very large. A continuous model schematizes such areas by using an energy in form (4) extending forms given in [2,[START_REF] Redlich | On the thermodynamics of solutions[END_REF] for compressible fluids.

For the same reasons than for a single fluid, motion is supposed to be isothermal, stationary and one dimensional with respect to x.

The acceleration of component i is:

Γ i = 1 2 d dx ( q 2 i ρ 2 i ).
and the equations of motion yield:

             Γ 1 = grad {C 1 ∆ρ 1 + D ∆ρ 2 -g o,ρ 1 } Γ 2 = grad {D ∆ρ 1 + C 2 ∆ρ 2 -g o,ρ 2 } or:                  C 1 ρ ′′ 1 + D ρ ′′ 2 = g o,ρ 1 + 1 2 q 2 1 ρ 2 1 + k 1 D ρ ′′ 1 + C 2 ρ ′′ 2 = g o,ρ 2 + 1 2 q 2 2 ρ 2 2 + k 2
(The derivatives along the motion axis are denoted by ′ ).

Combination of the last two equations yields the first integral:

g o + k 1 ρ 1 + k 2 ρ 2 - 1 2 q 2 1 ρ 1 - 1 2 q 2 2 ρ 2 -( 1 2 C 1 ρ ′ 2 1 + D ρ ′ 1 ρ ′ 2 + 1 2 C 2 ρ ′ 2 1 ) = k 3
In each bulk, densities have zero-gradients. Eliminating constants k i , dynamical conditions through the interfacial layer yield:

             [g ,ρ 1 (ρ 1 , ρ 2 )] = 0 [g ,ρ 2 (ρ 1 , ρ 2 )] = 0 [g -ρ 1 g ,ρ 1 -ρ 2 g ,ρ 2 ] = 0 (6)
where [ ] denotes the discontinuities through the layer and

g = g o - 1 2 q 2 1 ρ 1 - 1 2 q 2 2 ρ 2 .
In case of equilibrium (q 1 and q 2 are null), the minimum of the total free energy, with a given total mass for each constituent, yields conditions [START_REF] Peng | A new two-constant equation of state[END_REF].

By adding the term -1 2

q 2 1 ρ 1 - 1 2 q 2 2 ρ 2
to g o , the study of flat interfaces crossed by components of a mixture turns back to an equilibrium problem. In fact, a complete study of the thickness of the interfacial layer or of spherical interfaces of microscopic size requires the non-linear model.

It is more classical to use the variables v or c. The mapping (ρ 1 , ρ 2 , g) →

(v, c, G) where v = 1 ρ 1 + ρ 2 , c = ρ 2 ρ , G o = g o ρ and G = G o - 1 2 v 2 ( q 2 1 1 -c + q 2 2 c ) ,
allows to write

             [G ,v (v, c)] = 0 [G ,c (v, c)] = 0 [G -v G ,v -c G ,c ] = 0 (7)
Conditions ( 6) or [START_REF] Dunn | Interstitial working and a Nonclassical Continuum Thermodynamics[END_REF] express that the points corresponding to the bulks of the Gibbs surface (Σ) associated with the dynamical free energy g or G are contact points of a bitangent plane (Fig. 2).

Figure 2 The pressure of the mixture is p = -G o,v (v, c), the dynamical pressure is

P = -G ,v (v, c).
The z-coordinate of the point of intersection of the bitangent plane with the g-axis is -P . The chemical potential G ,c is denoted by Φ.

Contact points of the bitangent plane with surface (Σ) generate a curve. Along this curve, the straight line (D) connecting the contact points is the characteristic line of the plane. Along the line, the relation

(v -v 0 ) d G ,v + (c -c 0 ) d G ,c = 0
allows to generalize Clapeyron formula such that:

dP dΦ = c 1 -c 0 v 1 -v 0
where subscripts 0 and 1 are associated with the two bulks.

Maxwell rules for fluid mixtures

At a given temperature θ, the differential form dG = -P dv + Φ dc yields the relation:

(C) (Φ -Φ 0 ) dc -(P -P 0 ) dv = 0
where (C) is an arbitrary curve connecting the two bulks in the space (c, v).

The common values of Φ and P in the bulks are denoted by Φ 0 and P 0 .

We can calculate the integral along special paths:

Along the curve (C) given by the implicit function c(v) defined such that G ,c (c(v), v) = Φ 0 , the relation

∂ G o (v, c, θ) ∂v = -p (v, c, θ) (8)
Function p is determined except for an additional function of c and θ. With the physical assumption that G o is independent of c when v vanishes, the additional function depends only on θ.

The simplest model of equation of state is by van der Waals:

p = Rθ v -b - a v 2 (9)
where a and b are given by the mixing rules:

a 2 = (1 -c) a 2 1 + c a 2 2 et b = (1 -c)b 1 + c b 2 (10)
(Subscripts 1 and 2 are associated with the components 1 and 2).

One deduces:

G o = -R θ Ln (v -b) - a v .
Other Equations ( 7), ( 8), ( 9) and ( 10) lead to the system:

                 [ θ v -b + v 1 -c q 2 1 + v c q 2 2 - (b + 1) 2 v 2 ] = 0 [ θ{1 -Ln (v -b) + b 1 v -b } + 1 2 v 2 (1 -c) 2 q 2 1 -2 (b 1 + 1) (b + 1) v ] = 0 [ θ{1 -Ln (v -b) + b 2 v -b } + 1 2 v 2 c 2 q 2 2 -2 (b 2 + 1) (b + 1) v ] = 0
which is a linear system with respect to θ, q 2 1 , q 2 2 .

Conclusion

The study of fluid mixture motions by a model taking into account density gradients of components requires the knowledge of the global free energy. In fact, only physical experiments and molecular theories provide an equation of state (in van der Waals type). Isothermal motions can be studied by using such an equation and yield the jump conditions through interfacial layers. To carry the program one step further and to study non-isothermal motions, it is necessary to get additional knowledges (such that specific heats...).

by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation and in part by the French Foreign Office.

  P 0 ) dv = 0 represents the Mawxell equal area rule for the path abcd in the plan (P, v)where 2 P = -G ,v (c (v), v)Likely, for P = P 0 , the relation: (C) (Φ -Φ 0 ) dc = 0 represents the Mawxell equal area rule for the path abcd in the plan (c, Φ) where Φ = G ,c (c, v(c)) is such that v is expressed as a function of c by the relation G ,v (c, v(c)) = -P 0 .

Figure 3 These

 3 Figure 3

  equations of state are proposed for example p = R θ vb -a v(v + b) (particularly in petroleum industry [5,15,18]) or by changing the form of the mixing rules (other expressions than a 2 = n i=1 a 2 i c i and b = n i=1 b i c i can be considered).

(C) is not the curve associated with P = P (c, v) at c constant.

Acknowledgements: I am grateful to Professor P. Casal for his help and to the Institute of Mathematics and its Applications for hospitality and financial support in the month of October 1990. This research was supported in part