Bertrand Lods
email: lods@math.univ-bpclermont.fr

VARIATIONAL CHARACTERIZATIONS OF THE EFFECTIVE MULTIPLICATION FACTOR OF A NUCLEAR REACTOR CORE

Keywords: effective multiplication factor, nuclear reactor, neutron transport equation, energy-dependent diffusion equation, positivity. AMS SUBJECT CLASSIFICATION: 82D75, 35P15

We prove some inf-sup and sup-inf formulae for the so-called effective multiplication factor arising in the study of reactor analysis. We treat in a same formalism the transport equation and the energy-dependent diffusion equation.

INTRODUCTION

The aim of this paper is to give some variational characterizations of the effective multiplication factor arising in nuclear reactor theory. This work follows a very recent paper by M. Mokhtar-Kharroubi [START_REF] Mokhtar-Kharroubi | On the leading eigenvalue of neutron transport models[END_REF] devoted to the leading eigenvalue of transport operators.

In practical situations, the power distribution in a stable nuclear reactor core is determined as the steady-state solution φ of a linear transport equation for the neutron flux. Because of interactions between neutrons and fissile isotopes, a fission chain reaction occurs in the reactor core. Precisely, when an atom undergoes nuclear fission, some neutrons are ejected from the reaction and subsequently shall interact with the surrounding medium. If more fissile fuel is present, some may be absorbed and cause more fissions (see for details Refs. [START_REF] Bell | Nuclear reactor theory[END_REF][START_REF] Birkhoff | Reactor criticality in neutron transport theory[END_REF][START_REF] Duderstadt | Nuclear Reactor Analysis[END_REF][START_REF] Planchard | Méthodes mathématiques en neutronique[END_REF]). The linear stationary transport equation is therefore of non-standard type in the sense that the source term is itself a function of the solution. When delayed neutrons are neglected, this equation reads

v • ∇ x φ(x, v) + σ(x, v)φ(x, v) - V κ s (x, v, v ′ )φ(x, v ′ )dµ(v ′ ) = 1 k eff V κ f (x, v, v ′ )φ(x, v ′ )dµ(v ′ ) (1.1)
with free-surface boundary condition (i.e. the incoming flux is null). Here, the unknown φ(x, v) is the neutron density at point x ∈ D and velocity v ∈ V, where D is an open subset of R N (representing the reactor core) and the velocity space V is a closed subset of R N , dµ(•) being a positive Radon measure supported by V . For the usual cases, dµ(•) is either the Lebesgue measure on R N (continuous model) or on spheres (multigroup model). The transfer cross-sections κ s (•, •, •) and κ f (•, •, •) describe respectively the pure scattering and the fission process. The nonnegative bounded function σ(•, •) is the absorption cross-section [START_REF] Bell | Nuclear reactor theory[END_REF][START_REF] Duderstadt | Nuclear Reactor Analysis[END_REF][START_REF] Planchard | Méthodes mathématiques en neutronique[END_REF]. The positive ratio k eff is called the criticality eigenvalue (or the effective multiplication factor). It represents the average number of neutrons that go on to cause another fission reaction. The remaining neutrons either fail to induce fission, or never get absorbed and exit the system. Consequently, k eff measures the balance between the number of neutrons in successive generations (where the birth event separating generations is the fission process).

The interpretation of the effective multiplication factor k eff is related to the following three cases (see [START_REF] Bell | Nuclear reactor theory[END_REF][START_REF] Duderstadt | Nuclear Reactor Analysis[END_REF][START_REF] Planchard | Méthodes mathématiques en neutronique[END_REF]):

• If k eff = 1
, there is a perfect balance between production and removal of neutrons. The reactor is then said to be critical.

• The reactor is sub-critical when k eff < 1. This means that the removal of neutrons (at the boundary or due to absorption by the surrounding media) excesses the fission process and the chain reaction dies out rapidly.

• When k eff > 1, the fission chain reaction grows without bound and the reactor is said to be super-critical.

Up to now, in practical applications, the effective multiplication factor k eff was usually given by r σ (T -K s ) -1 K f = k eff , where the precise definition of the operators T , K s and K f is given subsequently and r σ [B] denotes the spectral radius of any generic bounded operator B: r σ [B] = lim n→∞ B n 1/n . Because its requires the computation of the resolvent (T -K s ) -1 , such a characterization makes the analysis of the effective multiplication factor quite difficult to handle. In particular, practical estimates of k eff as well as computational approximations are rather involved and merely rely on (direct or inverse) power method [START_REF] Allen | The inverse power method for calculation of multiplication factors[END_REF]. Our main objective in this paper is to provide tractable variational characterizations of k eff in terms of the different data of the system and suitable test functions. We hope that such a characterization shall be of interest for practical computations or for the homogenization of the above criticality transport equation in periodic media [START_REF] Allaire | Homogenization of the criticality spectral equation in neutron transport[END_REF][START_REF]First-order corrector for the homogenization of the criticality eigenvalue problem in the even parity formulation of the neutron transport[END_REF][START_REF] Mokhtar-Kharroubi | Homogenization of boundary value problems and spectral problems for neutron transport in locally periodic media[END_REF]. We also believe that our characterization can be useful in the delicate optimization problem of the assembly distribution in a nuclear reactor (see the recent contribution [START_REF] Thévenot | An optimality condition for the assembly distribution in a nuclear reactor[END_REF] based upon the homogenization method and where the criticality eigenvalue k eff plays a crucial role).

To be precise, we provide here variational characterizations of the effective multiplication factor of the type

1 k eff = min ϕ ∈W + p ess sup (x,v)∈D×V v • ∇ x ϕ(x, v) + σ(x, v)ϕ(x, v) - V Σ s (x, v, v ′ )ϕ(x, v ′ )dµ(v ′ ) V Σ f (x, v, v ′ )ϕ(x, v ′ )dµ(v ′ ) = max ϕ ∈W + p ess inf (x,v)∈D×V v • ∇ x ϕ(x, v) + σ(x, v)ϕ(x, v) - V Σ s (x, v, v ′ )ϕ(x, v ′ )dµ(v ′ ) V Σ f (x, v, v ′ )ϕ(x, v ′ )dµ(v ′ ) , (1.2)
where

W + p is a suitable class of positive test-functions in L p (D × V, dxdµ(v)) (1 p < ∞)
. This result (Theorem 3.3) holds true under compactness assumption on the full collision operator

K : ψ -→ V Σ s (x, v, v ′ ) + Σ f (x, v, v ′ ) ψ(x, v ′ ) dµ(v ′ )
and under positivity assumptions on the fission cross-section κ f (•, •, •). The main strategy to derive (1.2) is adapted from [START_REF] Mokhtar-Kharroubi | On the leading eigenvalue of neutron transport models[END_REF] where M. Mokhtar-Kharroubi proved similar variational characterizations for the leading eigenvalue of perturbed transport operators. Note that the above characterization still holds true for transport equations with general boundary conditions modeled by some nonnegative albedo operator (see Remark 3.6).

At this point, one recalls that, besides the critical eigenvalue k eff , it is also possible to investigate the reactivity of the nuclear reactor core through another physical parameter, namely, the leading eigenvalue s(A) of the operator A = T +K s +K f , also associated to positive eigenfunctions. The two parameters k eff and s(A) are related by the following: if s(A) < 0 then the reactor is subcritical (i.e. k eff < 1), while it is super-critical whenever s(A) > 0. The reactor is critical when s(A) = 0. The paper [START_REF] Mokhtar-Kharroubi | On the leading eigenvalue of neutron transport models[END_REF] provides a variational characterization of the leading eigenvalue of the transport operator A. However, for practical calculations in nuclear engineering, the critical eigenvalue k eff is a more effective parameter. Actually, the existence of the leading eigenvalue s(T + K s + K f ) is not always ensured but is related to the size of the domain D and the possibility of small velocities (for more details on this disappearance phenomenon, see e.g. [START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF]Chapter 5]). Since the existence of the effective multiplication factor is not restricted by such physical constraints, it appears more efficient to measure the reactivity of nuclear reactor cores by k eff . This is what motivated us to generalize the result of [START_REF] Mokhtar-Kharroubi | On the leading eigenvalue of neutron transport models[END_REF] and provide variational characterization of k eff .

In this paper, we also give a characterization of the criticality eigenvalue associated to the energydependent diffusion model used in nuclear reactor theory [START_REF] Bell | Nuclear reactor theory[END_REF][START_REF] Pao | Stability and instability of the energy dependent diffusion system in reactor dynamics[END_REF][START_REF] Planchard | Méthodes mathématiques en neutronique[END_REF]. For this description, the critical problem reads

-div x (D(x, ξ)∇ x ̺(x, ξ)) + σ(x, ξ)̺(x, ξ) - E Σ s (x, ξ, ξ ′ ) ̺(x, ξ ′ ) dξ ′ = 1 k eff E Σ f (x, ξ, ξ ′ ) ̺(x, ξ ′ ) dξ ′ , (1.3)
complemented by the Dirichlet boundary conditions ̺(•, ξ) |∂D = 0 a.e. ξ ∈ E. Here E is the set of admissible energies ξ = 1 2 mv 2 (m being the neutron mass and v the velocity

), i.e. E is a subset of [0, +∞[. The diffusion coefficient D(•, •) is a matrix-valued function over D × E and the unknown ̺(•, •) is nonnegative.
The derivation of diffusion-like models for some macroscopic distribution function ̺(x, ξ) (corresponding to some angular momentum of the solution φ to (1.1)) is motivated in nuclear engineering by the necessity to provide simplified models tractable numerically. Such a energy-dependent diffusion model can be derived directly from a phenomenological analysis of the scattering models or it can be derived from the above kinetic equation (1.1) through a suitable asymptotic procedure (see [START_REF] Degond | An infinite system of diffusion equations arising in transport theory: the coupled Spherical Harmonics Expansion model[END_REF] and the references therein for more details on that matter).

For this energy-dependent diffusion model, we give a variational characterization of k eff in terms of sup-inf and inf-sup criteria in the spirit of (1.2).

Actually, to treat the two above problems (1.1) and (1.3) it is possible to adopt a unified mathematical formalism. Precisely, let us denote by K s the integral operator with kernel κ s (•, •, •) and denote by K f the integral operator with kernel κ f (•, •, •). Then, problems (1.1) and (1.3) may be written in a unified abstract way:

T φ eff + K s φ eff + 1 k eff K f φ eff = 0, φ eff 0,
where the unbounded operator T refers to, according to the model we adopt:

• the transport operator:

T φ(x, v) = -v • ∇ x φ(x, v) -σ(x, v)φ(x, v),
associated to the absorbing boundary conditions φ |Γ -= 0.

• the energy-dependent diffusion operator:

T ψ(x, ξ) = div x (D(x, ξ)∇ x ψ(x, ξ)) -σ(x, ξ)ψ(x, ξ)
with Dirichlet boundary conditions ψ | ∂D (•, ξ) = 0 (a.e. ξ ∈ E).

The abstract treatment of the above problem is performed in Section 2 and relies mainly on positivity and compactness arguments. The main abstract result of this paper (Theorem 2.15) characterizes the criticality eigenvalue of a large class of (abstract) unbounded operators in L p -spaces. Besides this main analytical result, we also prove abstract results with their own interest. In particular, we provide in Theorem 2.12 an approximation resolution for the criticality eigenfunction φ eff which shall be hopefully useful for practical numerical approximations.

The outline of the paper is as follows. In Section 2, we describe the unified and abstract framework which allows us to treat in a same formalism Problems (1.1) and (1.3) with the aim of establishing general inf-sup and sup-inf formulae for the criticality eigenvalue of a class of unbounded operator. In Section 3, we are concerned with the characterization of the effective multiplication factor k eff associated to the transport problem (1.1). In Section 4, we investigate the effective multiplication factor associated to the energy-dependent diffusion model (1.3).

ABSTRACT VARIATIONAL CHARACTERIZATION

This section is devoted to several abstract variational characterizations of the criticality eigenvalue. It is this abstract material that shall allow us to treat in the same formalism Problems (1.1) and (1.3).

Setting of the problem and existence result.

Let us introduce the functional framework we shall use in the sequel. Given a measure space (Ω, ν) and a fixed 1 p < ∞, define

X p = L p (Ω, dν)
and denote by X q its dual space, i.e. X q = L q (Ω, dν) (1/p + 1/q = 1). We first recall several definitions and facts about positive operators. Though the various concepts we shall deal with could be defined in general complex Banach lattices, we restrict ourselves to operators in X p (1 p < ∞): Definition 2.1. A bounded operator B in X p is said to be irreducible if, for every nonnegative ϕ ∈ X p \ {0} and nonnegative ψ ∈ X q \ {0}, there exists n ∈ N such that

B n ϕ, ψ > 0,
where •, • is the duality pairing between X p and the dual space X q .

Let us denote the set of quasi-interiors elements of X p by X + p , i.e.

X + p = {f ∈ X p ; f (ω) > 0 dν -a.e. ω ∈ Ω}. Notice that, if f ∈ X +
p , then f, ψ > 0 for any nonnegative ψ ∈ X q \ {0}. Definition 2.2. A bounded operator B in X p will be said to be positivity improving if its maps nonnegative

f ∈ X p \ {0} into X + p , i.e. f ∈ X p \ {0}, f 0 =⇒ Bf ∈ X +
p . Remark 2.3. Notice that, given a bounded operator B in X p , if some power of B is positivity improving, then B is irreducible. This provides a practical criterion of irreducibility.

Recall also several fact about power-compact operators. Definition 2.4. A bounded operator B in a Banach space X is said to be power-compact if there exists n ∈ N such that B n is a compact operator in X.

The following fundamental result is due to B. De Pagter [START_REF] Pagter | Irreducible compact operators[END_REF].

Theorem 2.5. Let B be a bounded operator in a Banach space X. If B is irreducible and powercompact then r σ (B) > 0 where r σ (B) denotes the spectral radius of B.

Let T be a given densely defined unbounded operator

T : D(T ) ⊂ X p -→ X p such that s(T ) < 0 and (0 -T ) -1 (X + p ) ⊂ X + p .
(2.1) Let K s and K f be two nonnegative bounded operators in X p . We are interested in the abstract critical problem:

(T + K s + 1 k eff K f )φ eff = 0, φ eff ∈ D(T ) , φ eff 0, φ eff = 0. (2.2) Since s(T ) < 0, Problem (2.2) is equivalent to (0 -T ) -1 (K s + 1 k eff K f )φ eff = φ eff , φ eff 0, φ eff = 0.
Let us introduce the family of operators indexed by the positive parameter γ:

K(γ) = K s + 1 γ K f γ > 0.
Therefore, solving (2.2) is equivalent to prove the existence (and uniqueness) of k eff > 0 such that 1 is an eigenvalue of (0 -T ) -1 K(k eff ) associated to a nonnegative eigenfunction. Such an existence and uniqueness result can be found in [START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF]Theorem 5.30] (see also [START_REF] Marek | Frobenius theory of positive operators[END_REF]). We set K = K s + K f Theorem 2.6. Assume that (0 -T ) -1 K is power-compact and that (0 -T ) -1 K f is irreducible. Then, the spectral problem (2.2) admits a unique solution k eff > 0 associated with a nonnegative eigenfunction φ eff if and only if

lim γ→0 r σ [(0 -T ) -1 K(γ)] > 1 and r σ [(0 -T ) -1 K s ] < 1. (2.3)
Remark 2.7. Notice that our assumptions differs slightly from that of [START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF]. Actually, in [START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF], it is assumed that (0 -T ) -1 K(γ) is power-compact and irreducible for any γ > 0. Our assumption implies those of [START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF]. Indeed, in this case, there is an integer

N ∈ N such that (0 -T ) -1 K N is compact. Since, for any γ > 0, K(γ) max{1, 1/γ}K, one gets by a domination argument that (0 -T ) -1 K(γ)
N is compact for any γ > 0. This means that the power at which (0 -T

) -1 K(γ)
becomes compact is independent of γ > 0. In the same way, since (0 -T ) -1 K(γ) 1 γ (0 -T ) -1 K f for any γ > 0, our assumption implies the irreducibility of (0 -T ) -1 K(γ) for any γ > 0. Notice also that the result still holds if (0 -T ) -1 K s is irreducible. [START_REF] Kato | Perturbation Theory for linear operators[END_REF]Remark 3.3,p. 208]). By analyticity arguments (Gohberg-Shmulyan theorem), it is also strictly decreasing. Thus, k eff is characterized by

Remark 2.8. Under the assumptions of the previous Theorem, we point out that the mapping

γ > 0 → r σ [(0 -T ) -1 K(γ)] is continuous (see
r σ [(0 -T ) -1 K(k eff )] = 1.
Let us now give some variational characterizations of the criticality eigenvalue k eff appearing in Theorem 2.6.

Abstract variational characterization of k

eff . From now on, Assumption (2.3) is assumed to be fulfilled. Let W + p := D(T ) ∩ X + p .
(2.4) We start with the following characterization of k eff in terms of super-solution to the spectral problem (2.2). Proposition 2.9. Assume that (0 -T ) -1 K(γ) is power-compact and irreducible for any γ > 0.

For any ϕ ∈ W + p , let τ + (ϕ) := sup{γ > 0 such that (T + K(γ)) ϕ is nonnegative }
with the convention sup ∅ = 0. Then

k eff = sup ϕ∈W + p τ + (ϕ). Proof. Let ϕ ∈ X p be a nonnegative eigenfunction of (0 -T ) -1 K(k eff ) associated with the spectral radius r σ [(0 -T ) -1 K(k eff )] = 1, i.e. (0 -T ) -1 K(k eff ) ϕ = ϕ.
Clearly, ϕ ∈ D(T ). Moreover, since (0 -T ) -1 K(k eff ) is power-compact and irreducible, a wellknown consequence of Krein-Rutman Theorem is that

ϕ(ω) > 0 a.e. ω ∈ Ω i.e. ϕ ∈ W + p . Now (T + K(k eff )) ϕ = 0
is nonnegative so that τ + (ϕ) k eff and consequently

k eff sup ϕ∈W + p τ + (ϕ).
Assume now that k eff < sup

ϕ∈W + p τ + (ϕ) et let ψ ∈ W + p be such that τ + (ψ) > k eff . Denote γ = τ + (ψ). By definition (T + K(γ)) ψ 0, i.e. (0 -T ) -1 K(γ) ψ ψ. Thus, for any n ∈ N, [(0 -T ) -1 K(γ)] n ψ ψ so r σ [(0 -T ) -1 K(γ)] 1. Since, r σ [(0 -T ) -1 K(k eff )] = 1, Remark 2.8 ensures that γ k eff , which contradicts the choice of τ + (ψ) > k eff .
The following illustrates the fact that the extremal value to the above variational result is reached only by the nonnegative solution to the spectral problem (2.2).

Corollary 2.10. Assume that (0 -T ) -1 K(γ) is power-compact and irreducible for any γ > 0. Then, for any nonnegative ϕ ∈ D(T ) \ {0}

τ + (ϕ) = k eff if and only if (T + K(k eff )) ϕ = 0. Proof. Let ϕ ∈ D(T ) \ {0}, ϕ 0, be such that τ + (ϕ) = k eff . Then (T + K(k eff )) ϕ 0 i.e. (0 -T ) -1 K(k eff ) ϕ ϕ. (2.5) Suppose that ϕ = (0 -T ) -1 K(k eff ) ϕ, (2.6)
and let ψ ⋆ ∈ X q be a nonnegative eigenfunction of the dual operator

((0 -T ) -1 K(k eff )) ⋆ associated to r σ [((0 -T ) -1 K(k eff )) ⋆ ] = r σ [(0 -T ) -1 K(k eff )] = 1.
Then, by (2.5) and (2.6)

ϕ, ψ ⋆ < (0 -T ) -1 K(k eff ) ϕ, ψ ⋆ = ϕ, ((0 -T ) -1 K(k eff )) ⋆ ψ ⋆ = r σ [((0 -T ) -1 K(k eff )) ⋆ ] ϕ, ψ ⋆ = ϕ, ψ ⋆ , which leads to a contradiction. Hence, ϕ = (0 -T ) -1 K(k eff ) ϕ, i.e. (T + K(k eff ))ϕ = 0.
Conversely, if ϕ is a nonnegative eigenfunction of T + K(k eff ) associated to the null eigenvalue, then τ + (ϕ) k eff and the identity τ + (ϕ) = k eff follows from Proposition 2.9.

Remark 2.11. A careful reading of the proof here above shows that

, if ϕ ∈ X p is such that (0 - T ) -1 K(k eff ) ϕ ϕ 0, then ϕ ∈ D(T ) and (T + K(k eff )) ϕ = 0.
Let us denote by φ eff the unique critical eigenfunction with unit norm, i.e.

φ eff ∈ W + p satisfies (T + K(k eff ))φ eff = φ eff , φ eff = 1.
Then one can prove the following approximation resolution for the criticality eigenfunction φ eff whose proof is inspired by [START_REF] Mokhtar-Kharroubi | On the leading eigenvalue of neutron transport models[END_REF]Theorem 7]. Such a result shall be hopefully useful for practical numerical approximation of the critical mode φ eff of the reactor.

Theorem 2.12.

Let (ϕ k ) k ∈ D(T ) ∩ X + p be such that γ k := τ + (ϕ k ) → k eff . We assume ϕ k to be normalized by (0 -T ) -1 K(γ k ) N ϕ k = 1 (k ∈ N) (2.7)
where N is the integer given by Remark 2.7. Let us assume that 1 is a simple eigenvalue of (0 -T ) -1 K(k eff ). Moreover, in the case p = 1, let us assume that the dual operator (0 -T ) -1 K(k eff )

⋆ admits an eigenfunction associated to its spectral radius which is bounded away from zero. Then,

lim k→∞ ϕ k -φ eff = 0 where φ eff ∈ D(T ) ∩ X +
p is the unique positive eigenfunction of (0 -T ) -1 K(k eff ) associated to 1 and with unit norm .

Proof. According to the definition of γ

k := τ + (ϕ k ), (T + K(γ k ))ϕ k is nonnegative. Therefore, ϕ k (0 -T ) -1 K(γ k )ϕ k (2.8) and, iterating up to N ϕ k (0 -T ) -1 K(γ k ) N ϕ k (k ∈ N). (2.9)
This shows, according to (2.7), that ϕ k 1. Now, if 1 < p < ∞ there exists a subsequence (ψ k ) k which converges weakly to some ψ ∈ X p . If p = 1, the fact that γ k → k eff combined with the compactness of (0 -T ) -1 K(γ k )

N lead to the relative compactness of

(0 -T ) -1 K(γ k ) N ϕ k k in X 1 .
In particular, this sequence is equi-integrable and by domination (2.9), (ϕ k ) k is also equiintegrable. We can extract a subsequence (ψ k ) k converging weakly to some ψ ∈ X 1 . In both cases, the compactness of (0 -T ) -1 K(γ k ) N together with γ k → k eff yield to the strong convergence

(0 -T ) -1 K(γ k ) N ψ k → (0 -T ) -1 K(k eff ) N ψ so that (0 -T ) -1 K(k eff ) N ψ = 1.
In particular ψ = 0, and, taking the weak limit in (2.8), ψ (0 -T ) -1 K(k eff )ψ. Now, according to Remark 2.11, this last inequality is actually an equality, i. e.

ψ = (0 -T ) -1 K(k eff )ψ.
Iterating again, one gets

ψ = (0 -T ) -1 K(k eff ) N ψ = 1.
Now, since 1 is a simple eigenvalue of (0-T ) -1 K(k eff ), the set of eigenfunctions of (0-T ) -1 K(k eff ) with unit norm reduces to a singleton. This shows that ψ is the (unique) weak limit of any subsequence of (ϕ k ) k so that the whole sequence (ϕ k ) k converges weakly to ψ ∈ X p . The remainder of the proof consists in showing that the convergence actually holds in the strong sense.

Let us consider first the case 1 < p < ∞. To show now that ϕ k -ψ → 0, it suffices to prove that ϕ k → ψ . A consequence of the weak convergence leads to

1 = ψ lim inf k→∞ ϕ k .
Since ϕ k 1 for any k ∈ N, this proves the Theorem for 1 < p < ∞. Let us now assume p = 1. Then ϕ k → ψ strongly in X 1 if and only if the convergence holds in measure, i. e., for any Ξ ⊂ Ω with finite dν-measure and every ǫ > 0

lim k→∞ dν{ω ∈ Ξ ; |ψ k (ω) -ψ(ω)| > ǫ} = 0.
Arguing by contradiction, assume there exist Ξ ⊂ Ω with finite dν-measure, a subsequence still denoted (ψ k ) k and some δ > 0 and some ǫ 0 > 0 such that

dν{ω ∈ Ξ ; |ψ k (ω) -ψ(ω)| > ǫ 0 } δ for all m ∈ N. (2.10) Setting ψ k = (0 -T ) -1 K(γ k ) N ψ k one has lim k→∞ ψ k -ψ = 0 since ψ = (0 -T ) -1 K(γ k ) N ψ. Consequently, lim k→∞ dν{ω ∈ Ξ ; |ψ k (ω) -ψ(ω)| > ǫ 0 /2} = 0
and, one deduces immediately from (2.10) that

dν{ω ∈ Ξ ; |ψ k (ω) -ψ(ω)| > ǫ 0 /2} δ/2 for large k. (2.11)
Now, let ψ ⋆ ∈ L ∞ (Ω) be a positive eigenfunction of (0 -T ) -1 K(k eff ) ⋆ associated to the eigenvalue 1, with ψ ⋆ bounded away from zero. One has

ψ k -ψ k , ψ ⋆ {ω∈Ξ ; |ψ k -ψ k |>ǫ 0 /2} ψ k (ω) -ψ k (ω) ψ ⋆ (ω) dν(ω) inf w ψ ⋆ (ω) × ǫ 0 /2 × δ/2 = η > 0 or else (0 -T ) -1 K(γ k ) N ψ k , ψ ⋆ -ψ k , ψ ⋆ η for large k.
Equivalently

ψ k , {[(0 -T ) -1 K(γ k )] ⋆ } N ψ ⋆ -ψ k , ψ ⋆ η for large k. But, {[(0 -T ) -1 K(γ k )] ⋆ } N ψ ⋆ converges strongly to {[(0 -T ) -1 K(k eff )] ⋆ } N ψ ⋆ in L ∞ (Ω), which, combined with the weak convergence ψ k ⇀ ψ, implies ψ k , {[(0 -T ) -1 K(γ k )] ⋆ } N ψ ⋆ → ψ, {[(0 -T ) -1 K(k eff )] ⋆ } N ψ ⋆ = ψ, ψ ⋆ .
Now, the contradiction follows from the fact that ψ k , ψ ⋆ → ψ, ψ ⋆ .

The following characterizes k eff in terms of sub-solution to the spectral problem (2.2).

Proposition 2.13. Assume that (0 -T ) -1 K(γ) is power-compact and irreducible for any γ > 0. For any ϕ ∈ W + p , define τ -(ϕ) := inf{γ > 0 ; -(T + K(γ))ϕ ∈ X + p } with the convention inf ∅ = +∞. Then

k eff = inf ϕ∈W + p τ -(ϕ). Proof. Let ϕ ∈ W + p be such that τ -(ϕ) < +∞, and let γ > τ -(ϕ). Then -(T + K(γ)) ϕ ∈ X + p .
(2.12)

Since (0 -T ) -1 (X + p ) ⊂ X + p (see Eq. (2.1)), one gets that ϕ -(0 -T ) -1 K(γ) ϕ ∈ W + p . (2.13)
Now, let ψ ⋆ ∈ X q be a nonnegative eigenfunction of the dual operator ((0 -T ) -1 K(γ)) ⋆ associated with the spectral radius r σ [((0-T )

-1 K(γ)) ⋆ ] = r σ [(0-T ) -1 K(γ)]. Note that r σ [(0-T ) -1 K(γ)] > 0 according to Theorem 2.5. From Krein-Rutman theorem, ψ ⋆ (ω) > 0 dν -a.e. ω ∈ Ω. Thus, by (2.13) ϕ -(0 -T ) -1 K(γ) ϕ, ψ ⋆ > 0 i.e. ϕ, ψ ⋆ > (0 -T ) -1 K(γ) ϕ, ψ ⋆ = ϕ, ((0 -T ) -1 K(γ)) ⋆ ψ ⋆ = r σ [(0 -T ) -1 K(γ)] ϕ, ψ ⋆ .
Since ϕ, ψ ⋆ = 0, we get r σ [(0 -T ) -1 K(γ)] < 1. By Remark 2.8, this means that γ > k eff . Since γ > τ -(ϕ) is arbitrary, we obtain

k eff inf ϕ∈W + p τ -(ϕ).
Conversely, let γ > k eff . Then, r σ [(0 -T ) -1 K(γ)] < 1, and

[I -(0 -T ) -1 K(γ)] -1 = ∞ k=0 [(0 -T ) -1 K(γ)] k [(0 -T ) -1 K(γ)] n ∀ n ∈ N. Given ψ ∈ X + p , set ψ = (0 -T ) -1 ψ. Then, ψ ∈ W + p . Define ϕ = [I -(0 -T ) -1 K(γ)] -1 ψ.
(2.14)

Clearly, ϕ ∈ D(T ) and ϕ [(0 -T ) -1 K(γ)] n ψ (n ∈ N). Let ϕ ⋆ ∈ X q \ {0} be nonnegative. Since (0 -T ) -1 K(γ) is irreducible, there exists n ∈ N such that ϕ, ϕ ⋆ [(0 -T ) -1 K(γ)] n ψ, ϕ ⋆ > 0
Hence, ϕ, ϕ ⋆ > 0 for any nonnegative ϕ ⋆ ∈ X q \ {0}, so that ϕ ∈ X + p . Moreover, by (2.14),

-(T + K(γ)) ϕ = (0 -T ) ψ = ψ ∈ X + p . Hence, γ τ -(ϕ) which proves that k eff inf ϕ∈W + p τ -(ϕ).
The following result shows that only the solution of the criticality problem (2.2) realizes the above variational characterization.

Corollary 2.14. Assume that (0 -T ) -1 K(γ) is power-compact and irreducible for any γ > 0. Then, for any

ϕ ∈ W + p τ -(ϕ) = k eff if and only if (T + K(k eff )) ϕ = 0. Proof. Let ϕ ∈ W + p be such that τ -(ϕ) = k eff . Then, -(T + K(k eff )) ϕ 0 so ϕ (0 -T ) -1 K(k eff )ϕ.
Arguing as in the proof of Corollary 2.10, one can prove that ϕ = (0 -T ) -1 K(γ) ϕ, which means that (T + K(k eff )) ϕ = 0. The sufficient condition follows directly from Proposition 2.13.

We are now able to characterize the effective multiplication factor k eff by means of Inf-Sup and Sup-Inf criteria, where we recall that K = K s + K f . Theorem 2.15. Under the assumptions of Theorem 2.6, if K f (X + p ) ⊂ X + p then the criticality eigenvalue k eff is characterized by the following:

1 k eff = min ϕ∈W + p ess sup ω∈Ω -(T + K s )ϕ(ω) K f ϕ(ω) = max ϕ∈W + p ess inf ω∈Ω -(T + K s )ϕ(ω) K f ϕ(ω) .
Proof. Let ϕ ∈ W + p be given,

τ + (ϕ) = sup{γ > 0 ; (T + K s ) ϕ + 1 γ K f ϕ 0} = sup{γ > 0 ; -(T + K s ) ϕ 1 γ K f ϕ} = sup{γ > 0 ; -(T + K s ) ϕ(ω) 1 γ K f ϕ(ω) dν -a.e. ω ∈ Ω}. Since K f (X + p ) ⊂ X + p , one gets 1 τ + (ϕ) = ess sup ω∈Ω -(T + K s ) ϕ(ω) K f ϕ(ω)
. By Proposition 2.9,

1 k eff = inf ϕ∈W + p ess sup ω∈Ω -(T + K s ) ϕ(ω) K f ϕ(ω)
and the infimum is attained for the criticality eigenfunction. Similarly, let

ϕ ∈ W + p τ -(ϕ) = inf{γ > 0 ; -(T + K s ) ϕ - 1 γ K f ϕ 0} = inf{γ > 0 ; -(T + K s ) ϕ 1 γ K f ϕ} = inf{γ > 0 ; -(T + K s ) ϕ(ω) 1 γ K f ϕ(ω) dν -a.e. ω ∈ Ω}. So τ -(ϕ) = inf{γ > 0 ; -(T + K s ) ϕ(ω) K f ϕ(ω) 1 γ dν -a.e. ω ∈ Ω}, i.e. 1 τ -(ϕ) = ess inf ω∈Ω -(T + K s ) ϕ(ω) K f ϕ(ω) .
Using Proposition 2.13, one proves that

1 k eff = sup ϕ∈W + p 1 τ -(ϕ) = sup ϕ∈W + p ess inf ω∈Ω -(T + K s ) ϕ(ω) K f ϕ(ω) ,
which ends the proof.

The class of regular collision operators.

We end this section by recalling the class of regular collision operators introduced in kinetic theory by M. Mokhtar-Kharroubi [START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF]. This class of operators will also be useful to study diffusion problems of type (1.3). We assume here that the measure space (Ω, dν) writes as follows:

Ω = D × V, dν(ω) = dx ⊗ dµ(v), ω = (x, v) ∈ Ω
where dµ is a suitable Radon measure over V . Let K ∈ B(L p (Ω, dν)) be given by

K : ϕ -→ Kϕ(x, v) = V k(x, v, v ′ )ϕ(x, v ′ )dµ(v ′ ) ∈ L p (Ω, dν) (2.15)
where the kernel k(•, •, •) is measurable. For almost every x ∈ D, define

K(x) : ψ ∈ L p (V, dµ) -→ V k(x, v, v ′ )ψ(v ′ )dµ(v ′ ) ∈ L p (V, dµ)
and assume that the mapping K : x ∈ D → K(x) ∈ B(L p (V, dµ)) is strongly measurable and bounded, i.e. ess sup

x∈D K(x) B(L p (V,dµ)) < ∞.
The class of regular operators in L p spaces with 1 < p < ∞ is given by the following (see [ (1) For almost every

x ∈ D, K(x) ∈ B(L p (V, dµ)) is a compact operator, (2) { K(x) ; x ∈ D} is relatively compact in B(L p (V, dµ)).
In L 1 -spaces, the definition differs a bit. We have the following [START_REF] Lods | On linear kinetic equations involving unbounded cross-sections[END_REF] Definition 2.17. Let K be defined by (2.15). Then, K is said to be a regular operator whenever

{|k(x, •, v ′ )| , (x, v ′ ) ∈ D × V } is a relatively weakly compact subset of L 1 (V, dµ).
The main interest of that classes of operators relies to the following (see Ref. [START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF] for 1 < p < ∞ and Ref. [START_REF] Lods | On linear kinetic equations involving unbounded cross-sections[END_REF] for a similar result whenever p = 1): Proposition 2.18 (Approximation of regular operators). Let 1 < p < ∞ and let K defined by (2.15) be a regular operator in L p (D × V, dx ⊗ dµ(v)). Then, K can be approximated in the norm operator by operators of the form:

ϕ -→ i∈I α i (x)β i (v) V θ i (v ′ )ϕ(x, v ′ )dµ(v ′ )
where I is finite, α i ∈ L ∞ (D), β i ∈ L p (V, dµ) and θ i ∈ L q (V, dµ), 1/p + 1/q = 1.

THE CRITICAL TRANSPORT PROBLEM

3.1. Variational characterization. This section is devoted to the determination of the effective multiplication factor associated to the transport operator. We adopt the notations of Section 2.3, namely Ω = D × V and dν(x, v) = dx ⊗ dµ(v). Throughout this section, we assume D to be a convex and bounded open subset of R N while µ is the Lebesgue measure over R N or on spheres. In particular, our results cover continuous or multi-group neutron transport problems but do not apply to transport problems with discrete velocities. Let

Γ -:= (x, v) ∈ ∂D × V ; v • n(x) < 0
where n(x) denotes the outward unit normal at x ∈ ∂ D. Let T be the unbounded absorption operator

T : D(T ) ⊂ X p -→ X p ϕ -→ T ϕ(x, v) := -v • ∇ x ϕ(x, v) -σ(x, v)ϕ(x, v),
with domain

D(T ) = ψ ∈ X p ; v • ∇ x ψ ∈ X p and ψ |Γ -= 0 .
Here, the nonnegative function σ(•, •) ∈ L ∞ (D × V ) is the collision frequency. It is assumed to admit a positive lower bound σ(x, v) c > 0 a.e. (x, v) ∈ D × V.

(3.1) Define the (full) collision operator K as the bounded linear (partial) integral operator

K : ψ ∈ X p -→ Kψ(x, v) := V Σ(x, v, v ′ )ψ(x, v ′ )dµ(v ′ ) ∈ X p .
The collision kernel Σ(•, •, •) is assumed to be nonnegative. In nuclear reactor theory, in a fissile material, this collision kernel splits as

Σ(x, v, v ′ ) = Σ s (x, v, v ′ ) + Σ f (x, v, v ′ )
where Σ s (x, v, v ′ ) describes the pure scattering phenomena and Σ f (x, v, v ′ ) describes the fission processes. Define the corresponding linear operators

K s : ψ ∈ X p -→ K s ψ(x, v) := V Σ s (x, v, v ′ )ψ(x, v ′ )dµ(v ′ ) ∈ X p and K f : ψ ∈ X p -→ K f ψ(x, v) := V Σ f (x, v, v ′ )ψ(x, v ′ )dµ(v ′ ) ∈ X p .
As we told it in Introduction, we are interested here in the critical problem:

v • ∇ x ϕ(x, v) + σ(x, v)ϕ(x, v) - V Σ s (x, v, v ′ )ϕ(x, v ′ )dµ(v ′ ) = 1 k eff V Σ f (x, v, v ′ )ϕ(x, v ′ )dµ(v ′ ), (3.2)
where the eigenfunction ϕ is nonnegative and satisfies the boundary condition ϕ |Γ -= 0. We recall that the spectral bound of T is given by [START_REF] Voigt | Positivity in time dependent linear transport theory[END_REF]

s(T ) = -lim t→∞ inf (x,v)∈D×V t<τ (x,v) t -1 t 0 σ(x + sv, v)ds,
with τ (x, v) := inf{s > 0 ; x -sv / ∈ D}. Therefore, by (3.1), we have s(T ) < 0. Moreover,

(0 -T ) -1 ϕ(x, v) = τ (x,v) 0 exp - t 0 σ(x -vs, v)ds ϕ(x -vt, v)dt,
so that (0 -T ) -1 fulfills (2.1). Let us now recall the irreducibility properties of (0 -T ) -1 K f for the continuous and multigroup models. The following result may be found in Ref. [START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF], Theorem 5.15, Theorem 5.16, (see also [START_REF] Voigt | Positivity in time dependent linear transport theory[END_REF]).

Theorem 3.1. Let D be convex. Then, (0 -T ) -1 K f is irreducible in the two following cases:

(1) V is a closed subset of R N equipped with the Lebesgue measure dµ and there exist 0

< c 1 < c 2 < ∞, such that V 0 = {v ∈ R N ; c 1 < |v| < c 2 } ⊂ V with Σ f (x, v, v ′ ) > 0 a.e. (x, v, v ′ ) ∈ D × V × V 0 ∪ D × V 0 × V . (3.3)
(2) V is the union of k disjoint spheres (k 1),

V = k i=1 V i , V i = {v ∈ R N ; |v| = r i }, (r i > 0, i = 1, . . . , k)
and, on each sphere, dµ is the surface Lebesgue measure. Moreover, for any i, j ∈ {1, . . . , k}, there exists ℓ ∈ {1, . . . , m} such that

Σ f (x, v, v ′ ) > 0 a.e. (x, v, v ′ ) ∈ D × V i × V ℓ ∪ D × V ℓ × V j . (3.4)
Remark 3.2. In the above case [START_REF] Allaire | Homogenization of the criticality spectral equation in neutron transport[END_REF], corresponding to continuous models, it is possible to provide different criteria ensuring the irreducibility of (0 -T ) -1 K f (see for instance Ref. [START_REF] Greiner | Spectral properties and asymptotic behavior of the linear transport equation[END_REF]). In the second case [START_REF] Allen | The inverse power method for calculation of multiplication factors[END_REF], which corresponds to multigroup transport equation, several different criteria also exist [START_REF] Eds | One-parameter semigroups of positive operators[END_REF].

Using the notations of Section 2, we have the following characterization of the effective multiplication factor of the transport operator.

Theorem 3.3. Let us assume that K is a regular collision operator and that one of the hypothesis of Theorem 3.1 holds. The critical problem (3.2) admits a unique solution k eff if and only if

lim γ→0 r σ [(0 -T ) -1 K(γ)] > 1 and r σ [(0 -T ) -1 K s ] < 1.
Moreover,

1 k eff = min ϕ∈W + p ess sup (x,v)∈D×V v • ∇ x ϕ(x, v) + σ(x, v)ϕ(x, v) - V Σ s (x, v, v ′ )ϕ(x, v ′ ) dµ(v ′ ) V Σ f (x, v, v ′ )ϕ(x, v ′ ) dµ(v ′ ) = max ϕ∈W + p ess inf (x,v)∈D×V v • ∇ x ϕ(x, v) + σ(x, v)ϕ(x, v) - V Σ s (x, v, v ′ )ϕ(x, v ′ ) dµ(v ′ ) V Σ f (x, v, v ′ )ϕ(x, v ′ ) dµ(v ′ ) . (3.5)
Proof. Since K is a regular collision operator, one deduces from [19, Theorems 4.1 & 4.4] when 1 < p < ∞ (respectively [START_REF] Lods | On linear kinetic equations involving unbounded cross-sections[END_REF] if p = 1) that (0 -T ) -1 K is a power-compact operator in X p (1 p < ∞) under our assumptions on the measure µ. Moreover, in the continuous case, thanks to

(3.3), for any ϕ ∈ X + p K f ϕ(x, v) = V Σ f (x, v, v ′ ) ϕ(x, v ′ )dµ(v ′ ) V 0 Σ f (x, v, v ′ ) ϕ(x, v ′ )dµ(v ′ ) > 0 a.e. (x, v) ∈ D × V, i.e. K f (X + p ) ⊂ X + p .
Similarly, in the multigroup case, Eq. (3.4) implies K f (X + p ) ⊂ X + p . Now, the existence of k eff follows from Theorem 2.6 while (3.5) follows from Theorem 2.15.

Remark 3.4. Denote by φ eff the nonnegative solution of (3.2), one can check that φ eff ∈ W + p . Therefore, in (3.5), the supremum and the infimum are reached for φ = φ eff . Remark 3.5. Note that it is possible to provide practical criteria that are satisfied in nuclear reactor theory and that ensure the existence of k eff [START_REF]First-order corrector for the homogenization of the criticality eigenvalue problem in the even parity formulation of the neutron transport[END_REF][START_REF] Thévenot | An optimality condition for the assembly distribution in a nuclear reactor[END_REF]. Such criteria usually rely on dissipative properties of the pure scattering operator.

Remark 3.6. It is important to point out that the above characterization is not restricted to the case of absorbing conditions but also holds for general boundary conditions modeled by some suitable nonnegative albedo operator. Actually, if one considers a transport operator T H associated to general nonnegative albedo boundary operator H which relates the incoming and outgoing fluxes in D

[16], then the above theorem holds true provided (0 -T H ) -1 K is a power-compact operator in X p (1 p < ∞) when K s and K f are regular operators. This is always the case whenever 1 < p < ∞ by virtue of the velocity averaging lemma [START_REF] Latrach | Compactness results for transport equations and applications[END_REF]. The problem is more delicate in a L 1 -setting and is related to the geometry of the domain D [START_REF] Sanchez | The criticality eigenvalue problem for the transport with general boundary conditions[END_REF].

Necessary conditions of super-criticality and sub-criticality.

We shall use the result of the previous section to derive necessary conditions ensuring the reactor to be super-critical or sub-critical. Note that, for practical implications, a nuclear reactor can be operative and create energy only when slightly super-critical (i.e. 1 < k eff < 1 + δ with δ > 0 small enough), in this case, the whole chain fission being controlled by rods of absorbing matter. Throughout this section, we shall assume k eff to exist.

We shall provide lower and upper bounds on the effective multiplicative factor k eff only when the velocity space V is bounded away from zero. Recall that, since V is assumed to be closed, this means that 0 / ∈ V (see also Remark 3.14). For almost every x ∈ D, define K τ (x) as the following operator on L p (V, dµ):

K τ (x) : ψ ∈ L p (V, dµ) → V Σ(x, v, v ′ )τ (x, v ′ ) 1 + σ(x, v)τ (x, v) ψ(v ′ )dµ(v ′ ) ∈ L p (V, dµ)
where we recall that

Σ(x, v, v ′ ) = Σ s (x, v, v ′ ) + Σ f (x, v, v ′ ) and τ (x, v)
is the stay time in D. Then, one defines as in [START_REF] Mokhtar-Kharroubi | On the leading eigenvalue of neutron transport models[END_REF], the following

ϑ := inf ψ∈L p + (V,dµ) ess sup (x,v)∈D×V [K τ (x)ψ](v) ψ(v)
where

L p + (V, dµ) = {ψ ∈ L p (V, dµ) ; ψ(v) > 0 dµ -a.e. v ∈ V }.
Then, one has the following estimate:

Proposition 3.7. Under the assumptions of Theorem 3.3, if ϑ < 1, then k eff ϑ. Proof. Assume ϑ < 1. Given ϑ ∈ (ϑ, 1), let ψ 0 ∈ L p + (V, dµ) be such that ess sup (x,v)∈D×V [K τ (x)ψ 0 ](v) ψ 0 (v) ϑ.
Let us consider then the following test-function

ϕ 0 (x, v) = τ (x, v)ψ 0 (v). Since 0 / ∈ V , τ (•, •) is bounded and such an application ϕ 0 belongs to W + p since τ (x + tv, v) = τ (x, v) + t a.e. (x, v) ∈ D × V, t 0, implies v • ∇ x ϕ 0 (x, v) = ψ 0 (v).
Then, for any γ > 0, one sees that

-(T + K(γ))ϕ 0 (x, v) = ϑ -1 (1 + σ(x, v)τ (x, v)) ϑψ 0 (v) -[K τ (x)ψ 0 ](v) + (1 -ϑ) V Σ s (x, v, v ′ )τ (x, v ′ ) 1 + σ(x, v)τ (x, v) ψ 0 (v ′ )dµ(v ′ )+ + γ -ϑ γ V Σ f (x, v, v ′ )τ (x, v ′ ) 1 + σ(x, v)τ (x, v) ψ 0 (v ′ )dµ(v ′ ) . (3.6)
Since Σ s 0 and 1 -ϑ 0, one sees that

-(T + K(γ))ϕ 0 (x, v) ϑ -1 (1 + σ(x, v)τ (x, v)) ϑψ 0 (v) -[K τ (x)ψ 0 ](v) + γ -ϑ γ V Σ f (x, v, v ′ )τ (x, v ′ ) 1 + σ(x, v)τ (x, v) ψ 0 (v ′ )dµ(v ′ ) .
In particular, from the positivity of Σ f , one sees that, provided γ ϑ, -(T + K(γ))ϕ 0 (x, v) 0 for almost every (x, v) ∈ D × V . Then, from Proposition 2.13, this means that τ -(ϕ 0 ) ϑ and k eff ϑ.

Since ϑ > ϑ is arbitrary, one gets the result.

Remark 3.8. From the above result, one sees that the reactor is sub-critical whenever ϑ < 1. Note that the fact that ϑ < 1 implies k eff 1 is already contained in [START_REF] Mokhtar-Kharroubi | On the leading eigenvalue of neutron transport models[END_REF]Theorem 7].

The above result provides an upper bound of k eff leading to the sub-criticality of the reactor core. To get a lower bound of k eff , one defines a similar quantity ϑ = sup

ψ∈L p + (V,dµ) ess inf (x,v)∈D×V [K τ (x)ψ](v) ψ(v) .
Proposition 3.9. Under the assumptions of Theorem 3.3, if ϑ > 1, then k eff ϑ. In particular, for a reactor core to be sub-critical, it is necessary that ϑ 1.

Proof. The proof is very similar to that of Prop. 3.7. Namely, assume ϑ > 1. For any ϑ ∈ (1, ϑ), let

ψ 0 ∈ L p + (V, dµ) be such that [K τ (x)ψ 0 ](v) ϑψ 0 (v) for almost every (x, v) ∈ D × V . Then, the function ϕ 0 (x, v) = τ (x, v)ψ 0 (v) belongs to W +
p and, arguing as in Prop. 3.7, thanks to Eq. (3.6), one sees that, (T + K(γ))ϕ is nonnegative for any γ ϑ. Consequently, τ + (ϕ) ϑ and Prop. 2.9 implies that k eff ϑ for any ϑ ∈ (1, ϑ). Remark 3.10. To the author's knowledge, the identity ϑ = ϑ is an open question. Notice however that, according to I. Marek's result, Ref. [START_REF] Marek | Frobenius theory of positive operators[END_REF], Theorem 3.2, for any x ∈ D, one has the identity

sup ψ∈L p + (V,dµ) ess inf v∈V [K τ (x)ψ](v) ψ(v) = inf ψ∈L p + (V,dµ) ess sup v∈V [K τ (x)ψ](v) ψ(v) = r σ [K τ (x)],
where we recall that K τ (x) is an operator in L p (V, dµ).

In the same spirit, for almost every x ∈ D, define K τ f (x) as the following operator on L p (V, dµ):

K τ f (x) : ψ ∈ L p (V, dµ) → V Σ f (x, v, v ′ )τ (x, v ′ ) 1 + σ(x, v)τ (x, v) ψ(v ′ )dµ(v ′ ) ∈ L p (V, dµ)
and let us define, as in [START_REF] Mokhtar-Kharroubi | On the leading eigenvalue of neutron transport models[END_REF], the set I f of all β 0 for which there exists [START_REF] Mokhtar-Kharroubi | On the leading eigenvalue of neutron transport models[END_REF]Lemma 4] the set I is closed so that, if one defines

ψ ∈ L p + (V, dµ) \ {0} such that [K τ f (x)ψ](v) βψ(v), for almost every (x, v) ∈ D × V. According to
β f := sup{β, β ∈ I} then, there exists ψ f ∈ L p + (V, dµ)\{0} such that [K τ f (x)ψ f ](v) β f ψ f (v) for almost every (x, v) ∈ D × V.
When the velocity space is bounded away from 0 then, β f provides a lower bound for k eff : Proposition 3.11. Under the assumptions of Theorem 3.3, one has k eff β f .

Proof. Set ϕ f (x, v) = τ (x, v)ψ f (v) where ψ f ∈ L p + (V, dµ
) is defined here above. Arguing as in the proof of Proposition 3.7, one sees that, since 0 / ∈ V , ϕ f ∈ W + p . Therefore, Theorem 3.3 ensures that

1 k eff ess sup (x,v)∈D×V v • ∇ x ϕ f (x, v) + σ(x, v)ϕ f (x, v) - V Σ s (x, v, v ′ )ϕ f (x, v ′ ) dµ(v ′ ) V Σ f (x, v, v ′ )ϕ f (x, v ′ ) dµ(v ′ )
.

As in the proof of Prop. 3.7 and since Σ s and ϕ s are nonnegative, one gets

1 k eff ess sup (x,v)∈D×V ψ f (v) [K τ f (x)ψ f ](v) 1 β f
which ends the proof.

Remark 3.12. The above Proposition provides a lower bound of the criticality eigenvalue k eff that depends only on the fission collision operator K f . In particular, a sufficient condition for the reactor to be super-critical is β f > 1.

Proposition 3.13. Let v 0 := inf{|v| ; v ∈ V } and let d be the diameter of D. Define

Λ f := sup ψ∈L p + (V,dµ) ess inf (x,v)∈D×V 1 ψ(v) V Σ f (x, v, v ′ )τ (x, v ′ )ψ(v ′ )dµ(v ′ ).
Then, under the assumptions of Theorem 3.3,

1 k eff 1 + σ d v 0 Λ f , (3.7)
where σ := ess sup

(x,v)∈D×V σ(x, v).
In particular, if V bounded then

k eff 1 1 + σ d v 0 ess inf (x,v)∈D×V V Σ f (x, v, v ′ )τ (x, v ′ )dµ(v ′ ). (3.8)
Proof. Let us consider again test-functions of the form ϕ(x, v) = τ (x, v)ψ(v) where ψ ∈ L p + (V, dµ). Then, as above, according to (3.5)

1 k eff ess sup (x,v)∈D×V 1 + σ(x, v)τ (x, v) - 1 ψ(v) V Σ s (x, v, v ′ )τ (x, v ′ )ψ(v ′ )dµ(v ′ ) 1 ψ(v) V Σ f (x, v, v ′ )τ (x, v ′ )ψ(v ′ )dµ(v ′ ) 1 + ess sup (x,v)∈D×V σ(x, v)τ (x, v) ess inf (x,v)∈D×V 1 ψ(v) V Σ f (x, v, v ′ )τ (x, v ′ )ψ(v ′ )dµ(v ′ )
.

Since such an inequality holds for arbitrary ψ(v) > 0 and since τ (x, v) d/|v| d/v 0 for almost every (x, v) ∈ D × V, one gets (3.7). To prove (3.8), it suffices to consider the test-function ψ(v) = 1 (v ∈ V ), which belongs to L p + (V, dµ) provided V is bounded. Remark 3.14. We dealt in this section with the case of velocities bounded away from zero. For practical use in nuclear engineering, this is no major restriction. However, it should also be possible to derive explicit bounds of k eff when 0 ∈ V . In such a case, the exit time τ (x, v) is not bounded anymore but behave as 1 |v| for small |v|. Therefore, test-functions of the form ϕ(x, v) = τ (x, v)ψ(v) belong to W + p if and only if ψ(v) |v| ∈ L p + (V, dµ).

THE CRITICAL PROBLEM FOR THE ENERGY-DEPENDENT DIFFUSION MODEL

4.1. Variational characterization. In this section, we are concerned with the following

-div x (D(x, ξ)∇ x ̺(x, ξ)) + σ(x, ξ)̺(x, ξ) - E Σ s (x, ξ, ξ ′ ) ̺(x, ξ ′ ) dξ ′ = 1 k eff E Σ f (x, ξ, ξ ′ ) ̺(x, ξ ′ ) dξ ′ , (4.1)
where the unknown ̺(•, •) is assumed to be nonnegative and to satisfy the Dirichlet boundary conditions

̺ |∂D (•, ξ) = 0 a.e. ξ ∈ E,
where D is C 2 open bounded and connected subset of R N and E is an interval of ]0, ∞[. We will assume throughout this section that there exist some constants σ i > 0 (i = 1, 2) such that

0 < σ 1 σ(x, ξ) σ 2 < ∞, a.e. (x, ξ) ∈ D × E. (4.2)
Moreover, we assume the measurable matrix-valued application D(•, •) satisfies the following (uniform) ellipticity property ess inf

(x,ξ)∈D×E N i,j=1 d ij (x, ξ)η i η j d 1 |η| 2 (η ∈ R N ) (4.3)
and regularity assumption d ij (•, ξ) ∈ W 1,2 loc (D) for almost every ξ ∈ E. We will study Problem (4.1) in a Hilbert space setting for simplicity. Namely, set

X 2 = L 2 (D × E, dxdξ).
Let us assume the kernels Σ s (•, •, •) and Σ f (•, •, •) to be nonnegative and define the scattering operator

K s : ψ ∈ X 2 -→ K s ψ(x, ξ) = E Σ s (x, ξ, ξ ′ )ψ(x, ξ ′ )dξ ′ ∈ X 2 ,
and the fission operator

K f : ψ ∈ X 2 -→ K f ψ(x, ξ) = E Σ f (x, ξ, ξ ′ )ψ(x, ξ ′ )dξ ′ ∈ X 2 .
We will assume K s and K f to be bounded operators in X 2 . Define then the full collision operator

K : ψ ∈ X 2 -→ Kψ(x, ξ) = E Σ(x, ξ, ξ ′ )ψ(x, ξ ′ )dξ ′ ∈ X 2 , where Σ(x, ξ, ξ ′ ) = Σ s (x, ξ, ξ ′ ) + Σ f (x, ξ, ξ ′ ) (x, ξ, ξ ′ ) ∈ D × E × E.
Let us introduce the diffusion operator

T : D(T ) ⊂ X 2 -→ X 2 ̺ -→ T ̺(x, ξ) = div x (D(x, ξ)∇ x ̺(x, ξ)) -σ(x, ξ)̺(x, ξ),
with domain

D(T ) = {ψ ∈ X 2 ; ψ(•, ξ) ∈ H 1 0 (D) ∩ H 2 (D)
a.e. ξ ∈ E and T ψ ∈ X 2 } where H 1 0 (D) and H 2 (D) are the usual Sobolev spaces. With these notations, the spectral problem (4.1) reads

(T + K s + 1 k eff K f )̺ eff = 0, ̺ eff ∈ D(T ) , ̺ eff 0, ̺ eff = 0.
According to the strong maximum principle, it is clear that s(T ) < 0 and (0 -T ) -1 (X + 2 ) ⊂ X + 2 . In order to apply Theorem 2.15, one has to make sure that (0 -T ) -1 K is power-compact and that (0 -T ) -1 K f is irreducible. Let us begin with the following compactness result which is similar to the usual velocity averaging lemma (see [START_REF] Golse | Regularity of the moments of the solution of a transport equation[END_REF] and [START_REF] Mokhtar-Kharroubi | Mathematical Topics in Neutron Transport Theory[END_REF]Chapter 2]) for transport equations and is based on some consequence of the Sobolev embedding Theorem [START_REF] Brezis | Analyse fonctionnelle et applications[END_REF].

Theorem 4.1. If K ∈ B(X 2 ) is regular then K(0 -T ) -1 is a compact operator in X 2 .
Proof. By Proposition 2.18, it suffices to prove the result for a collision operator K such that

K : ̺ ∈ X 2 -→ K̺(x, ξ) = α(x)h(ξ) E f (ξ ′ )̺(x, ξ ′ )dξ ′ ∈ X 2 where α ∈ L ∞ (D), h ∈ L 2 (E, dξ) and f ∈ L 2 (E, dξ).
Moreover, by a density argument, one can also assume f and h to be continuous functions with compact support in E. Let us split K(0 -T ) -1 as:

K(0 -T ) -1 = ΘM(0 -T ) -1 where Θ : ̺ ∈ L 2 (D, dx) -→ [Θ̺](x, ξ) = α(x)h(ξ)̺(x) ∈ X 2 ,
and M is the averaging operator

M : ψ ∈ X 2 -→ Mψ(x) = E f (ξ ′ )ψ(x, ξ ′ )dξ ′ ∈ L 2 (D).
It is enough to prove that M(0 -T ) -1 : X 2 → L 2 (D) is compact. Let B be a bounded subset of X 2 . One has to show that {Mg ; g ∈ (0 -T ) -1 (B)} is a relatively compact subset of L 2 (D). For any ϕ ∈ B, set g(x, ξ) = (0 -T ) -1 ϕ(x, ξ).

For almost every ξ ∈ E, g(•, ξ) ∈ H 1 0 (D). One extends g to the whole space R N by

g(x, ξ) = g(x, ξ) if x ∈ D 0 else.
Clearly, for almost every ξ ∈ E, g(•, ξ) ∈ H 1 (R N ). Consequently, according to [7, Proposition IX.3], for a. e. ξ ∈ E and any h

∈ R N τ h g(•, ξ) -g(•, ξ) L 2 (D) |h| ∇ x g(•, ξ) L 2 (D) , where τ h f (x) = f (x + h) (x ∈ D, h ∈ R N ), i.e. D |g(x + h, ξ) -g(x, ξ)| 2 dx |h| 2 D |∇ x g(x, ξ)| 2 dx. (4.4)
Now, recall that

-div x (D(x, ξ)∇ x g(x, ξ)) + σ(x, ξ)g(x, ξ) = ϕ(x, ξ) (x, ξ) ∈ D × E.
Multiplying this identity by g(x, ξ) and integrating by parts yield, thanks to the ellipticity property (4.3),

d 1 D×E |∇ x g(x, ξ)| 2 dxdξ D×E |g(x, ξ)||ϕ(x, ξ)|dxdξ.
In particular, since B is bounded, by Cauchy-Schwarz inequality, there exists c > 0 such that

sup g∈(0-T ) -1 (B) D×E |∇ x g(x, ξ)| 2 dxdξ c. (4.5)
Then, (4.5) together with (4.4) yield

D×E |g(x + h, ξ) -g(x, ξ)| 2 dxdξ c|h| 2 .
By Hölder's inequality, since f is continuous with compact support

D |M g(x + h) -M g(x)| 2 dx = D dx E (g(x + h, ξ) -g(x, ξ))f (ξ)dξ 2 C D×E |g(x + h, ξ) -g(x, ξ)| 2 dxdξ |h| 2 C,
where C > 0 does not depend on g. In particular,

lim h→0 sup g∈(0-T ) -1 (B) D |M g(x + h) -M g(x)| 2 dx = 0.
Now, using that M g = M g one deduces the conclusion from Riesz-Fréchet-Kolmogorov Theorem [START_REF] Brezis | Analyse fonctionnelle et applications[END_REF].

We are now in position to prove the main result of this section where the notations of Section 2 are adopted :

Theorem 4.2. Let K ∈ B(X 2 ) be regular. Assume there exists an open subset E 0 ⊂ E such that Σ f (x, ξ, ξ ′ ) > 0 a.e. (x, ξ, ξ ′ ) ∈ D × E × E 0 . (4.6)
Then, the problem (4.1) admits a effective multiplication factor k eff > 0 if, and only if,

lim γ→0 r σ [(0 -T ) -1 K(γ)] > 1 and r σ [(0 -T ) -1 K s ] < 1. B. LODS Moreover, k eff is characterized 1 k eff = min ϕ∈W + p ess sup (x,v)∈D×E -div(D(x, ξ)∇ x ϕ(x, ξ)) + σ(x, ξ)ϕ(x, ξ) - E Σ s (x, ξ, ξ ′ )ϕ(x, ξ ′ )dξ ′ E Σ f (x, ξ, ξ ′ )ϕ(x, ξ ′ )dξ ′ = max ϕ∈W + p ess inf (x,ξ)∈D×V -div(D(x, ξ)∇ x ϕ(x, ξ)) + σ(x, ξ)ϕ(x, ξ) - E Σ s (x, ξ, ξ ′ )ϕ(x, ξ ′ )dξ ′ E Σ f (x, ξ, v ′ )ϕ(x, ξ ′ )dξ ′ . (4.7)
Proof. From (4.6), K f (X + 2 ) ⊂ X + 2 . Now, for almost every ξ ∈ E, define T ξ as the following operator on L 2 (D):

T ξ : ̺ ∈ D(T ξ ) → T ξ ̺(x) = div x (D(x, ξ)∇ x ̺(x)) -σ(x, ξ)̺(x),
where D(T ξ ) = H 1 0 (D) ∩ H 2 (D) turns out to be independent of ξ. Since D is connected, the (elliptic) maximum principle implies that (0 -T ξ ) -1 is irreducible (see [START_REF] Davies | Heat kernels and spectral theory[END_REF]Theorem 3.3.5] or [START_REF] Arendt | Heat Kernels, 9th. Internet Seminar[END_REF]Section 11.2]). Actually, since T ξ is the generator of a holomorphic semigroup, this implies that (0 -T ξ ) -1 is positivity improving (see [23, p. 306

]), i.e. (0 -T ξ ) -1 ̺(x) > 0 for almost every x ∈ D provided ̺ ∈ L 2 (D), ̺(x) 0 for almost every x ∈ D and ̺ = 0. Now, let ψ ∈ X 2 , ψ(x, ξ) 0 for almost every (x, ξ) ∈ D × E, ψ = 0. Then, K f ψ 0 and (0 -T ξ ) -1 K f ψ(x, ξ) > 0 for almost every (x, ξ) ∈ D × E.
It is easy to see that this exactly means that (0 -T ) -1 K f ψ(x, ξ) > 0 for almost every (x, ξ) ∈ D × E and the irreducibility of (0 -T ) -1 K f follows. Since (0 -T ) -1 K is a compact operator by Theorem 4.1, the conclusion follows from Theorems 2.6 and 2.15.

Explicit bounds.

In this section, we derive explicit bounds for the effective multiplication factor k eff . As we did in Section 3.2, the strategy consists in applying Theorem 4.7 to suitable test-functions. We assume the hypothesis of Theorem 4.2 to be met. Moreover, we assume here that the diffusion coefficient D(•, •) is degenerate, i.e. It is well-known [START_REF] Davies | Heat kernels and spectral theory[END_REF] that λ 0 > 0 and that there exists a positive eigenfunction ̺ 0 solution to (4.8). Set E + = {ψ ∈ L 2 (E, dξ) ; ψ(ξ) > 0 a.e. ξ ∈ E}. In the spirit of Section 3.2, for almost every x ∈ D, define K λ 0 f (x) as the following operator on L 2 (E, dξ):

K λ 0 f (x) : ψ ∈ L 2 (E, dξ) → E Σ f (x, ξ, ξ ′ ) λ 0 d 1 (ξ) + σ(x, ξ) ψ(ξ ′ )dξ ′ ∈ L 2 (E, dξ)
and let I f be the set of all β 0 for which there exists ψ ∈ E + such that [K λ 0 f (x)ψ](ξ) βψ(ξ), for almost every (x, ξ) ∈ D × E. Then, one has the following bounds of k eff , in the spirit of Propositions 3.7 & 3.9. Proof. The proof is very similar to that of Prop. 3.7 & 3.9. We only prove the first part of the result, the second part proceeding along the same lines. Assume thus that ϑ > 1. For any ϑ ∈ (1, ϑ), let ψ 0 ∈ E + be such that ess inf (x,ξ)∈D×E where K λ 0 f (x) has been already defined and the definition of K λ 0 s (x) is similar (Σ s replacing Σ f ). Then, from the positivity of Σ s and Σ f , the assumption ϑ > 1 implies that (T +K(γ))ϕ is nonnegative for any γ ϑ. Consequently, τ + (ϕ) ϑ and Prop. 2.9 implies that k eff ϑ. Since ϑ ∈ (1, ϑ) is arbitrary, one obtains k eff ϑ.

Whenever E is of finite Lebesgue measure, one has the following practical criteria, already stated by C. V. Pao [24, Theorem 5.3] using completely different arguments. then, the reactor core is non super-critical, i.e. k eff 1.

Proof. Since E is of finite Lebesgue measure, the constant function ψ = 1 E such that ψ(ξ) = 1 for any ξ ∈ E belongs to E + . Then, assumption (4.9) means exactly that [K λ 0 (x)1 E ](ξ) > 1 E (ξ) for almost any (x, ξ) ∈ D × E. Therefore, ϑ > 1 and the conclusion follows from Prop. 4.4.

D

  (x, ξ) = D 0 (x)d 1 (ξ), (x, ξ) ∈ D × E, where D 0 (•) is a matrix-valued application satisfying the ellipticity condition (4.3), D(•) ∈ W 1,2 loc (D) and d 1 (•) is a bounded real-valued application with ess inf ξ∈E d 1 (ξ) > 0.Let λ 0 be the principal eigenvalue of the following elliptic problem in L 2 (D)div x (D 0 (x)∇̺(x)) + λ 0 ̺(x) = 0, (x ∈ D) ̺ |∂D (x) = 0 (x ∈ ∂D).(4.8)

Proposition 4 . 3 .

 43 Setting β 0 := sup{β, β ∈ I}, one has k eff β 0 . In particular, a necessary condition to the reactor to be sub-critical is β 0 < 1.Proof. As in Section 3.2, the set I is closed. Therefore, there existsψ ∈ E + such that [K λ 0 f (x)ψ](ξ) β 0 ψ(ξ) for almost every (x, ξ) ∈ D × E. Now, set ϕ 0 (x, ξ) = ̺ 0 (x)ψ(ξ), then, ϕ ∈ W + 2 and -div x (D(x, ξ)∇ x ϕ(x, ξ)) = -d 1 (ξ)ψ(ξ)div x (D 0 (x)∇̺ 0 (x)) = λ 0 d 1 (ξ)ψ(ξ)̺ 0 (x) (x, ξ) ∈ D × E.Consequently, thanks to (4.7) one has1 k eff ess sup (x,ξ)∈D×E [λ 0 d 1 (ξ) + σ(x, ξ)]̺ 0 (x)ψ(ξ) -̺ 0 (x) E Σ s (x, ξ, ξ ′ )ψ(ξ ′ )dξ ′ ̺ 0 (x) E Σ f (x, ξ, ξ ′ )ψ(ξ ′ )dξ ′ ess sup (x,ξ)∈D×E [λ 0 d 1 (ξ) + σ(x, ξ)]ψ(ξ) E Σ f (x, ξ, ξ ′ )ψ(ξ ′ )dξ ′In the same spirit, for almost every x ∈ D, define K λ 0 (x) as the operator on L 2 (E, dξ) given byK λ 0 (x) : ψ ∈ L 2 (E, dξ) → E Σ(x, ξ, ξ ′ ) λ 0 d 1 (ξ) + σ(x, ξ) ψ(ξ ′ )dξ ′ ∈ L 2 (E, dξ)where we set Σ(x, ξ, ξ ′ ) = Σ s (x, ξ, ξ ′ ) + Σ f (x, ξ, ξ ′ ). As in Section 3.2, set ϑ := inf ψ∈E + ess sup (x,ξ)∈D×E [K λ 0 (x)ψ](ξ) ψ(ξ) , and ϑ := sup ψ∈E + ess inf (x,ξ)∈D×E [K λ 0 (x)ψ](ξ) ψ(ξ) .

Proposition 4 . 4 .

 44 Under the assumptions of Theorem 4.2, if ϑ > 1, then k eff ϑ. On the other hand, if ϑ > 1, then k eff ϑ.

[K λ 0

 0 (x)ψ 0 ](ξ) ψ 0 (ξ) ϑ. Choose then the test-function ϕ(x, ξ) = ̺ 0 (x)ψ 0 (ξ). Such an application ϕ belongs to W + 2 and, as in the above proof, for any γ > 0, one sees that(T + K(γ))ϕ(x, ξ) = ̺ 0 (x) ϑ (λ 0 d 1 (ξ) + σ(x, ξ)) -ϑψ 0 (ξ) + [K λ 0 (x)ψ 0 ](ξ)+ (ϑ -1)[K λ 0 s (x)ψ 0 ](ξ) + ϑ -γ γ [K λ 0 f (x)ψ 0 ](ξ)

Corollary 4 . 5 .

 45 Assume E to be of finite Lebesgue measure. Ifλ 0 d 1 (ξ) + σ(x, ξ) < E [Σ s (x, ξ, ξ ′ ) + Σ f (x, ξ, ξ ′ )]dξ ′ (x, ξ) ∈ D × E,(4.9)

Remark 4 . 6 .

 46 Notice that, under the above assumption, one hask eff ess inf (x,ξ) E [Σ s (x, ξ, ξ ′ ) + Σ f (x, ξ, ξ ′ )]dξ ′ λ 0 d 1 (ξ) + σ(x, ξ) .

Acknowledgment. The author is grateful to Luisa Arlotti for a careful reading of a first draft of the manuscript and her valuable suggestions.


Arnaud Turbin-Orger

Laurent Chaunier

Hubert H Chiron

Guy G Della Valle

Guy G Della

Alveolar structure of bread dough and rheological properties of its constitutive phases

AACCI Annual Meeting 2012

Abstract submission

Alveolar structure of bread dough and rheological properties of its constitutive phases.

A. Turbin-Orger # , L. Chaunier, H. Chiron, G. Della Valle Bread doughs of various compositions were prepared and the evolution of their cellular structure during proofing was studied by computed X-ray microtomography (XRT) with high resolution (5 µm). The cellular structure was followed during the last stage of proofing, from 40 min to 180 min, to focus on the films separating bubbles and their possible coalescence. Images analysis allowed to determine the porosity, the connectivity index, the bubbles size and the cell walls thickness distributions which were fitted by usual mathematical functions. Images showed that different compositions led to different cellular structures. The kinetics of porosity and connectivity had sigmoid shapes with 0.7 and 1 as asymptotic value, respectively. For an standard formula, mean cell diameter grew from 410 µm to 675 µm and the mean cell walls thickness grew from 200 µm to 230 µm. At the end of proofing most bubbles were interconnected, i.e. they were separated by films of thickness lower than 5 µm. Changes of recipe led to different structures and were discussed regarding the influence of the liquid fraction and the sugar content on the coalescence of bubbles.

These results suggested that at the end of proofing, the dough could be considered as a three-phases co-continuous medium: gas / liquid / viscoelastic matrix. In this context, the aim of this study was also to determine the role of each on the evolution of the cellular structure, particularly the role of thin films, separating the cells. Elongational behavior of dough was characterized by lubricated squeezing flow. Elongational properties, consistency and flow indexes of the dough, were influenced by the formulation. The volume fraction of liquid increases the heterogeneity of the cellular structure.

Connectivity of the gas cells that may arise during proofing before steric hindrance did not limit their free growth. The elongational properties of dough, alone, did not explain the influence of formulation on the cellular structure. The liquid phase of dough seemed to have an important role too. The dough liquor (DL), could be considered as a good model of this liquid phase. To clarify its role, it was extracted from the dough by ultracentrifugation and its rheological properties were analyzed for various amount of water, sugar and oil. Its density (≈ 1.1) was less than that of the hydrated starchgluten matrix (≈ 1.25). DL is a yield stress fluid, its viscosity was essentially modified by the sugar content, while its surface tension was close to 40 mN / m and its foaming properties underlined the role of fat and sugar on the foam stability.

Keywords: bubbles, films, X-ray microtomography, image analysis, liquid phase, bread dough.