Bertrand Lods

Mustapha Mokhtar-Kharroubi
email: mustapha.mokhtar-kharroubi@univ-fcomte.fr

CONVERGENCE TO EQUILIBRIUM FOR LINEAR SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION WITH HARD AND SOFT POTENTIALS: A SEMIGROUP APPROACH IN L 1 -SPACES

Keywords: . Linear Boltzmann equation, soft and hard potentials, positive semigroup, ergodic projection

Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft

potentials: a semigroup approach in L 1 -spaces.

INTRODUCTION

We investigate in the present work the large time behavior of solutions to the linear Boltzmann equation for both hard and soft potentials, under some cut-off assumption. We consider solutions in some weighted L 1 -spaces and our approach is based on functional analytic results regarding positive semigroups in such spaces. To keep the presentation and results simple, we investigate only here the spatially homogeneous Boltzmann equation which exhibits already a quite rich behavior. Extension to spatially inhomogeneous problems is planned for future investigations.

1.1. The kinetic model. Before entering the details of our results, let us formulate the problem we aim to address here: we shall consider the spatially homogeneous linear Boltzmann equation (BE in the sequel)

∂ t f (t, v) = Lf (t, v), f (0, v) = f 0 (v) 0
(1.1) in which L is the linear Boltzmann operator given by

Lf = Q(f, M) (1.2)
where Q(f, g) denotes the bilinear Boltzmann operator

Q(f, g) = R d ×S d-1 B(v -v * , σ) f (v ′ )g(v ′ * ) -f (v)g(v * ) dv * dσ (1.3)
where v ′ and v ′ * are the pre-collisional velocities which result, respectively, in v and v * after elastic collision

v ′ = v + v * 2 + |v -v * | 2 σ, v ′ * = v + v * 2 - |v -v * | 2 σ. (1.4)
Here f and g are nonnegative functions of the velocity variable v ∈ R d and B(q, n) is a nonnegative function . We will assume throughout this paper that the distribution function M appearing in (1.2) is a given Maxwellian function:

M(v) = ̺ 2πΘ d/2 exp - |v -u| 2 2Θ , v ∈ R d , (1.5)
where u ∈ R d is the given bulk velocity and Θ > 0 is the given effective temperature of the host fluid. Notice that, by Galilean invariance, there is no loss of generality in assuming

̺ = Θ = 1, u = 0.
We shall investigate in this paper several collision operators L = L B corresponding to various interactions collision kernels B = B(v -v * , σ). Typically, we shall consider the case

B(v -v * , σ) = Φ(|v -v * |) b(cos θ), cos θ = v -v * |v -v * | , σ
where b : [-1, 1] → R + and Φ : R + → R + are measurable. We shall consider models with angular cut-off, i.e. we assume

ℓ b := S d-1 b(cos θ)dσ < ∞ ∀v, v * ∈ R d ; (1.6)
which, in turns, reads

ℓ b = |S d-2 | 1 -1 1 -s 2 d-3 2 b(s)ds < ∞.
Notice that, without loss of generality, we can assume that ℓ b = 1. Typically, we shall consider the case of power-like potentials Φ(•):

Φ(r) = r γ ∀r > 0
where we distinguish between two cases:

(1) Hard potentials for which γ 0;

(2) Soft potentials for which -d < γ < 0. A very peculiar model, particularly relevant for physical applications, is the one of hard-spheres in dimension d = 3 for which

B(v -v * , σ) = B hs (v -v * , σ) = b 0 |v -v * |
i.e. Φ(r) = r and b(s) = b 0 is constant.

Notice that, independently of the interaction kernel B, it is very easy to check that L(M) = 0 and that, in any reasonable spaces, the kernel of L is spanned by M, i.e. M is the only equilibrium with unit density.

Notice that, thanks to the cut-off assumption (1.6), the Boltzmann operator L can be split into a gain part and a loss part:

Lf = Kf -T f where the gain part

Kf (v) = L + f (v) := R d ×S d-1 B(v -v * , σ)f (v ′ )M(v ′ * )dv * dσ v ∈ R d (1.7)
while the loss part T = L -is a multiplication operator

T f = R d ×S d-1 B(v -v * , σ)f (v)M(v * )dv * dσ = Σ(v) f (v)
where the collision frequency

Σ(v) = R d ×S d-1 B(v -v * , σ)M(v * )dv * dσ = ℓ b (Φ * M) (v)
where * denotes the convolution in R d . For power-like potentials, i.e. whenever Φ(r) = r γ , there exist two positive constants σ 1 , σ 2 > 0 such that

σ 1 (1 + |v|) γ Σ(v) σ 2 (1 + |v|) γ ∀v ∈ R d . (1.8)
The non-local part K admits an integral form, based upon the Carleman representation of Q + . Namely, there exists some measurable kernel k = k B (v, w) such that

Kf = L + B f (v) = R d k B (v, w)f (w)dw ∀f ∈ L 1 (Σ(v), dv) = L 1 ((1 + |v|) γ , dv) .
(1.9)

The explicit form of the kernel k B (v, w) can be derived, following the lines of [START_REF] Carleman | Problèmes mathématiques dans la théorie cinétique des gaz[END_REF]. Details of such computations are given in Appendix A.

1.2. Presentation of the results and strategy. We provide here an unified approach to the problem of convergence towards equilibrium for solutions to (1.1). As already said, we will deal with solutions to (1.1) in suitable weighted L 1 -spaces. Namely, we shall consider here functional space of the type

X = L 1 (R d , m -1 (v)dv)
endowed with its natural norm where the weight function m = m(v) positive and such that m -1 (v) 1 for any v ∈ R d . We shall consider here two different types of weights:

(Exponential weights) m(v) = exp(-a|v| s ) a 0 , s ∈ (0, 1],

(1.10) or

(Algebraic weights) m(v) = 1 + |v| β -1 , β 0. (1.11)
More precisely, for hard-potentials, our approach will require the weights m to be non trivial, i.e a > 0 for exponential weights or β > 0 for algebraic weight where as, for soft potentials, we will deal with "unweighted" spaces, i.e. m(v) = 1 for any v ∈ R d which corresponds to a = 0 or β = 0.

The approach we present here is, by some aspects, much simpler than the afore-mentioned ones and relies only on very general results about positive semigroups in L 1 -spaces.

We wish in particular to emphasize that one of the main interesting features of our approach is that provides directly both the well-posedness (in the semigroup sense) of (1.1) and the asymptotic behavior of the solutions. This is a major contrast with respect to the recent quantitative approach [START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-theorem[END_REF] based upon space enlargement and factorisation for which the wellposedness of the associated problem is always given as a first initial assumption. The strategy we adopt is very natural from the viewpoint of classical perturbation theory of semigroups. It consists, roughly speaking, in considering the non-local operator K as a perturbation of the multiplication operator T associated to Σ and showing that K is weakly compact with respect to T in X. Notice that, for hard-potentials, K is an unbounded operator and one has to invoke perturbation theorem for semigroup which allows for unbounded perturbation. Such generation result is recalled in the next section as well as the basic results concerning positive semigroup in L 1 -spaces.

For hard potential, our main result is the following:

D(L) = Y =: L 1 (R d , (1 + |v|) γ m -1 (v)dv) ⊂ X is the generator of a positive C 0 -semigroup (V (t)) t 0 on X. Moreover, there exist C > 0 and λ ⋆ > 0 such that the C 0 -semigroup (V (t)) t 0 generated by (L, D(L)) in X satisfies V (t)f 0 -̺ 0 M X C exp(-λ ⋆ t) f 0 X for all nonnegative f 0 ∈ X, ∀t 0 where ̺ 0 = R d f 0 (v)dv for any f 0 ∈ X.
The above result asserts that (1.1) is well-posed in the semigroup sense and that the associated solutions converge exponentially fast towards the unique equilibrium with same mass as f 0 . Notice that the convergence of V (t) as t → ∞ holds in operator norm. The proof of the Theorem is perturbative in nature and consists in looking L and the associated semigroup (V (t)) t 0 as perturbations of the multiplication operator T and the associated semigroup (U (t)) t 0 . Notice that the spectrum of T is given by

S(T ) = Range(-Σ) = (-∞, -η]
where η = inf v∈R d Σ(v) and Range(-Σ) denotes the closure (in C) of the range of -Σ(•). The main steps to prove the above Theorem are the following:

i) The main tool in the proof is the fact that the collision operator K is positive and weakly compact as an operator from Y → X. Since Y is also the domain of the multiplication operator T , it shows that K is T -relatively weakly compact. In particular, this implies that the spectrum of L = K + T consists in some essential spectrum which is included in the spectrum of T and eigenvalues of finite algebraic multiplicities which can accumulate only in -η. In particular, this shows the existence of some spectral positive spectral gap λ ∈ (0, η) for L. Remember that, in general and because of the absence of some spectral mapping theorem for semigroups, this is not enough in general to obtain the decay prescribed by Theorem 1.1. However, since we are dealing with positive semigroups in L 1 -spaces, the decay can be obtained by the following arguments:

ii) From the above point and since we are dealing here with positive operators in L 1 -spaces, Desch's theorem directly ensures that L = T + K is the generator of a positive semigroup (V (t)) t 0 on X which is a perturbation of the C 0 -semigroup (U (t)) t 0 generated by T . Moreover, these two semigroups share the essential spectrum. iiii) Finally, since 0 is an algebraically simple eigenvalue of L the decay estimate follows from general results concerning positive semigroups in L 1 -spaces. In the above steps, the most technical part of the proof consists in showing the weakcompactness property of K : Y → X. To do so, we exploit the integral nature of K given by (1.9) together with suitable comparison techniques that allow us to restrict ourselves to the properties of K whenever γ = d -2 (corresponding to hard-spheres interactions). Notice in particular that, for such a technical step, the introduction of the weight m is definitely necessary since it can be shown that, without weight, the operator K is not weakly-compact (see [START_REF] Lods | On linear kinetic equations involving unbounded cross-sections[END_REF] and Remark 3.4).

For soft potentials, our main contribution are the following two results. First, a non quantitative convergence result is establish

Theorem 1.2. Assume now that X = L 1 (R d , dv). If γ ∈ (-d, 0) and b(•) ∈ L 1 (S d-1 ) then X = Ker(L) ⊕ Im(L) and the C 0 -semigroup (V (t)) t 0 generated by L in X satisfies lim t→∞ V (t)f -̺ f M X = 0 (1.12)
where

̺ f = R d f (v)dv for any nonnegative f ∈ X.
Second, whenever f ∈ Ker(L) ⊕ Im(L), we can explicit the rate of convergence: Theorem 1.3. Under the assumptions of Theorem 1.2, given some nonnegative

f ∈ Ker(L) ⊕ Im(L),
then for any c ∈ (0, 1),

V (t)f -̺ f M X = O ϑ -1 log (ct) as t → ∞
where ϑ -1 log is the inverse of the increasing mapping ϑ log : r > 0 → ϑ(r) log 1 + ϑ(r) r and

ϑ(r) = 1 r 1 1 -Σmax √ r 2 +Σ 2 max , Σ max = sup v∈R d Σ(v).
In particular, for any ε > 0, there exists

C = C(f, ε) > 0 such that V (t)f -̺ f M X C (1 + t) -1 3+ε ∀t 0.
Again, the above result asserts that (1.1) is well-posed in the semigroup sense and that the associated solutions converge towards the unique equilibrium with same mass as the initial datum. The proof of the above result is a consequence of a general result concerning the strong convergence to ergodic projection for perturbed substochastic semigroups [START_REF] Mokhtar-Kharroubi | On strong convergence to ergodic projection for perturbed substochastic semigroups, Semigroups of Operators, Theory and Applications[END_REF]. Notice that such a result relies on the so-called "0-2" law for C 0 -semigroups by G. Greiner [1, p. 346] for the nonquantitative version (1.12). To get a quantitative convergence rate, we resort to recent results about non-exponential convergence rate for semigroups [START_REF] Chill | Quantified versions of Ingham's theorem[END_REF] related to Ingham's theorem (see also [START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF][START_REF] Batty | Perturbations of generators of C0-semigroups and resolvent decay[END_REF]). Notice that, because of the absence of a spectral gap for L, we do not expect any exponential relaxation to equilibrium. 1.3. Related literature. As already mentioned, we investigate here both the cases of hard and soft potentials. As well-documented, at least in an Hilbert setting, these two cases exhibit very different behaviour and usually require different methods of investigation. Let us describe here the known results regarding the large time behavior of solutions to (1.1). A first general comment is that, due to its paramount importance for the study of close-to-equilibrium solutions to the nonlinear Boltzmann equation, the linearized Boltzmann equation received much more attention than the linear one. Let us recall here that the operator associated to the linearized Boltzmann equation is given by

L f = Q(f, M) + Q(M, f )
whereas we recall that Lf = Q(f, M). One sees in particular that both the operators are expected to share many spectral properties. However, two very important differences have to be emphasised:

(1) First, while the linear operator L admits a single equilibrium state with unit mass given by M, the linearized operator L is such that

L (M) = L (v i M) = L (|v| 2 M) = 0, ∀i = 1, . . . , d.
This means that, in any reasonable functional space, 0 is a simple eigenvalue of L with Ker(L) = span(M) while the kernel of L is d + 2-dimensional:

Ker(L ) = span{M v i M |v| 2 M ; i = 1, . . . , d}.
(2) Second and more important for the analysis we shall perform here, the semigroup associated to the linear Boltzmann operator L is positive whereas the one associated to L is not. This is very natural from the physical point of view since equation (1.1) is aimed to describe the evolution particles interacting with an host medium while the evolution equation associated to L aims to describe the fluctuation around the equilibrium of the solution to the nonlinear Boltzmann equation. As already said, we will take much benefit of the fact that the semigroup generated by L is positive. Let us now describe in more details the existing results devoted to the large time behaviour of solution to (1.1). We shall distinguish here the two cases of hard and soft potentials.

• As far as hard potentials are concerned, it is by now well understood that, in the Hibert space

H = L 2 (R d , M -1 (v)dv) (1.
13) the linear Boltzmann operator (with its natural domain) is negative, self-adjoint. This means that

E (f, f ) := - R d f Lf M -1 dv 0 ∀f ∈ D(L) ⊂ H.
Moreover, 0 is a simple eigenvalue associated to the eigenfunction M and there exists λ 2 > 0 such that

E (f, f ) λ 2 f H ∀f ∈ D(L),
f ⊥M where of course the orthogonality is meant with respect to the natural inner product of H.

Clearly, this proves that L admits a spectral gap in H of size at least λ 2 . Since L is self-adjoint in H, this spectral result for the generator L translates "for free" to the associated C 0 -semigroup (exp(tL)) t 0 yielding an exponential trend to equilibrium with rate exp(-λ 2 t). The first proof of such a result can be traced back to D. Hilbert himself [START_REF] Hilbert | Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen[END_REF]. General proof of such an exponential trend to equilibrium are available in the literature. Historically, the first proof is based upon Weyl's Theorem and consists in showing that the non-local operator K is a compact perturbation of the multiplication operator T , ensuring that the essential spectrum of L coincide with that of T (see [START_REF] Grad | Principles of the kinetic theory of gases[END_REF][START_REF] Hilbert | Grundzüge einer Allgemeinen Theorie der Linearen Integralgleichungen[END_REF]). Only recently, such an approach has been made quantitative and explicit estimate of the spectral gap λ 2 have been provided [START_REF] Lods | Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models[END_REF] for hard-spheres interactions.

The above approach seems at first sight purely hilbertian and not well-adapted to the study of (1.1) in spaces of physical interest like L 1 -spaces. In such spaces indeed, the problem has received much less attention. We mention here the seminal work of Suhadolc [START_REF] Suhadolc | Linearized Boltzmann equation in L 1 space[END_REF] who introduced the spectral study of L for hard-spheres interactions in weighted spaces like

L 1 (R d , exp(α|v| 2 )dv).
In particular, it can be proven then that, in such a space, L is the generator of a positive semigroup. More recently, a careful study of the linearized Boltzmann operator has been performed in weighted L 1 -spaces with stretched exponential weights [START_REF] Mouhot | Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials[END_REF]. Such an approach is based upon the knowledge of the spectral properties of the linearized operator in the Hilbert space H combined with recent results concerning the stability of the spectrum in enlarged functional spaces [START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-theorem[END_REF]. This approach has been applied recently for the linear Boltzmann operator with hardspheres interactions in [START_REF] Bisi | Uniqueness in the weakly inelastic regime of the equilibrium state to the Boltzmann equation driven by a particle bath[END_REF] and provides quantitative estimates of the spectral gap in weighted L 1 -spaces. Such results are presented in the Appendix B of the paper for the sake of completeness. We also mention the recent result [START_REF] Bisi | Entropy dissipation estimates for the linear Boltzmann operator[END_REF] in which the exponential convergence of the solutions to (1.1) is measured in relative entropy and an explicit convergence rate is provided by means of entropy/entropy production estimate.

• The linear Boltzmann equation associated to cut-off soft potentials received much less attention. In the seminal paper [START_REF] Caflisch | The Boltzmann equation with a soft potential, I. Linear, spatially homogeneous[END_REF], the linearized operator has been studied in the space H defined by (1.13). Adapting the argument of [START_REF] Caflisch | The Boltzmann equation with a soft potential, I. Linear, spatially homogeneous[END_REF] to the linear operator L, one checks easily that, in this case, the linear operator L is still self-adjoint and negative in H but then its continuous spectrum extends to the origin so that L has no spectral gap in the space H. It is then clear that, at the semigroup level, no exponential convergence towards equilibrium can occur. We mention here that, in [START_REF] Caflisch | The Boltzmann equation with a soft potential, I. Linear, spatially homogeneous[END_REF], almost exponential convergence to equlibrium is obtained for the linearized Boltzmann for initial datum belonging to the subspace of H made of functions bounded above by some suitable Maxwellian (see Remark 4.7 for more details). Let us mention here that, using argument similar to those of Theorem 1.3, it is possible to obtain algebraic rate of convergence (like t -1 ) for the linear Boltzmann operator in the space H (see Remark 4.7). To our knowledge, the only convergence results available for the linear Boltzmann equation are all based upon entropy inequalities [START_REF] Pettersson | On weak and strong convergence to equilibrium for solutions to the linear Boltzmann equation[END_REF] and consists in showing that, since the relative entropy (with respect to the equilibrium solution M) is continuously decreasing along the solutions to (1.1) and since M is the only equilibrium solution (with unit mass), a variant of classical LaSalle invariance principle implies the convergence of f (t, v) towards M. Such an approach requires at least that the initial state f 0 is of finite relative entropy, i.e.

R d f 0 (v) log f 0 (v) M(v) dv < ∞ (1.14)
which, roughly speaking, would correspond to convergence of f (t, v) in some suitable Orliczspace L log L.

1.4. Organization of the paper. After this long introduction, the paper is organized as follows.

Section 2 contains all the material from semigroup theory and spectral analysis needed for the rest of the paper. The results presented there are not new but maybe some of them are not very well-known in the kinetic community. We emphasise in particular the crucial role played by positivity in our analysis. Section 3 is devoted to the study of the linear BE in the case of hard potentials and is culminating with the proof of Theorem 1.1 following the above steps i)-iii) here above. Section 4 deals with the convergence result for soft potential interactions and provides the proof of Theorem 1.2. In Appendix A, we derive the expression of the kernel k B (v, w) for various type of interactions B as well as several of their important properties. The material of this Appendix is the basic toolbox for the proof of the weak compactness of K used in the proof of Theorem 1.1. Finally, in Appendix B, we provide some quantitative estimates of the spectral gap λ * appearing in Theorem 1.1 using some of the results established recently in [START_REF] Bisi | Uniqueness in the weakly inelastic regime of the equilibrium state to the Boltzmann equation driven by a particle bath[END_REF] which use the enlargement and factorisation techniques of [START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-theorem[END_REF].

REMINDER ABOUT POSITIVE SEMIGROUPS IN L 1 -SPACES

We review here some known result about the asymptotic properties of stochastic semigroups as t → ∞ in L 1 -spaces. We will focus here only on the asymptotic stability of positive semigroups, i.e. the convergence towards equilibrium density. For such questions, one should distinguish between two important cases: a first one for which the existence of spectral gap implies exponential convergence towards equilibrium; a second one in which the generator does not exhibit a spectral which is related to ergodic theorem. We begin with general properties of positive semigroups in L 1 -spaces.

2.1. Spectral properties of semigroups in L 1 . Let us consider a Borel measure µ over some given space Ω and let X = L 1 (Ω, dµ). Given a nonnegative mapping

M : Ω → R the multiplication operator A : D(A) ⊂ X → X defined by Af (x) = -M (x)f (x) for any f ∈ D(A) = {f ∈ X ; M f ∈ X} is the generator of a positive C 0 -semigroup (U (t)) t 0 in X given by [U (t)f ](x) = exp(-M (x)t)f (x) ∀t 0.
Remark 2.1. The semigroup is positive in the sense that it preserves the positive cone of X, i.e. f ∈ X, f 0 µ-a.e. implies U (t)f 0 µ-a.e.

Remark 2.2. Notice that, if equipped with the graph norm f

A := f + Af , the space D(A) can be identified with L 1 (Ω, (1 + M (x))dµ(x)).
General result about semigroup theory ensures that, as a multiplication operator, the spectrum of A is given by the so-called essential range of -M , i.e.

S(A) = {λ ∈ C ; µ ({x ∈ Ω , |λ + M (x)| < ε}) = 0∀ε > 0} and, if M (•) is continuous, one actually has S(A) = Range(-M ) ⊂ R -.
Since, for any t 0, U (t) is also a multiplication operator, its spectrum is given by the essential range of exp(-M (•)t) and, in particular,

S(U (t)) = exp (S(A)t)

∀t 0

where the closure is meant of course in C. In particular, if we define

M 0 := µ -ess-inf x∈Ω M (x), then, then one has S(A) ⊂ (-∞, -M 0 ]
and the type of the semigroup (U (t)) t 0 is less than -M 0 . Recall the type of a semigroup (U (t)) t 0 is defined as

ω 0 (U ) = inf{ω ∈ R , ∃C 1 U (t) C exp(ωt) ∀t 0}.
A fundamental property of positive semigroups in L 1 -spaces is the following (see [START_REF] Engel | One parameters semigroups for linear evolution equations[END_REF])

Proposition 2.3. If X = L 1 (Ω, µ
) and (S(t)) t 0 is a positive C 0 -semigroup of X then its type coincide with the spectral bound of its generator G, i.e.

ω 0 (S) = s(G).
Let us consider now the influence of unbounded perturbations. Namely, consider now a positive (integral) operator B : D(A) ⊂ X → X and assume that, for any λ > -M 0 ,

BR(λ, A) : X → X is a weakly compact operator in X
where R(λ, A) = (λ -A) -1 is the resolvent of A which exists at least for λ > -M 0 . In such a case, even if B is not a bounded operator, one knows 'for free' that the sum A + B is the generator of a positive C 0 -semigroup in X. Precisely, one has:

Theorem 2.4. [24, Theorem 6] If B is a positive operator such that BR(λ, A) is a weakly compact operator in X then, T = A + B (with domain D(A)) is the generator of a C 0 -semigroup (V (t)) t 0 in X. Moreover, the essential spectrum of (V (t)) t 0 is such that S ess (U (t)) = S ess (V (t)) ∀t 0
where we adopted here the notion of essential spectrum due to Schechter.

Remark 2.5. Since X is an L 1 -spaces, for an (un)bounded operator T : D(T ) ⊂ X → X the essential spectrum of S is exactly the part of the spectrum left invariant by weakly-compact operators, i.e.

S ess (T ) = K∈W (X)

S(T + K)

where W (X) is the ideal of B(X) made of the weakly compact operators in X. We refer to [START_REF] Latrach | Essential spectra on spaces with the Dunford-Pettis property[END_REF] for more details on that matter.

Remark 2.6. The semigroup (V (t)) t 0 in the above theorem is given by a Dyson-Phillips series

V (t) = ∞ j=0 U j (t), t 0
where U 0 (t) = U (t) is the unperturbed semigroup and, for any j 0,

U j+1 (t) = t 0 U j (t -s)BU 0 (s)ds. (2.1)
Remark 2.7. Since B is a weakly-compact perturbation of A, one has S ess (B) = S ess (A) ⊂ (-∞, -M 0 ] and

S(B) ⊂ (-∞, -η] ∪ {λ k } k
where (λ k ) k is a sequence of eigenvalues of B (maybe finite or empty). In particular, exp(λ k t) is an eigenvalue of V (t) for any t 0 and any k.

2.2. Exponential trend to equilibrium in presence of a spectral gap. We begin here with a general result which holds for any C 0 -semigroup in some Banach space X.

Theorem 2.8. [12, Theorem 3.7, Chapter V] Let (V (t)) t 0 be a C 0 -semigroup with generator A in some Banach space X such that ω ess (V ) < 0

where ω ess (V ) denotes the essential type of (V (t)) t 0 . Then the set {λ ∈ S(A) ; Reλ 0} is finite (or empty) and consists of isolated eigenvalues λ 1 , . . . , λ m of A with finite algebraic multiplicity. If P 1 , . . . , P m denote the corresponding spectral projections and k 1 , . . . , k m the corresponding orders of poles of R(•, T ) then

V (t) = V 1 (t) + . . . + V m (t) + R m (t), t 0 (2.2)
where

V n (t) = exp(λ n t) kn-1 j=0 t j j! (A -λ n ) j P n n = 1, . . . , m
and there exists δ > 0 and C 1 such that

R m (t) B(X) C exp(-δt) ∀t 0.
The above result is important in particular when the above set {λ ∈ S(A) ; Reλ 0} reduces to a singleton λ 1 which becomes then the dominant eigenvalue of T . In such a case, the conclusion reads

V (t) -V 1 (t) C exp(-δt) ∀t 0.
If moreover λ 1 is a simple eigenvalue, i.e. k 1 = 1, then the rescaled semigroup exp(-λ 1 t)V (t) converges exponentially fast towards a rank-one projection. Sufficient conditions for this to holds are explicit whenever X is a Banach lattice and when the C 0 -semigroup (V (t)) t 0 is positive. For simplicity, we state here our result in the case in which X is some L 1 -space: Proposition 2.9. Assume X = L 1 (Ω, µ) where (Ω, F) is a measure space and µ a given measure over (Ω, F). Let (V (t)) t 0 be a positive C 0 -semigroup with generator A in X satisfying the assumption of Theorem 2.8. Then, there exists ε > 0,

S(A) ∩ {z ∈ C ; Rez s(A) -ε} = {s(A)}
where s(A) is an isolated eigenvalue of A with finite algebraic multiplicity k 1 and the splitting (2.2) holds for m = 1 and any s(A) -ε < δ < s(A).

Again, the above can be made more precise if the semigroup is irreducible. We recall that, in L 1 (Ω, µ), a C 0 -semigroup (V (t)) t 0 is said to be irreducible if, given f ∈ X, φ ∈ X ⋆ , f, φ > 0 µ-a. e., then there exists t 0 0 such that

V (t 0 )f , φ X,X ⋆ > 0
where •, • X,X ⋆ denotes the duality bracket between X and its dual space X ⋆ = L ∞ (Ω, µ).

One has then the following:

Π = φ ⊗ f where φ ∈ X ⋆ is a positive eigenvector of A ⋆ , f ∈ X is a positive eigenvector of A and u, φ X,X ⋆ = 1.
Remark 2.11. Notice that, if (V (t)) t 0 is irreducible with generator A, then there exists κ 0 such that {λ ∈ S(A) ; Reλ = s(A)} = s(A) + κ i Z. We refer to [12, Theorem 1.12, Chapter VI] for further details.

2.3.

Convergence to equilibrium without spectral gap. In the above section, we described the situation in which the generator of the semigroup admits a spectral gap and its consequences on the exponential convergence. In this section, we describe a completely different situation in which no spectral gap estimate is available. In such a case, we do not expect the semigroup to converge exponentially to some "equilibrium". We present here some recent result, due to the second author, which provides in L 1 -spaces sufficient condition for the semigroup to converge as time goes to infinity in absence of spectral gap. Such result will be then applied to the BE for soft potential in the last part of the paper. Theorem 2.12. Let (U (t)) t 0 be a substochastic C 0 -semigroup in L 1 (Ω, µ) with generator A and let B : D(A) → L 1 (Ω, dµ) be positive and satisfy

lim λ→+∞ r σ B(λ -A) -1 < 1
and

Ω Af + Bf dµ 0 ∀f ∈ D(B) ; f 0
where r σ (•) denotes the spectral radius. Let (V (t)) t 0 be the substochastic C 0 -semigroup generated by

T = A + B : D(T ) = D(A) → L 1 (Ω, dµ). We assume that (V (t)) t 0 is irreducible and that Ker(T ) = {0}. If lim t→0 U (t)f L 1 (Ω,µ) = 0 ∀f ∈ L 1 (Ω, µ)
and if there exists m ∈ N such that the mapping

t 0 → R m (t) = ∞ j=m U j (t) ∈ B(L 1 (Ω, dµ))
is continuous in operator norm (where (U j (t)) j has been defined in (2.1)), then

lim t 0 V (t)f = Pf
where P denotes the ergodic projection on Ker(A).

Remark 2.13. Recall that, under the above hypothesis, Ker(A) = span(ϕ) is one dimensional with ϕ > 0, Ker(A * ) = span(ϕ * ) is one dimensional where A * denotes the adjoint of A and ϕ * > 0. In such a case, the C 0 -semigroup (V (t)) t 0 is mean ergodic with ergodic projection P given by

Pf = Ω ϕ * f dµ ϕ ∀f ∈ L 1 (Ω, dµ)
where ϕ * and ϕ are chosen such that Ω ϕ * ϕdµ = 1 (see [START_REF] Mokhtar-Kharroubi | On strong convergence to ergodic projection for perturbed substochastic semigroups, Semigroups of Operators, Theory and Applications[END_REF] for details).

EXPONENTIAL TREND TO EQUILIBRIUM FOR HARD POTENTIALS

We consider in this section the case of hard-potentials and aim to prove Theorem 1.1. We recall that we shall consider here a positive weight function m = m(v) such that m -1 (v) 1 for any v ∈ R d and will work in the space

X = L 1 (m -1 (v)dv).
The typical weights we shall consider are exponential weights given by (1.10) and algebraic weights given by (1.11).

Since we are interested here in the properties of the linear BE in the space X for hard potential interactions, we shall consider the collision interactions

B(v -v * , σ) = |v -v * | γ b(cos θ), cos θ = v -v * |v -v * | ; n
with b nonnegative satisfying the cut-off assumption (1.6) and moreover γ 0. Introduce then

Y = Y γ = L 1 R d ; (1 + |v|) γ m -1 (v)dv ⊂ X and let T f (v) = Σ(v)f (v), D(T ) = {f ∈ X ; Σf ∈ X} = Y γ . Clearly, T generates a C 0 -semigroup (U (t)) t 0 in X defined by U (t)f (v) = exp(-Σ(v)t)f (v) t 0, f ∈ X.
The linear Boltzmann operator is then defined as

Lf (v) = Kf (v) -Σ(v)f (v) = (K + T ) f (v), ∀f ∈ D(T )
where the gain part K is given by (1.7).

A first very important observation is the following:

Lemma 3.1. The linear operator K depends continuously on b ∈ L 1 (S d-1 ): more precisely, if

(b n ) n ⊂ L 1 (S d-1
) is a sequence of kernels such that

lim n→∞ b n -b L 1 (S d-1 ) = lim n→∞ S d-1 |b n (cos θ) -b(θ)| dσ = 0 then lim n→∞ K n -K B(Y,X) = 0 where (K n ) n is the sequence of collision operators associated to (b n ) n .
Proof. The proof relies on standard computations for the Boltzmann operator (see for instance [23, Proposition 2.1] for similar considerations). Let γ 0 be given and let (b n ) n ⊂ L 1 (S d-1 ) be given. Using the explicit expression (1.7), for any n ∈ N and any f ∈ Y it holds

K n f -Kf X R 2d ×S d-1 |B(v -v * , σ) -B n (v -v * , σ)| |f (v ′ )| M(v ′ * )m -1 (v)dvdv * dσ
where we set

B n (v -v * , σ) = |v -v * | γ b n (cos θ), B(v -v * , σ) = |v -v| γ b(cos θ). Using the pre- post collisional change of variables (v ′ , v ′ * , σ) → (v, v * , σ) (see [29, Chapter 1, Section 4.5]), we obtain K n f -Kf X R 2d ×S d-1 |B(v -v * , σ) -B n (v -v * , σ)| |f (v)| M(v * )m -1 (v ′ )dvdv * dσ = R 2d |b n (cos θ) -b(cos θ)| |v -v * | γ |f (v)| M(v * )m -1 (v ′ )dvdv * dσ R d (1 + |v|) γ |f (v)|dv R d M(v * )(1 + |v * |) γ dv * S d-1 |b n (cos θ) -b(cos θ)| m -1 (v ′ )dσ where we used that |v -v * | γ (1 + |v|) γ (1 + |v * |) γ for any v, v * ∈ R 2d . Now, if m is an exponential weight of type (1.10), since |v ′ | s = (|v ′ | 2 ) s/2 (|v ′ | 2 +|v ′ * | 2 ) s/2 ) = (|v| 2 + |v * | 2 ) s/2 |v| s + |v * | s for any v, v * , σ, we get that m -1 (v ′ ) m -1 (v)m -1 (v * ) ∀v, v * , σ ∈ R 2d × S d-1
to that the above inequality reads

K n f -Kf X b n -b L 1 (S d-1 ) R d (1 + |v|) γ |f (v)|m -1 (v) dv R d M(v * )(1 + |v * |) γ m -1 (v * ) dv * and, since R d M(v * )(1 + |v * |) γ m -1 (v * ) dv * = C(m) < ∞ we get that K n f -Kf X C(m) b n -b L 1 (S d-1 ) f Y ∀f ∈ Y
which proves the result.

If the weight m is an algebraic weight of type (1.11), as above we see that, for any

v, v * , σ ∈ R 2d × S d-1 , m -1 (v ′ ) = 1 + |v ′ | β 1 + |v| 2 + |v * | 2 β/2 1 + 2 β/2-1 |v| β + |v * | β so that m -1 (v ′ ) C β m -1 (v) m -1 (v * )
with C β = max(1, 2 β/2-1 ). One concludes as above that

K n f -Kf X C(β) b n -b L 1 (S d-1 ) f Y ∀f ∈ Y
for some positive constant C(β) depending only on β.

Remark 3.2.

As we shall see later on, since the angular collision kernel maybe approximated by some sequence of bounded kernels over S d-1 , to investigate the weak-compactness properties of K, it will be enough to assume b to be bounded.

For simplicity, we denote K γ the gain part associated to the variable hard-spheres collision kernel

B γ (v -v * , σ) = |v -v * | γ , i.e. K γ f (v) = R d ×S d-1 |v -v * | γ f (v ′ )M(v ′ * )dv * dσ ∀f ∈ Y
and, clearly, for any nonnegative f , one sees that

Kf (v) b L ∞ K γ f (v) ∀v ∈ R d .
One has the following fundamental result Proof. Recall that (see Appendix A)

K γ f (v) = R d k γ (v, w)f (w)dw
where

k γ (v, w) |v -w| γ-(d-2) k d-2 (v, w) and k d-2 (v, w) = C|v -w| -1 exp - 1 8 |v -w| + |v| -|w| |v -w| 2 ∀v, w ∈ R d × R d .
Let us prove the weak compactness property. Let dν(v) = m -1 (v)dv and let B be the unit ball of Y. Since X = L 1 (dν), according to Dunford-Pettis Theorem, the weak compactness of K d-2 amounts to prove that

sup f ∈B A |K γ f (v)| dν(v) -→ 0 as ν(A) → 0 (3.1)
and

sup f ∈B |v|>r |K γ f (v)| dν(v) -→ 0 as r → ∞. (3.2)
Using the representation of K γ as an integral operator, it is easy to check that (3.1) and (3.2) will follow if one is able prove that

sup w∈R d m(w) (1 + |w|) γ A k γ (v, w)m -1 (v)dv -→ 0 as ν(A) → 0 (3.3)
and

sup w∈R d m(w) (1 + |w|) γ |v|>r k γ (v, w)m -1 (v)dv -→ 0 as r → ∞. (3.4)
Let us prove (3.3). Let A ⊂ R d be a given Borel subset and let w ∈ R d be fixed. Let

B w = {v ∈ R d , |v -w| < 1}. Since k γ (v, w) C|v -w| γ+1-d one has A k γ (v, w)m -1 (v)dv C A |v -w| γ+1-d dν(v) = C A∩Bw |v -w| γ+1-d dν(v) + A∩B c w |v -w| γ+1-d dν(v) . Clearly, since γ + 1 -d -1 A∩B c w |v -w| γ+1-d dν(v) ν(A)
while, for any p > 1, 1/q + 1/p = 1, one has . Now, if m(v) = exp(-a|v| s ) for a > 0 and s ∈ (0, 1), one gets that

m -1 (v) m -1 (w) exp(a|v -w| s ) exp(a)m -1 (w) ∀w ∈ R d , ∀v ∈ B w while, for algebraic weights, i.e. if m(v) = (1 + |v|) -β , β > 0 then m -1 (v) = (1 + |v -w + w| β ) 1 + 2 β-1 |v -w| β + 2 β-1 |w| (1 + 2 β-1 )(1 + |w| β ) ∀w ∈ R d , v ∈ B w .
One sees from this that, in both cases, there exists some positive constant C > 0 such that

A∩Bw |v -w| p(γ+1-d) m -1 (v)dv C m -1 (w) Bw |v -w| p(γ+1-d) dv. Choosing now 1 p < d d-γ-1 , one sees that Bw |v -w| p(γ+1-d) dv = |S d-1 | 1 0 d̺ ̺ 1-d-p(1+γ-d)
is finite and is independent of w. Thus, there exists C = C(p) > 0 such that

A k γ (v, w)m -1 (v)dv C m -1 p (w)ν(A) 1/q + ν(A) ∀w ∈ R 3 .
Since p > 1, this proves that (3.3) holds true. Let us now prove (3.4). One first notices that

sup |w| r/2 m(w) (1 + |w|) γ |v|>r k γ (v, w)m -1 (v)dv -→ 0 as r → ∞. (3.5)
Indeed, one notices that

k γ (v, w) C |v -w| γ+1-d exp 1 4 |w| 2 -|v| 2 with γ + 1 -d -1.
Therefore, if |w| r/2 and |v| > r, one gets

k γ (v, w) C r 2 γ+1-d exp -3 16 |v| 2
where we used that |v| > 2|w|. Then, (3.5) follows easily since, in both the considered cases,

R d exp -3 16 |v| 2 dν(v) < ∞.
Consequently, to prove (3.4), it is enough to show that

sup |w|>r/2 m(w) (1 + |w|) γ |v|>r k γ (v, w)m -1 (v)dv -→ 0 as r → ∞. (3.6)
According to Proposition A.1, for both the considered weights, there exist C 0 > 0 and α > 0 such that

R d k γ (v, w)m -1 (v)dv C 0 1 + |w| γ-α m -1 (w) with α = s if m(v) = exp(-a|v| s ) s ∈ (0, 1] or α = 2 if m(v) = (1 + |v| -β ), β > 0. Therefore, for any r > 0, m(w) (1 + |w|) γ |v|>r k γ (v, w)m -1 (v)dv C 0 1 + |w| γ-α (1 + |w|) γ
and (3.6) follows since sup |w|>r/2 1 + |w| γ-α (1 + |w|) γ → 0 as r → ∞. This achieves to prove that K γ : Y → X is weakly-compact.

Remark 3.4. Notice that the above result does not hold true without weight m

-1 (i.e. if a = 0 or β = 0). Indeed, it is shown in [20, Proposition 3.22] that K d-2 : L 1 (R d ; (1 + |v|)dv) → L 1 (R d ; dv)
is not weakly compact. The same method shows that actually, K γ is not weakly compact for γ ∈ [0, d -2]. Notice that, in the absence of weight, it is (3.6) which is violated while (3.3) and (3.5) still hold true. Remark 3.5. Notice also that almost all the above computations are still valid for γ < 0. In particular, (3.5) and (3.3) still hold true for -d γ < 0. However, the final argument in our proof does not hold true for γ < 0, since clearly Proof. The fact that KR(λ, T ) is positive is obvious. Given λ > 0, the range of R(λ, T ) is Y and one has to prove that K : Y → X is weakly-compact.

lim r→∞ sup |w|>r/2 1 + |w| γ-α (1 + |w|) γ = ∞.
Let (b n ) n be a sequence of angular kernel such that b n ∈ L ∞ (S d-1 ) and b n -b L 1 (S d-1 ) → 0 as n → ∞ and let (K n ) n be the sequence of collision operator associated to

(b n ) n . According to Lemma 3.1, lim n K n -K B(Y,X) = 0. In particular, if K n is weakly compact for any n ∈ N, so is K. It suffices therefore to prove that K : Y → X is weakly-compact whenever b ∈ L ∞ (S d-1 ). Now, for b ∈ L ∞ (S d-1 ), one clearly has Kf (v) b L ∞ (S d-1 ) K γ f (v) ∀v ∈ R d , ∀f ∈ Y, f 0,
i.e. K is dominated by the operator b L ∞ (S d-1 ) K γ . By Dunford-Pettis Theorem, the weak compactness of K γ given by Proposition 3.3 ensures that of K.

From now on, we shall always consider that the angular kernel b ∈ L 1 (S d-1 ).

According to the previous Proposition, Theorem 2.4 applies to this case yielding the following

S(T ) = Range(-Σ) = (-∞, -η]
where η = ess -inf v∈R d Σ(v) > 0 is explicit. Moreover, the essential spectrum of (L, D(L)) is then

S ess (L) = S ess (T ) ⊂ (-∞, -η]
and S(L) = S ess (T ) {λ k } k 1 where (λ k ) k are eigenvalues of (L, D(L)) with finite algebraic multiplicities.

Remark 3.9. Notice that, since the kernel k(•, •) is positive:

k(v, w) > 0 for a. e. v, w ∈ R d × R d the semigroup (V (t)) t 0 generated by (L, D(L)) is irreducible in X. Actually, if f ∈ X, f 0, f = 0, then [V (t)f ](v) > 0 for a. e. v ∈ R d .
Indeed, the semigroup being given by a Dyson-Phillips expansion series

V (t)f = ∞ n=0 V n (t) where V 0 (t) = U (t) and V n+1 (t)f = t 0 V n (t -s)KU (s)f ds for any f ∈ X. Since the kernel k(•, •) is positive, it is easy to check that [V n (t)f ](v) > 0 for a. e. v ∈ R d if f ∈ X, f 0 non identically equal to zero.
Notice that 0 is a simple eigenvalue of L with eigen-space spanned by M. In particular, the spectral bound s(L) is such that s(L) 0. Finally, using Proposition 2.3 one has Proposition 3.10. The type ω 0 (V ) of the C 0 -semigroup (V (t)) t 0 is zero, i.e.

ω 0 (V ) = s(L) = 0.
Proof. Let •; • X,X ⋆ denotes the duality bracket between X and the dual space X ⋆ . Now, since ω ess (V ) < 0 s(L), one knows (see Remark 3.8) that s(L) is an isolated eigenvalue of L with finite algebraic multiplicity. According to Proposition 2.10, since the C 0 -semigroup (V (t)) t 0 is irreducible, s(L) is also an eigenvalue of the adjoint operator L ⋆ in X ⋆ associated to a positive eigenfunction g ⋆ , g ⋆ (v) > 0 for a.e. v ∈ R d . Then, since L(M) = 0, one has

0 = L(M) ; g ⋆ X,X ⋆ = M ; L ⋆ (g ⋆ ) X,X ⋆ = s(L) M ; g ⋆ X,X ⋆
and, since both g ⋆ and M are positive, one has M ; g ⋆ X,X ⋆ > 0 from which necessarily s(L) = 0. The conclusion follows from the well-know fact that, being the C 0 -semigroup (V (t)) t 0 positive, one has ω 0 (V ) = s(L). Remark 3.11. In the above proof, we used the fact that s(L) is an eigenvalue of the adjoint operator (L ⋆ , D(L ⋆ )) associated to a positive eigenfunction g ⋆ . Actually, thanks to the conservative properties of L one can now be more precise. Namely, one notices that

R d Lf (v)dv = 0 ∀f ∈ D(L). (3.7)
In particular, with the notations of the above proof, one deduces that

Lf ; m X,X ⋆ = f ; L ⋆ m X,X ⋆ = R d Lf (v)dv = 0 ∀f ∈ D(L).
In particular, L ⋆ m = 0 and s(L) = 0 is an eigenvalue of the adjoint operator (L ⋆ , D(L ⋆ )) associated to the eigenfunction g ⋆ = m. Moreover, using again Proposition 2.10, the spectral projection Π 0 associated to the eigenvalue s(L) = 0 of L is given by Π 0 = m ⊗ M, i.e.

Π 0 f = f, m X,X ⋆ M = ̺ f M ; with ̺ f = R d f (v)dv ; ∀f ∈ X. (3.8)
We are in position to prove Theorem 1.1

Proof of Theorem 1.1. The fact that L is the generator of a positive semigroup in X has been already proven in Theorem 3.7 so we focus here only on the large time behaviour of (V (t)) t 0 .

Combining Theorem 3.7 with Proposition 3.10, one has

ω ess (V ) = ω ess (U ) -η < 0 = ω 0 (V ) = s(L).
We get therefore the conclusion using Proposition 2.9, Theorem 2.8 together with the expression of Π 0 given by (3.8).

CONVERGENCE TO EQUILIBRIUM FOR LINEAR BE WITH SOFT POTENTIALS

We investigate in this section the case of soft potentials. Our analysis is performed in the "unweighted" space

X = L 1 (R d , dv)
since we shall apply results pertaining to the "substochastic theory" of semigroups. We recall here that, in such a case, the collision frequency is bounded by virtue of (1.8). We use the notations of the previous section. Using again the decomposition

L = K -T with T f (v) = Σ(v)f (v), one has D(T ) = X and, consequently, the C 0 -semigroup (U (t)) t 0 in X defined by U (t)f (v) = exp(-Σ(v)t)f (v) t 0, f ∈ X,
is uniformly continuous in the following sense:

lim t→0 + U (t) -I L (X) = 0 (4.1)
where L (X) stands for the Banach space of all bounded operators in X endowed with the uniform operator topology induced by • L (X) whereas I is the identity operator in X. Indeed, one checks easily that, for any f ∈ X:

U (t)f -f X R d |exp(-tΣ(v)) -1| |f (v)|dν(v) t R d Σ(v) |f (v)|dν(v) σ 2 t f X , from which
U (t) -I L (X) tσ 2 ∀t > 0
where σ 2 > 0 is the positive constant appearing in (1.8). Then, (4.1) follows. Concerning the collision operator K, we have the following: Proposition 4.1. Assume m to be given by (1.10) or (1.11).

If γ ∈ (-d, 0] and b ∈ L 1 (S d-1 ) is nonnegative, then K : X → X is bounded.
Proof. The proof uses some of the computations made in Lemma 3.1. Namely, given f ∈ X and γ ∈ (-d, 0], b ∈ L 1 (S d-1 ), one sees that

Kf X R 2d ×S d-1 B(v -v * , σ) |f (v)| M(v * )dvdv * dσ = R 2d b(cos θ) |v -v * | γ |f (v)| M(v * )dvdv * dσ.
Therefore, we get

Kf X b L 1 (S d-1 ) R 2d |f (v)| M(v * ) |v -v * | γ dv * dv.
Since it clearly holds that M ∈ L q (R d ) for any q > 1, one deduces from Hardy-Littlewood-Sobolev inequality (see [START_REF] Lieb | Analysis, Second Edition[END_REF]Theorem 4.3]) that

R 2d |f (v)| M(v * ) |v -v * | γ dv * dv C d (γ) f L 1 (R d ) M L q (R d )
with some universal constant C d (γ) depending only on γ and with q > 1 such that γ/d = 1/q -1.

Thish proves the result.

With this in hands, one can prove a first non quantitative convergence theorem

Proposition 4.2. If γ ∈ (-d, 0) and b(•) ∈ L 1 (S d-1 ) then L generates a C 0 -semigroup (V (t)) t 0 in X which satisfies lim t→∞ V (t)f 0 -̺ 0 M X = 0 where ̺ 0 = R d f 0 (v)dv for any f 0 ∈ X.
Proof. Since both T and K are bounded operator, one has L ∈ L (X) and the C 0 -semigroup (V (t)) t 0 generated by L is uniformly continuous. Since the same holds for the C 0 -semigroup

(U (t)) t 0 , defining R 1 (t) = V (t) -U (t)
it is clear that the mapping t 0 → R 1 (t) ∈ L (X) is continuous (with respect to the operator norm). Since moreover Ker(L) = {0} (since M ∈ X and L(M) = 0) and the semigroup (V (t)) t 0 is irreducible, one concludes with Theorem 2.12.

We can make the above result more precise and provide here a quantitative version of the above convergence result. Recall that the gain operator K can be written as an integral operator

Kf (v) = R d k(v, w)f (w)dw ∀f ∈ X
for some nonnegative measurable kernel k(v, w) such that

Σ(v) = R d k(w, v)dw ∀v ∈ R d . (4.2)
Recalls that Σ ∈ L ∞ (R d ) and inf v Σ(v) = 0. The above identity will play a crucial role in the proof of the following:

Proposition 4.3. Define the mapping ϑ : R + → R + by ϑ(r) := 1 r 1 1 -Σmax √ r 2 +Σ 2 max (r > 0) (4.3)
where

Σ max := sup v Σ(v) < ∞. Then, one has i R \ {0} ⊂ ̺(L) and R(iα, L) L (X) ϑ (|α|) ∀α ∈ R \ {0}.
Proof. Consider the resolvent equation

λf -Lf = g, g ∈ X, f ∈ D(L) = X
where λ / ∈ Range(-Σ) = S(T ) where T is the multiplication operator by -Σ. Since L = K + T and λ ∈ ̺(T ), such a problem is then equivalent to φ -KR(λ, T )φ = g where φ = (λ -T )f . In particular, for such λ, one will get λ ∈ ̺(L) if

KR(λ, T ) L (X) < 1 and then φ = ∞ n=0 (KR(λ, T )) n g and f = ∞ n=0 KR(λ, T ) (KR(λ, T )) n g. (4.4)
Let us now consider λ = iα with α ∈ R, α = 0. Then, α / ∈ Range(-Σ) and R(iα, T ) is the multiplication operator by 1 iα+Σ . Consequently,

KR(iα, T )φ = R d k(v, w) iα + Σ(w) φ(w)dw ∀φ ∈ X. Consequently KR(iα, T )φ X R d dv R d k(v, w) α 2 + Σ 2 (w) |φ(w)| dw = R d Σ(w) α 2 + Σ 2 (w) |φ(w)|dw
where we used Fubini's theorem and the identity (4.2). Therefore

KR(iα, T ) L (X) sup w Σ(w) α 2 + Σ 2 (w) = Σ max α 2 + Σ 2 max < 1
since the mapping r 0 → r √ α 2 +r 2 is nondecreasing. Then, iα ∈ ̺(L) and from (4.4), we deduce that

R(iα, L) L (X) 1 |α| ∞ n=0 Σ max α 2 + Σ 2 max n = 1 |α| 1 1 -Σmax √ α 2 +Σ 2 max
where we used the fact that R(iα, T )

L (X) = sup v |iα + Σ(v)| -1 = |α| -1
. This proves the result.

Introduce the mapping

ϑ log (r) := ϑ(r) log 1 + ϑ(r) r r > 0.
Notice that, since ϑ(•) is strictly decreasing with

lim r→+∞ ϑ(r) = 0, lim r→0 + ϑ(r) = +∞.
the same holds true for ϑ log (•) and ϑ -1 log : R + → R + denotes its inverse mapping.

Recall that the C 0 -semigroup (V (t)) t 0 generated by L in X is such that

lim t→∞ V (t)f 0 -Pf 0 X = 0
where the projection P is given by

Pf 0 = R d f 0 (v)dv M for any f 0 ∈ X.
Moreover, P is the spectral projection over Ker(L) and, as such, the following holds

X = Ker(L) ⊕ Im(L).
Denote then by Z = Ker(L) ⊕ Im(L). It is easy to check that Z is dense in X and one deduces from the spectral properties of L established in Proposition 4.3 the following Theorem 4.4. Assume that f ∈ Z, then, for any c ∈ (0, 1),

V (t)f -Pf X = O ϑ -1 log (ct) as t → ∞.
Proof. Since its generator L is bounded, the C 0 -semigroup (V (t)) t 0 is analytic and therefore asymptotically analytic [START_REF] Engel | One parameters semigroups for linear evolution equations[END_REF]. Then, combining Proposition 4.3 with [11, Corollary 2.12], we deduce that, for any c ∈ (0, 1)

V (t)L R(1, L) L (X) = O(ϑ -1 log (c t)) as t → ∞. (4.5)
In particular,

V (t)h = O(ϑ -1 log (c t)) as t → ∞
for any h ∈ Im(L) from which we get the result since Pf = V (t)Pf for any t 0.

One easily deduces the following algebraic rate of convergence: Corollary 4.5. Let f ∈ Z. Then, for any ε > 0, there exists C > 0 such that

V (t)f -Pf X C (1 + t) -1 3+ε ∀t 0. (4.6)
Proof. The proof consists simply in noticing that, since ϑ(r) ≃ 2Σ 2 max r 3 whenever r ≃ 0 + , for any ε > 0, there is some universal constant κ ε > 0 such that ϑ log (r) κ ε r 3+ε for r ≃ 0 + .

One deduces from that that

ϑ -1 log (ct) = O(t -1 3+ε ) for t → ∞
and the result follows.

Remark 4.6. Notice that the above convergence result, combined with Datko-Pazy's Theorem shows in particular that Im(L) is not closed in X. Indeed, if it were closed, we would have Z = X and the above convergence theorem would imply that for, say p > 4

∞ 0 V (t)f -Pf p X dt < ∞ ∀f ∈ X.
According to Datko-Pazy's Theorem [START_REF] Engel | One parameters semigroups for linear evolution equations[END_REF]Theorem V.1.8], the semigroup (V (t)(I -P)) t 0 would then be asymptotically exponentially stable. In particular, the spectrum of its generator L(I -P) would be contained in {z ∈ C ; Rez -λ * } for some λ * > 0 and 0 would be an isolated eigenvalue of L. We know that this cannot be the case so Z = X.

Remark 4.7. The study of the decay rate of the Boltzmann linearized operator in the hilbert space

H = L 2 (R d , M -1 (v)dv)
has been performed by R. Caflisch in the seminal paper [START_REF] Caflisch | The Boltzmann equation with a soft potential, I. Linear, spatially homogeneous[END_REF] in the dimension d = 3 for γ ∈ (-1, 0). The same techniques can be performed for the linear operator L, and, adopting the notations of the previous result, [9, Theorem 3.1] would read as follows: for any f ∈ H such that

f -̺ f M ∞,a = sup v∈R d exp(a |v| 2 ) |f (v) -̺ f M(v)| < ∞ for some 0 < a < 1 4 (4.7)
there exists C > 0 and λ > 0 (explicit and depending on γ, f α ) such that

V (t)f -̺ f M H C f -̺ f M ∞,a exp -λt 2 2-γ ∀t 0.
Notice that, here again, H splits as H = Ker(L) ⊕ Im(L). One can get rid of the very restrictive pointwise decay (4.7) using an approach similar to the one we performed for the proof of Theorem 4.4. Indeed, recall that L is self-adjoint in the space H [START_REF] Caflisch | The Boltzmann equation with a soft potential, I. Linear, spatially homogeneous[END_REF]) so that, classical spectral analysis of self-adjoint operators [START_REF] Engel | One parameters semigroups for linear evolution equations[END_REF] ensures that

R(iα , L) L (H) 1 |α| ∀α ∈ R \ {0}.
and then, setting u = rn + z, z ⊥ n, r ∈ R, we get 2) dz .

K γ f (v) = 2 d-2 R×S d-1 f (v + rn)|r| d-2 drdn n ⊥ M(v + z) |z + rn| γ-(d-
Setting now w = v + rn, dw = 2|r| 1-d drdn, we get

K γ f (v) = 2 d-1 R d f (w) dw |v -w| (v-w) ⊥ M(v + z) |z + w -v| γ-(d-2) dz i.e. K γ f (v) = R d k γ (v, w) f (w)dw with k γ (v, w) = 2 d-1 |v -w| -1 (v-w) ⊥ M(v + z) |z + w -v| γ-(d-2) dz.
Let us compute the above kernel in a more precise way. Let v, w ∈ R d be given. We denote the center of mass by V = v + w 2 . Then, for any z ∈ (v -w) ⊥ , one has

|v + z| 2 = |V + z| 2 + 1 4 |v -w| 2 + 1 2 |v| 2 -|w| 2
and, setting V = V 0 + V ⊥ where V ⊥ ⊥ v -w and V 0 parallel to (v -w) we get Therefore

|V + z| 2 = |V 0 | 2 + |V ⊥ + z| 2 with |V 0 | 2 = |v| 2 -|w|
k γ (v, w) = 2 d-1 (2π) -d/2 |v -w| -1 exp - 1 8 |v -w| + |v| 2 -|w| 2 |v -w| 2 (v-w) ⊥ exp - |V ⊥ + z| 2 2 |z + w -v| γ-(d-2) dz.
We denote by I γ (v, w) this last integral:

I γ (v, w) = (v-w) ⊥ exp - |V ⊥ + z| 2 2 |z + w -v| γ-(d-2) dz = (v-w) ⊥ exp - |ξ| 2 2 |V ⊥ -ξ + v -w| γ-(d-2) dξ
where we recall that V ⊥ is orthogonal to v -w. One sees in particular that, if γ = d -2, the last integral is constant (independent of v, w):

I d-2 (v, w) = R d-1 exp - |x| 2 2 dx = (2π) d-1 2 , so that k d-2 (v, w) = 2 d-1 (2π) -1/2 |v -w| -1 exp - 1 8 |v -w| + |v| 2 -|w| 2 |v -w| 2 ∀v, w. (A.1)
For general γ d -2, one notices that, being both V ⊥ and ξ orthogonal to v -w:

I γ (v, w) = (v-w) ⊥ exp - |ξ| 2 2 |V ⊥ -ξ| 2 + |v -w| 2 γ-d+2 2 dξ so that I γ (v, w) |v -w| γ-(d-2) I d-2 (v, w) ∀v, w ∀γ d -2 (A.2) and k γ (v, w) |v -w| γ-(d-2) k d-2 (v, w) ∀γ d -2. (A.3)
Such a rough estimate is enough to prove a general property of k γ (already established in [START_REF] Bisi | Uniqueness in the weakly inelastic regime of the equilibrium state to the Boltzmann equation driven by a particle bath[END_REF]Proposition A.1] in the case of hard-spheres in dimension d = 3):

Proposition A.1. The following holds.

(1) Assume m(v) = exp(-a|v| s ), a > 0, s ∈ (0, 1] and set

H γ (w) = R d k γ (v, w)m -1 (v)dv, w ∈ R d , γ d -2.
Then, there exists a positive constant C 0 > 0 such that

H γ (w) C 0 (1 + |w| γ-s ) m -1 (w) ∀w ∈ R d .
(2) If m(v) = 1 + |v| β -1 for β > 0 and let H γ be defined as above. Then, there exists a positive constant C 1 > 0 such that

H γ (w) C 1 (1 + |w| γ-2 ) m -1 (w) ∀w ∈ R d .
Proof. In both cases, recall that, according to (A.3), one has

H γ (w) R d |v -w| γ-(d-2) k d-2 (v, w)m -1 (v)dv
where k d-2 (v, w) is given by (A.1). Taking into account that |v| 2 -|w| 2 |v-w| -|v -w| = 2 v-w |v-w| • w we may rewrite (A.1) as

k d-2 (v, w) = 2 d-1 (2π) -1/2 |v -w| -1 exp - 1 2 |v -w| + v -w |v -w| • w 2 . (A.4)
Assume now that m is an exponential weight, i.e. m(v) = exp(-a|v| s ) for a > 0 and s ∈ (0, 1].

Performing the change of variables u = v -w and using spherical coordinates (with ̺ = |u| and ̺|w| cos θ = u • w) one gets therefore

H γ (w) C d |S d-2 | ∞ 0 d̺ π 0 G(̺, θ)dθ
for some universal constant C d > 0 and where

G(̺, θ) = ̺ γ exp - 1 2 ̺ + |w| cos θ 2 + a ̺ 2 + |w| 2 + 2̺|w| cos θ s/2 (sin θ) d-2 . Setting now A = [0, ∞) × [0, π],
we split A into the two regions of integration:

A 1 = {(̺, θ) ∈ A ; 3|w| cos θ -2̺} and A 2 = A \ A 1 .
Notice first that, since cos θ 1 and s ∈ (0, 1] exp a(̺ 2 + |w| 2 + 2̺|w| cos θ) s/2 exp (a(̺ + |w|) s ) exp(a̺ s ) exp(a|w| s ) ∀(̺, y) ∈ A.

Moreover, since ̺ + |w| cos θ ̺/3 for any (̺, θ) ∈ A 1 we have

A 1 G(̺, θ)d̺dθ exp(a|w| s ) ∞ 0 ̺ γ d̺ π 0 exp - 1 18 ̺ 2 + a̺ s (sin θ) d-2 dθ C 1 exp(a|w| s ) = C 1 m -1 (w) (A.5)
since the integral is convergent. Let us estimate now the integral over A 2 . For any (̺, θ) ∈ A 2 , one notices first that

̺ 2 + |w| 2 + 2̺|w| cos θ < |w| 2 -̺ 2 /3 and ̺ (3/2)|w|, so that A 2 G(̺, θ)d̺dθ (3/2)|w| 0 ̺ γ exp a |w| 2 -̺ 2 /3 s/2 d̺ π 0 exp - 1 2 (̺ + |w| cos θ) 2 (sin θ) d-2 dθ (A.6)
To carry out the θ-integral, perform the change of variables z = ̺ + |w| cos θ to get

π 0 exp - 1 2 (̺ + |w| cos θ) 2 (sin θ) d-2 dθ 1 |w| ∞ -∞ exp -z 2 /2 dz = C 2 |w|
for some explicit C 2 > 0 1 . Plugging this in (A.6) we obtain

A 2 G(̺, θ)d̺dθ C 2 |w| (3/2)|w| 0 ̺ γ exp a |w| 2 -̺ 2 /3 s/2 d̺.
1 notice that this proof needs d 3

Arguing now as above, one gets that

A 2 G(̺, θ)d̺dθ 3 2 |w| 2 0 ̺ γ   1 + |w| 2 - ̺ 2 3 β 2   d̺ π 0 exp - 1 2 (̺ + |w| cos θ) 2 (sin θ) d-2 dθ C |w| 3 2 |w| 0 ̺ γ   1 + |w| 2 - ̺ 2 3 β 2   d̺
for some positive constant C > 0. One computes the above integral as in the previous part and obtain that there exists a positive constant C 2 > 0 such that

3 2 |w| 0 ̺ γ   1 + |w| 2 - ̺ 2 3 β 2   d̺ C 2 |w| γ-1 |w| 2 |w| 2 4 1 + x β/2 dx C 2 |w| γ-1 m -1 (w).
Gathering all together, we get that there exists

C 3 > 0 such that A 2 G(̺, θ)d̺dθ C 3 |w| γ-2 m -1 (w). (A.10)
Combining (A.9) and (A.10) we get the desired conclusion.

Remark A.2. Notice that the above computations are valid for any γ ∈ (-d, ∞) (i.e. it is valid for both hard and soft potentials). However, it turns out to be particularly relevant for hard potentials γ ∈ [0, d -2]. For soft potentials, it will become inoperative as explained by the following reasoning.

Recalling that X = L 1 (R d , dν(v)) with dν(v) = m -1 (v)dv, one can write K γ as an integral operator for the measure dν

K γ f (v) = R d k γ (v, w)f (w)dν(w) by simply setting k γ (v, w) = k γ (v, w)m(w), ∀v, w ∈ R d .
In such a case, the above Proposition asserts that

R d k γ (v, w)dν(v) = m(w)H γ (w) C 0 1 + |w| γ-α where α = s if m(v) = exp(-a|v| s ), s ∈ (0, 1] and α = 2 if m(v) = (1 + |v| β ) -1 (β > 0). Since, for γ 0 the mapping w → 1 + |w| γ-α (1 + |w|) γ is bounded we see that, for hard potentials there exists ϑ ∈ L ∞ (R d ) such that R d k γ (v, w)dν(v) ϑ(w) Σ(w) for a. e. w ∈ R d
i.e. [START_REF] Mokhtar-Kharroubi | New generation theorems in transport theory[END_REF]Inequality (5)] is satisfied. Notice however that nothing guarantees that the subcriticality condition [START_REF] Mokhtar-Kharroubi | New generation theorems in transport theory[END_REF]Inequality (6)] is met in X. It it not clear if the above is still true for soft potential, i.e. whenever γ < 0.

We go on with properties of the k γ (v, w) for positive γ. First of all, the detailed balance law holds true:

k γ (v, w)M(w) = k γ (w, v)M(v) ∀v, w ∈ R d × R d
This leads us to define

p γ (v, w) = M -1/2 (v)k γ (v, w)M 1/2 (w), v, w ∈ R d × R d , so that p γ (v, w) = p γ (w, v). Notice that, p γ (v, w) |v -w| γ-d+2 M -1/2 k d-2 (v, w)M 1/2 (w)
from which we easily see that

p γ (v, w) C |v -w| γ-d+1 exp - 1 8 |v -w| 2 + |v| 2 -|w| 2 2 |v -w| 2 ∀v, w ∈ R 2d .
This yields to the following whose prove is a direct adaptation of that given in [2, Lemma 3.2].

Lemma A.3. For any γ > d-2

2 , there exists C > 0 such that

R d |p γ (v, w)| 2 dw C (1 + |v|) , ∀v ∈ R d .
Proof. We use the above bound to get that there exists C > 0 such that

R d p γ (v, w) 2 dw C R d |v -w| 2γ+2-2d exp - 1 4 |v -w| 2 + |v| 2 -|w| 2 2 |v -w| 2 dw C R d |u| 2γ+2-2d exp - 1 4 |u| 2 + |v| 2 -|u -v| 2 2 |u| 2 du
where we set u = v -w. Using spherical coordinates u = ̺ dn and denoting by θ the angle between u and v, one readily gets

R d p γ (v, w) 2 dw C ∞ 0 ̺ 2γ+1-d d̺ S d-1 exp - 1 4 ̺ 2 + (2|v| cos θ -̺) 2 dn = C ∞ 0 ̺ 2γ+1-d d̺ π 0 sin d-2 (θ) exp - 1 4 ̺ 2 + (2|v| cos θ -̺) 2 dθ.
Notice that the assumption 2γ + 1 -d > 0 implies sup

v∈R d R d p γ (v, w) 2 dw Cπ ∞ 0 ̺ 2γ+1-d exp - ̺ 2 4 d̺ < ∞.
Set for simplicity

I(v) = ∞ 0 ̺ 2γ+1-d d̺ π 0 sin d-2 (θ) exp - 1 4 ̺ 2 + (2|v| cos θ -̺) 2 dθ, v ∈ R d .
As in [2, Lemma 3.1], perform the change of variables x = ̺/|v| -2 cos θ, y = ̺/|v|, one gets

I(v) = |v| 2γ+2-d 2 Ω y 2γ+1-d 1 - (x -y) 2 4 d-3 2 exp - |v| 2 4 x 2 + y 2 dxdy |v| 2γ+2-d 2 Ω y 2γ+1-d exp - |v| 2 4 x 2 + y 2 dxdy
where Ω = {(x, y) ∈ R 2 ; y 0 ; |x -y| 2}. Changing the variables again as t = |v|x/2 and s = |v|y/2 one finally gets

I(v) 2 2γ+2-d |v| R dt ∞ 0 exp -t 2 + s 2 s 2γ+1-d ds
and using again that 2γ + 1 -d one sees that this last integral is convergent which gives the conclusion.

APPENDIX B. QUANTITATIVE ESTIMATES OF THE SPECTRAL GAP FOR HARD-POTENTIALS

We present here a quantitative version of Theorem 1.1 which relies on recent advances on the so-called factorisation method and enlargement of the functional space, initiated in [START_REF] Mouhot | Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials[END_REF] for the nonlinear Boltzmann equation and extended to an abstract framework in [START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-theorem[END_REF]. The method consists in using the fact that, in the hilbert space

H = L 2 (R d , M -1 dv)
the spectral gap of the linear Boltzmann operator L is explicit (at least for hard-spheres interactions [START_REF] Lods | Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models[END_REF] but the approach given then can be extended easily to the general hard potential dealt with in this paper). In particular, since L is self-adjoint in this space, the decay of the associated semigroup is directly deduced from the spectral structure of the generator.

To avoid confusion, in this section, we shall denote L 2 the linear Boltzmann in the Hilbert space H while L 1 will stand for the linear Boltzmann operator in X where X has been defined in the Introduction:

X = L 1 (R d , m -1 (v)dv)
where the weight m is given by (1.10) or (1.11). We also denote by (V 2 (t)) t 0 and (V 1 (t)) t 0 the C 0 -semigroups generated respectively by L 2 and L 1 .

Notice that we restrict ourselves, for simplicity, to the case in which b ∈ L ∞ (S d-1 ), which, as seen in Section 3 is no restriction. Then, again, there is no loss in generality in assuming that b is constant, b ≡ 1. In such a case, the linear operator K corresponds simply to K γ .

We begin by recalling the following result (see [START_REF] Lods | Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models[END_REF][START_REF] Levermore | Compactness of the gain parts of the linearized Boltzmann operator with weakly cutoff kernels[END_REF]) Moreover, Ker(L 2 ) = span(M) and there exists some explicit µ 2 > 0 such that -

R d f L 2 f M -1 dv µ 2 f H ∀f ∈ D(L 2 ) with R d f (v)dv = 0.
As a consequence, In order to extend the above result to the weighted L 1 -space X, we shall resort to the following that we state here for the above operators L The above Theorem asserts that, provided L 1 admits the above splitting (i)-(ii) in the weighted space L 1 , the second eigenvalue of L 1 in X is λ 2 = -µ 2 . This, combines with the results of Section 3 would implies that the constant λ * appearing in Theorem 1.1 can be chosen as µ 2 + δ for any δ > 0. Since µ 2 is explicit, this provides a quantitative estimate of the speed of convergence in Theorem 1.1.

V 2 (t)f -̺ f M H exp(-µ 2 t)
Remark B.3. Notice that it would be possible to invoke also results from [START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-theorem[END_REF], especially, [START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-theorem[END_REF]Theorem 2.13], to deduce decay properties for the semigroup (V 1 (t)) t 0 from that of (V 2 (t)) t 0 under the assumptions of Theorem B.2. We refer also to [START_REF] Mischler | Spectral analysis of semigroups and growth-fragmentation equations[END_REF] for more results in this direction and to [START_REF] Cañizó | Exponential convergence to equilibrium for subcritical solutions of the Becker-Döring equations[END_REF]Theorem 3.1] for a much less general but simpler version of this kind of results.

It remains to prove that L 1 can be split as

L 1 = A 1 + B 1
where both the operators A 1 and B 1 satisfy the assumptions of Theorem B.2. To do so, for any R > 0, let

A 1 f = K(χ B R f ), B 1 = L 1 -A 1
where B R is the open ball in R d with radius R > 0 and center 0 and χ B R denotes the indicator function of B R .

Our proof is very similar to the one of [START_REF] Bisi | Uniqueness in the weakly inelastic regime of the equilibrium state to the Boltzmann equation driven by a particle bath[END_REF] and is based upon the results of Appendix A. One has therefore, with the notations of Section 3

A 1 f (v) = B R k(v, w)f (w)dw, B 1 f (v) = -Σ(v)f (v) +
R d k 2 γ (v, w)M -1 (v)dv CM -1 (w)(1 + |w|) -1 CM -1 (w) ∀w ∈ R 3 .
Thus,

A 1 f H √ C |w| R
|f (w)|M -1/2 (w)dw and, since the domain of integration is bounded, there is some positive constant c R > 0 such that A 1 f H c R f X which proves point.

Theorem 1 . 1 .

 11 If the weight function is given by (1.10) or (1.11), then, for any γ ∈ [0, d -2] and any b ∈ L 1 (S d-1 ), the linear Boltzmann operator (L, D(L)) with

Proposition 3 . 3 .

 33 If the weight function m are given by (1.10) or (1.11) then, for any γ ∈ [0, d-2], K γ : Y → X is a positive weakly-compact operator.

  A∩Bw |v -w| p(γ+1-d) m -1 (v)dv 1/p

Corollary 3 . 6 .

 36 For any γ ∈ (0, d -2] and any b(•) ∈ L 1 (S d-1 ), let K denote the collision operator associated toB(v -v * , σ) = |v -v * | γ b(cos θ) defined by (1.7). Then K R(λ, T ) : X → Xis positive and weakly compact for any λ > 0.

Theorem 3 . 7 .Remark 3 . 8 .

 3738 If the weight function is given by (1.10) or(1.11), then, for any γ ∈ [0, d -2], the linear Boltzmann operator (L, D(L)) with D(L) = Y is the generator of a positive C 0 -semigroup (V (t)) t 0 on X with moreover S ess (V (t)) = S ess (U (t)) ∀t 0. Notice that the spectrum S(T ) of T in X is given by

Theorem B. 1 .

 1 For γ ∈ [0, d -2], the linear operator (L 2 , D(L 2 ) is negative self-adjoint in H where D(L 2 ) = {f ∈ H ; Σf ∈ H} = L 2 (R d , (1 + |v| γ ) M -1 (v)dv).

  f H for all f ∈ H, ∀t 0 where ̺ f = R d f (v)dv for any f ∈ H.Notice that, the spectrum of L 2 in H can be expressed as followsS(L 2 ) = (-∞, -η] ∪ {λ n } n where λ 1 = 0 > λ 2 > λ n > λ n+1 (n ∈ R) are nonpositive eigenvalues of L 2 with finite algebraic multiplicities and lim n→∞ λ n = -η = -inf v∈R d Σ(v) > 0.For any n ∈ N, we denote by Π L 2 ,n the spectral projection associated to λ n . Notice that µ 2 = -λ 2 .

2 and L 1 :

 1 Theorem B.2.[START_REF] Gualdani | Factorization for non-symmetric operators and exponential H-theorem[END_REF] Theorem 2.1] Assume that the linear operator (L 1 , D(L 1 )) in X admits the following splitting:L 1 = A 1 + B 1 where (i) A 1 : X → H is bounded; (ii) the operator B 1 : D(B 1 ) → X (with D(B 1 ) = D(L 1 ) = Y) is β-dissipative for some positive β > 0, i.e. R d signf (v)B 1 f (v)m -1 (v)dv -β f Y ∀f ∈ Y. (B.1)Then, for any ε > 0,S(L 1 ) ∩ {z ∈ C Rez -η + ε} = S(L 2 ) ∩ {z ∈ C ; Rez -η + ε} = {λ 1 , . . . , λ Nε }for some finite N ε ∈ N. Moreover, for any k ∈ {1, . . . , N ε }, λ k is an eigenvalue of L 1 with algebraic multiplicity which is finite and equal to the one of λ k as an eigenvalue of L 2 and the associated spectral projection Π L 1 ,k is given by Π L 1 ,k | H = Π L 2 ,k .

Proposition B. 4 .A 1 :A 1

 411 {|w|>R} k(v, w)f (w)dw With the above notations, for any R > 0, X → H is bounded.Proof. Let f ∈ X be given and let k(v, w) be the kernel of K. Recall that K = K γ with kernel k γ (v, w). Using Minkowski's integral inequality (with measures M -1 (v)dv and |f (w)|dw) one gets easily the notations of Appendix A, one hasR d k 2 γ (v, w)M -1 (v)dv = M -1 (w) R d p 2 γ (v, w)dvand, using Lemma A.3, there is some positive constant C > 0 such that

  Proposition 2.10. [1, C.III. Proposition 3.5, p. 310] Assume X = L 1 (Ω, µ) where (Ω, F) is a measure space and µ a given measure over (Ω, F). Let (V (t)) t 0 be a irreducible C 0 -semigroup with generator A in X. Then the following holds[START_REF] Arendt | One-parameter semigroups of positive operators[END_REF] Every positive eigenvector f of A is a quasi-interior point, i.e. f > 0 µ-a. e.(2) Every positive eigenvector of the adjoint A ⋆ is strictly positive.

(3) If Ker(s(A) -A ⋆ ) contains a positive eigenelement, then dim(Ker(s(A) -A)) 1. (4) If s(A) is a pole of the resolvent of A, then it has algebraic multiplicity one. The corresponding spectral projection is then of the form

References 1

Then, according to [START_REF] Batty | Fine scales of decay of operator semigroups[END_REF]Theorem 1.5], such an estimate is equivalent to

Arguing then as in the proof of Theorem 4.4, one conclude that, if f ∈ Ker(L) ⊕ Im(L), then there exists C > 0 such that

Again, as in Remark 4.6, Im(L) is not closed.

APPENDIX A. COMPUTATION OF THE KERNEL k(v, w)

This section is devoted to the computation of the gain part operator K γ for the special case in which the collision kernel is given by

and, in particular, does not depend on the angular cross-section. In such a case,

We aim to prove that K γ admits the integral representation

and shall resort to the so-called Carleman representation to compute the kernel k γ (v, w). Namely, one considers an alternative parametrization of post-collisional velocities as

for which

where

We refer to [START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF]Chapter 2A. 4] for details. Since 2) and

2 exp(ax s/2 )dx.

for some explicit C 3 > 0 depending only on γ. We observe now that, for any r > 0,

Using this in (A.7) for r = |w| 2 we get

Putting together (A.5) and (A.8) we finally obtain the result. Assume now that m -1 (v) = 1 + |v| β for β > 0. One argues in a similar way. Namely, as above,

for some universal constant C d > 0 and where now

As above, one splits the region A = [0, ∞) × [0, π] into the two above regions A 1 and A 2 . One clearly has

from which, as above,

Now, we prove that R > 0 can be chosen in such a way that

satisfies the above point (ii). For any f ∈ Y, set

dv. Now, one has the following Proposition B.5. Assume that m is given by (1.10) with 1 -γ < s < 1 There exists R > 0 large enough so that

Proof. Let f ∈ Y be fixed.

where we used the fact that Σ(v) σ 0 (1 + |v|) γ for some positive constant σ 0 > 0 and set, as in Appendix A,

Let us assume now that m is an exponential weight given by (1.10).Then, using Proposition A.1, there is some positive constant C > 0 such that

In other words,

We choose now R > 0 such that C(1 + |v| γ-s ) -σ 0 (1 + |v|) γ -σ 0 for all |v| > R (which can be done since s > 0), so that

i.e. B 1 satisfies (B.1) with β = σ 0 .

Assume now that m is an algebraic weight given by (1.11). Then, as above, using Proposition A.1, there is some positive constant C > 0 such that

and, repeating the above argument, one can choose R > 0 large enough so that C(1 + |v| γ-s )σ 0 (1 + |v|) γ -σ 0 for all |v| > R to see that B 1 satisfies (B.1) with β = σ 0 .


Humberto Yáñez-Godoy

Sidi Mohammed Elachachi

Effects of Spatial Soil Heterogeneities on Structural Behavior of a Steel Sheet Pile

This article aims to introduce spatial variability effect on the behavior of extended structures as the sheet piles. The importance of these aspects can provide elements of response to experts in charge of the standards of design for these structures. Two main problems are identified, the first one is about the most appropriate way to model the variability of soil and its interaction with the sheet pile. The second one is to take into account the uncertainties and their propagation through a model. A 2-D mechanical model of the sheet pile has served to propagate the soil variability and the system-parameter uncertainties and allowed to study the response of the different structural elements of the structure. The spatial soil heterogeneities show clearly its effect on the behavior of the structure. This analysis has opened up new possibilities to advance in researches on longitudinal aspects for these structures.

INTRODUCTION

A large proportion of structural damages to extended structures, as sheet piles, can result from three factors:

• the partial and or the lack of knowledge of geotechnical characteristics (behavior of soilstructure interaction), • the variability of soil (properties and mechanical characteristics), • the non-inclusion of longitudinal dimension in standard codes and design guidelines.

In this way, for the design of sheet piles, NF EN 1997-1, (2014) considers only the crosssection for the ultimate and serviceability limit states and neglects then the effect of the redistribution of the internal strains of the structure due to longitudinal differential displacements. On the other hand, the approach considered by standards is quasi-deterministic and nonetheless incomplete. Some studies can cover these kinds of limitation and play a practical complementary role in design situations not covered by standards (see for example [START_REF] Low | Reliabilitybased design and its complementary role to Eurocode 7 design approach[END_REF]). Furthermore, a probabilistic approach seems to be more suited to integrate both soil uncertainties and spatial variability [START_REF] Ahmed | Probabilistic analysis of strip footings resting on a spatially random soil using subset simulation approach[END_REF]).

With regard to sheet piles, although several studies (e.g. [START_REF] Wang | MCS-based probabilistic design of embedded sheet pile walls[END_REF], [START_REF] Low | Reliabilitybased design and its complementary role to Eurocode 7 design approach[END_REF]) have dealt with both soil uncertainties and spatial variability by considering a combination of probabilistic approach and limit equilibrium method (method often used for a preliminary design), we note very few studies in which numerical models as finite-element methods or subgrade reaction methods take into account spatial soil variability.

Furthermore, some studies are divided on the question of taking into account the interaction soil-structure. For example, [START_REF] Schweiger | Reliability Analysis in Geotechnics with Deterministic Finite Elements -Theoretical Concepts and Practical Application[END_REF]) shows that geotechnical uncertainties not taken into account can result in undersized structures. In contrast, [START_REF] Schweckendiek | Reliability of sheet pile walls and the influence of corrosion: structural reliability analysis with finite elements[END_REF]), notes that the effect of soil variability is negligible but it would require being associated to other factors such as corrosion. Both of these studies don't consider spatial soil variability.

In this article a first exploratory 2-D approach to introduce spatial soil variability for the sheet piles is performed. The physical model of the interaction soil-structure together with a stochastic model characterizing the variability of the soil is then developed. A 2-D mechanical finite-element model of the sheet pile serves to propagate the soil variability and the systemparameter uncertainties and allows to study the response of the different structural elements of the structure.

A variability effect analysis is then done using the direct Monte Carlo numerical simulation method in order to evaluate both a confidence interval of the bending moment along the sheet pile and a probability of exceeding a reference value. A discussion is then given on the comparison of the results obtained by considering two different scenarios: homogeneous soil (uncertain model parameters as input data are considered, this is unstructured variability) and heterogeneous soil (random fields to represent spatial variability are considered, this is structured variability).

STRUCTURAL COMPUTATIONAL MECHANICS OF A STEEL SHEET PILE SEAWALL

Mechanical modeling description

The example adapted from the one presented in [START_REF] Boéro | Time-function reliability of harbour infrastructures from stochastic modelling of corrosion[END_REF] is representative of quay walls and considers a structure with U shaped piles, anchored through one level of passive tie rods. A recapitulation of the principal geometrical and nominal mechanical characteristics of the quay is shown in Figure 1. We assume here that the soil correlation along x is negligible after 20 meters and that the sheet pile can be modeled with a 2-D model. We therefore consider the stability of a sheet pile 2 meter wide, the distance between tie-rods. A load q of 50kPa is applied to the quay. The water level of sea is equals to groundwater table. The construction phases of the quay are not taken into account. The steel (present in the main wall, the pile anchorage and the tie-rods) is assumed to behave in an isotropic and linearly elastic manner. We consider in this paper to use a mechanical finite-element model that let to take into account the interaction soil-structure by a Mohr-Coulomb model. The model (see Figure 2) has been developed within the CAST3M finite element computer code (http://www.cast3m.cea.fr) developed by the Atomic Energy Commission (CEA).

Soil-structure model

The soil (non cohesive soil type) is modeled as a homogeneous powder material in one case (see 2.3 and 4.1) and as a heterogeneous powder material by taking into account spatial variability in a second case (see 4.2) and behaves in a perfectly elasto-plastic manner (Mohr-Coulomb criterion). The friction angle ϕ' is considered as a random variable for the first case and as a random field for the second case. The elements of interface, laid out around the main wall, are described by the Coulomb friction criterion. The angle of the interface is approximately equal to 2/3 ϕ' (this angle varies then in function of ϕ' value), and the contact cohesion is negligible. The resistance of the tension elements in the interface soil-sheet pile is zero, which allows for relative movement between the nodes of the various interfaces.

Deterministic computations

The deterministic computations are carried out for 3 different values of ϕ': 24°, 35° and 40°. In fact, these values correspond respectively to minimum, most likely and maximum values for a sand soil [START_REF] Rackwitz | JCCS Probabilistic Model Code, Section 3.7: Soil properties[END_REF]).

First at all, in order to obtain reference values to compare to the probabilistic analysis presented in section 3, the influence of deterministic values of ϕ' on the wall bending moment is analyzed. Figure 3 shows the results. The lowest value of ϕ' (24°) makes the wall more flexible. Indeed, the maximum moment at over -10 m (this is the permanent immersion zone) is 2 times higher than bending moment for ϕ' = 35°. The bending moment at the level of the passive tie-rods is also increased by a factor of 2.5 times in rapport to the other values.

These different situations corresponding to three nominal values are not certainly a confidence interval of the possible mechanical stress on a structure for a given problem. In fact, by knowing a prior the most likely value of the nominal value of ϕ' (that supposes a good knowledge of intrinsic properties of the embankment soil) we can then introduce the random nature of soil (random variables, random fields) in the mechanical model by taking into account the available information.

MODELLING OF UNCERTAIN PARAMETERS AND SPATIAL VARIABILITY IN SHEET PILES

Uncertain model parameters

The only variable geotechnical parameter considered in the probabilistic analysis presented in this study is the friction angle ϕ'. In fact, this parameter is a dominant random variable as shown in [START_REF] Boéro | Time-function reliability of harbour infrastructures from stochastic modelling of corrosion[END_REF] or [START_REF] Cherubini | Probabilistic approach to the design of anchored sheet pile walls[END_REF].

The values considered for the mean µ and the coefficient of variation COV are respectively: µμ !! = 35° and COV !! = 10% (indicative minimum value for ϕ', drained soil, according to [START_REF] Rackwitz | JCCS Probabilistic Model Code, Section 3.7: Soil properties[END_REF]). The scenario proposed for modeling ϕ' is a beta probability distribution constructed under the constraints defined by the available (published) information (MaxEnt principle): min !! = 24°, µμ !! = 35° and max !! = 40°. The beta probability distribution is written as (see [START_REF] Desceliers | Robustness analysis of an uncertain computational model to predict well integrity for geologic CO 2 sequestration[END_REF]):

B !! = max ! ! -min ! ! 𝑍 + min ! ! (1)
where Z is a beta random variable with values in [0, 1] and with parameters a and b defined by:

𝑎 = ! ! ! ! ! ! 1 -𝑚 ! -𝑚 ! (2) 𝑏 = 1 -𝑚 ! ! ! (!!! ! ) ! ! ! -1 (3)
in which the mean value 𝑚 ! and the variance 𝜎 ! ! of random variable Z are written as

𝑚 ! = (µμ !! - min ! ! )/(max ! ! -min ! ! ) and 𝜎 ! = σ !! / (max ! ! -min ! ! ).
A random sample of the beta probability distribution of ϕ' is shown in Figure 4.

Variability of soil

order to represent the variability of a homogeneous soil the friction angle ϕ' is modeled as a 1-D normal stationary random field (through the soil depth) by using the spectral approach that takes advantage of the Fast Fourier Transform technique [START_REF]Simulation of random envelope processes[END_REF]). This random field is characterized by its mean µμ !! variance σ !! ! and correlation function ρ τ , where τ is the lag. A single exponential correlation function is used:

ρ τ = 𝑒𝑥𝑝 - !! ! (4)
where 𝛿 is the correlation length (CL) of ϕ'.

Concerning the fluctuation values of CL, they depend on the soil type, the geotechnical parameter and the direction of variability (vertical or horizontal) [START_REF] Phoon | Characterization of geotechnical variability[END_REF]. Concerning vertical fluctuation, it appears generally that vertical CL is around a few meters. We will also focus on higher values to study their influence on the behavior of the structure.

In order to introduce the continuous random field profile on the nodes of the finite element mesh (see Figure 2), we discretized it into N vertical elements. This approach let us to maintain continuity in the random field over the total depth z. Indeed, for M realizations of random field, the 𝜑′ !!! ! values for each vertical element is obtained by a polynomial of order l:

𝜑' 𝑧 !!! ! !!! ! = 𝑎 !! + 𝑎 ! ! 𝑧 ! ! ! !!! (5)
An example for one realization of a continuous random field and its discretized profile by polynomial interpolation is shown in Figure 5.

Propagation of uncertainty

The propagation of uncertainty via the finite element model developed within the CAST3M finite element computer code (see section 2.1) is performed by direct Monte-Carlo numerical simulation method using the MATLAB Statistics Toolbox. Figure 6 shows the flowchart of propagation of uncertainty using the coupling system MATLAB-CAST3M. For a homogeneous soil a random sample is generated from beta probability distribution of friction angle using MATLAB, and is then introduced into the mechanical model.

For a heterogeneous soil with spatial variability several realizations of a Gaussian random field of friction angle are generated using the spectral approach (implemented in MATLAB) discussed in section 3.2 and are then introduced into the finite element model.

PROBABILISTIC ANALYSIS OF THE STRUCTURAL BEHAVIOUR

Case 1: Homogeneous soil

The effect of input variable in section 3.1 on the mechanical model is now studied by analyzing the output distributions for a homogeneous soil. Direct simulations are made with the finite element numerical model and with random variable ϕ'. 1000 simulations are carried out and for each one a vector of the bending moment in the steel sheet pile is computed.

Results, sorted in quantiles (2.5%, 50%, 97.5%), are shown in Figure 7, where a beta probability distribution of ϕ' is considered. We have retained for this scenario a COV = 10%. Confidence interval at the permanent immersion zone, notably at -9 m, is proportionally larger than the one at the level of the passive tie-rods, at -3.5 m. Indeed, maximum bending moment for quantile 2.5% at -9 m is about 100% more important than median value. On the other hand, we can see this same rate for maximum bending moment for quantile 97.5% at -3.5 m with respect to median value. We also note that the asymmetry in these results, quantiles 2.5% and 97.5%, against quantile 50% can be explained by the nature of beta distribution.

Case 2: Heterogeneous soil with spatial variability

In order to appreciate the influence of spatial variability, we limit the study to the normal random field. Direct simulations are made with the finite element numerical model and with the random field of ϕ'. 1000 simulations are carried out and for each one a vector of the bending moment in the steel sheet pile is computed.

Results, sorted in quantiles (2.5%, 50%, 97.5%), are shown in Figure 8, where four different correlations lengths, CL, of ϕ' are considered: 1 m (Figure 8 The same observations, with regard to the evolution of the confidence interval at the level of the passive tie-rods for the four values of CL, can be drawn from these made for the maximum bending moment.

The influence of these soil heterogeneities on the structural behavior are presented in section 5.

VARIABILITY EFFECT ASSESSMENT

Exceedance probability function

We are interested in compare the probability of exceeding a reference value along the sheet pile.

More precisely, we consider this reference value as the maximum bending moment, 𝑀 ! = 199 kN.m/ml, corresponding to the nominal value ϕ' = 35° (see Figure 3) at the permanent immersion zone. The probability 𝑃 ! is calculated as:

𝑃 ! = 𝑃(𝑀 !"# (𝑨) > 𝑀 ! ) (6)
where 𝑀 !"# (𝑨) is the computed maximum bending moment according to the random variable A = {ϕ'} for a homogeneous soil and according to the random field A = µμ !! + 𝜑′(𝑧) for a heterogeneous soil with spatial variability. The purpose to compute these probabilities is to know the order of magnitude with respect to a deterministic computation when one considers the random character and spatial variability of sensitivity parameters as the angle of friction.

Influence of spatial soil variability

Results of computed probabilities with Eq. ( 6) are shown in Table 1. Spatial variability when CL = 1 m, shows to have a greater influence on the bending moment than the other cases. We observe a smoothing effect for CL = 10, 16 and 100, compared to CL = 1 m, indeed, these values remain very close to the one of beta distribution.

CONCLUSIONS

We have presented in this paper the effects of soil variability on sheet piles in order to study its global behavior. This structural behavior is random in nature due to the interaction of the structural system and its environment (soilstructure interaction) and implies a significant number of uncertainties. We have limited our study to only one geotechnical parameter. The angle of friction is a parameter very sensitive and a first deterministic parametric computation presented in this paper has confirmed this assumption.

We have then introduced the soil variability by considering two approaches: uncertain model parameters (unstructured variability) and random fields (structured spatial variability). The two cases of application let us to estimate a confidence interval, which allowed to appreciate the sensitivity of the maximum moment, notably at the permanent immersion zone and at the level of the passive tie-rods. The vertical correlation length for the values between 1 m and 10 m seemed to be more sensitive on the effect of the maximum moment than the values beyond 10 m. This last observation is verified in the computation of the exceedance probability function, being the spatial variability influence for the vertical correlation length equals to 1 m more important than the others. We have then showed that spatial variability associated to sheet piles can be not negligible for specific parameters as the friction angle ϕ'.

The effect of spatial soil heterogeneities open up new possibilities to advance in our researches on longitudinal aspects such as, for example, the longitudinal differential displacements which could impact sensitive structural elements as the tie rods. These future researches can provide elements of response to experts in charge of the standards of design for these structures.

Figure 1 :

 1 Figure 1: Geometrical and mechanical characteristics of the steel sheet pile seawall.

Figure 2 :

 2 Figure 2: Finite element mesh used.

Figure 3 :

 3 Figure 3: Bending moment in the steel sheet pile seawall.

Figure 4 :

 4 Figure 4: Probability distribution function of friction angle.

Figure 5 :

 5 Figure 5: One continuous simulated realization and one profile discretization of the normal random field of ϕ'over the total depth of the soil: 𝜇 !! = 35°, COV = 10%, correlation length CL = 10 m, single exponential correlation function.

Figure 6 :

 6 Figure 6: Coupling system MATLAB-CAST3M.

Figure 7 :

 7 Figure 7: Bending moment in the steel sheet pile seawall for a homogeneous soil: beta probability distribution of friction angle, N = 1000 simulations, COV = 10%.

  (a)), 10 m (Figure 8 (b)), 16 m (Figure 8(c)) and 100 m (Figure 8(d)). We have retained for these four scenarios a COV = 10%. A high sensitivity is observed for the maximum bending moment M max at -8.5 m in relation to deterministic computation presented in Figure 3 (M max = -199 kN.m/ml for ϕ' = 35°). Indeed, for interval LC = [1 m; 10 m] (Figure 8(a) and 8(b)), quantile 2.5% indicates a significant increase of the moment from 12% to 30%. Nonetheless, for higher values than LC = 10 m (Figure 8(c) and 8(d)), we observe that differences in relation to CL = 10 m are smaller. Concerning quantile 97.5%, for interval LC = [1 m; 10 m] (Figure 8(a) and 8(b)) we note a decreased value of the maximum bending moment, in relation to deterministic value, from 8% to 25%. We also notice smaller differences for higher values than LC = 10 m in relation to CL = 10 m (Figure 8(c) and 8(d)).

Figure 8 :

 8 Figure 8: Bending moment in the steel sheet pile seawall for a heterogeneous soil: random field of

Table 1 :

 1 Probabilities computed for the different cases, COV = 10%.

	Homogeneous soil		Heterogeneous soil
	A = {ϕ'}		A = 𝝁 𝝋! + 𝝋′(𝒛) CL
	Beta distribution	1	10	16	100

th International Conference on Applications ofStatistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015

ACKNOWLEDGMENTS

The first author would like to thank the AUGC (Association Universitaire de Génie Civil -France) for its funding contribution.