JOS Marzia Bisi
email: marzia.bisi@unipr.it

Jose Carrillo
email: carrillo@mat.uab.es

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

É A Carrillo

D I Dipartimento

Matematica

ICREA (INSTITUCI Ó CATALANA DE RECERCA I ESTUDIS Antonio Carrillo

Avanc ¸ats

EQUILIBRIUM SOLUTION TO THE INELASTIC BOLTZMANN EQUATION DRIVEN BY A PARTICLES THERMAL BATH

whether they are published or not. The documents may come

INTRODUCTION

The dynamics of rapid granular flows is commonly modelled by a suitable modification of the Boltzmann equation for inelastic hard-spheres interacting through binary collisions [START_REF] Brilliantov | Kinetic theory of granular gases[END_REF][START_REF] Villani | Mathematics of granular materials[END_REF]. As well-known, in absence of energy supply, inelastic hard spheres are cooling down and the energy continuously decreases in time. In particular, the Boltzmann collision operator for inelastic hard spheres does not exhibit any non trivial steady state. This is no more the case if the spheres are forced to interact with an external agency (thermostat) and, in such a case, the energy supply may lead to a non-equilibrium steady state. For such driven system (in a space homogeneous setting), the time-evolution of the one-particle distribution function f (v, t), v ∈ R 3 , t > 0 satisfies the following

∂ t f = τ Q(f, f ) + G(f ), (1.1)
where τ 0 is a given constant, Q(f, f ) is the inelastic Boltzmann collision operator, expressing the effect of binary collisions of particles, while G(f ) models the forcing term.

There exist in the literature several physical possible choices for the forcing term G in order to avoid the cooling of the granular gas: stochastic heating, particles heating or scaled variables to study the cooling of granular systems and even a nonlinear forcing term given by the quadratic elastic Boltzmann operator has been taken into account [START_REF] Fournier | A spatially homogeneous Boltzmann equation for elastic, inelastic, and coalescing collisions[END_REF]. These options have been studied first in the case of inelastic Maxwell models [START_REF] Bobylev | On some properties of kinetic and hydrodynamic equations for inelastic interactions[END_REF][START_REF] Carrillo | Steady states of a Boltzmann equation for driven granular media[END_REF][START_REF] Bobylev | Moment equations for a granular material in a thermal bath[END_REF][START_REF] Bobylev | Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions[END_REF][START_REF] Spiga | The dissipative linear Boltzmann equation[END_REF][START_REF] Bisi | Contractive Metrics for a Boltzmann equation for granular gases: Diffusive equilibria[END_REF][START_REF] Bisi | Decay rates in probability metrics towards homogeneous cooling states for the inelastic Maxwell model[END_REF][START_REF] Carrillo | Contractive probability metrics and asymptotic behavior of dissipative kinetic equations[END_REF][START_REF] Bobylev | On the self-similar asymptotics for generalized non-linear kinetic Maxwell models[END_REF]. The most natural one is the pure diffusion thermal bath for which

G(f ) = µ ∆f (1.2)
where µ > 0 is a constant, studied in [START_REF] Gamba | On the Boltzmann equation for diffusively excited granular media[END_REF][START_REF] Mischler | Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited inelastic hard spheres[END_REF] for hard-spheres. Such a forcing term corresponds to the physical situation in which granular beads receive random kicking in their velocity, like air-levitated disks [START_REF] Bizon | Transport coefficients for granular media from molecular dynamics simulations[END_REF]. Another example is the thermal bath with linear friction

G(f ) = µ ∆f + λdiv(v f ), (1.3)
where λ and µ are positive constants. We also have to mention the fundamental example of anti-drift forcing term which is related to the existence of self-similar solution to the inelastic Boltzmann equation: G(f ) = -κ div(vf ), κ > 0.

(1.4) This problem has been treated in [START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem[END_REF][START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF][START_REF] Mischler | Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres[END_REF] for hard-spheres. For all the forcing terms given by (1.2), (1.3), (1.4) it is possible to prove the existence of a non-trivial stationary state F 0 such that τ Q(F, F ) + G(F ) = 0.

Moreover, such a stationary state can be chosen to be smooth, i.e. F ∈ C ∞ (R 3 ). Finally, even if the uniqueness (in suitable class of functions) of such a stationary state is an open problem, it can be shown for all these models that, in the weakly elastic regime in which the restitution coefficient is close to unity, the stationary state is unique. For an exhaustive survey of the "state of art" on the mathematical results for the evolution of granular media see [START_REF] Villani | Mathematics of granular materials[END_REF].

We are concerned here with a similar question when the forcing term G is given by a linear scattering operator. This corresponds to a situation in which the system of inelastic hard spheres is immersed into a so called particles thermal bath, i. e. G is given by a linear Boltzmann collision operator of the form:

G(f ) = B[f, F 1 ]
where F 1 stands for the distribution function of the host fluid and B[•, •] is a given collision operator for (elastic or inelastic) hard-spheres. The precise definition of G is given in Subsection 2.1. This kinetic model has already been tackled for instance in [START_REF] Bisi | Fluid-dynamic equations for granular particles in a host-medium[END_REF][START_REF] Bisi | Dilute granular flows in a medium[END_REF] in order to derive closed macroscopic equations for granular powders in a host medium. Let us also mention the work [START_REF] Biben | Stationary state of thermostated inelastic hard spheres[END_REF] that investigates the case of a particles thermal bath made of elastic hard-spheres at thermodynamical equilibrium (i.e. F 1 is a suitable Maxwellian). The deviations of the steady state (which is there assumed to exist) from the Gaussian state are analyzed numerically. For inelastic Maxwellian molecules, the existence of a steady state for a particles thermal bath has been obtained in [START_REF] Carrillo | Contractive probability metrics and asymptotic behavior of dissipative kinetic equations[END_REF]. To our knowledge, the existence of a stationary solution of (1.1) for particles bath heating and inelastic hard-spheres is an open problem and it is the main aim of this paper.

Our strategy, inspired by several works in the kinetic theory of granular gases [START_REF] Gamba | On the Boltzmann equation for diffusively excited granular media[END_REF][START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF] or for coagulation-fragmentation problems [START_REF] Bagland | Self-similar solutions to the Oort-Hulst-Safronov coagulation equation[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF], is based on a dynamic proof of the existence of stationary states, see [START_REF] Carrillo | Contractive probability metrics and asymptotic behavior of dissipative kinetic equations[END_REF]Lemma 7.3] for a review. The exact "fixed point theorem" used here is reported in Subsection 2.2. The identification of a suitable Banach space and of a convex subset that remains invariant during the evolution, is achieved by controlling moments and L p -norms of the solutions. In Section 3, we present regularity properties of the gain part of both collision operators Q and G in (1.1). Then, in Section 4 we get at first uniform bounds for the moments and the Lebesgue norms; in addition, we prove the strong continuity of the semi-group associated to (1.1), and the existence and uniqueness of a solution to the Cauchy problem. All this material allows to obtain, in Section 5, existence of non-trivial stationary states. Finally, Section 6 contains the study of regularity of stationary solutions. Many technical estimates involving the quadratic dissipative operator Q(f, f ) are based on results presented in [START_REF] Bouchut | A proof of the smoothing properties of the positive part of Boltzmann's kernel[END_REF][START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF][START_REF] Mischler | Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres[END_REF][START_REF] Mouhot | Regularity theory for the spatially homogeneous Boltzmann equation with cut-off[END_REF] and in the references therein, but their extension to the linear inelastic operator G(f ) is not trivial at all for the following reasons. First, since G is not quadratic, it induces a lack of symmetry particularly relevant in the study of propagation of L p norms. Second, since the microscopic collision mechanism is affected by the mass ratio of the two involved media (thermal bath and granular material), Povzner-like estimates for G are not straightforward consequences of previous results from [START_REF] Gamba | On the Boltzmann equation for diffusively excited granular media[END_REF]. Let us finally mention that our analysis also applies to linear scattering model which corresponds to the case τ = 0. For such a linear Boltzmann operator, we obtain the existence of an equilibrium solution, generalizing the results of [START_REF] Martin | Thermalization of a particle by dissipative collisions[END_REF][START_REF] Lods | The dissipative linear Boltzmann equation for hard spheres[END_REF][START_REF] Spiga | The dissipative linear Boltzmann equation[END_REF] to non-necessarily Maxwellian host distribution.

PRELIMINARIES

Let us introduce the notations we shall use in the sequel. Throughout the paper we shall use the notation

• = 1 + | • | 2 .
We denote, for any η ∈ R, the Banach space

L 1 η = f : R 3 → R measurable ; f L 1 η := R 3 |f (v)| v η dv < +∞ .
More generally we define the weighted Lebesgue space

L p η (R 3 ) (p ∈ [1, +∞), η ∈ R) by the norm f L p η (R 3 ) = R 3 |f (v)| p v pη dv 1/p . The weighted Sobolev space W k,p η (R 3 ) (p ∈ [1, +∞), η ∈ R and k ∈ N) is defined by the norm f W k,p η (R 3 ) =   |s| k ∂ s v f p L p η   1/p
where ∂ s v denotes the partial derivative associated with the multi-index s ∈ N N . In the particular case p = 2 we denote H k η = W k,2 η . Moreover this definition can be extended to H s η for any s 0 by using the Fourier transform.

2.1. The kinetic model. We assume the granular particles to be perfectly smooth hard spheres of mass m = 1 performing inelastic collisions. Recall that, as usual, the inelasticity of the collision mechanism is characterized by a single parameter, namely the coefficient of normal restitution 0 < ǫ < 1. To define the collision operator we write

Q(f, f ) = Q + (f, f ) -Q -(f, f ), (2.1)
where the "loss" term

Q -(f, f ) is Q -(f, f ) = f (f * |v|), (2.2)
and the "gain" term Q + (f, f ) is given by

Q + (f, f ) = 1 4πǫ 2 R 3 S 2 |v -w|f ( ′ v)f ( ′ w) dσ dw,
where the pre-collisional velocities read as

′ v = v + ζ 2ǫ (|q|σ -q), ′ w = w - ζ 2ǫ (|q|σ -q), (2.3)
with ζ = 1+ǫ 2 . Notice that we always have 1 2 < ζ < 1. Its weak formulation will be the main tool in the rest and it reads as

R 3 Q + (f, f )(v) ψ(v) dv = 1 4π R 3 R 3 f (v) f (w) |q| S 2 ψ(v ′ ) dσ dw dv, (2.4)
where q = v -w is the relative velocity of two particles about to collide, and v ′ is the velocity after the collision. The collision transformation that puts v and w into correspondence with the post-collisional velocities v ′ and w ′ can be expressed as follows:

v ′ = v + ζ 2 (|q|σ -q), w ′ = w - ζ 2 (|q|σ -q). (2.5)
Combining (2.2) and (2.4) and using the symmetry that allows us to exchange v with w in the integrals we obtain the following symmetrized weak form

R 3 Q(f, f )(v) ψ(v) dv = 1 2 R 3 R 3 f (v) f (w) |q|A ζ [ψ](v, w) dw dv, (2.6)
where

A ζ [ψ](v, w) = 1 4π S 2 (ψ(v ′ ) + ψ(w ′ ) -ψ(v) -ψ(w)) dσ. (2.7)
The inelastic Boltzmann operator Q(f, f ) satisfies the basic conservation laws of mass and momentum, obtained by taking ψ = 1, v in the weak formulation (2.6)

, since A ζ [1] = A ζ [v] = 0.
On the other hand, in the modelling of dissipative kinetic equations, conservation of energy does not hold. In fact, we obtain

A ζ [|v| 2 ] = -1-ǫ 2
4 |v -w| 2 from which we deduce

R 3 Q(f, f )(v) |v| 2 dv = - 1 -ǫ 2 8 R 3 R 3 |v -w| 3 f (v)f (w) dv dw, (2.8)
where we observe the dissipation of kinetic energy. In the absence of any other source of energy, the system cools down as t → ∞ following Haff's law as proved in [START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF].

As already said in Introduction, the forcing term G arising in the kinetic equation (1.1) is chosen to be a linear scattering operator, corresponding to the so called particles bath heating,

G(f ) := L(f ) = 1 2πλ R 3 S 2 |q • n| e -2 f (v ⋆ )F 1 (w ⋆ ) -f (v)F 1 (w) dw dn (2.9)
where λ is the mean free path, q = v -w is the relative velocity, v ⋆ and w ⋆ are the pre-collisional velocities which result, respectively, in v and w after collision. The collision mechanism related to the linear scattering operator is characterized by

(v -w) • n = -e(v ⋆ -w ⋆ ) • n, (2.10)
where n ∈ S 2 is the unit vector in the direction of impact and 0 < e < 1 is the constant restitution coefficient (possibly different from ǫ). Here, we will consider a similar separation of the operator into gain and loss terms, L(f ) = L + (f ) -L -(f ), with obvious definitions. Here the host fluid is made of hard-spheres of mass m 1 (possibly different from the traced particles mass m = 1) and the distribution function F 1 of the host fluid fulfils the following: Assumption 2.1. F 1 is a nonnegative normalized distribution function with bulk velocity u 1 ∈ R 3 and temperature Θ 1 > 0. Moreover, F 1 is smooth in the following sense,

F 1 ∈ H s δ (R 3 ), ∀s, δ 0
and of finite entropy Lemma 4] that there exists some χ > 0 such that

R 3 F 1 (v) log F 1 (v) dv < ∞. Remark 2.2. Notice that, since F 1 ∈ L 1 2 is of finite entropy, it is well-known [2,
ν(v) := 1 2π λ R 3 S 2 |(v -w) • n|F 1 (w) dw dn χ 1 + |v| 2 ∀v ∈ R 3 .
(2.11)

A particular choice of the distribution function F 1 , corresponding to a host fluid at thermodynamical equilibrium, is the following Maxwellian distribution

F 1 (v) = M 1 (v) = m 1 2πΘ 1 3/2 exp - m 1 (v -u 1 ) 2 2Θ 1 , v ∈ R 3 , (2.12)
Notice however that our approach remains valid for more general distribution function.

For particles of mass m = 1 colliding inelastically with particles of mass m 1 , the restitution coefficient being constant, the expressions of the pre-collisional velocities (v ⋆ , w ⋆ ) are given by [START_REF] Brilliantov | Kinetic theory of granular gases[END_REF][START_REF] Spiga | The dissipative linear Boltzmann equation[END_REF]

v ⋆ = v -2α 1 -β 1 -2β (q • n) n, w ⋆ = w + 2(1 -α) 1 -β 1 -2β (q • n) n,
where α is the mass ratio and β denotes the inelasticity parameter

α = m 1 1 + m 1 ∈ (0, 1), β = 1 -e 2 ∈ [0, 1/2).
The post-collisional velocities are given by

v ⋆ = v -2α(1 -β) (q • n) n, w ⋆ = w + 2(1 -α)(1 -β) (q • n) n. (2.13)
This linear operator can also be represented in a form closer to (2.6). By making use of the following identity [START_REF] Bobylev | Fourier transform method in the theory of the Boltzmann equation for Maxwellian molecules[END_REF][START_REF] Desvillettes | About the use of the Fourier transform for the Boltzmann equation[END_REF],

S 2 (q • n) + ϕ(n(q • n)) dn = 1 4 S 2 ϕ q -|q|σ 2 dσ
for any function ϕ, with q = q/|q|, we can rewrite the operator as

L(f ) = 1 4πλ R 3 S 2 |q| e -2 f (ṽ ⋆ )F 1 ( w⋆ ) -f (v)F 1 (w) dw dσ (2.14) with ṽ⋆ = v -α 1 -β 1 -2β (q -|q|σ) , w⋆ = w + (1 -α) 1 -β 1 -2β (q -|q|σ) .
For such a description, the post-collisional velocities are

ṽ⋆ = v -α(1 -β) (q -|q|σ) , w⋆ = w + (1 -α)(1 -β) (q -|q|σ) . (2.15)
We consider Eq. (1.1) in the weak form: for any regular ψ = ψ(v), one has

d dt R 3 f (v, t)ψ(v) dv = τ 2 R 3 R 3 f (v, t) f (w, t) |q|A ζ [ψ](v, w) dw dv + 1 λ R 3 R 3 |q|f (v, t)F 1 (w)J e [ψ](v, w) dv dw (2.16)
where

J e [ψ](v, w) = 1 2π S 2 |q • n| (ψ(v ⋆ ) -ψ(v)) dn = 1 4π S 2 (ψ(ṽ ⋆ ) -ψ(v)) dσ.
2.2. Proof of stationary states: basic tools and strategy. As stated in the Introduction, the final purpose of this paper is to prove the existence of a non-trivial regular stationary solution F 0 to (1.1). Namely, we look for

F ∈ L 1 , F 0 such that τ Q(F, F ) + G(F ) = 0.
(2.17)

Remark 2.3. Notice that such a problem is trivial in the elastic case ǫ = 1 and whenever F 1 is the Maxwellian distribution (2.12). Indeed, in such a case, the Maxwellian equilibrium distribution M ♯ of L provided by [START_REF] Martin | Thermalization of a particle by dissipative collisions[END_REF][START_REF] Lods | The dissipative linear Boltzmann equation for hard spheres[END_REF][START_REF] Spiga | The dissipative linear Boltzmann equation[END_REF] is a stationary solution to (1.1) since

Q(M ♯ , M ♯ ) = 0 (elastic Botlzmann equation) and L(M ♯ ) = 0.
The main ingredients are to show the existence of fixed points for the flow map at any time, and thus a continuity in time argument of the semi-group that allows to identify this one-parameter family of fixed points as a stationary point of the flow. Contraction estimates were used in [START_REF] Bisi | Contractive Metrics for a Boltzmann equation for granular gases: Diffusive equilibria[END_REF][START_REF] Carrillo | Contractive probability metrics and asymptotic behavior of dissipative kinetic equations[END_REF] while in the hard-sphere case the Tykhonov Fixed Point Theorem was the tool needed [START_REF] Gamba | On the Boltzmann equation for diffusively excited granular media[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF][START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF][START_REF] Bagland | Self-similar solutions to the Oort-Hulst-Safronov coagulation equation[END_REF].

The exact result that will be used can be summarized as:

Lemma 2.4 (Dynamic proof of stationary states).

Let Y be a Banach space and (S t ) t 0 be a continuous semi-group on Y such that i) there exists Z a nonempty convex and weakly (sequentially) compact subset of Y which is invariant under the action of S t (that is S t z ∈ Z for any z ∈ Z and t 0); ii) S t is weakly (sequentially) continuous on Z for any t > 0. Then there exists z 0 ∈ Z which is stationary under the action of S t (that is S t z 0 = z 0 for any t 0).

The strategy is therefore to identify a Banach space Y and a convex subset Z ⊂ Y in order to apply the above result. To do so, one shall prove that

• for any f 0 ∈ Y , there is a solution f ∈ C ([0, ∞), Y ) to Eq. (1.1) with f (t = 0) = f 0 ; • the solution f is unique in Y and if f 0 ∈ Z then f (t) ∈ Z for any t 0;
• the set Z is (weakly sequentially) compactly embedded into Y ;

• solutions to (1.1) have to be (weakly sequentially) stable, i.e., for any sequence

(f n ) n ⊂ C ([0, ∞), Y ) of solutions to (1.1) with f n (t) ∈ Z for any t 0, then, there is a subse- quence (f n k ) k which converges weakly to some f ∈ C ([0, ∞), Y ) such that f is a solution to (1.1).
If all the above points are satisfied by the evolution problem (1.1), then one can apply Lemma 2.4 to the semi-group (S t ) t 0 which to any f 0 ∈ Y associates the unique solution f (t) = S t f 0 to (1.1). Moreover, the regularity properties of the gain part of the operators [START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF] shall provide us the needed regularity to show the existence of smooth stationary states.

REGULARITY OF GAIN OPERATORS

We recall the following result, taken from [32, Theorem 2.5, Proposition 2.6] and based on [START_REF] Bouchut | A proof of the smoothing properties of the positive part of Boltzmann's kernel[END_REF][START_REF] Lu | A direct method for the regularity of the gain term in the Boltzmann equation[END_REF], on the regularity properties of the gain part operator Q + (g, f ) that we state here only for hard-spheres interactions in space dimension N = 3. Proposition 3.1 (Regularity of the gain term Q + ). For all s, η > 0, we have

Q + (g, f ) H s+1 η C(s, η, ǫ) g H s η+2 f H s η+2 + g L 1 η+2 f L 1 η+2
where the constant C(s, η, ǫ) > 0 only depends on the restitution coefficient ǫ ∈ (0, 1], s and η. Moreover, for any p ∈ [1, ∞) and δ > 0, there exist θ ∈ (0, 1) and a constant C δ > 0, only depending on p, ǫ and δ, such that

R 3 Q + (f, f ) f p-1 dv C δ f 1+pθ L 1 f p(1-θ) L p + δ f L 1 2 f p L p 1/p .
On the other hand, the linear operator L(f ) is quite similar to the quadratic Boltzmann operator associated to hard-spheres interactions and constant restitution coefficient e by fixing one of the distributions. In fact, it is possible to obtain the following similar result: Proposition 3.2 (Regularity of the gain term L + ). For all s, η > 0, we have

L + (f ) H s+1 η C(s, η, e) F 1 H s η+2 f H s η+2 + F 1 L 1 η+2 f L 1 η+2 (3.1)
where the constant C(s, η, e) > 0 only depends on the restitution coefficient e ∈ (0, 1], s and η. Moreover, for any p ∈ (1, ∞) and δ > 0, there exist q < p and a constant K δ > 0, only depending on p, e and δ, such that

R 3 L + (f ) f p-1 dv K δ F 1 L q f p-1 L p f L 1 + δ F 1 L 1 2 f p L p 1/p + F 1 L p 1/p f L 1 2 f p-1 L p 1/p . (3.2)
Proof. The proof of these two estimates relies on the same steps as in Sections 2.2, 2.3 and 2.4 of [START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF], see also [START_REF] Mouhot | Regularity theory for the spatially homogeneous Boltzmann equation with cut-off[END_REF]. We need just to have the same basic estimates as in their case. We start with the proof of (3.1). An expression of the Fourier transform of L + can be obtained as:

F L + (f ) (ξ) := R 3 exp(-iξ • v)L + (f )(v) dv = 1 4πλ S 2 G(ξ + , ξ -) dσ with G(v, w) = |v -w|f (v)F 1 (w)
, G its Fourier transform with respect to (v, w) and

ξ + = (1 -α(1 -β))ξ + α(1 -β)|ξ|σ, ξ -= α(1 -β)ξ -α(1 -β)|ξ|σ.
With this expression at hand, it is immediate to generalize to L + the regularity result in [32, Theorem 2.5, Proposition 2.6] giving (3.1). Now, let us prove the second result. We first notice that, as in [START_REF] Arlotti | Integral representation of the linear Boltzmann operator for granular gas dynamics with applications[END_REF], the gain operator L + admits an integral representation. Actually, even if it is assumed in [START_REF] Arlotti | Integral representation of the linear Boltzmann operator for granular gas dynamics with applications[END_REF] that F 1 is given by the Maxwellian distribution (2.12), a careful reading of the calculations of [START_REF] Arlotti | Integral representation of the linear Boltzmann operator for granular gas dynamics with applications[END_REF] yields

L + f (v) = R 3 f (w)k(v, w) dw, (3.3)
where

k(v, w) = 1 2e 2 γ 2 |v -w| V 2 •(w-v)=0 F 1 v + V 2 + 1 -2γ 2γ (w -v) dV 2 with γ = α 1-β 1-2β and γ = (1 -α) 1-β 1-2β .
Arguing as in [START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF], we define the operator T related to the Radon transform:

T : g ∈ L 1 (R 3 , dv) → T g(v) = 1 |v| z⊥v g(µv + z) dz where µ = 1 -1-2γ 2γ . For any h ∈ R 3 , let τ h denote the translation operator τ h f (v) = f (v -h), for any v ∈ R 3 . Then, for any g ∈ L 1 (R 3 , dv), one sees that (τ w • T )(g)(v) = 1 |v -w| z⊥(v-w) g(µ(v -w) + z) dz = 1 |v -w| z⊥(v-w) g v -w + z + 1 -2γ 2γ (w -v) dz, ∀v, w ∈ R 3 .
Choosing g = τ -w F 1 leads to the following expression of the kernel k(v, w):

k(v, w) = 1 2ǫ 2 γ 2 τ w • T • τ -w (F 1 )(v), v, w ∈ R 3 .
This previous computation is at the heart of the arguments of [32, Theorem 2.2], from which one gets a version of Lions' Theorem [START_REF] Lions | Compactness in Boltzmann's equation via Fourier integral operators and applications I, II, III[END_REF] for a suitable regularized cut-off kernel with collision frequency of the form

B m,n (|q|, q • σ) = Φ Sn (|q|) b Sm (q • σ),
L + Sm,n (f ) = 1 4πλ e 2 R 3 S 2 B m,n (|q|, q • σ)f (v ⋆ )F 1 (w ⋆ ) dw dσ (3.4)
then, for any η ∈ R + and any p > 1, there is C(p, η, m, n) > 0 depending only on p, η and (m, n), such that

L + Sm,n (f ) L p η C(p, η, m, n) F 1 L q η f L 1 2|η| (3.5)
for some q < p given by q = 5p 3+2p if p ∈ (1, 6] while q = p 3 if p ∈ [6, +∞) (see [32, Corollary 2.4]). In particular, Hölder's inequality leads to

R 3 L + Sm,n (f )f p-1 dv R 3 f p dv p-1 p L + Sm,n (f ) L p C(m, n) f L 1 F 1 L q f p-1 L p
for some explicit constant C(m, n) > 0.

Similarly, one can define the remainder part of L + which splits as

L + -L + Sm,n =: L + Rm,n = L + RSm,n + L + SRm,n + L + RRm,n with L + RSm,n (f ) = 1 4πλ e 2 R 3 S 2 Φ Rn (|q|) b Sm (q • σ)f (v ⋆ )F 1 (w ⋆ ) dw dσ, L + SRm,n (f ) = 1 4πλ e 2 R 3 S 2 Φ Sn (|q|) b Rm (q • σ)f (v ⋆ )F 1 (w ⋆ ) dw dσ, L + RRm,n (f ) = 1 4πλe 2 R 3 S 2 Φ Rn (|q|) b Rm (q • σ)f (v ⋆ )F 1 (w ⋆ ) dw dσ, where Φ Rn (|q|) = |q| -Φ Sn (|q|) and b Rm (q • σ) = 1 -b Sm (q • σ), q ∈ R 3 , σ ∈ S 2 . Hölder's inequality provides R 3 L + Rm,n (f ) f p-1 dv f p-1 L p 1/p L + Rm,n (f ) L p -1/p ′
with p ′ such that

L + SRm,n (f ) + L + RRm,n (f ) L p η ε(m) F 1 L 1 |1+η|+|η| f L p 1+η + f L 1 |1+η|+|η| F 1 L p 1+η
for some explicit constant ε(m) that, since the angular part of the collision kernel is such that lim m→∞ b R,m L 1 (S 2 ) = 0, converges to 0 as m goes to infinity. It remains to estimate the norm of L + RSm,n (f ). We follow now the lines of [36, Chapter 9, p. 395] (which differs slightly from [32, Proposition 2.6] and is more adapted to the linear case). Precisely, we split f as

f = f r + f r c = f (v)χ {|v| r} + f (v)χ {|v|>r}
for some r > 0. Then, as in [36, p. 395], there is some positive constant C > 0 such that

L + RSm,n (f r ) L p η C r n F 1 L 1 |2+η|+|η| f L p 1+η while L + RSm,n (f r c ) L p η C m λ r f L 1 |2+η|+|η| F 1 L p 1+η with λ > 0.
Gathering all the above estimates we get, for η = -1/p ′ ,

R 3 L + Rm,n (f ) f p-1 dv C f p-1 L p 1/p r n F 1 L 1 2 f L p 1/p + m λ r f L 1 2 F 1 L p 1/p + ε(m) F 1 L 1 1 f p L p 1/p + F 1 L p 1/p f L 1 1 f p-1 L p 1/p C r n + ε(m) F 1 L 1 2 f p L p 1/p + C m λ r + ε(m) F 1 L p 1/p f L 1 2 f p-1 L p 1/p .
The proof follows then by choosing first m large enough then r large enough and subsequently n big enough.

̺(t) = R 3 f (v, t) dv, u(t) = 1 ̺(t) R 3 vf (v, t) dv
and the temperature

Θ(t) = 1 3̺(t) R 3 |v -u(t)| 2 f (v, t) dv, ∀t 0.
Note that Eq. (2.16) for ψ = 1 leads to the mass conservation identity ̺(t) = 0 i.e.

̺(t) = ̺(0) := 1. Now, Eq. (2.16) for ψ(v) = v yields u(t) = - α(1 -β) λ R 3 R 3 |v -w|(v -w)f (v, t)F 1 (w) dv dw, ∀t 0
which illustrates the fact that the bulk velocity is not conserved. To estimate the second order moment of f , let us introduce the auxiliary function:

F (t) = R 3 R 3 |v -w| 2 f (v, t)F 1 (w) dv dw.
Notice that

F (t) = R 3 |v -u 1 | 2 f (v, t) dv + 3 m 1 Θ 1 = 3 Θ(t) + |u(t) -u 1 | 2 + 3 m 1 Θ 1 . (4.1)
In particular, to obtain uniform in time bounds of the mean velocity and the temperature, it is enough to provide uniform in time estimates of F (t). With the special choice

ψ(v) = |v -u 1 | 2 one has A ζ [ψ](v, w) = ζ(1 -ζ)|q| 4π S 2 (σ • q -|q|) dσ = -ζ(1 -ζ)|q| 2 = - 1 -ǫ 2 4 |q| 2 while J e [ψ](v, w) = 2α 2 (1 -β) 2 |q| 2 -2α(1 -β) q, v -u 1 = -2κ(1 -κ)|q| 2 -2κ q, w -u 1 , v, w ∈ R 3 with κ = α(1 -β) = α 2
(1 + e) ∈ (0, 1) and •, • denoting the scalar product. It is easy to see that

Ḟ (t) = - (1 -ǫ 2 )τ 8 R 3 R 3 f (v, t)f (w, t)|q| 3 dv dw - 2κ(1 -κ) λ R 3 R 3 |v -w| 3 f (v, t)F 1 (w) dv dw + 2κ λ R 3 R 3 |q| q, u 1 -w f (v, t)F 1 (w) dv dw. (4.2)
Now, since

R 3
f (v, t) dv = 1 for any t 0, Jensen's inequality yields

R 3 f (w, t)|q| 3 dw v - R 3 wf (w, t) dw 3 = |v -u(t)| 3
and consequently

R 3 R 3 f (v, t)f (w, t)|q| 3 dv dw R 3 |v -u(t)| 3 f (v, t) dv R 3 |v -u(t)| 2 f (v, t) dv 3/2 = 3 Θ(t) 3/2
where we used again Jensen's inequality. In the same way,

R 3 R 3 |q| 3 f (v, t)F 1 (w) dv dw R 3 R 3 |v -w| 2 f (v, t)F 1 (w) dv dw 3/2 = F (t) 3/2 .
Finally, the third integral in (4.2) is estimated as

R 3 R 3 |q| q, u 1 -w f (v, t)F 1 (w) dv dw R 3 R 3 |q| 2 |u 1 -w|f (v, t)F 1 (w) dv dw 2 R 3 |v -u 1 | 2 f (v, t) dv R 3 |w -u 1 |F 1 (w) dw + 2 R 3 |w -u 1 | 3 F 1 (w) dw C 0 F (t)
where

C 0 = 2 max        R 3 |w -u 1 |F 1 (w) dw , R 3 |w -u 1 | 3 F 1 (w) dw R 3 |w -u 1 | 2 F 1 (w) dw        .
In conclusion, we obtain

Ḟ (t) - (1 -ǫ 2 )τ 8 3Θ(t) 3/2 - 2κ(1 -κ) λ F (t) 3/2 + 2C 0 κ λ F (t) -γ 1 F (t) 3/2 +γ 2 F (t) (4.3)
where

γ 1 = 2κ(1 -κ) λ > 0 and γ 2 = 2C 0 κ λ > 0.
A simple use of the maximum principle shows that

F (t) max γ 2 γ 1 2 , F (0) , ∀t 0.
Because of (4.1), this leads to explicit upper bounds of the temperature Θ(t) and the velocity

|u(t) -u 1 |, namely sup t 0 3Θ(t) + |u(t) -u 1 | 2 max γ 2 γ 1 2 , F (0) < ∞. (4.4)
4.2. Propagation of moments. To extend the previous basic estimates, in the the spirit of [START_REF] Bobylev | Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions[END_REF], we deduce from Povzner-like estimates some useful inequalities on the moments

r (t) = R 3 f (v, t)|v| 2r dv, t 0, r 1
where f (t) is a solution to (1.1) with unit mass. One sees from (1.1) that

d dt r (t) = τ Q r (t) + L r (t), where Q r (t) = R 3 Q(f, f )(v, t)|v| 2r dv, L r (t) = R 3 L(f )(v, t)|v| 2r dv.
The calculations provided in [START_REF] Gamba | On the Boltzmann equation for diffusively excited granular media[END_REF][START_REF] Bobylev | Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions[END_REF] allow to estimate, in an almost optimal way, the quantity Q r . One has to do the same for L r (t) given by

L r (t) = 1 λ R 3 R 3 f (v, t)F 1 (w)|v -w|J e [ | • | 2r ](v, w) dv dw.
To do so, let us derive Povzner-like estimates for L in the spirit of [START_REF] Gamba | On the Boltzmann equation for diffusively excited granular media[END_REF]. The application of the result of [START_REF] Gamba | On the Boltzmann equation for diffusively excited granular media[END_REF] is not straightforward since, obviously, L is not quadratic and because of the influence of the mass ratio α = m 1 m+m 1 in the collision mechanism. Here, we will write the mass of particles m even if taken as unity for the sake of the reader. To be precise, we are looking for estimates of

J e [| • | 2r ](v, w) = 1 2π S 2 |q • n| |v ⋆ | 2r -|v| 2r dn, r 1.
To do so, it shall be convenient to write

J e [| • | 2r ](v, w) = 1 2πm r S 2 |q • n| Ψ m|v ⋆ | 2 -Ψ m|v| 2 dn (4.5)
where Ψ(x) = x r , r 1. We adopt the strategy used in [START_REF] Gamba | On the Boltzmann equation for diffusively excited granular media[END_REF] and write

Ψ m|v ⋆ | 2 -Ψ m|v| 2 = q e (Ψ)(v, w) + Ψ(m 1 |w| 2 ) -Ψ(m 1 |w ⋆ | 2 ) (4.6)
where

q e (Ψ)(v, w) = Ψ m|v ⋆ | 2 + Ψ m 1 |w ⋆ | 2 -Ψ m|v| 2 -Ψ m 1 |w| 2 . Now, q e (Ψ)(v, w) = p e (Ψ)(v, w) -n e (Ψ)(v, w) with p e (Ψ)(v, w) = Ψ m|v| 2 + m 1 |w| 2 -Ψ m|v| 2 -Ψ m 1 |w| 2 n e (Ψ)(v, w) = Ψ m|v| 2 + m 1 |w| 2 -Ψ m|v ⋆ | 2 -Ψ m 1 |w ⋆ | 2 .
Applying [START_REF] Gamba | On the Boltzmann equation for diffusively excited granular media[END_REF]Lemma 3.1] to the function Ψ with x = m|v| 2 and y = m 1 |w| 2 , we see that there exists A > 0 such that

p e (Ψ)(v, w) A m|v| 2 Ψ ′ m 1 |w| 2 + m 1 |w| 2 Ψ ′ m|v| 2 (4.7)
while, since Ψ is nondecreasing and

m|v| 2 + m 1 |w| 2 m|v ⋆ | 2 + m 1 |w ⋆ | 2 , there exists b > 0 such that n e (Ψ)(v, w) b m m 1 |v ⋆ | 2 |w ⋆ | 2 Ψ ′′ m|v ⋆ | 2 + m 1 |w ⋆ | 2 . One can then write n e (Ψ)(v, w) b ∆(v ⋆ , w ⋆ ) m|v ⋆ | 2 + m 1 |w ⋆ | 2 2 Ψ ′′ m|v ⋆ | 2 + m 1 |w ⋆ | 2 where ∆(v ⋆ , w ⋆ ) = m|v ⋆ | 2 m 1 |w ⋆ | 2 (m|v ⋆ | 2 + m 1 |w ⋆ | 2 ) 2 .
To estimate better the above term ∆(v ⋆ , w ⋆ ), it will be convenient to parametrize the post-collisional velocities in the center of mass-relative velocity variables, which, with respect to the usual transformation (see e.g. [26, Eq. (3.10)]) depend on the masses m and m 1 . Namely, let us set

v ⋆ = z + m 1 ℓ|q|̟ m + m 1 , w ⋆ = z -mℓ|q|̟ m + m 1
where z = mv +m 1 w, q = v -w and ̟ is a parameter vector on the sphere S 2 . The parameter ℓ is positive and such that v ⋆ -w ⋆ = ℓ|v -w|̟. In particular, one sees from the representation (2.13) that 0 < ℓ 1. In this representation, one has

|v ⋆ | 2 = 1 (m + m 1 ) 2 |z| 2 + m 2 1 ℓ 2 |q| 2 + 2ℓm 1 |q||z| cos µ and |w ⋆ | 2 = 1 (m + m 1 ) 2 |z| 2 + m 2 ℓ 2 |q| 2 -2ℓm|q||z| cos µ ,
where µ is the angle between z and ̟. One has then

m|v ⋆ | 2 + m 1 |w ⋆ | 2 = 1 m + m 1 |z| 2 + ℓ 2 m m 1 |q| 2 . (4.8)
One can check that

(m|v ⋆ | 2 ) (m 1 |w ⋆ | 2 ) = m m 1 (m + m 1 ) 4 |z| 2 + ℓ 2 mm 1 |q| 2 2 -|z| 2 -ℓ 2 mm 1 |q| 2 2 cos 2 µ + ℓ(m 1 -m)|z||q| + |z| 2 -ℓ 2 mm 1 |q| 2 cos µ 2 -4ℓ 2 mm 1 |z| 2 |q| 2 cos 2 µ , i.e. (m|v ⋆ | 2 ) (m 1 |w ⋆ | 2 ) m m 1 (m + m 1 ) 4 |z| 2 + ℓ 2 mm 1 |q| 2 2
(1 -cos 2 µ) .

Therefore ∆(v ⋆ , w ⋆ ) m m 1 (m + m 1 ) 2 sin 2 µ .
We obtain then an estimate similar to the one obtained in [START_REF] Gamba | On the Boltzmann equation for diffusively excited granular media[END_REF]. Moreover, it is easy to see from (4.8)

that m|v ⋆ | 2 + m 1 |w ⋆ | 2 ℓ 2 m|v| 2 + m 1 |w| 2
and, arguing as in [START_REF] Gamba | On the Boltzmann equation for diffusively excited granular media[END_REF], there exists some constant η > 0 such that

n e [Ψ](v, w) η sin 2 µ m|v| 2 + m 1 |w| 2 2 Ψ ′′ m|v| 2 + m 1 |w| 2 . (4.9)
This allows to prove the following:

Ψ(m|v ⋆ | 2 ) -Ψ(m|v| 2 ) q e [Ψ](v, w) + Ψ m 1 |w| 2 = q e [Ψ](v, w) + m r 1 |w| 2r .
Then, integrating (4.7) and (4.9) with respect to the angle n ∈ S 2 , one obtains, as in [26, Lemma 3.3.] and [START_REF] Gamba | On the Boltzmann equation for diffusively excited granular media[END_REF]Lemma 3.4], that there are A r and k r > 0 such that, for any v, w ∈ R 3 :

|v -w| 1 2πm r S 2 q e (Ψ)(v, w)|q • n| dn A r |v||w| 2r + |v| 2r |w| -k r |v| 2r+1 + |w| 2r+1 ,
and this concludes the proof.

The above Lemma (restoring m = 1) together with the known estimates for Q r (t) allow to formulate the following Proposition 4.2 (Propagation of moments). Let f (t) be a solution to (1.1) with unit mass. For any r 1, let

r (t) = R 3 f (v, t)|v| 2r dv, t 0.
Then, there are positive constants A r , K r and C r that depend only on r, α, β, τ, λ and the moments of F 1 such that

d dt r (t) C r + A r r (t) -K r 1+1/2r r (t), ∀t 0.
As a consequence, if r (0) < ∞, then sup t 0 r (t) < ∞.

Proof. Recall that d dt r (t) = τ Q r (t) + L r (t), where Q r (t) = R 3 Q(f, f )(v, t)|v| 2r dv, L r (t) = R 3 L(f )(v, t)|v| 2r dv.
According to [26, Lemma 3.4], there exist A r > 0 and k r > 0 such that

Q r (t) A r 1/2 (t) r (t) -k r r+1/2 (t), t 0. Now, from Lemma 4.1 λL r (t) A r M r 1/2 (t) + A r M 1/2 r (t) + m r 1 R 3 R 3 |v -w||w| 2r f (v, t)F 1 (w) dw dv -k r r+1/2 (t) -k r M r+1/2 ,
where M s = R 3

|w| 2s F 1 (w) dw, s 1. One has

R 3 R 3 |v -w||w| 2r f (v, t)F 1 (w) dw dv M r 1/2 (t) + M r+1/2
and, denoting c 1/2 := sup t 0 1/2 (t) < ∞, one has

L r (t) C r + A r M 1/2 λ r (t) - k r λ r+1/2 (t)
where

C r = c 1/2 A r M r + c 1/2 m r 1 M r + m r 1 M r+1/2
/λ is a positive constant depending only on α, β, λ, r 1 and the moments of F 1 . Gathering all these estimates leads to

d dt r (t) C r + A r r (t) -K r r+1/2 (t)
where A r = τ A r c 1/2 + 1 λ A r M 1/2 > 0 and K r = τ k r + kr λ > 0. Now, thanks to the mass conservation and Hölder's inequality, one gets r+1/2 (t) 1+1/2r r (t) which leads to the desired result.

Remark 4.3. We see from the definition of the positive constants A r , C r and K r that the above Proposition still holds true whenever τ = 0 (i.e. for the linear problem).

Propagation of Lebesgue norms.

Let us consider now an initial condition f 0 ∈ L 1 2 ∩ L p for some 1 < p < ∞. We compute the time derivative of the L p norm of the solution f (v, t) to (1.1):

1 p d dt R 3 f p (v, t) dv = τ R 3 Q + (f, f )f p-1 dv -τ R 3 f p-1 Q -(f, f ) dv + R 3 L + (f )f p-1 dv - R 3 L -(f )f p-1 dv.
Using the fact that

R 3 f p-1 Q -(f, f ) dv 0 and L -(f )(v) = ν(v)f (v)
where the collision frequency ν(v) is given by

ν(v) = 1 2πλ R 3 S 2 |(v -w) • n|F 1 (w) dw dn,
we obtain the estimate:

1 p d dt R 3 f p (v, t) dv τ R 3 Q + (f, f )f p-1 dv + R 3 L + (f )f p-1 dv - R 3 ν(v)f p (v, t) dv.
Using the lower bound (2.11), we get

1 p d dt f (t) p L p τ R 3 Q + (f, f )f p-1 dv + R 3 L + (f )f p-1 dv -χ f p L p 1/p
. Proposition 3.1 and the conservation of mass imply that, for any δ > 0, there is θ > 0 and some C δ such that

R 3 Q + (f, f )f p-1 (v, t) dv C δ f (t) p(1-θ) L p + δ f (t) L 1 2 f (t) p L p 1/p .
Moreover, Proposition 3.2 implies that, for any δ > 0,

R 3 L + (f, f ) f p-1 dv C 1 f (t) p-1 L p + C 2 δ f (t) p L p 1/p + f (t) L 1 2 f (t) p-1 L p 1/p
, for some constants C 1 , C 2 > 0 that depend only on p, δ, η, α, e and the norms of F 1 in the spaces involved in (3.2). Recall that there is some M 2 such that

sup t 0 f (t) L 1 2 = 1 + sup t 0 R 3 |v| 2 f (v, t) dv M 2 < ∞.
Now, using Young's inequality, xy p-1 1 p x p + p-1 p y p , for any x, y 0, we have

R 3 L + (f, f ) f p-1 dv C 1 f (t) p-1 L p + C 3 δ f (t) p L p 1/p + M p
for some constant C 3 > 0. Collecting all the bounds above, we get the estimate

1 p d dt f (t) p L p τ C δ f (t) p(1-θ) L p + δτ M 2 f (t) p L p 1/p + C 1 f (t) p-1 L p + C 3 δ f (t) p L p 1/p + M p 2 -χ f (t) p L p 1/p τ C δ f (t) p(1-θ) L p + δ(τ M 2 + C 3 ) - χ 2 f (t) p L p 1/p + C 1 f (t) p-1 L p + C 3 δ M p 2 - χ 2 f (t) p L p , since • L p 1/p • L p . Choosing now δ such that δ(τ M 2 + C 3 ) < χ/2,
1 p d dt f (t) p L p C 4 f (t) p(1-θ) L p + C 5 f (t) p-1 L p -C 6 f (t) p L p + C 3 M p 2 δ.
It is not difficult to get then that sup t 0 f (t) L p < ∞. This can be summarized in the following

t 0 f (t) L 1 2 + f (t) L p < ∞.
Remark 4.5. Notice that the fact that F 1 is of finite entropy (see Assumption 2.1) has been used here above, via the lower bound (2.11), in order to control from below L p norms involving the loss operator L -. Whenever τ > 0, it is possible then to replace such estimates involving L -by others that involve Q -. Notice also that, whenever τ = 0 (i.e. in the linear case), only the above constant C 4 vanishes and we still have

sup t 0 f (t) L p < ∞.
As a corollary, we deduce as in [START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF]Section 3.4], see also [START_REF] Carrillo | Intermediate asymptotics beyond homogeneity and self-similarity: long time behavior for ut = ∆φ(u)[END_REF], the following non-concentration result: Proposition 4.6 (Uniform non-concentration). Let f 0 be given with unit mass. Assume that there exists some p ∈ (1, ∞) such that f 0 ∈ L 1 2 ∩ L p . Then, there exists some positive constant ν 0 such that

ν 0 R 3 |v -u(t)| 2 f (v, t) dv 1/ν 0 , ∀t 0, where f (v, t) is the solution to (1.1) with f (0) = f 0 and u(t) = R 3 vf (v, t) dv, t 0.
Proof. Let f (t) be the solution to (1.1) with f (0) = f 0 . From the above Proposition, there exists C p > 0 such that sup t 0 f (t) L p C p , and Hölder's inequality implies that, for any r > 0, sup

t 0 {|v-u(t)|<r} f (v, t) dv C p 4π 3 r 3 p-1 p
.

Accordingly, there is some r 0 > 0 such that

{|v-u(t)|<r 0 } f (v, t) dv 1 2 , ∀t 0.
Then, for any t 0, recalling that

R 3 f (v, t) dv = 1 for any t 0, R 3 f (v, t)|v -u(t)| 2 dv {|v-u(t)| r 0 } f (v, t)|v -u(t)| 2 dv r 2 0 {|v-u(t)| r 0 } f (v, t) dv r 2 0 1 - {|v-u(t)|<r 0 } f (v, t) dv r 2 0 2
which concludes the proof.

4.4. L 1 -stability. As in [START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF], in order to prove the strong continuity of the semi-group (S t ) t 0 associated to (1.1), one has to provide an estimate of f (t) -g(t) for two solutions f (t) and g(t) of (1.1) with initial conditions f (0), g(0) in some subspace of L 1 . This is the object of the following stability result, inspired by [32, Proposition

Proposition 4.7 (L 1 -stability). Let f 0 , g 0 be two nonnegative functions of L 1 3 and let

f (t), g(t) ∈ C(R + , L 1 2 ) ∩ L ∞ (R + , L 1
3 ) be the associated solutions to (1.1). Then, there is Λ > 0 depending only on

sup t 0 f (t) + g(t) L 1 3 such that f (t) -g(t) L 1 2 f 0 -g 0 L 1 2 exp(Λt), ∀t 0. Proof. Let h(t) = f (t) -g(t)
. Then, h satisfies the following equation:

∂ t h(v, t) = τ Q(f, f ) -Q(g, g) + L(h), h(0) = f 0 -g 0 . (4.10)
As in [START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem[END_REF][START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF], the proof consists in multiplying (4.10) by ψ(v, t) = sgn(h(v, t)) v 2 and integrating over R 3 . We get

d dt R 3 |h(v, t)| v 2 dv = I(t) + L(t)
where

I(t) = τ R 3 Q(f, f ) -Q(g, g) ψ(v, t) dv and L(t) = R 3 L(h)(v, t)ψ(v, t) dv.
To estimate the integral I(t) we resume the arguments of [31, Proposition 3.4] that we shall need again later. According to (2.6)

I(t) = τ 2 R 3 R 3 (f (v, t) f (w, t) -g(v, t)g(w, t)) |q|A ζ [ψ(t)](v, w) dw dv.
The change of variables (v, w) → (w, v) implies that

I(t) = τ 2 R 3 R 3 (f (v, t) -g(v, t)) (f (w, t) + g(w, t)) |q|A ζ [ψ(t)](v, w) dw dv.
Moreover, it is easily seen from the definition of ψ that

(f (v, t) -g(v, t))A ζ [ψ(t)](v, w) 1 4π |f (v, t) -g(v, t)| S 2 v ′ 2 + w ′ 2 -v 2 + w 2 dσ 2 |f (v, t) -g(v, t)| w 2
where we used the fact that

A ζ [ • 2 ](v, w) = -1-ǫ 2 4 |q| 2 0. Therefore, I(t) τ R 3 R 3 |q| |f (v, t) -g(v, t)| (f (w, t) + g(w, t)) w 2 dv dw τ R 3 |f (v, t) -g(v, t)| v 2 dv R 3 (f (w, t) + g(w, t)) w 3 dw i.e. I(t) τ f (t) + g(t) L 1 3 f (t) -g(t) L 1 2 , ∀t 0. (4.11)
On the other hand, recalling that

L(h)(v, t) = L + (h)(v, t) -ν(v)h(v, t), from formula (3.3) one has L(t) = R 3 ψ(v, t) dv R 3 h(w, t)k(v, w) dw - R 3 ν(v)|h(v, t)| v 2 dv R 3 v 2 dv R 3 |h(w, t)|k(v, w) dw - R 3 ν(v)|h(v, t)| v 2 dv, i.e. L(t) R 3 L(|h|)(v, t) v 2 dv. Now, since R 3
L(|h|)(v, t) dv = 0 for any h, one gets that

L(t) R 3 L(|h|)(v, t) |v| 2 dv.
Resuming the calculations performed in Section 4.1 (see Eq. (4.2)), one gets that

R 3 L(|h|)(v, t) |v| 2 dv - 2κ(1 -κ) λ R 3 R 3 |v -w| 3 |h(v, t)|F 1 (w) dv dw + 2κ λ R 3 R 3 |q| q, -w |h(v, t)|F 1 (w) dv dw 2κ λ R 3 R 3 |q| 2 |w||h(v, t)|F 1 (w) dv dw.
This leads to

L(t) 2κ λ 2 R 3 |v| 2 |h(v, t)| dv R 3 |w|F 1 (w) dw + 2 R 3 |h(v, t)| dv R 3 |w| 3 F 1 (w) dw and, setting c + = 4κ λ max R 3 |w|F 1 (w) dw, R 3 |w| 3 F 1 (w) dw , we get L(t) c + R 3 |h(v, t)| v 2 dv = c + f (t) -g(t) L 1 2 .
Gathering (4.11) together with the latter estimate and denoting then Λ = τ sup t 0 f (t)+g(t) L 1 3 + c + , we get the estimate

d dt f (t) -g(t) L 1 2 Λ f (t) -g(t) L 1 2 , t 0
and the proof is achieved.

4.5.

Well-posedness of the Cauchy problem. We are in position to prove that the Boltzmann equation (1.1) admits a unique regular solution in the following sense:

Theorem 4.8 (Existence and uniqueness of solution to the Cauchy problem). Take an initial datum f 0 ∈ L 1 3 . Then, for all T > 0, there exists a unique solution

f ∈ C([0, T ]; L 1 2 ) ∩ L ∞ (0, T ; L 1 3 ) to the Boltzmann equation (1.1) such that f (v, 0) = f 0 (v). Proof. Let T > 0 be fixed. The uniqueness in C([0, T ]; L 1 2 ) ∩ L ∞ (0, T ; L 1
3 ) trivially follows from Proposition 4.7. The proof of the existence is made in several steps, following the lines of [START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part I: The Cauchy problem[END_REF]Section 3.3], see also [START_REF] Mischler | On the spatially homogeneous Boltzmann equation[END_REF][START_REF] Fournier | A spatially homogeneous Boltzmann equation for elastic, inelastic, and coalescing collisions[END_REF].

Step 1. Let us first consider an initial datum f 0 ∈ L 1 4 , and define the "truncated" collision operators

R 3 Q n (f, f )(v) ψ(v) dv = 1 2 R 3 R 3 1 {|q| n} |q|f (v) f (w) A ζ [ψ](v, w) dw dv, R 3 L n (f )(v) ψ(v) dv = 1 λ R 3 R 3 1 {|q| n} |q|f (v)F 1 (w)J e [ψ](v, w) dv dw (4.12)
for any regular test function ψ. The operators Q n and L n are bounded in any L 1 q , and they are Lipschitz in L 1 2 on any bounded subset of L 1 2 . Therefore, following [START_REF] Arkeryd | On the Boltzmann equation[END_REF], we can use the Banach fixed point theorem to get the existence of a solution 0

f n ∈ C([0, T ]; L 1 2 ) ∩ L ∞ (0, T ; L 1 4 ) to the Boltzmann equation ∂ t f = τ Q n (f, f ) + L n (f ).
Thanks to the uniform propagation of moments in Proposition 4.2, there exists a constant C T > 0 (that does not depend on n) such that

sup [0,T ] f n L 1 4 C T , ∀n ∈ N.
Step 2. Let us prove that the sequence (f n ) n is a Cauchy sequence in C([0, T ]; L 1 2 )∩L ∞ (0, T ; L 1 4 ). For any m n, writing down the equation satisfied by f m -f n and multiplying it by ψ(v, t) = sgn(f m (v, t) -f n (v, t)) v 2 as in the proof of Proposition 4.7, we get

d dt R 3 |f n (v, t) -f m (v, t)| v 2 dv = I m,n (t) + J m,n (t)
where

I m,n (t) = τ R 3 Q m (f m , f m ) -Q n (f n , f n ) ψ(v, t) dv and J m,n (t) = R 3 L m (f m )(v, t) -L n (f n )(v, t) ψ(v, t) dv.
We begin by estimating I m,n (t). It is easy to see that I m,n (t) = I 1 m,n (t) + I 2 m,n (t) where

I 1 m,n (t) = τ 2 R 3 R 3 f m (v, t) f m (w, t) -f n (v, t)f n (w, t) B m (q)A ζ [ψ(t)](v, w) dw dv, while I 2 m,n (t) = τ 2 R 3 R 3 (B m (q) -B n (q)) f n (v, t)f n (w, t)A ζ [ψ(t)](v, w) dw dv,
where B n (q) = |q|1 {|q| n} . Arguing as in the proof of (4.11), we get easily that

I 1 m,n (t) τ f n (t) + f m (t) L 1 3 f n (t) -f m (t) L 1 2 , ∀t 0.
The estimate of I 2 m,n (t) is more involved. One observes first that

B m (q) -B n (q) = |q|1 {n |q| m} |q| 1 {|v| n/2} + 1 {|w| n/2} .
As in the proof of Proposition 4.7, one has

A ζ [ψ(t)](v, w) 1 4π S 2 v ′ 2 + w ′ 2 + v 2 + w 2 dσ 2 v 2 + w 2
and, since |q| v w , one gets

I 2 m,n (t) τ R 3 R 3 f n (v, t)f n (w, t)|q| 1 {|v| n/2} + 1 {|w| n/2} v 2 + w 2 dw dv τ R 3 R 3 f n (v, t)f n (w, t) v w v 2 + w 2 1 { v n/2} + 1 { w n/2} dw dv.
It is not difficult to deduce then that

I 2 m,n (t) 4τ R 3 f n (v, t) v 3 dv R 3 f n (v, t) v 3 1 { v n/2} dv . Since sup [0,T ] f n (t) L 1 4
C T for any n ∈ N, the latter integral is estimated as

R 3 f n (v, t) v 3 1 { v n/2} dv R 3 f n (v, t) v 4 1 { v n/2} dv v 2C T n
and we get

I 2 m,n (t) 4τ R 3 f m (v, t) v 3 dv 2C T n 8C 2 T τ n , ∀t ∈ [0, T ], m n.
Therefore,

I m,n (t) 2τ C T f n (t) -f m (t) L 1 2 + 8C 2 T τ n , ∀t ∈ [0, T ], m n. (4.13)
We proceed in the same way with J m,n (t). First, we notice that J m,n (t) splits as J m,n (t) = J 1 m,n (t) + J 2 m,n (t) with

J 1 m,n (t) = 1 λ R 3 R 3 B m (q) [f m (v, t) -f n (v, t)] F 1 (w)J e [ψ(t)] (v, w) dv dw and J 2 m,n (t) = 1 λ R 3 R 3 [B m (q) -B n (q)] f n (v, t)F 1 (w)J e [ψ(t)] (v, w) dv dw.
Arguing as in the proof of Proposition 4.7, we get

J 1 m,n (t) R 3 L m (|f n -f m |)(v, t) v 2 dv 2κ λ R 3 R 3 |q| 2 |w||(f n -f m )(v, t)|F 1 (w) dv dw
and there exists a positive constant c + such that

J 1 m,n (t) c + f n (t) -f m (t) L 1 2 , ∀t ∈ [0, T ].
Let us now estimate J 2 m,n (t). As above,

J 2 m,n (t) = 1 λ R 3 R 3 |q|1 {n |q| m} f n (v, t)F 1 (w)J e [ψ(t)] (v, w) dv dw
and

J e [ψ(t)] (v, w) 1 2π S 2 |q • n| v ⋆ 2 + v 2 dn = J e • 2 + 2 2π S 2 |q • n| v 2 dn.
Calculations already performed lead then to

J e [ψ(t)] (v, w) -2κ q, w + 2 v 2 2 v w 2 + v 2 , ∀ v, w ∈ R 3 .
Finally,

J 2 m,n (t) 2 λ R 3 R 3 |q|1 {|q| n} f n (v, t)F 1 (w) v w 2 + v 2 dw dv 2 λ R 3 R 3 1 {|q| n} f n (v, t)F 1 (w) v 3 w + v 2 w 3 dw dv.
Now, arguing as we did for I 2 m,n (t), there exists some constant CT that depends only on

F 1 L 1 4 and sup n sup [0,T ] f n (t) L 1 4 such that J 2 m,n (t) CT n , ∀m n, t ∈ [0, T ].
Gathering all these estimates, we obtain the existence of constants C 1 (T ) and C 2 (T ) that do not depend on m, n such that

d dt R 3 |f n (v, t)-f m (v, t)| v 2 dv C 1 (T ) f n (t)-f m (t) L 1 2 + C 2 (T ) n , ∀t ∈ [0, T ], m n.
This ensures that (f n ) n is a Cauchy sequence in C([0, T ]; L 1 2 ). Denoting by f its limit, we obtain that f ∈ C([0, T ]; L 1 2 ) ∩ L ∞ (0, T ; L 1 4 ) is a solution to the Boltzmann equation (1.1) (with the actual collision operators Q and L).

Step 3. When the initial datum f 0 ∈ L 1 3 , we introduce the sequence of initial data f 0,j := f 0 1 |v| j . Since f 0,j ∈ L 1 4 , we have the existence of a solution f j ∈ C([0, T ]; L 1 2 ) ∩ L ∞ (0, T ; L 1 4 ) to the Boltzmann equation associated to the initial datum f 0,j . Moreover, there exists C T such that sup [0,T ] f j L 1 3 C T . We establish again that (f j ) j is a Cauchy sequence in C([0, T ]; L 1 2 ) by using the L 1 -stability in Proposition 4.7.

EXISTENCE OF NON-TRIVIAL STATIONARY STATE

All the material of the previous sections allows us to state our main result: Proof. As already announced, the existence of stationary solution to (1.1) relies on the application of Lemma 2.4 to the evolution semi-group (S t ) t 0 governing (1.1). Namely, for f 0 ∈ L 1 , let f (t) = S t f 0 denote the unique solution to (1.1) with initial state f (0) = f 0 . The continuity properties of the semi-group are proved by the study of the Cauchy problem, recalled in Section 4. Let us fix p o ∈ (1, ∞). On the Banach space Y = L 1 2 , thanks to the uniform bounds on the L 1 3 and L po norms, the nonempty convex subset

Z = 0 f ∈ Y, R 3 f dv = 1 and f L 1 3 + f L po M is
stable by the semi-group provided M is big enough. This set is weakly compact in Y by Dunford-Pettis Theorem, and the continuity of S t for all t 0 on Z follows from Proposition 4.7. Then, Lemma 2.4 shows that there exists a nonnegative stationary solution to (1.1) in L 1 3 ∩ L po with unit mass. In fact, the uniform in time L p bounds also imply the boundedness of F in L p for all p ∈ (1, ∞).

As a corollary of Theorem 5.1, choosing τ = 0 allows us to prove the existence of a steady state to the linear inelastic scattering operator L when the distribution function of the background is not a Maxwellian, generalizing the result of [START_REF] Martin | Thermalization of a particle by dissipative collisions[END_REF][START_REF] Lods | The dissipative linear Boltzmann equation for hard spheres[END_REF][START_REF] Spiga | The dissipative linear Boltzmann equation[END_REF]. Corollary 5.2. Let F 1 satisfy Assumption 2.1. Then, the linear inelastic scattering operator L defined by (2.9) admits a unique nonnegative steady state F ∈ L 1 2 ∩ L p , p ∈ (1, ∞), with unit mass and positive temperature.

Proof. The existence of a nonnegative equilibrium solution F ∈ L 1 2 is a direct application of Theorem 5.1 with τ = 0. A simple application of Krein-Rutman Theorem implies the uniqueness of the stationary solution F within the range of nonnegative distributions with unit mass.

Remark 5.3 (H-Theorem and trend towards equilibrium).

As in [START_REF] Lods | The dissipative linear Boltzmann equation for hard spheres[END_REF], it is possible to prove a linear version of the classical H-Theorem for the linear inelastic Boltzmann equation (1.1) with τ = 0 :

∂ t f = L(f ), f (t = 0) = f 0 ∈ L 1 . (5.1)
Namely, for any convex C 1 function Φ : R + → R, let

H Φ (f |F ) = R 3 F (v)Φ f (v) F (v) dv, f ∈ L 1 .
Arguing as in [START_REF] Lods | The dissipative linear Boltzmann equation for hard spheres[END_REF], it is easy to prove that, if the initial state f 0 has unique mass and finite entropy

H Φ (f 0 |F ) < ∞, then d dt H Φ (f (t)|F ) 0 (t 0) (5.2)
where f (t) stands for the (unique) solution to (5.1). Moreover, still arguing as in [START_REF] Lods | The dissipative linear Boltzmann equation for hard spheres[END_REF], one proves that if moreover

R 3 1 + v 2 + | log f 0 (v)| f 0 (v) dv < ∞, then lim t→∞ R 3 |f (v, t) -F (v)| dv = 0.

REGULARITY OF THE STEADY STATE

In this final section, our aim is to establish the existence of some smooth stationary solution to (1.1). Namely, adopting the strategy of [32, Section 4.1], we prove Theorem 6.1 (Regularity of stationary solutions). There exists a stationary solution F to the Boltzmann equation

τ Q(F, F ) + L(F ) = 0 that belongs to C ∞ (R 3 ).
We shall follow the same lines of [37, Theorem 5.5] and [32, Section 3.6], from which we deduce the exponential decay in time of singularities and thus the smoothness of stationary solutions. This proof needs the following ingredients:

i) The stability result already proved in Proposition 4.7.

ii) An estimate on the Duhamel representation [START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF]Proposition 3.4] of the solution to (1.1) (see Proposition 6.2). iii) A result of propagation of Sobolev norms (see Proposition 6.3).

Let us first extend the regularity estimate of [START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF]Proposition 3.4] to our situation. For any

f ∈ L 1 , let Σ(f )(v) = τ (| • | * f ) (v) + ν(v) = τ R 3 |v -w|f (w) dw + ν(v).
It is easy to see that, for f 0 ∈ L 1 3 , the unique solution f (t) to (1.1) is given by the the following Duhamel representation:

f (v, t) = f 0 (v) e -t 0 Σ(f )(v,s) ds + t 0 τ Q + (f, f ) + L + (f ) (v, s) e -t s Σ(f )(v,r) dr ds = f 0 (v)G(v, 0, t) + t 0 τ Q + (f, f ) + L + (f ) (v, s)G(v, s, t) ds (6.1)
where we set

G(v, s, t) = exp - t s Σ(f )(v,
L(f )(v) = R 3
|v -w|f (w) dw.

It is clear that

Σ(f )(v) = L(τ f + F 1 )(v), ∀f ∈ L 1 .
Now, according to [START_REF] Duduchava | On estimates of the Boltzmann collision operator with cut-off[END_REF]Lemma 4.3], for any given k 0 and any δ > 3/2, the linear operator

L : H k δ -→ W k+1,∞ -1
is bounded, i.e. for any δ > 3/2 and any k 0, there exists C k,δ such that , ∀λ > 0, ∀ℓ, r ∈ R .

L(g) W k+1,∞ -1 C k,
As a consequence we have for any k 0,

t 0 τ Q + (f, f ) + L + (f (•, s) G(•, s, t) ds H k+1 η C t 0 e K(t-s) τ Q + (f, f )(•, s) 2 H k+1 η+1 + L + (f )(•, s) 2 H k+1 η+1 G(•, s, t) 2 W k+1,∞ -1 ds 1/2
.

Recall now the so-called Bouchut-Desvillettes-Lu regularity result in Propositions 3.1 and 3.2:

Q + (f, f ) H k+1 η+1 C f 2 H k η+3 + f 2 L 1 η+3 and L + (f ) H k+1 η+1 C F 1 H k η+3 f H k η+3 + F 1 L 1 η+3 f L 1 η+3
.

Arguing now as in [37, Proposition 5.2] and using the estimate (6.4) with the choice δ = η + 3, we get which proves (6.3). The proof of (6.2) is similar.

t 0 τ Q + (f, f ) + L + (f ) (•, s) G(•, s, t) ds
A direct consequence of the previous result together with the uniform L 2 bounds is the uniform in time propagation of Sobolev norms. The proof is carried on exactly as in [START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF]Proposition 3.5]. Proposition 6.3. Let F 1 satisfy Assumption 2.1. Let f 0 ∈ L 1 2 , f 0 0 with unit mass and let f be the unique solution of the Boltzmann equation (1.1) in C(R + ; L 1 2 ) ∩ L ∞ (R + ; L 1 3 ) associated with f 0 . Then, for all s > 0 and η 1, there exists w(s) > 0 such that

f 0 ∈ H s η+w =⇒ sup t 0 f (•, t) H s η < +∞ .
The previous ingredients allow to proof the following theorem, see [START_REF] Mouhot | Regularity theory for the spatially homogeneous Boltzmann equation with cut-off[END_REF]Theorem 5.5] for the proof. 2 ) ∩ L ∞ (R + ; L 1 3 ) associated with f 0 . Let F 1 satisfy Assumption 2.1. Let s 0, q 0 be arbitrarily large. Then f splits into the sum of a regular and a singular part

f = f R + f S where    sup t 0 f R (t) H s q ∩L 1 2 < +∞ , f R 0 ∃λ > 0 : f S (t) L 1 2 = O e -λt .
Proof. The proof is easily adapted from [START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres, Part II: Self-similar solution and tail behavior[END_REF]Theorem 3.6] since the L 1 -stability result (Proposition 4.7), the Duhamel representation (Proposition 6.2), the uniform propagation of Sobolev norms (Proposition 6.3) allow to adapt directly [START_REF] Mouhot | Regularity theory for the spatially homogeneous Boltzmann equation with cut-off[END_REF]Theorem 5.5].

Finally, Theorem 6.4 allows to prove the main Theorem 6.1.

Lemma 4 . 1 ( 1 m

 411 Povzner-like estimates for L). Let Ψ(x) = x r , r > 1. Then, there exist positive constants k r and A r such that |v -w|J e [| • | 2r ](v, w) A r |v||w| 2r + |v| 2r |w| + m r r |v -w||w| 2r -k r |v| 2r+1 + |w| 2r+1 , for any v, w ∈ R 3 . Proof. Bearing in mind that J e [| • | 2r ](v, w) is provided by (4.5) and (4.6), first of all, since Ψ m 1 |w ⋆ | 2 0, we note that

Proposition 4 . 4 (

 44 Propagation of L p -norms). Let p ∈ (1, ∞) and f 0 ∈ L 1 2 ∩ L p with unit mass.Then, the solution f (t) to (1.1) satisfies the following uniform bounds

  sup

Theorem 5 . 1 (

 51 Existence of stationary solutions). For any distribution function F 1 (v) satisfying Assumption 2.1 and any τ 0, there exists a nonnegative F ∈ L 1 2 ∩ L p , p ∈ (1, ∞) with unit mass and positive temperature such that τ Q(F, F ) + L(F ) = 0.

Theorem 6 . 4 (

 64 Exponential decay of singularities). Let f 0 ∈ L 1 2 ∩ L 2 with unit mass and let f be the unique solution of the Boltzmann equation (1.1) in C(R + ; L 1

  with Φ Sn smooth and with compact support 2 n , n , and b Sm smooth and supported in -1 + 2 m , 1 -2 m . More precisely, defining the smoothed-out operator in angular and radial variables L + Sm,n as in [32, Section 2.4]:

  we get the existence of positive constants C 4 , C 5 and C 6 such that

  There are some positive constants C Duh , K such that for any k ∈ N and η 0Q + (f, f ) + L + (f ) (•, s) dsThe proof is quite similar to [37, Proposition 5.2]. Here, for simplicity we have done it for natural k, although it is simple to generalize it to k > 0 by interpolation. Precisely, for any f ∈ L 1 define

	Proposition 6.2. we have
	f 0 (•)G(•, 0, t) H k+1 η	C Duh e -Kt f 0 H k+1 η+1	sup 0 r t	f (•, r) 2 H k η+3 + sup 0 r t	f (•, r) k+3 η+3 H k	(6.2)
	and
	t
	0	η G(•, s, t) τ H k+1	(6.3)
			C Duh sup 0 r t	f (•, r) 2 H k η+3	+ sup 0 r t	f (•, r) k+3 η+3 H k	.
	Proof.
				r) dr		0 s t,	v ∈ R 3 .

  δ g H k Let us fix now k ∈ N and δ > 3/2. Since F 1 ∈ H k δ due to Assumption 2.1, one deduces that Σ(f ) W k+1,∞where, as in the rest of the proof, we shall denote any positive constant independent of f and possibly dependent on F 1 by C. Setting F (v, s, t) = according to Assumption 2.1 and (2.11), we see that there exists some constant χ > 0 such thatΣ(f )(v) L(F 1 )(v) χ, ∀f ∈ L 1 , f 0, ∀v ∈ R 3 .By taking the successive derivatives of G(v, s, t) = exp(-F (v, s, t)), one gets as in [37, Proposition 5.2] < K < χ. Then, we shall use the following estimate[START_REF] Mouhot | Regularity theory for the spatially homogeneous Boltzmann equation with cut-off[END_REF] Lemma 5.3] that allows to exchange a time integral and a Sobolev norm:

	δ C f + F 1 H k , δ t s Σ(f )(v, r) dr, ∀g ∈ H k δ . , ∀f ∈ H k δ f (•, r) 2 H k δ dr 1/2 + C(t -s) F 1 H k δ t -s t Ce -χ(t-s) √ -1 C √ t -s s 0 for any g (k+1)/2 t s f (•, r) 2 H k δ dr + (t -s) F 1 H k , 0 s t. δ + 1 Ce -K(t-s) 1 + sup s r t f (•, r) k+1 H k δ , (6.4) 0, G(•, s, t) W k+1,∞ one sees that F (•, s, t) W k+1,∞ -1 Now, since L(g) -1 for some 0 t 0 Z(•, s) ds H r ℓ 1 √ λ 0 H r ℓ ds e λ(t-s) Z(•, s) 2 t 1/2

Acknowledgements: JAC acknowledges the support from DGI-MEC (Spain) FEDER-project MTM2005-08024. JAC and MB acknowledge partial support of the Acc. Integ. program HI2006-0111. MB acknowledges support also from MIUR (Project "Non-conservative binary interactions in various types of kinetic models"), from GNFM-INdAM, and from the University of Parma. JAC acknowledges IPAM-UCLA where part of this work was done. Finally, BL wishes to thank Clément Mouhot for fruitful discussions.


domain: math info sdv

text:
Anne Goelzer

Philippe Bessières

Vincent Fromion

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Global reconstruction of genetic and metabolic regulatory networks of Bacillus subtilis

Anne Goelzer, Philippe Bessières and Vincent Fromion

Mathématique, Informatique et Génome (MIG), INRA, 78352 Jouy-en-Josas, France

In these last years, the reconstruction of genetic and metabolic regulatory networks at the cell level and their analysis have become an active area of research in (systems) biology, because of its mandatory role. One of the challenging problems that reconstruction can help to solve, consists in identifying the ways that the bacteria "senses" its physiological state and environment (mainly through metabolite pools and fluxes), and how the cell coordinates its genetic network versus metabolic network to adapt to environmental changes. Few tools for automatic reconstruction exist, or they only allow the reconstruction of genetic interaction networks and metabolic pathways [START_REF] Karp | The pathway tools software[END_REF]. Regulatory network are mainly reconstructed from literature [START_REF] Herrgard | Reconstruction of microbial transcriptional regulatory networks[END_REF] and specific databases [START_REF] Makita | DBTBS: database of transcriptional regulation in Bacillus subtilis and its contribution to comparative genomics[END_REF]. Such a reconstruction exists for several model organisms such as Saccharomyces cerevisiae [START_REF] Herrgard | Integrated analysis of regulatory and metabolic networks reveals novel regulatory mechanisms in Saccharomyces cerevisiae[END_REF], but not for Bacillus subtilis.

In order to understand the global regulation of the cell, and the central role of metabolic pools as physiological sensors, we have reconstructed the genetic and metabolic regulatory networks of Bacillus subtilis, with a special attention to metabolites involved in regulation. Our model not only integrates the known transcription factors and their metabolic effectors, but also the known allosteric regulations of enzymes. We have functionally identified some key metabolites that seem central to the cell. The genetic and metabolic networks appear to be organized in modules, coupled by central metabolites (ATP, GTP, ppGpp) whose level should correlate to environmental conditions (nutrient availability in particular).

The main goal of this talk is to present the reconstruction of the regulatory network of Bacillus subtilis for the main metabolic pathways, and to point the state of knowledge available in literature and databases for this bacterium [START_REF] Goelzer | MIG internal report[END_REF]. We will also present some graphic representations that summarize genetic and metabolic regulations for each pathway, and the modular organization of metabolic pathways.


domain: sdv sde

text:
Mathilde M Causse

Marion Prudent

Nadia Bertin

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Combining fruit appearance and organoleptic quality is a new challenge for tomato breeders. Tomato flavor is primarily influenced by sweetness, which is related to sugar content in fruit. High sugar and acid contents are found in wild tomato species, but associated with small fruit size. Increasing sugar content is thus often hampered by this negative relationship. Several experiments were performed in order to analyze this antagonism. First at the genetic level, QTL mapping in several progenies revealed frequent colocalisations of QTLs controlling these two traits with opposite effects. Fine mapping experiments were then performed and, in some cases, close linkage of QTLs in repulsion was shown. In order to try to characterize the molecular bases of these QTLs, transcriptome and proteome analyses of near isogenic lines differing only in the QTL region were performed and revealed several genes putatively involved in the traits. In a parallel approach, in order to understand which biological processes were underlying this antagonism, a fruit removal experiment was achieved, modifying the source : sink ratio of the plants and plant behaviors under two carbohydrate supply conditions were examined. The role of cell number in fruit size determination was shown and different behaviors of the genotypes identified.

INRA, UR1052 Génétique et amélioration des fruits et légumes, F-84000 Avignon, France

INRA, UR1115 Plantes et systèmes de culture horticoles, F-84000 Avignon, France