Stéphane Mischler
email: mischler@ceremade.dauphine.fr

Kinetic equations with Maxwell boundary conditions

Keywords: Mathematics Subject Classification (2000): 76P05 Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05] Vlasov-Poisson, Boltzmann and Fokker-Planck equations, Maxwell or diffuse reflection, nonlinear gas-surface reflection laws, Darrozès-Guiraud information, trace Theorems, renormalized convergence, biting Lemma, Dunford-Pettis Lemma. 6 Boltzmann, Vlasov-Poisson and Fokker-Planck equations

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction and main results

Let Ω be an open and bounded subset of R N and set O = Ω × R N . We consider a gas confined in Ω ⊂ R N . The state of the gas is given by the distribution function f = f (t, x, v) ≥ 0 of particles, which at time t ≥ 0 and at position x ∈ Ω, move with the velocity v ∈ R N . The evolution of f is governed by a kinetic equation written in the domain (0, ∞) × O and it is complemented with a boundary condition that we describe now.

We assume that the boundary ∂Ω is sufficiently smooth. The regularity that we need is that there exists a vector field n ∈ W 2,∞ (Ω; R N ) such that n(x) coincides with the outward unit normal vector at x ∈ ∂Ω. We then define Σ x ± := {v ∈ R N ; ± v • n(x) > 0} the sets of outgoing (Σ x + ) and incoming (Σ x -) velocities at point x ∈ ∂Ω as well as Σ = ∂Ω × R N and

Σ ± = {(x, v) ∈ Σ; ±n(x) • v > 0} = {(x, v); x ∈ ∂Ω, v ∈ Σ x ± }.
We also denote by dσ x the Lebesgue surface measure on ∂Ω and by dλ k the measure on (0, ∞) × Σ defined by dλ k = |n(x) • v| k dtdσ x dv, k = 1 or 2.

The boundary condition takes into account how the particles are reflected by the wall and thus takes the form of a balance between the values of the trace γf of f on the outgoing and incoming velocities subsets of the boundary:

(γ -f )(t, x, v) = R x (γ + f (t, x, .))(v) on (0, ∞) × Σ -, (1.1)
where γ ± f := 1 (0,∞)×Σ± γf . The reflection operator is time independent, local in position but can be local or nonlocal in the velocity variable. In order to describe the interaction between particles and wall by the mean of the reflection operator R, J. C. Maxwell [START_REF] Maxwell | On stresses in rarefied gases arising from inequalities of temperature[END_REF] proposed in 1879 the following phenomenological law by splitting the reflection operator into a local reflection operator and a diffuse (also denominated as Maxwell) reflection operator (which is nonlocal in the velocity variable):

R = (1 -α) L + α D. (1.2)
Here α ∈ (0, 1] is a constant, called the accommodation coefficient. The local reflection operator L is defined by

(L x φ) (v) = φ(R x v), with R x v = -v (inverse reflection) or R x v = v -2 (v • n(x)
) n(x) (specular reflection). The diffuse reflection operator D = (D x ) x∈∂Ω according to the Maxwellian profile M with temperature (of the wall) Θ > 0 is defined at the boundary point x ∈ ∂Ω for any measurable function φ on Σ x + by

(D x φ)(v) = M (v) φ(x),
where the normalized Maxwellian M is

M (v) = (2 π) 1-N 2 Θ -N +1 2 e -|v| 2 2 Θ , (1.3)
and the out-coming flux of mass of particles φ(x) is

φ(x) = v ′ •n(x)>0 φ(v ′ ) v ′ • n(x) dv ′ = Σ x + φ M dµ x . (1.4)
It is worth emphasizing that the normalization condition (1.3) is made in order that the measure dµ x (v) := M (v) |n(x) • v| dv is a probability measure on Σ x ± for any x ∈ ∂Ω. Moreover, for any measurable function φ on Σ x + there holds

Σ x - R x φ |n(x) • v| dv = Σ x - L x φ |n(x) • v| dv = Σ x - D x φ |n(x) • v| dv = Σ x + φ n(x) • v dv, (1.5)
which means that all the particles which reach the boundary are reflected (no particle goes out of the domain nor enters in the domain).

The reflection law (1.2) was the only model for the gas/surface interaction that appeared in the literature before the late 1960s. In order to describe with more accuracy the interaction between molecules and wall, other models have been proposed in [START_REF] Cercignani | The Boltzmann equation and its application[END_REF], [START_REF] Cercignani | Scattering kernels for gas/surface interaction[END_REF], [START_REF] Kuščer | Phenomological aspects of gas-surface interaction[END_REF] where the reflection operator R is a general integral operator satisfying the so-called non-negative, normalization and reciprocity conditions, see [START_REF] Cercignani | The mathematical theory of dilute gases[END_REF] and Remark 6.4. We do not know whether our analysis can be adapted to such a general kernel. However, the boundary condition can be generalized in an other direction, see [START_REF] Cercignani | a new scattering kernel in kentic theory of gases[END_REF], [START_REF] Bogdanov | Interaction of gases with surfaces[END_REF], and we will sometimes assume that the following nonlinear boundary condition holds

R φ = (1 -α) L φ + α D φ, α = α( φ), (1.6)
where α : R + → R + is a continuous function which satisfies 0 < ᾱ ≤ α(s) ≤ 1 for any s ∈ R + .

In the domain, the evolution of f is governed by a kinetic equation

∂f ∂t + v • ∇ x f = I(f ) in (0, ∞) × O, (1.7)
where I(f ) models the interactions of particles each one with each other and with the environment. Typically, it may be a combination of the quadratic Boltzmann collision operator (describing the collision interactions of particles by binary elastic shocks), the Vlasov-Poisson operator (describing the fact that particles interact by the way of the two-body long range Coulomb force) or the Fokker-Planck operator (which takes into account the fact that particles are submitted to a heat bath). More precisely, for the nonlinear boundary condition (1.6) we are able to deal with Fokker-Planck type equations, in particular the Fokker-Planck-Boltzmann equation (FPB in short) and the Vlasov-Poisson-Fokker-Planck system (VPFP in short), while for a constant accommodation coefficient we are able to deal with Vlasov type equations such as the Boltzmann equation and the Vlasov-Poisson system (VP in short). We refer to section 6 where these models are presented. It is worth mentioning that the method presented in this paper seems to fail for the Vlasov-Maxwell system.

Finally, we complement these equations with a given initial condition f (0, .) = f in ≥ 0 on O, (1.8) which satisfies the natural physical bounds of finite mass, energy and entropy

O f in (1 + |v| 2 + | log f in |) dxdv =: C 0 < ∞. (1.9)
We begin with a general existence result that we state deliberately in an imprecise way and we refer to section 6 (and Theorem 6.2) for a more precise statement. (1.6) or the boundary value problem associated to the Boltzmann equation or the VP system with constant accommodation coefficient (1.2). For any non-negative initial datum f in with finite mass, energy and entropy ((1.9) holds) there exists at least one (renormalized) solution f ∈ C([0, ∞); L 1 (O)) with finite mass, energy and entropy to the kinetic equation (1.7) associated to the initial datum f in and such that the trace function γf fulfills the boundary condition (1.1).

Theorem 1.1 Consider the initial boundary value problem (1.1)-(1.7)-(1.8) associated to the FPB equation or the VPFP system with possibly mass flux depending accommodation coefficient

The Boltzmann equation and the FPB equation for initial data satisfying the natural bound (1.9) was first studied by R. DiPerna and P.-L. Lions [START_REF] Diperna | On the Fokker-Planck-Boltzmann equation[END_REF][START_REF] Diperna | On the Cauchy problem for Boltzmann equation: global existence and weak stability[END_REF][START_REF] Diperna | Global solutions of Boltzmann's equation and entropy inequality[END_REF] who proved stability and existence results for global renormalized solutions in the case of the whole space (Ω = R N ). Afterwards, the corresponding boundary value problem with reflection boundary conditions (1.1) and constant accommodation coefficient has been extensively studied in the case of the Boltzmann model [START_REF] Hamdache | Weak solutions of the Boltzmann equation[END_REF], [START_REF] Arkeryd | A global existence theorem for initial-boundary-value problem for the Boltzmann equation when the boundaries are not isothermal[END_REF], [START_REF] Arkeryd | On the solvability and assymptotics of the Boltzmann equation in irregular domains[END_REF], [START_REF] Arkeryd | On diffuse reflection at the boundary for the Boltzmann equation and related equations[END_REF], [START_REF] Arkeryd | Asymptotics of the Boltzmann equation with diffuse reflection boundary conditions[END_REF], [START_REF] Cercignani | On the initial value problem for the Boltzmann equation[END_REF], [START_REF] Goudon | Sur quelques questions relatives à la théorie cinétique des gaz et à l'équation de Boltzmann[END_REF], [START_REF] Heintz | Boundary value problems for nonlinear Boltzmann equation in domains with irregular boundaries[END_REF], [START_REF] Cercignani | Initial boundary value problems for the Boltzmann equation[END_REF], [START_REF] Mischler | On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[END_REF]. It has been proved, in the partial absorption case γ -f = θ Rγ + f with θ ∈ [0, 1) and in the completely local reflection case (i.e. (1.1) holds with α ≡ 0), that there exists a global renormalized solution. But in the most interesting physical case (when θ ≡ 1 and α ∈ (0, 1]), it has only been proved in [START_REF] Arkeryd | On diffuse reflection at the boundary for the Boltzmann equation and related equations[END_REF] that the following boundary inequality condition

γ -f ≥ R(γ + f ) on (0, ∞) × Σ - (1.10)
holds, instead of the boundary equality condition (1.1). However, it is worth mentioning that if the renormalized solution built in [START_REF] Arkeryd | On diffuse reflection at the boundary for the Boltzmann equation and related equations[END_REF] is in fact a solution to the Boltzmann equation in the sense of distributions, then that solution satisfies the boundary equality condition (1.1) (a result that one deduces thanks to the Green formula by gathering the fact that the solution is mass preserving and the fact that the solution already satisfies the boundary inequality condition (1.10)). Also, the Boltzmann equation with nonlinear boundary conditions has been treated in the setting of a strong but non global solution framework in [START_REF] Goudon | Existence of solutions of transport equations with non linear boundary conditions[END_REF].

With regard to existence results for the initial value problem for the VPFP system set in the whole space, we refer to [START_REF] Bouchut | Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions[END_REF], [START_REF] Bouchut | Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system[END_REF], [START_REF] Bouchut | On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with coulombic and newtonian potentials[END_REF], [START_REF] Castella | The Vlasov-Poisson-Fokker-Planck system with infinite kinetic energy[END_REF], [START_REF] Carrillo | On the initial value problem for the VPFP system with initial data in L p spaces[END_REF], [START_REF] Carrillo | On the Vlasov-Poisson-Fokker-Planck equation with measures in Morrey spaces as initial data[END_REF], [START_REF] Diperna | Solutions globales d'équations du type Vlasov-Poison[END_REF], [START_REF] Rein | Generic global classical solutions of the Vlasov-Fokker-Planck-Poisson system in three dimensions[END_REF], [START_REF] Victory | On classical solutions of Vlasov-Poisson-Fokker-Planck systems[END_REF], [START_REF] Carrillo | Asymptotic behaviour and selfsimilarity for the three dimensional Vlasov-Poisson-Fokker-Planck system[END_REF], [START_REF] Soler | Asymptotic behavior for the Vlasov-Fokker-Planck-Poisson system[END_REF] as well as [START_REF] Chandrasekhar | Stochastic problems in physics and astronomy[END_REF] for physical motivations. The initial boundary value problem has been addressed in [START_REF] Bonilla | Asymptotic behavior of an initial boundary value problem for the Vlasov-Poisson-Fokker-Planck System[END_REF], [START_REF] Carrillo | Global weak solutions for the initial-boundary value problems to the Vlasov-Poisson-Fokker-Planck system[END_REF]. We also refer to [START_REF]Weak solutions of the Vlasov-Poisson initial boundary value problem[END_REF], [START_REF] Abdallah | Weak solutions of the Vlasov-Poisson initial boundary value problem[END_REF], [START_REF] Guo | Global weak solutions of the Vlasov-Poisson system with boundary conditions[END_REF], [START_REF] Poupaud | Boundary value problems for the stationary Vlasov-Poisson system[END_REF], [START_REF] Weckler | Vlasov-Poisson initial boundary value problem[END_REF] for the initial boundary value problem for the VP system and to [START_REF] Poupaud | Boundary value problems for the stationary Vlasov-Poisson system[END_REF] for the corresponding stationary problem. We emphasize that in all these works only local reflection or prescribed incoming data are treated, and to our knowledge, there is no result concerning the diffuse boundary condition for the VP system or for the VPFP system.

We also mention that there is a great deal of information for the boundary value problem in an abstract setting in [START_REF] Voigt | Fonctional analytic treatment of the initial boundary value problem for collisionless gases[END_REF], [START_REF] Greenberg | Boundary value problems in abstract kinetic theory[END_REF] with possibly nonlinear boundary conditions [START_REF] Beal | Abstract time dependent transport equations[END_REF], [START_REF] Petterson | On solutions to the linear Boltzmann equation with general boundary conditions and infiniterange forces[END_REF].

In short, the present work improves the already known existence results for kinetic equations with diffusive boundary reflection into three directions.

• On the one hand, we prove that (1.1) is fulfilled, while only the boundary inequality condition (1.10) was previously established.

• On the other hand, we are able to consider a large class of kinetic models (including Vlasov-Poisson term) while only the Boltzmann equation (or linear equations) could be handled with earlier techniques.

• Finally, we are able to handle some nonlinear boundary condition in the case of Fokker-Planck type equation.

We do not present the proof of Theorem 1.1 (nor the proof of its accurate version Theorem 6.2) because it classically follows from a sequential stability or sequential compactness result that we present below and a standard (but tedious) approximation procedure, see for instance [START_REF] Mischler | On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[END_REF] or the above quoted references. We deliberately state again the sequential stability result in an imprecise way, referring to section 6 for a more accurate version. (1.6) or the boundary value problem associated to the Boltzmann equation or the VP system with constant accommodation coefficient (1.2). Let then (f n ) be a sequence of (renormalized) solutions to that equation and assume that (f n ) and the trace sequence (γf n ) satisfy the natural physical a priori bounds (to be specified for each model). If f n (0, .) converges to f in weakly in L 1 (O) then, up to the extraction of a subsequence, f n converges (at least) weakly in L 1 ([0, T ] × O) for all T ∈ (0, ∞) to a (renormalized) solution f to the kinetic equation (1.7) with initial value f in . Furthermore, for any ε > 0 and T > 0 there exists a measurable set A ⊂ (0, T ) × ∂Ω such that meas ((0, T ) × ∂Ω \ A) < ε and

Theorem 1.2 Consider the initial boundary value problem (1.1)-(1.7)-(1.8) associated to the FPB equation or the VPFP system with possibly mass flux depending constant accommodation coefficient

γ + f n ⇀ γ + f weakly in L 1 (A × R N , dλ 1 ), (1.11)
the convergence being strong in the case of the Fokker-Planck type equations. As a consequence we can pass to the limit in the reflection boundary condition (1.1)-(1.2) (and (1.1)- (1.6) in the case of Fokker-Planck type equations), so that the reflection boundary condition (1.1) is fulfilled.

Let us briefly explain the main steps and difficulties in the proof of the stability result.

• The first step consists in collecting the physical estimates available on the solution f to the equation (1.1)-(1.7)-(1.8) and on its trace γf . In the interior of the domain the a priori bounds satisfied by f strongly depend on the model considered but they are the same than those available in the case of the whole space. In general, for the trace, we are only able to prove that

∀ T T 0 ∂Ω E γ + f M dσ x dt ≤ C T , (1.12)
with C T only depending on C 0 and T , where the functional E = E x is the Darrozès-Guiraud information defined by

E φ := Σ x + h(φ) dµ x -h Σ x + φ dµ x , h(s) = s log s, (1.13)
and where we recall that dµ x (v) := M (v) |n(x) • v| dv is a probability measure on Σ x + so that E(φ) ≥ 0 thanks to the Jensen inequality. Let us emphasize that additionally to the a priori bound of the Darozès-Guiraud information (1.12), we can prove an L 1 a priori bound in the case of the Boltzmann equation (and of the FPB equation) and only an L 1/2 a priori (but also a posteriori) bound in the case of the VP system (and the FPVP system): in both cases, we do not have any a priori information on the trace which guaranties uniform local equiintegrability on the trace functions of a sequence of solutions. The main difficulty is thus the lack of a good a priori bound on the trace.

• The next step consists in specifying the sense of the equations. The physical a priori estimates on f make possible to give a sense to (1.7) in a renormalized sense as introduced by DiPerna and Lions. What is then the meaning of the trace γf of f ? That so-called trace problem has been studied in [START_REF] Bardos | Problèmes aux limites pour les E.D.P. du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport[END_REF], [START_REF] Cessenat | Théorèmes de trace L p pour les espaces de fonctions de la neutronique[END_REF], [START_REF] Agoshkov | On the existence of traces of functions in spaces used in transport theory problems (in Russian)[END_REF], [START_REF] Ukaï | Solutions of the Boltzmann equations, Paterns and Waves-Qualitative Analysis of Nonlinear Differential Equations[END_REF], [START_REF] Greenberg | Boundary value problems in abstract kinetic theory[END_REF], [START_REF] Cannone | On the trace theorem in kinetic theory[END_REF] for the Vlasov equation with a Lipschitz force field and extended to the Vlasov-Fokker-Planck equation in [START_REF] Carrillo | Global weak solutions for the initial-boundary value problems to the Vlasov-Poisson-Fokker-Planck system[END_REF]. In the case of the VP and the VPFP systems, the a priori estimate on the force field does not guarantee Lipschitz regularity but only Sobolev regularity. A trace theory has been developed in [START_REF] Mischler | On the trace problem for solutions of the Vlasov equation[END_REF][START_REF] Mischler | On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[END_REF] for the (possibly renormalized) solutions of the Vlasov equation with a force field in Sobolev space that we extend here to the solutions of the Vlasov-Fokker-Planck equation. The trace of a solution is here defined by a Green formula written on the renormalized equation.

• In a last step, we have to pass to the limit in a sequence of solutions which satisfy the "natural physical bounds". For the equation satisfied by f in the interior of the domain, the proofs have been done already by DiPerna-Lions [START_REF] Diperna | Solutions globales d'équations du type Vlasov-Poison[END_REF][START_REF] Diperna | On the Fokker-Planck-Boltzmann equation[END_REF][START_REF] Diperna | On the Cauchy problem for Boltzmann equation: global existence and weak stability[END_REF] and Lions [START_REF] Lions | Compactness in Boltzmann equation via Fourier integral operators and applications Part I[END_REF], and nothing has to be changed.

The main difficulty solved here is to handle the boundary condition which is made up of two equations:

(1) the renormalized Green formula which links together the solution f in the interior of the domain with its trace function γf ;

(2) the boundary equality condition (1.1) which connects together the incoming velocity particles density γ -f with the outgoing velocity particles density γ + f . Let us emphasize that using only the L 1 boundedness information (as it is available for the Boltzmann equation for instance) on a sequence (γf n ) of the trace of solutions to a kinetic equation satisfying the boundary condition (1.1) it is only possible to prove the boundary inequality condition (1.10). Indeed, on the one hand as in [START_REF] Arkeryd | On diffuse reflection at the boundary for the Boltzmann equation and related equations[END_REF] we may use that, up to the extraction of a subsequence, f n ⇀ f weakly in L 1 and γ ± f n ⇀ η ± in the weak sense of measures for some measures η ± ≥ 0. Then the limit boundary densities η ± fulfill the boundary equality condition (1.1), η -= R η + , whereas they are not the trace functions associated to f but they are their regular parts with respect to the lebesgue measure: γf ± = dη± dλ1 . Putting together these two informations yields to the boundary inequality condition (1.10). On the other hand, as in [START_REF] Goudon | Sur quelques questions relatives à la théorie cinétique des gaz et à l'équation de Boltzmann[END_REF], we may use that, up to the extraction of a subsequence, f n ⇀ f weakly in L 1 and γ ± f n ⇀ g ± in the biting L 1 -weak sense (see below) for some measurable functions g ± ≥ 0. Then the limit boundary densities g ± are the trace functions associated to f , g ± = γ ± f , whereas the reflection operator is only l.s.c. with respect to the biting L 1 -weak convergence, R g + ≤ lim inf R γ + f n . Again, these two informations only imply the boundary inequality condition (1.10).

• In this paper, we prove some L 1 -weak (L 1 -strong in the case of FP models) convergence in the velocity variable for the sequence (γ + f n ) (as stated in Theorem 1.2) which is strong enough to conclude. Our proof is based on the use of notions of weak-weak convergences, namely the renormalized convergence (r-convergence) and the biting L 1 -weak convergence (b-convergence). We say weak-weak convergences in order to express the fact that they are extremely weak sense of convergence (weaker, for instance, to the L 1 -weak convergence and to the a.e. convergence) and which are not furthermore associated to any topological structure, see Proposition A.2. On the one hand, thanks to the trace theory, we prove that the sequence of trace functions (γf n ) r-converges to γf (as well as a.e. in the case of FP models). Next, thanks to some additional L 1 a priori bounds, or because the r-convergence is almost equivalent to the b-convergence when the limit function belongs to L 0 , we deduce that γ + f n b-converges to γ + f . Finally, that information and the boundedness of the Darrozès-Guiraud information leads to (1.11).

Let us now briefly outline the contents of the paper. In section 2, we consider the free transport equation for which we apply the above strategy. We present for this very simple case the different tools (renormalized and biting L 1 -weak convergence, trace theory and Darrozès-Guiraud information), we state a first velocity L 1 -weak compactness result and then we prove the corresponding version of the stability Theorem 1.2. In Section 3, we develop the notion of renormalized convergence in a more general framework and we prove some more accurate version of biting L 1 -weak convergence and velocity L 1 -weak compactness. In Section 4, we present the trace theory for the Vlasov-Fokker-Planck equation with Sobolev regularity on the force field. In Section 5, putting together the results from Section 3 and Section 4, we establish the renormalized convergence and the almost everywhere convergence of trace functions sequences. In Section 6 we present the models and we establish the main stability (up to the boundary) results. Finally, in the Appendix, we come back to the notion of renormalized convergence for which we give several relevant examples and counterexamples.

2 An illuminating example: the free transport equation.

In this section we assume that f is governed by the free transport equation

∂f ∂t + v • ∇ x f = 0 in (0, ∞) × O, (2.1)
complemented with the initial condition (1.8) and the boundary reflection condition (1.1) with constant restitution coefficient α ∈ (0, 1]. Our aim is to adapt the DiPerna-Lions stability theory to that simple boundary value problem. We follow the strategy expounded in the introduction. We first collect the a priori bounds satisfied by a solution to the boundary value problem (2.1)-(1.1)-(1.8) with initial datum satisfying (1.9). We next present some general functional analysis tools which roughly speaking make possible to deduce the L 1 weak convergence in the v variable of a sequence which is uniformly bounded in L 1 and for which the associated Darrozès-Guiraux information is uniformly bounded. We finally state and prove the stability result associated to the boundary value problem (2.1)-(1.1)-(1.8).

Remark 2.1 It is worth mentioning that the proof of the corresponding stability result for the Boltzmann equation is essentially the same as for the free transport equation. We refer to section 6 where that model is handled. However, the reader who is only interested in the Boltzmann model may easily adapt the proof below with the arguments introduced in [START_REF] Mischler | On the trace problem for solutions of the Vlasov equation[END_REF] (it will be more elementary than the proof presented in Section 3 to Section 6 which is made in order to also deal with a Vlasov-Poisson term and/or with a Fokker-Planck term).

A priori bounds.

Lemma 2.2 For any non-negative initial datum f in such that (1.9) holds and any time T ∈ (0, ∞) there exists a constant C T (only depending on C 0 and T ) such that any sufficiently regular and decreasing at the infinity solution f to the initial boundary value problem (2.1)

-(1.1)-(1.8) satisfies sup [0,T ] O f (1 + |v| 2 + | log f |) dxdv + α T 0 ∂Ω E γ + f M dσ x dt ≤ C T , (2.2)
where E is defined in (1.13), and

α T 0 Σ γf (1 + |v| 2 ) |n(x) • v| dvdσ x dt ≤ C T . (2.3)
Proof of Lemma 2.2. We consider a solution f of (1.1)-(2.1)-(1.8), which is sufficiently regular and decreasing at the infinity in such a way that all the integrations by parts in our arguments are legitimate.

Step 1. Mass conservation. Integrating the free transport equation (2.1) over x, v, using the Green formula and the identity (1.5), we obtain the mass conservation

∀ t ≥ 0 O f (t, .) dvdx = O f in dvdx.
Step 2. Relative entropy.

Multiplying the free transport equation (2.1) by h ′ (f /M ), with h(s) = s log s, and integrating it over x, v, we have

d dt O h(f /M ) M dvdx = Σ h(γf /M ) M v • n(x) dvdσ x . (2.4)
The Darrozès-Guiraud inequality states that the entropy boundary flux at the right hand side of equation (2.4) is non-negative. That is a straightforward consequence of the Jensen inequality taking advantage that dµ x (v) = M |v • n(x)| dv is a probability measure. We present now the proof of an accurate version of the Darrozès-Guiraud inequality which make precise how much that term is non-negative. From the boundary reflection condition (1.1), the convexity of h and the expression (1.2) of the reflection operator, we have

R N h(γf /M ) dµ x (v) = Σ + x h(γ + f /M ) dµ x (v) - Σ - x h(Rγ + f /M ) dµ x (v) (2.5) ≥ α Σ + x h(γ + f /M ) dµ x (v) - Σ - x h(Dγ + f /M ) dµ x (v) +(1 -α) Σ + x h(γ + f /M ) dµ x (v) - Σ - x h(Lγ + f /M ) dµ x (v) = α Σ + x h(γ + f /M ) dµ x (v) -h( γ + f ) = α E x ( γ + f M ),
where we have performed the change of variables L x : v → R x v in the second term with jac L x = 1, so that this term vanishes, and where the Darrozès-Guirraud information functional E x is defined in (1.13) and γ + f is defined in (1.4). Gathering (2.4) and (2.5), we get

d dt O h(f /M ) M dvdx + α ∂Ω E x (γ + f ) dσ x ≤ 0.
Finally, using the elementary estimates, that one can find in [START_REF] Lions | Conditions at infinity for Boltzmann's equation[END_REF] for instance,

R N f |v| 2 4 Θ + | log f | dv ≤ C M + R N h(f /M ) M dv, (2.6) and R N h(f in /M ) M dv ≤ R N f in |v| 2 4 Θ + | log f in | dv + C M , (2.7)
for some constant C M ∈ (0, ∞), we obtain that (2.2) holds.

Step 3. Additional L 1 estimates. For the sake of completeness we sketch the proof of the L 1 a priori bound (2.3) already established in [START_REF] Arkeryd | On diffuse reflection at the boundary for the Boltzmann equation and related equations[END_REF][START_REF] Mischler | On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[END_REF]. We multiply the free transport equation (2.1) by n(x) • v and we integrate it over all variables, to get

T 0 Σ γf (n(x) • v) 2 dvdσ x dt = O f n(x) • v dvdx 0 T + T 0 O f v • ∇ x n(x)v dvdxdt,
so that, thanks to (2.2) and because n ∈ W 1,∞ (Ω),

T 0 Σ γf (n(x) • v) 2 dvdσ x dt ≤ C T . (2.8)
We then remark that for the constant

C 1 := M (v) (n(x) • v) 2 -1 L 1 (Σ x -) we have γ + f = C 1 Σ x - M (v) γ + f (n(x) • v) 2 dv = C 1 Σ x - γ -f (n(x) • v) 2 dv, (2.9)
and that for the constant

C 2 := M (v) (1 + |v| 2 ) |n(x) • v| L 1 (Σ x -) we have Σ x - γ -f (1 + |v|) 2 |n(x) • v| dv = Σ x - M (v) γ + f (1 + |v| 2 ) |n(x) • v| dv = C 2 γ + f . (2.10)
Finally, we come back to the equation (2.1) that we multiply by |v| 2 and that we integrate in all variables. We obtain

O f (T, .) |v| 2 dvdx + α T 0 Σ+ γ + f |v| 2 n(x) • v dvdσ x dt (2.11) = O f in |v| 2 dvdx + α T 0 Σ- γ -f |v| 2 |n(x) • v| dvdσ x dt.
Estimate (2.3) follows gathering (2.9), (2.8) and (2.10), (2.11). ⊓ ⊔ 2.2 Biting L 1 -weak convergence and L 1 -weak compactness in the velocity variable.

In this section we present some functional analysis results which make possible to obtain the L 1weak convergence in the v variable of a sequence which satisfies a L 1 bound and a uniform bound of its Darrozès-Guiraud information. We state the result in some more general setting because we believe that it may have its own interest (outside the applications to the trace theory for kinetic equations). For that purpose, we introduce a first notion of weak-weak convergence, namely the biting L 1 -weak convergence. It seems to have been introduced by Kadec and Pelzyński [START_REF] Kadec | Bases, lacunary sequence and complemented subspaces in the space Lp[END_REF] and rediscovered and developed in a L 1 and bounded measure framework by Chacon and Rosenthal in the end of the 1970's, see [START_REF] Gaposkhin | Convergences and limit theorems for sequences of random variables[END_REF], [START_REF] Brooks | Continuity and compactness of measures[END_REF]. Let us first recall the definition of the biting L 1 -weak convergence that we extend to a "L framework".

In the sequel Y = (Y, ν) stands for a separable and σ-compact topological space, i.e. Y = ∪ k Y k where (Y k ) is an increasing sequence of compact sets, endowed with its σ-ring of Borel sets and with a locally finite Borel measure ν. We denote by L(Y ) the space of all measurable functions φ : Y → R and by L 0 (Y ) the subset of all measurable and ν-almost everywhere finite functions. In order to simplify the presentation, we will be only concerned with non-negative functions of L and L 0 . Thus, in this section, we also denote by L and L 0 the cone of non-negative functions in these spaces, and we do not specify it anymore.

Definition 2.3 We say that a sequence (ψ n ) of L(Y ) converges in the biting L 1 -weak sense (or b-converges) to ψ ∈ L(Y ), denoted ψ n b ⇀ ψ, if for every k ∈ N we can find A k ⊂ Y k in such a way that (A k ) is increasing, ν(Y k \A k ) < 1/k, ψ n ∈ L 1 (A k ) for all n large enough and ψ n ⇀ ψ weakly in L 1 (A k ). In particular, that implies ψ ∈ L 0 (Y ).
The fundamental result concerning the biting L 1 -weak convergence is the so-called biting Lemma that we recall now. We refer to [START_REF] Cembrabos | Banach spaces of vector-valued functions[END_REF], [START_REF] Ball | Remarks on Chacon's biting lemma[END_REF], [START_REF] Brooks | Continuity and compactness of measures[END_REF], [START_REF] Gaposkhin | Convergences and limit theorems for sequences of random variables[END_REF] and [START_REF] Kadec | Bases, lacunary sequence and complemented subspaces in the space Lp[END_REF] for a proof of this Lemma. We also refer to [START_REF] Acerbi | Semi-continuity problems in the calculus of variations[END_REF] and [START_REF] Coifman | Compensated compactness and Hardy spaces[END_REF] for other developments related to the biting L 1 -weak convergence. Extension of this theory to multi-valued functions has been done by Balder, Castaing, Valadier and others; we refer to [START_REF] Saadoune | Extracting a good subsequence from a bounded sequence of integrable functions[END_REF] for precise references.

Theorem 2.4 (biting Lemma). Let (ψ n ) be a bounded sequence of L 1 (Y ). There exists ψ ∈ L 1 (Y ) and a subsequence (

ψ n ′ ) such that (ψ n ′ ) b-converges to ψ and ψ L 1 ≤ lim inf ψ n ′ L 1 .
Our first result is a kind of intermediate result between the biting Lemma and the Dunford-Pettis Lemma. More precisely, we prove the L 1 -weak compactness in the v variable for sequences (φ n ) which are bounded in L 1 and such that the associated Darrozès-Guiraud information is uniformly (in n) bounded. It is based on the biting Lemma, the Dunford-Pettis Lemma and a convexity argument.

Theorem 2.5 Consider j : R + → R a convex function of class C 2 (0, ∞) such that j(s)/s → +∞ when s ր +∞ and such that the application J from (R + ) 2 to R defined by J(s, t) = (j(t) - j(s)) (t -s) is convex, ω a non-negative function of R N such that ω(v) → ∞ when |v| → ∞ and, for any y ∈ Y , a probability measure µ y on R N . Assume that (φ n ) is a sequence of non-negative measurable functions on Y × R N such that Y R N φ n (y, v) (1 + ω(v)) + E(φ n (y, .)) dµ y (v) dν(y) ≤ C 1 < ∞, (2.12)
where E = E j,y is the non-negative Jensen information functional defined by

E(φ) = R N j(φ) dµ y -j R N φ dµ y if 0 ≤ φ ∈ L 1 (R N , dµ y ).
Then, there exists φ ∈ L 1 (Y × R N ) and a subsequence (φ n ′ ) such that for every k ∈ N we can find

A k ⊂ Y k in such a way that (A k ) is increasing, ν(Y k \A k ) < 1/k and φ n ′ ⇀ φ weakly in L 1 (A k × R N ; dν dµ).
Furthermore, E is a convex and weakly L 1 l.s.c. functional, and thus

Y R N φ(y, v) (1 + ω(v)) + E(φ(y, .)) dµ y (v) dν(y) ≤ C 1 . (2.13)
Proof of Theorem 2.5. From the bound (2.12) and the biting Lemma we know that there exists a subsequence n ′ such that for every k ∈ N we can find a Borel set

A = A k ⊂ Y k with ν(Y k \A) < 1/k such that R N φ n ′ dµ y (v) weakly converges in L 1 (A). (2.14)
Thanks to (2.14), the Dunford Pettis Lemma and the De La Vallée-Poussin uniform integrability criterion there is a convex function Φ = Φ k such that Φ(s)/s → ∞ when s → ∞ and

A Φ R N φ n ′ dµ y (v) dν(y) ≤ C 2 = C 2 (k) < ∞. Furthermore, we can assume that Φ(0) = 0, Φ ′ = a m in [m, m + 1] with j ′ (s 0 ) ≤ a m ր +∞, where s 0 ∈ N ⋆ is such that j(s 0 ) ≥ 0 and j ′ (s 0 ) ≥ 0.
Then we define Ψ = Ψ k by Ψ(s) = j(s) for s ∈ [0, s 0 ] and by induction on m ∈ N, we consider

t m such that j ′ (t m ) = a m -Ψ ′ (s m ) + j ′ (s m ) and we set s m+1 = [t m ] + 1, Ψ ′′ := j ′′ on [s m , t m ] and Ψ ′′ := 0 on [t m , s m+1 ] so that t m ≥ s m ≥ m and Ψ ′ (s m+1 ) ≥ a m ≥ Ψ ′ (s m ). Therefore, we have built a convex function Ψ such that the function s → j(s) -Ψ(s) is convex, Ψ(s)/s ր ∞ since Ψ ′ (s) ր ∞, and Ψ ≤ Φ since Ψ ′ ≤ Φ ′ , so that A Ψ R N φ n ′ dµ dν ≤ C 2 . (2.15)
The Jensen inequality, written for the function s → j(s) -Ψ(s), gives

R N Ψ(φ n ′ ) dµ -Ψ R N φ n ′ dµ ≤ E(φ n ′ ),
and combining it with (2.12) and (2.15) we get

A×R N Ψ(φ n ′ ) dµ dν ≤ C 1 + C 2 ,
and thus

A×R N Ψ + (φ n ′ ) dµ y dν ≤ C 1 + C 2 + A×R N Ψ -(φ n ′ ) dµ y dν (2.16) ≤ C 3 (k) := C 1 + C 2 + ν(A) sup j -< ∞.
Thanks to estimates (2.12), (2.16) and the Dunford-Pettis Lemma we get that (φ n ′ ) falls in a relatively weakly compact set of L 1 (A k × R N ) for any k ∈ N. We conclude, by a diagonal process, that there is a function φ ∈ L 1 (Y × R N ) and a subsequence (φ n ′′ ) which converges to φ in the sense stated in Theorem 2.5.

In order to prove that E is a convex functional, we begin by assuming that j

∈ C 1 (R + , R), so that E is Gâteaux differentiable. By definition of the G-differential ∇E(φ) • ψ := lim t→0 E(φ + t ψ) -E(φ) t = R N j ′ (φ) ψ dµ -j ′ R N φ dµ R N ψ dµ, for any 0 ≤ φ, ψ ∈ L ∞ (R N )
. Therefore, by the Jensen inequality, we have

∇E(ψ) -∇E(φ), ψ -φ = R N J(φ, ψ) dµ -J R N φ dµ, R N ψ dµ ≥ 0, so that ∇E is monotone and thus E is convex on L ∞ (R N ): for any 0 ≤ φ, ψ ∈ L ∞ (R N ) and any t ∈ (0, 1) E(φ + (1 -t) ψ) ≤ t E(φ) + (1 -t) E(ψ). (2.17) When j / ∈ C 1 (R + , R)
we define, for any ε > 0, the function j ε (s) = j(s + ε) -j(ε) which belongs to C 1 (R + , R), and the above computation for the associated functional E ε is correct, so that inequality (2.17) holds for E replaced by E ε . Then, writing inequality (2.17) for E ε and fixed 0 ≤ φ, ψ ∈ L ∞ (R N ), t ∈ (0, 1) and passing to the limit ε → 0 we obtain that

E is convex on L ∞ (R N ). Now let us fix 0 ≤ φ, ψ ∈ L 1 (R N ), t ∈ (0, 1). If j(φ) or j(ψ) / ∈ L 1 (R N ) then t E(φ) + (1 -t) E(ψ) = +∞
and the convex inequality (2.17) obviously holds. In the other case, we have j(φ), j(ψ) ∈ L 1 (R N ), and we can choose two sequences 0 ≤ (φ n ), (ψ n ) of L ∞ (R N ) such that φ n ր φ and ψ n ր ψ a.e.. Passing to the limit ε → 0 in the convex inequality (2.17) written for φ ε and ψ ε we get, by the Lebesgue convergence dominated Theorem and the Fatou Lemma,

R N j(t φ + (1 -t) ψ) ≤ lim inf ε→0 R N j(t φ ε + (1 -t) ψ ε ) ≤ t E(φ) + (1 -t) E(ψ) + j R N t φ + (1 -t) ψ , which exactly means that E is a convex functional in L 1 (R N ). Finally, if 0 ≤ φ, ψ ∈ L 1 (Y × R N )
and t ∈ (0, 1), then φ(y, .), ψ(y, .) ∈ L 1 (R N ) for almost every y ∈ Y and, integrating the convex inequality (2.17), we obtain that the functional

0 ≤ φ ∈ L 1 (Y × R N ) → F (φ) = Y E(φ) dν
is convex. Furthermore, by Fatou Lemma, F is l.s.c. for the strong convergence in L 1 , for the weak σ(L 1 , L ∞ ) convergence and for the biting L 1 -weak convergence, so that (2.13) holds.

⊓ ⊔

We introduce a second kind of weak-weak convergence, namely the renormalized convergence, which is the very natural notion of convergence when we deal with sequences of trace functions, as we will see below. We now present the definition (in a simplified case) and a first elementary result that we will use in the next subsection. More about the renormalized convergence is presented in section 3.

Definition 2.6 Let us define the sequence (T

M ) by setting T M (s) := s ∧ M = min(s, M ) ∀ s, M ≥ 0. We say that a sequence (φ n ) of L(Y ) converges in the renormalized sense (or r-converges) if there exists a sequence ( TM ) of L ∞ (Y ) such that T M (φ n ) ⇀ TM σ(L ∞ (Y ), L 1 (Y )) ⋆ and TM ր φ a.e. in Y. Lemma 2.7 For any sequence (φ n ) of L(Y ) and φ ∈ L 0 (Y ) such that φ n b ⇀ φ in the biting L 1 -weak sense, there exists a subsequence (φ n ′ ) such that φ n ′ r ⇀ φ in the renormalized sense.
Proof of Lemma 2.7. We follow the proof of [START_REF] Ball | Remarks on Chacon's biting lemma[END_REF] where that result is established in a L 1 framework. By assumption, for any k ∈ N, there exists a Borel set A k such that ν (Y k \A k ) < 1/k and φ n ⇀ φ weakly in L 1 (A k ). Thanks to Dunford-Pettis Lemma, there is a function δ k : R + → R + such that δ k (M ) → 0 when M → +∞ and

A k φ n 1 {φn≥M} dy ≤ δ k (M ) ∀ n, M, k ∈ N * . (2.18)
Moreover, there exists a subsequence (φ n ′ ) of (φ n ) and a sequence ( TM ) of L ∞ (Y ) such that for any M ∈ N there holds

T M (φ n ′ ) ⇀ TM σ(L ∞ (Y ), L 1 (Y )).
We obviously have that ( TM ) is an increasing sequence in L ∞ (Y ) and TM ≤ φ a.e. because that is true on any A k . Observe that

0 ≤ φ n -T M (φ n ) ≤ (φ n -M ) 1 φn≥M a.
A k |φ -TM | dν = lim n ′ →∞ A k (φ n ′ -T M (φ n ′ )) [sign(φ -TM )] dν ≤ lim inf n ′ →∞ A k φ n ′ 1 {φ n ′ ≥M} dy ≤ δ k (M ).
That proves TM → φ a.e. in Y when M → ∞, and then

φ n ′ r ⇀ φ. ⊓ ⊔ 2.3
The trace theorem and the stability result.

Let us recall the following trace theorem which makes precise the meaning of the trace of a solution.

Theorem 2.8 [56] Let g ∈ L ∞ (0, T ; L 1 (O)) satisfy Λ g := ∂ t g + v • ∇ x g = 0 in D ′ ((0, T ) × O).
There exists γg ∈ L 1 loc ((0, T ) × Σ; dλ 2 ) and g 0 ∈ L 1 (O) which satisfy the renormalized Green formula

T 0 O β(g) Λφ dvdxdt = T 0 Σ β(γ g) φ n(x) • v dvdσ x dt - O β(g 0 ) φ dxdv, (2.20)
for all β ∈ W 1,∞ (R) and all test functions φ ∈ D([0, T ) × Ō), as well as for all β ∈ W 1,∞ loc (R), with β ′ ∈ L ∞ (R), and all test functions φ ∈ D([0, T ) × Ō) such that φ = 0 on [0, T ) × Σ 0 .

We may then state our first main result. Theorem 2.9 Let f n ∈ L ∞ (0, ∞; L 1 (O)) be a sequence of solutions to the initial boundary value problem (2.1)-(1.1)- (1.8) such that both (f n ) and the trace sequence (γf n ) satisfy the associated natural a priori bounds: for any T > 0 there is a constant

C T sup [0,T ] O f n 1 + |v| 2 + | log f n | dvdx ≤ C T (2.21) and T 0 Σ γf n (1 + |v| 2 ) |n(x) • v| dvdσ x dt + T 0 ∂Ω E(γ + f n ) dσ x dt ≤ α -1 C T . (2.22)
On the one hand, there exists f ∈ L ∞ (0, ∞; L 1 (O)) satisfying (2.2) and f in ∈ L 1 (O) satisfying (1.9) such that, up to the extraction of subsequences,

f n ⇀ f σ(L 1 , L ∞ ), f n (0, .) ⇀ f in σ(L 1 , L ∞ ), (2.23)
and f is a solution to the free transport equation (2.1)-(1.8) with initial datum f in .

On the other hand, there exists η ± ∈ L 1 (0, T ) × Σ ± , dλ 1 ) for all T ∈ (0, ∞), which furthermore satisfies (2.2)-(2.3) (with γ ± f replaced by η ± ), such that, up to the extraction of a subsequence, for any T, ε > 0 there exists a measurable set A ⊂ (0, T ) × ∂Ω such that meas ((0, T ) × ∂Ω \ A) < ε and

γ ± f n ⇀ η ± weakly in L 1 (A × R N , dλ 1 ), (2.

24)

As a consequence, γ ± f = η ± and the reflection boundary condition (1.1) holds.

Proof of Theorem 2.9. First, from (2.21) and the Dunford-Pettis lemma we deduce (2.23). Then, thanks to Lemma 2.7, extracting again a subsequence if necessary, we deduce that f n r ⇀ f or more precisely, there exists two sequences ( TM ) and ( T 0

M ) such that T M (f n ) ⇀ TM σ(L ∞ , L 1 ) ⋆ and TM ր f a.e., (2.25) T M (f n (0, .)) ⇀ T 0 M σ(L ∞ , L 1 ) ⋆ and T 0 M ր f in a.e.. (2.26)
Next, from (2.22) and Theorem 2.5 (with (2.24). That implies that for any T, ε > 0 there exists a measurable set A ⊂ (0, T ) × ∂Ω such that meas ((0, T ) × ∂Ω \ A) < ε and

φ n = γ + f n /M , j(s) = s log s, ω(v) = |v| 2 , dν(y) = dσ x dt, dµ y (v) = |n(x) • v| M (v) dv) we deduce that γ + f n ⇀ η + in the sense stated in
γ + f n ⇀ η + weakly in L 1 (A), so that D(γ + f n ) ⇀ D(η + ) in the sense stated in (2.24). That also implies that for any φ ∈ L ∞ (R N v ) R N γ + f n (t, x, v) φ(R x v) n(x) • v dv ⇀ R N η + (t, x, v) φ(R x v) n(x) • v dv weakly in L 1 (A),
which means nothing but L(γ + f n ) ⇀ L(η + ) in the sense stated in (2.24). Gathering these two convergence results, we get γ -f n ⇀ η -in the sense stated in (2.24) with η -:= R(η + ). Finally, thanks to Lemma 2.7 again, extracting a subsequence if necessary, we deduce that γf n r ⇀ η := η + 1 (0,∞)×Σ+ + η -1 (0,∞)×Σ-or more precisely, there exists a sequence (γ M ) such that

T M (γf n ) ⇀ γM σ(L ∞ , L 1 ) ⋆
and γM ր η a.e.. (2.27) We write then the Green renormalized formula (2.20) for the free transport equation

T 0 O T M (f n ) Λϕ dvdxdt = T 0 Σ T M (γ f n ) ϕ n(x) • v dvdσ x dt - O T M (f n (0, .)) ϕ dxdv,
for any ϕ ∈ D([0, T ) × Ō). Using (2.25), (2.26) and (2.27), we may pass twice two the limit in the above equation, first when n → ∞, next when M → ∞, and we get

T 0 O f Λϕ dvdxdt = T 0 Σ η ϕ n(x) • v dvdσ x dt - O f in ϕ dxdv.
In other words, f is a solution to the free transport equation and γ ± f = η ± thanks to the trace Theorem 2.8. We conclude by gathering that information with the equation satisfied by η ± . ⊓ ⊔

3 On the convergence in the renormalized sense.

Basic properties.

We present the main basic properties concerning the notion of convergence in the renormalized sense. More about renormalized convergence is set out in the appendix section. In that section the framework and notations are the same as those of subsection 2.2, and again, we only deal with non-negative functions of L = L(Y ), but we do not specify it anymore.

Definition 3.1 We say that α is a renormalizing function if α ∈ C b (R) is increasing and 0 ≤ α(s) ≤ s for any s ≥ 0. We say that (α M ) is a renormalizing sequence if α M is a renormalizing function for any M ∈ N and α M (s) ր s for all s ≥ 0 when M ր ∞. Given any renormalizing sequence (α M ), we say that (φ n ) (α M )-renormalized converges to φ (or we just say that

(φ n ) r-converges to φ) if there exists a sequence (ᾱ M ) of L ∞ (Y ) such that α M (φ n ) ⇀ ᾱM σ(L ∞ (Y ), L 1 (Y )) ⋆ and ᾱM ր φ a.e. in Y.
Notice that the renormalized convergence as defined in definition 2.6 is nothing but the (T M )renormalized convergence.

Proposition 3.2 1. The (α M )-renormalized limit in the definition 3.1 does not depend on the renormalizing sequence (α M ), but only on the sequence (φ n ). In other words, given two renormalizing sequences (α M ) and (β M ), if (φ n ) (α M )-renormalized converges to φ α and (β M )-renormalized converges to φ β then φ α = φ β . 2. For any sequence (φ n ) of L there exists a subsequence (φ n ′ ) of (φ n ) and a function φ ∈ L such that (φ n ′ ) (α M )-renormalized converges to φ for any renormalizing sequence (α M ).

3. A sequence (φ n ) which converges to φ a.e. or strongly in L p , p ∈ [1, ∞], also r-converges to φ. From a sequence (φ n ) which converges to φ weakly in L p , p ∈ [1, ∞], or in the biting L 1 -weak sense, we may extract a subsequence (φ n ′ ) which r-converges to φ. Remark 3.3 1. The definition of the (α M )-renormalized convergence with α M = T M is important in order to obtain the renormalized convergence of the trace functions sequence in Theorem 5.2. Indeed, T M is not smooth enough in order to be taken as a renormalizing function for the VFP equation and we have to introduce the "smooth" renormalizing functions α := Φ M,θ .

2. Because of Proposition 3.2 we will often make the abuse of language by not specifying the renormalizing sequence (α M ) used to define the (α M )-renormalized convergence and by saying that (φ n ) r-converges (to φ) when it is only a subsequence of (φ n ) which r-converges (to φ).

3. Let us notice that in general we can not exclude that the limit φ ≡ +∞, since for instance the sequence (φ n ) defined by φ n = n belongs to L and r-converges to φ ≡ ∞.

Proof of the Proposition 3.2.

Step 0. We first claim that for any sequence (φ n ) of L and any renormalizing sequence (α M ) there exists a subsequence (φ n ′ ) of (φ n ) and φ ∈ L such that (φ n ′ ) (α M )-renormalized converges to φ. Indeed, for any M we can find a subsequence (n

M k ) k and ᾱM ∈ L ∞ such that α M (φ n M k ) ⇀ ᾱM weakly in L ∞ .
By a diagonal process we can obtain a unique subsequence (n ′ ) such that the above weak convergence holds for any M ∈ N. Furthermore, since (α M ) is increasing, we get that (ᾱ M ) is an increasing sequence of non-negative measurable functions, so that it converges to a limit φ ∈ L.

Step 1. Assume that for a renormalizing sequence (α K ) we have α K (φ n ) ⇀ ᾱK ր ψ. Thanks to step 0, there exits a sub-sequence (φ n ′ ), a sequence TM ∈ L ∞ and a function φ ∈ L such that

T M (φ n ′ ) ⇀ TM ր φ. It is clear that ∀K, M ∈ N ∀ε > 0 there is k M,ε , m K ∈ N such that α K ≤ T mK and T M ≤ α kM,ε + ε. Therefore, writing that α K (φ n ) ≤ T mK (φ n ) and T M (φ n ) ≤ α kM,ε (φ n ) + ε,
and passing to the limit n → +∞, we get ᾱK ≤ TmK ≤ φ and TM ≤ ᾱkM,ε + ε ≤ ψ + ε.

Then passing to the limit M, K ր ∞ we obtain that ψ ≤ φ ≤ ψ + ε for any ε > 0, and finally passing to the limit ε → 0 we conclude that ψ = φ.

Step 2. Let us remark that the class of renormalizing functions is separable for the uniform norm of C(R + ). For instance, the family

A = {α k } of functions α such that 0 ≤ α(s) ≤ s and α ′ (s) = J j=1 θ j 1 [aj ,aj+1[ (s), a j , θ j ∈ Q +
is countable and dense. By a diagonal process and thanks to step 0, we can find a subsequence (φ n ′ ) in such a way that for any α ∈ A there exists ᾱ ∈ L ∞ such that α(φ n ′ ) ⇀ ᾱ. Let us fix now (β M ) a renormalizing sequence. On one hand, for any M there exists a sequence (α k ) of A such that α k ≤ β M ≤ α k + 1/k for any k ∈ N and α k ր β M . We already know that α k (φ n ′ ) ⇀ ᾱk . Since (ᾱ k ) is not decreasing, it converges a.e., and we set β * M = lim ᾱk . On the other hand, thanks to Step 0, there exists a subsequence (φ n ′′ ) and a function βM such that β M (φ n ′′ ) ⇀ βM . That implies ᾱk ≤ βM ≤ ᾱk + 1/k. Passing to the limit k → ∞, we get βM = β * M . Therefore, by uniqueness of the limit, it is the all sequence β M (φ n ′ ) which converges to β * M . Finally, thanks to the usual monotony argument we deduce that φ n ′ converges in the (β M )-renormalized sense and its limit is necessary φ thanks to Step 1.

Step 3. If φ n → φ a.e. then clearly α M (φ n ) ⇀ α M (φ) L ∞ -weak and α M (φ) ր φ for any renormalizing sequence (α M ), so that φ n r ⇀ φ. If (φ n ) converges strongly or weakly in L p , p ∈ [1, ∞], then it obviously converges in the biting L 1 -weak sense and we may apply Lemma 2.7.

⊓ ⊔

Let us now define the limit superior and the limit inferior in the renormalized sense.

Definition 3.4 Let (φ n ) be a sequence of L. Consider I the set of all the increasing applications ı : N → N such that the subsequence (φ ı(k) ) k≥0 of (φ n ) n≥0 converges in the renormalized sense and note φ ı = r-lim φ ı(k) . Thanks to the Proposition 3.2.2, we know that I is not empty. We defined the limit superior and the limit inferior of (φ n ) in the renormalized sense by

∀ a, b, M ≥ 0 M ∧ (a + b) ≤ M ∧ a + M ∧ b ≤ (2 M ) ∧ (a + b), we deduce w-lim [M ∧ (φ n + ψ n )] ≤ w-lim [M ∧ φ n ] + w-lim [M ∧ ψ n ] ≤ w-lim [(2 M ) ∧ (φ n + ψ n )] so that r-lim (φ n + ψ n ) = φ + ψ. Next, from the elementary identity ∀ a, b, M ≥ 0 (a b) ∧ M = a b ∧ (M/a)
and because for any ε > 0 there holds 0 < λ -ε ≤ λ n ≤ λ + ε for n large enough, we have

(λ -ε) φ n ∧ M λ -ε ≤ (λ n φ n ) ∧ M ≤ (λ + ε) φ n ∧ M λ + ε .
We deduce that for a subsequence (λ

n ′ φ n ′ ) ∀ ε > 0 (λ -ε) T M λ-ε ≤ w-lim n ′ →∞ T M (λ n ′ φ n ′ ) ≤ (λ + ε) T M λ+ε ,
so that, passing to the limit ε → 0 and using that T M/(λ+ε) ≤ T M/(λ-ε) ,

λ TM = w-lim n ′ →∞ T M (λ n ′ φ n ′ ).
Passing to the limit M → ∞, we conclude that r-lim (λ n φ n ) = λ φ.

Step 2. We know that

β(s) = inf ℓ≥β ℓ(s),
where the inf is taken over all real values affine functions ℓ(t) = a t + b which satisfy a, b ≥ 0 and β(t) ≤ ℓ(t) for any t ≥ 0. Furthermore, for any ℓ and M , there clearly exists K M such that

T M (ℓ(s)) ≤ ℓ(T K (s)) and ℓ(T M (s)) ≤ T K (ℓ(s)) for all K ≥ K M , s ≥ 0.
We deduce that for any ℓ ≥ β, we have

T M (β(φ n )) ≤ ℓ(T K (φ n )).
Therefore, we get lim sup

n T M (β(φ n )) ≤ ℓ(lim n T K (φ n )) ≤ ℓ(φ)
and finally lim sup

n T M (β(φ n )) ≤ β(φ)
for any M, which exactly means that r-limsup β(φ n ) ≤ β(φ).

Step 3. For any subsequence (n

′ ) such that β(φ n ′ ), β(φ n ′ /2+φ/2)) and β(φ n ′ /2+φ/2))-β(φ n ′ )/2- β(φ)/2 ≥ 0 converge in the renormalized sense, we have r-lim β φ n ′ + φ 2 - β(φ n ′ ) 2 - β(φ) 2 + β(φ) 2 + r-lim β(φ n ′ ) 2 = r-lim β φ n ′ + φ 2 ,
thanks to step 1. As a consequence, we get

0 ≤ r-lim β φ n ′ + φ 2 - β(φ n ′ ) 2 - β(φ) 2 = r-lim β φ n ′ + φ 2 - β(φ) 2 -r-lim β(φ n ′ ) 2 ≤ β(φ) - β(φ) 2 - β(φ) 2 = 0,
thanks to step 2 and because φ n ′ /2 + φ/2 r ⇀ φ. Therefore, for any k, we have

0 ≤ lim n→∞ Y k T 1 β φ n ′ + φ 2 - β(φ n ′ ) 2 - β(φ) 2 dν ≤ Y k r-limsup β φ n ′ + φ 2 - β(φ n ′ ) 2 - β(φ) 2 dν = 0,
so that, up to extraction a subsequence,

β φ n ′ + φ 2 - β(φ n ′ ) 2 - β(φ) 2
→ 0 a.e. on Y and φ n ′ → φ a.e. on Y.

Step 4. Fix χ ∈ C c (Y ), the space of continuous functions on Y with compact support, such that 0 ≤ χ ≤ 1. Since T M (φ n ) χ ⇀ TM χ weakly in L 1 , we have

S(T M (φ n ) χ) ⇀ S( TM χ) weakly in L 1 . (3.1)
We deduce, using

T K (S(T M (φ n ) χ)) ≤ T K (S(φ n )) and Proposition 3.2.3 that S( TM χ) = r-liminf n→∞ S(T M (φ n ) χ) ≤ r-liminf n→∞ S(φ n ).
We conclude letting χ ր 1 and M → +∞. ⊓ ⊔

From renormalized convergence to weak convergence.

We give now a kind of extension of the biting Lemma in the L 0 framework.

Definition 3.6

We say that a sequence (ψ n ) is asymptotically bounded in L 0 (Y ) if for any k ∈ N there exists δ k : R + → R + such that δ k (M ) ց 0 when M ր +∞ and for any M there is

n k,M such that meas {y ∈ Y k , ψ n (y) ≥ M } ≤ δ k (M ) ∀k ∈ N, ∀n ≥ n k,M . (3.2)
Theorem 3.7 Let (ψ n ) be a sequence of L 0 (Y ) which r-converges to ψ with ψ ∈ L 0 (Y ). Then (ψ n ) is asymptotically bounded in L 0 (Y ) and there exists a subsequence (ψ n ′ ) which b-converges to ψ.

Remark 3.8 In the L 1 framework, J. Ball & F. Murat [START_REF] Ball | Remarks on Chacon's biting lemma[END_REF] have already proved that the biting L 1weak convergence implies, up to the extraction of a subsequence, the convergence in the renormalized sense, as it has been recalled and extended to the L 0 framework in Lemma 2.7. As a consequence, combining Ball & Murat's result with Theorem 3.7, we get the equivalence between the biting L 1weak convergence and the renormalized convergence. More precisely, considering a sequence (ψ n ) of L(Y ), it is equivalent to say that, up to the extraction of a subsequence,

ψ n b ⇀ ψ in the biting L 1 -weak sense (so that ψ ∈ L 0 (Y )), (3.3) ψ n r ⇀ ψ in the renormalized sense and ψ ∈ L 0 (Y ). (3.4)
Furthermore, in both cases, the full sequence (ψ n ) is asymptotically bounded in L 0 . Again, we refer to the appendix where some complements about r-convergence and b-convergence are given.

Proof of Theorem 3.7. Step 1. Proof of the asymptotic boundedness in L 0 . We argue by contradiction. For an arbitrary ε > 0 we know that there exists B ⊂ Y k such that ν(Y k \B) < ε/2 and ψ ∈ L 1 (B). If there is no m ∈ N such that meas {y ∈ B, ψ n (y) ≥ m} < ε/2 for all n large enough, this means that there exists an increasing sequence (n m ) such that meas {y ∈ B, ψ nm (y) ≥ m} ≥ ε/2 ∀m ≥ 0.

Therefore, for any ℓ ∈ N and any m ≥ ℓ we have

B T ℓ (ψ nm ) ≥ ℓ meas {y ∈ B, ψ nm (y) ≥ ℓ} ≥ ℓ meas {y ∈ B, ψ nm (y) ≥ m} ≥ ℓ ε 2 ,
and passing to the limit m → ∞, we get

B ψ ≥ B w-lim m→∞ T ℓ (ψ nm ) ≥ ℓ ε 2 ∀ℓ ≥ 0.
Letting ℓ ր ∞ we get a contradiction with the fact that ψ ∈ L 1 (B). As a conclusion, we have proved that for any ε > 0 there exists m ε and n ε such that meas {y ∈ Y k , ψ n ≥ m ε } < ε for any n ≥ n ε , and (3.2) easily follows.

Step 2. Proof of the convergence in the biting L 1 -weak sense. As in Step 1, for any k ∈ N we can choose B such that ν(Y k \B) < 1/3k and ψ ∈ L 1 (B). Setting

B ψ dy = C 0 , we construct a sequence (n ℓ ) such that B T ℓ (ψ n ℓ ) dy ≤ C 0 + 1 ℓ . (3.5)
From (3.5), Theorem 2.4 (biting Lemma) and Lemma 2.7, we may extract a subsequence, still denoted by (ψ n ℓ ), which b-converges and r-converges to a limit denoted by ψ * ∈ L 1 (B). On the one hand, for any M ∈ N we have T M (ψ n ℓ ) ≤ T ℓ (ψ n ℓ ) for ℓ ≥ M so that, passing to the limit ℓ → ∞, we get w-lim T M (ψ n ℓ ) ≤ ψ * and thus ψ ≤ ψ * . On the other hand, from Theorem 2.4 (biting Lemma) again, we have

ψ * L 1 ≤ lim inf T ℓ (ψ n ℓ ) L 1 ≤ C 0 = ψ L 1 .
Gathering these two inequalities, we have proved

T ℓ (ψ n ℓ ) ⇀ ψ weakly in L 1 (B).
Furthermore, since (ψ n ) is asymptotically bounded in L 0 (Y ) we have, up to the extraction of a subsequence again,

meas{ψ n ℓ = T ℓ (ψ n ℓ )} = meas{ψ n ℓ > ℓ} ≤ δ k (ℓ) -→ ℓ→∞ 0.
Therefore, we can choose an other subsequence, still noted (ψ n ℓ ), such that

Z L := {∀ℓ ≥ L / ψ n ℓ = T ℓ (ψ n ℓ )} satisfies meas(Z L ) ≤ ℓ≥L meas{ψ n ℓ > ℓ} -→ L→∞ 0.
Finally, choosing L large enough such that meas (Z L ) < 1/3k and setting

A k := B ∩ Z c L , we have |Y k \A| < 1/k, ψ n ℓ ∈ L 1 (A) for all ℓ ≥ L and ψ n ℓ = T ℓ (ψ n ℓ ) ⇀ ψ weakly in L 1 (A).
We conclude thanks to a diagonal process.

⊓ ⊔

A simple but fundamental consequence of Theorem 2.5 and Theorem 3.7 is the following.

Theorem 3.9 Consider a function m : R N → R and a family of measures d̟ y on R N such that

R N m(v) d̟ y (v) = 1, R N m(v) 1/4 d̟ y (v) ≤ C 4 ∀y and m(0) ≥ m(v) -→ |v|→∞ 0. Let (φ n ) be a sequence of L 0 (Y × R N ) which satisfies Y E φ n (y, .) m(.) dν(y) ≤ C 1 < ∞,
with E just like in Theorem 2.5 with dµ y (v) = m(v) d̟ y (v), and assume that

ψ n (y) := R N φ n (y, v) d̟ y (v) r ⇀ ψ with ψ ∈ L 0 (Y ). (3.6)
Then, there exists φ ∈ L 1 (Y × R N , dνd̟) and a subsequence (φ n ′ ) such that for every k ∈ N we can find

A k ⊂ Y k in such a way that (A k ) is increasing, ν(Y k \A k ) < 1/k and φ n ′ ⇀ φ weakly in L 1 (A k × R N , dνd̟). As a consequence ψ = R N φ d̟ and E(φ/m) ∈ L 1 (Y ).
Proof of Theorem 3.9. From (3.6), Theorem 3.7 and Definition 2.3 we know that there exists a subsequence (ψ n ′ ) such that for every k ∈ N we can find

A = A k ⊂ Y k satisfying (A k ) is increasing, ν(Y k \A k ) < 1/k and ψ n ′ is weakly compact in L 1 (A).
Next, we come back to estimate (2.16) in the proof of Theorem 2.5, which written with the new notation, becomes

A×R N φ n ′ Ξ φ n ′ m(v) d̟ y dν ≤ C 3 , (3.7)
where we have set Ξ(s) := Ψ + (s)/s. Of course, we can assume without loss of generality that Ξ is not decreasing, Ξ(s) ր ∞ when s ր ∞ and Ξ(s) ≤ s 1/2 . From (3.7) we deduce

A×R N φ n ′ Ξ φ n ′ m(0) d̟ y dν ≤ C 3 , (3.8)
as well as

A×R N φ n ′ Ξ(m(v) -1/2 ) d̟ y dν ≤ (3.9) ≤ A×R N φ n ′ Ξ(m(v) -1/2 ) 1 {φ n ′ ≤m(v) 1/2 } + 1 {φ n ′ ≥m(v) 1/2 } d̟ y dν ≤ A×R N m(v) 1/4 d̟ y dν + A×R N φ n ′ Ξ φ n ′ m(v) d̟ y dν ≤ C 4 |Y k | + C 3 .
Gathering (3.8) and (3.9), we deduce thanks to the Dunford-Pettis Lemma that (φ n ′ ) belongs to a weak compact set of L 1 (A× R N , dνd̟), and we conclude as in the end of the proof of Theorem 2.5. ⊓ ⊔

4 Trace theorems for solutions of the Vlasov-Fokker-Planck equation.

Statement of the trace theorems

In this section we recall the trace results established in [START_REF] Mischler | On the trace problem for solutions of the Vlasov equation[END_REF], [START_REF] Mischler | On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[END_REF] for the Vlasov equation (which corresponds to the case ν = 0 in the Theorem below) and we extend them to the VFP equation. Given a vector field E = E(t, x, v), a source term G = G(t, x, v), a constant ν ≥ 0 and a solution g = g(t, x, v) to the Vlasov-Fokker-Planck equation

Λ E g = ∂g ∂t + v • ∇ x g + E • ∇ v g -ν ∆ v g = G in (0, T ) × O, (4.1)
we show that g has a trace γg on the boundary (0, T ) × Σ and a trace γ t g on the section {t} × O for any t ∈ [0, T ]. These trace functions are defined thanks to a Green renormalized formula. We write indifferently γ t g = g(t, .).

The meaning of equation (4.1) is of two kinds. In the first case, we assume that g ∈ L ∞ (0, T ;

L p loc ( Ō)), with p ∈ [1, ∞]
, is a solution of (4.1) in the sense of distributions, i.e.,

T 0 O (g Λ ⋆ E φ + G φ) dvdxdt = 0, (4.2)
for all test functions φ ∈ D((0, T ) × O), where we have set

Λ ⋆ E φ = ∂φ ∂t + v • ∇ x φ + E • ∇ v φ + ν ∆ v φ + (div v E) φ.
In this case we assume

E ∈ L 1 0, T ; W 1,p ′ loc ( Ō) , div v E ∈ L 1 0, T ; L p ′ loc ( Ō) , G ∈ L 1 loc ([0, T ] × Ō), (4.3)
where p ′ ∈ [1, ∞] stands for the conjugate exponent of p, given by 1/p + 1/p ′ = 1, and we make one of the two additional hypothesis

ν T 0 OR |∇ v g| 2 dvdxdt ≤ C T,R (4.4) or ν T 0 OR |∇ v g| 2 1 {M≤|g|≤M+1} dvdxdt ≤ C T,R ∀M ≥ 0. (4.5)
Remark 4.1 The bound (4.5) is the natural bound that appears when we consider, for example, the initial value problem with initial datum g 0 ∈ L p (O) when Ω = R N or when Ω is an open subset of R N and specular reflections are imposed at the boundary.

In the second case, we assume that g is a renormalized solution of (4.1). In order to make precise the meaning of such a solution, we must introduce some notations. We denote by B 1 the class of functions β ∈ W 2,∞ (R) such that β ′ has a compact support and by B 2 the class of functions β ∈ W 2,∞ loc (R) such that β ′′ has a compact support. Remark that for every u ∈ L(Y ) and

β ∈ B 1 one has β(u) ∈ L ∞ (Y ). We shall write g ∈ C([0, T ]; L(O)) if β(g) ∈ C([0, T ]; L 1 loc ( Ō)) for every β ∈ B 1 .
We say that g ∈ L((0, T ) × O) is a renormalized solution of (4.1) if for all β ∈ B 1 we have

E ∈ L 1 0, T ; W 1,1 loc ( Ō) , β ′ (g) G ∈ L 1 loc ([0, T ] × Ō), ν β ′′ (g) |∇ v g| 2 ∈ L 1 loc ([0, T ] × Ō), (4.6) and β(g) is solution of Λ E β(g) = β ′ (g) G -ν β ′′ (g) |∇ v g| 2 in D ′ ((0, T ) × O). (4.7)
We can now state the trace theorems for the Vlasov-Fokker-Planck equation (4.1).

Theorem 4.2 (The case p = ∞). Let g ∈ L ∞ ([0, T ] × O) be a solution of equation ( 4.2)-( 4.3)- (4.4). There exists γg defined on (0, T ) × Σ and for every t ∈ [0, T ] there exists

γ t g ∈ L ∞ (O) such that γ t g ∈ C([0, T ]; L a loc ( Ō)) ∀a ∈ [1, ∞) and γ g ∈ L ∞ ((0, T ) × Σ), (4.8)
and the following Green renormalized formula

t1 t0 O β(g) Λ ⋆ E φ + (β ′ (g) G -ν β ′′ (g) |∇ v g| 2 ) φ) dvdxdt = (4.9) = O β(g(t, .)) φ dxdv t1 t0 + t1 t0 Σ β(γ g) φ n(x) • v dvdσ x dt, holds for all t 0 , t 1 ∈ [0, T ], all β ∈ W 2,∞ loc (R) and all test functions φ ∈ D([0, T ] × Ō).
Remark 4.3 A fundamental point, which is a consequence of the Green formula (4.9), is the possibility of renormalizing the trace function, i.e.

γ β(g) = β(γ g) (4.10) for all β ∈ W 2,∞ (R). More generally, (4.10) holds as soon as γ β(g) is defined. This is the property that will allow us to define the trace of a renormalized solution.

2)-(4.3)-(4.5)

. There exists γg defined on (0, T ) × Σ and for every t ∈ [0, T ] there exists

γ t g ∈ L p (O) such that γ t g ∈ C([0, T ]; L 1 loc (O)) and γ g ∈ L 1 loc [0, T ] × Σ, dλ 2 , (4.11)
and satisfy the Green formula (4.9) for every t 0 , t 1 ∈ [0, T ], every β ∈ B 1 and every test functions φ ∈ D([0, T ] × Ō), as well as for every t 0 , t 1 ∈ [0, T ], every β ∈ B 2 and every test functions φ ∈ D 0 ([0, T ] × Ō), the space of functions φ ∈ D([0, T ] × Ō) such that φ = 0 on (0, T ) × Σ 0 .

Theorem 4.5 (The renormalized case). Let g ∈ L((0, T ) × O) satisfy the bound condition (4.6) and the equation (4.7). Then there exists γg ∈ L([0, T ] × Σ) and for every t ∈ [0, T ] there exists γ t g ∈ C([0, T ]; L(O)) which satisfy the Green formula (4.9) for all t 0 , t 1 ∈ [0, T ], all β ∈ B 1 and all test functions φ ∈ D([0, T ]× Ō). Furthermore, if (4.7) makes sense for at least one function

β such that β(s) ր ∞ when s ր ∞, then γ t g ∈ L 0 (O) for any t ∈ [0, T ] and γg ∈ L 0 ([0, T ] × Σ).

Proof of the trace theorems

We begin with some notations. For a given real R > 0, we define andL a,b loc the space L a (0, T ; L b loc ( Ō)) or L a (0, T ; L b loc ( Ω)). Proof of Theorem 4.2. First step: a priori bounds. In this step we assume that g is a solution of (4.1) and is "smooth". Precisely, g ∈ W 1,1 0, T ; W 1,∞ (Ω; W 2,∞ (R N )) , in such a way that the Green formula (4.9) holds. The trace γg in (4.9) is defined thanks to the usual trace theorem in the Sobolev spaces. We shall prove two a priori bounds on g. Let us define

B R = {y ∈ R N / |y| < R}, Ω R = Ω ∩ B R , O R = Ω R × B R and Σ R = (∂Ω ∩ B R ) × B R . We also denote by L a,b R the space L a (0, T ; L b (O R )) or L a (0, T ; L b (Ω R )),
β ∈ W 2,∞ loc (R) by β(s) = |s| -1/2 if |s| ≥ 1 s 2 /2 if |s| ≤ 1 so that β ′ (s) = 1 if s ≥ 1 s if |s| ≤ 1 -1 if s ≤ -1 and β ′′ (s) = 0 if |s| ≥ 1 1 if |s| ≤ 1 ,
and thus β ∈ B 1 . Fix R > 0 and consider χ ∈ D( Ō) such that 0 ≤ χ ≤ 1, χ = 1 on O R and supp χ ⊂ ŌR+1 . We set φ = χ n(x) • v. The Green formula (4.9) gives

T 0 Σ β(γ g) χ (n(x) • v) 2 dvdσ x dt = - O β(g(t, .)) φ dxdv T 0 + T 0 O β(g) Λ ⋆ E φ + (β ′ (g) G -ν β ′′ (g) |∇ v g| 2 ) φ) dvdxdt.
We deduce from it a first a priori bound: there are some constants γ R and C R such that

γ R T 0 ΣR |γ g| (n(x) • v) 2 dvdσ x dt ≤ T 0 ΣR β(γ g) (n(x) • v) 2 dvdσ x dt ≤ C R T 0 OR+1 g 2 (1 + |E|) + |G| + ν |∇ v g| 2 dvdxdt (4.12) +C R OR+1 g 2 (0, .) + g 2 (T, .) dxdv,
where we have used the fact that for u

∈ L ∞ (Y R ) with Y R = O R or Σ R there holds γ R YR |u| ≤ YR β(u) ≤ γ -1 R YR u 2 .
Let K ⊂ O be a compact set and consider φ ∈ D(O) such that 0 ≤ φ ≤ 1, φ = 1 on K and R > 0 such that supp φ ⊂ O R . We fix t 0 ∈ [0, T ]. The Green formula (4.9) implies

+ t1 t0 O β(g) Λ ⋆ E φ + (β ′ (g) G -ν β ′′ (g) |∇ v g| 2 ) φ) dvdxdt,
and we get a second a priori bound

γ R K |g|(t 1 , .) dxdv ≤ C R OR g 2 (t 0 , .) dxdv (4.14) + C R T 0 OR g 2 (1 + |E|) + |G| + ν |∇ v g| 2 dvdxdt.
Second step: regularization and passing to the limit. Let us now consider a function g which satisfies the assumptions of Theorem 4.2. We define the mollifier ρ k by

ρ k (z) = k N ρ(k z) ≥ 0, k ∈ N ⋆ , ρ ∈ D(R N ), supp ρ ⊂ B 1 , R N ρ(z) dz = 1,
and we introduce the regularized functions g k = g ⋆ x,k ρ k * v ρ k , where * stands for the usual convolution and ⋆ x,k for the convolution-translation defined by

(u ⋆ x,k h k )(x) = [τ 2 n(x)/k (u * h k )](x) = R N u(y) h k (x - 2 k n(x) -y) dy, for all u ∈ L 1 loc ( Ω) and h k ∈ L 1 (R N ) with supp h k ⊂ B 1/k . Lemma 4.6 With this notation one has g k ∈ W 1,1 0, T ; W 1,∞ (Ω; W 2,∞ (R N ))
and

Λ E g k = G k in D ′ ((0, T ) × O),
with G k ∈ L 1 loc ((0, T ) × Ō) for all k ∈ N. Moreover, the sequences (g k ) and (G k ) satisfy

(g k ) is bounded in L ∞ ((0, T ) × O), g k -→ g a.e. in (0, T ) × O, ∇ v g k -→ ∇ v g in L 2 loc ([0, T ] × Ō) and G k -→ G in L 1 loc ([0, T ] × Ō). (4.15)
The proof of Lemma 4.6 is similar to the proof of [55, Lemma 1] and of [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF]Lemma II.1] to which we refer. From Lemma 4.6 we have that for all k, ℓ ∈ N ⋆ the difference g k -g ℓ belongs to W 1,1 0, T ;

W 1,∞ (Ω; W 2,∞ (R N )) and is a solution of Λ E (g k -g ℓ ) = G k -G ℓ in D ′ ((0, T ) × O).
We know, thanks to (4.15), that g k (t, .) converges to g(t, .) in L 2 loc ( Ō) for a.e. t ∈ [0, T ]; we fix t 0 such that g k (t 0 , .) → g(t 0 , .). Moreover, up to a choice for the continuous representation of g k , we can assume that g k ∈ C([0, T ], L 1 loc ( Ō)). Therefore, the estimate (4.13) applied to g k -g ℓ in t 0 and the convergence (4.15) imply that for all compact sets K ⊂ O we have

sup t∈[0,T ] (g k -g ℓ )(t, .) L 1 (K) -→ k,ℓ→+∞ 0. (4.16)
We deduce from this, that there exists, for any time t ∈ [0, T ], a function γ t g such that g k (t, .)

converges to γ t g in C([0, T ]; L 1 loc (O)); in particular, g(t, x, v) = γ t g(x, v) for a.e. (t, x, v) ∈ (0, T ) × O.
Thus, we also have g k (t, .) = (γ t g) ⋆ x,k ρ k * v ρ k a.e. in (0, T ) × O, and since these two functions are continuous, the equality holds for all (t, x, v) ∈ [0, T ] × Ō and k ∈ N ⋆ , so that g k (t, .) → γ t g in L 2 loc ( Ō) for all t ∈ [0, T ]. Using now the estimate (4.12), applied to g k -g ℓ , and the convergence (4.15) and (4.16) we get that

T 0 ΣR |γg k -γg ℓ | (n(x) • v) 2 dvdσ x dt -→ k,ℓ→+∞ 0,
for all R > 0. We deduce that there exists a function γg

∈ L 1 loc ([0, T ] × Σ, (n(x) • v) 2 dvdσ x dt), which is the limit of γg k in this space. Moreover, since γg k L ∞ ≤ g k L ∞ is bounded, we have γg ∈ L ∞ ((0, T ) × O).
Finally, we obtain the Green formula (4.9) writing it first for g k and then passing to the limit k → ∞ thanks to the convergence previously obtained. Uniqueness of the trace function follows from the Green formula.

⊓ ⊔

Proof of Theorem 4.5. The proof is based on Theorem 4.2 and on a monotony argument. This is exactly the same as the one presented in [START_REF] Mischler | On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[END_REF] in the case of Vlasov equation. Let (β M ) M≥1 be a sequence of odd functions of B 1 such that

β M (s) = s if s ∈ [0, M ] M + 1/2 if s ≥ M + 1, and |β M (s)| ≤ |s| for all s ∈ R. The function α M (s) := β M (β -1 M+1 (s)), with the convention α M (s) = M + 1/2 if s ≥ M + 3/2,
is well defined, odd and also belongs to B 1 . We will construct the trace function γg as the limit of (γβ M (g)) when M → ∞, that one being defined thanks to Theorem 4.2. Indeed, the condition (4.5) implies that

∇ v g 1 |g|≤M+1 ∈ L 2 loc ([0, T ] × Ō),
and then

∇ v β M (g) = β ′ M (g) ∇ v g ∈ L 2 loc ([0, T ] × Ō)
in such a way that β M (g) satisfies the assumption on Theorem 4.2. We define Γ

(±) M = {(t, x, v) ∈ (0, T ) × Σ, ±γβ M (g)(t, x, v) > 0} and Γ (0) M = {(t, x, v) ∈ (0, T ) × Σ, γβ M (g)(t, x, v) = 0}.
Thanks to the definition of α M and the renormalization property (4.10) of the trace, one has γ β M (g) = γ α M (β M+1 (g)) = α M (γ β M+1 (g)). We deduce that, up to a set of measure zero,

Γ (+) M = Γ (+) 1 , Γ (-) M = Γ (-) 1 and Γ (0) M = Γ (0) 1 for all M ≥ 1.
Therefore the sequence (γ β M (g)) M≥1 is increasing on Γ (+) 1 and decreasing on Γ (-) 1 . This implies that γ β M (g) converges a.e. to a limit denoted by γg which belongs to L([0, T ] × Σ). Obviously, if (4.7) holds for one function β such that β(s) ր +∞ when s ր ±∞, then β(γg) ∈ L 1 ((0, T ) × Σ, dλ 2 ) and γg ∈ L 0 ((0, T ) × Σ). In order to establish the Green formula (4.9) we fix β ∈ B 1 and φ ∈ D((0, T ] × Ō). We write the Green formula for the function β(β M (g)), and using the fact that

γ β • β M (g) = β(γβ M (g)), we find T 0 O β • β M (g) ( ∂φ ∂t + v • ∇ x φ + E • ∇ v φ) + (β • β M ) ′ (g) G φ dvdxdt = = T 0 Σ β(γ β M (g)) φ n(x) • v dvdσ x ds.
We get (4.9) by letting M → ∞ and noticing that β

• β M (s) → β(s) for all s ∈ R. ⊓ ⊔ Remark 4.7 Theorem 4.
4 is now a quite simple consequence of Theorem 4.5 using the a priori bounds stated in the proof of Theorem 4.2. Let us emphasize that with the additional assumption (4.4) in hands, it is possible to give a direct proof of Theorem 4.4 (following the proof of Theorem 4.2) instead of passing through the renormalization step. See [START_REF] Mischler | On the trace problem for solutions of the Vlasov equation[END_REF] for details.

Proof of Theorem 4.4. For all β ∈ B 1 it is clear that β(g) ∈ L ∞ , ∇ v β(g) ∈ L 2
and that β(g) is solution of (4.7) using Lemma 5.5 below (we just have to multiply equation (5.20), in the case µ ≡ 0, by β ′ (g k ) and to pass to the limit k → ∞). Thanks to Theorem 4.5, we already know that g has a trace γ t g ∈ L(O) and γg ∈ L((0, T ) × O) which satisfies the Green formula (4.9) for all β ∈ B 1 and φ ∈ D([0, T ] × Ō). We just have to prove that γg and γ t g belong to the appropriate space. On one hand, for all β ∈ B 1 such that |β(s)| ≤ |s| one has

β(γ t g) L p R ≤ sup k sup [0,T ] β(g k (t, .)) L p R ≤ sup [0,T ] g k (t, .) L p R ≤ g L ∞,p R ,
and thus, choosing β = β M , defined in the proof of Theorem 4.5, one gets, passing to the limit

M → ∞, sup [0,T ] γ t g L p R ≤ g L ∞,p R < ∞.
In the same way and using (4.12), we show that

γg L 1 ([0,T ]×ΣR,dλ2) < ∞.
We still have to prove that γ t g ∈ C([0, T ], L 1 loc ( Ō)), which is an immediate consequence of the following Lemma.

⊓ ⊔

Lemma 4.8 Let (u n ) be a bounded sequence of L 1 loc (O) such that β(u n ) ⇀ β(u) in C c (O) ′ for all β ∈ B 2 . Then u n → u in L 1 loc (O).
Proof of Lemma 4.8. We fix j : R → R a non-negative function of class C 2 , strictly convex on the interval [-M, M ] and such that j ′′ (t) = 0 for all t / ∈ [-M, M ]; in particular j ∈ B 2 . We also

consider χ ∈ C c (O) such that 0 ≤ χ ≤ 1. By assumption O j(u n ) χ → O j(u) χ (4.17)
and by convexity of j one also has lim inf

n→∞ O j u n + u 2 χ ≥ O j(u) χ since u n + u 2 ⇀ u in C c (O) ′ . (4.18) Remarking that 1 2 j(t) + 1 2 j(s) -j t + s 2 ≥ 0 ∀t, s ∈ R, (4.19)
we deduce from (4.17) and (4.18) that

O 1 2 j(u n ) + 1 2 j(u) -j u n + u 2 χ → 0. (4.20)
From the fact that in (4.19) the inequality is strict whenever t, s ∈ [-M, M ] and t = s, we obtain from (4.20) that there exists a subsequence (u

n k ) such that u n k → u a.e. on supp χ ∩ [ |u| < M ].
The preceding argument being valuable for arbitrary M and χ, we obtain, by a diagonal process, a subsequence of (u n ), still denoted by (u n k ), such that u n k → u a.e. in O.

We now set j ± (s) = s ± . We first remark that we can write j ± = j ±,1 + j ±,2 with j ±,1 ∈ B 2 and j ±,2 ∈ W 2,∞ (R) in such a way that

O j ± (u n k ) χ → O j ± (u) χ.
On the other hand, the elementary inequality b -|a -b| ≤ a ∀a, b ≥ 0 and the dominated convergence Theorem imply j

± (u n k ) -|j ± (u n k ) -j ± (u)| → j ± (u) in L 1 loc (O). It follows that lim sup k→∞ O j ± (u n k ) -j ± (u) χ = O j ± (u) χ -lim k→∞ O j ± (u n k ) χ = 0.
We conclude that

u n k = j + (u n k ) -j -(u n k ) → j + (u) -j -(u) = u strongly in L 1 loc (O)
and that, in fact, it is the whole sequence (u n ) which converges.

⊓ ⊔

5 Renormalized convergence for the trace functions sequence.

We present now a quite general stability result in both the interior and up to the boundary for a sequence of renormalized solutions to the Vlasov-Fokker-Planck equation in a domain. This will be a key argument in the proof of Theorem 1.2. In some sense, this result says that renormalized convergence, as well as the a.e. convergence, can be propagated from the interior to the boundary. Notice that it is not clear that a similar result holds for the L 1 -weak convergence.

Theorem 5.1 Define B 3 as the class of functions of W 1,∞ loc (R) such that |β ′ (s)| (1+s) -1 ∈ L ∞ (R). Consider three sequences (g n ), (E n ) and (G n ), with G n = G + n -G - n , G ± n ≥ 0,
which satisfy for any renormalizing sequence (α M ) in B 3 and for any β ∈ B 3 the convergence assumptions

g n ⇀ g weakly in L ∞ (0, T : L 1 (O)), (5.1) E n → E strongly in L 1 ((0, T ) × Ω), uniformly bounded in L 1 (0, T ; W 1,1 (Ω)), (5.2) α ′ M (g n ) G ± n ⇀ Ḡ± M weakly in L 1 ((0, T ) × O R ), (5.3) with Ḡ± M ր G ± a.e. and β ′ (g) G ± ∈ L 1 ((0, T ) × O R ),
as well as the renormalized Vlasov equation

Λ En β(g n ) = β ′ (g n ) G n in D ′ ((0, T ) × O), (5.4)
for which each term clearly makes sense thanks to (5.1)- (5.3)

. Then g ∈ L ∞ (0, T ; L 1 (O)) is a solution of Λ E β(g) = β ′ (g) G in D ′ ((0, T ) × O), G = G + -G -, (5.5)
for any β ∈ B 3 . Furthermore, the traces γg n and γg defined thanks to the Theorem 4.5 satisfy γg n r ⇀ γg in the renormalized sense. (5.6) Proof of Theorem 5.1. The proof is essentially the same as Step 2 in the proof of [START_REF] Mischler | On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[END_REF]Proposition 5] and as the proof of Theorem 2.9. Nevertheless, for the sake of completeness, we sketch the main arguments.

Step 1. Up to the extraction of a subsequence, we have g n r ⇀ g thanks to (5.1) and Lemma 2.7, and there exists η ∈ L((0, T ) × Σ) such that γg n r ⇀ η thanks to Proposition 3.2. More precisely, there exists two sequences (ᾱ M ) and (γ M ) and such that

α M (g n ) ⇀ ᾱM σ(L ∞ , L 1 ) ⋆
and ᾱM ր g a.e., (5.7) α M (γg n ) ⇀ γM σ(L ∞ , L 1 ) ⋆ and γM ր η a.e.. (5.8) The Green formula (4.9) associated to the equation (5.4)

with β = α M implies T 0 O α M (g n ) Λ * E ϕ + α ′ M (g n ) G n ϕ dvdxdt = T 0 Σ α M (γ g n ) ϕ n(x) • v dvdσ x dt,
for any ϕ ∈ D((0, T ) × Ō). Passing to the limit M → ∞ with the help of (5.7), (5.2), (5.3) in the above identity, we obtain

Λ E ᾱM = ḠM := Ḡ+ M -Ḡ- M in D ′ ((0, T ) × O), (5.9)
and γ ᾱM = γM thanks to the trace Theorem 4.2 and the convergence (5.8).

Step 2.

For a given function β ∈ B 3 ∩ L ∞ , we write the renormalized Green formula (4.9) associated to the equation (5.9) as

T 0 O β(ᾱ M ) Λ * E ϕ + β ′ (ᾱ M ) ḠM ϕ dvdxdt = T 0 Σ β(γ M ) ϕ n(x) • v dvdσ x dt, (5.10)
for any ϕ ∈ D((0, T ) × Ō). Using that (ᾱ M ), (G ± M ) and (γ M ) are a.e. increasing sequences we have

β(ᾱ M ) ր β(g), β ′ (ᾱ M ) Ḡ± M ր β ′ (g) G ± in L 1 ((0, T ) × O) (5.11)
as well as β(γ M ) ր β(η) a.e. and uniformaly bounded in L ∞ ((0, T ) × O). (5.12)

Passing to the limit in (5.10) with the help of (5.11) and (5.12) we obtain

T 0 O β(g) Λ * E ϕ + β ′ (g) G ϕ dvdxdt = T 0 Σ β(η) ϕ n(x) • v dvdσ x dt,
which precisely means that η = γg. We conclude by gathering that information with (5.8). ⊓ ⊔ Theorem 5.2 Consider three sequences (g n ), (E n ) and (G n ) which satisfy, for all β ∈ B 4 the class of functions of

W 2,∞ loc (R) such that |β ′ (s)| (1 + s) -1 ∈ L ∞ (R) and |β ′′ (s)| (1 + s) -2 ∈ L ∞ (R),
g n → g strongly in L 1 ((0, T ) × O) and is uniformly bounded in L ∞ (0, T ; L 1 (O)), (5.13) E n ⇀ E weakly in L 1 (0, T ; W 1,1 loc ( Ō)), (5.14) (5.16) as well as the renormalized Vlasov-Fokker-Planck equation (5.17) for which each term makes sense thanks to (5.13)- (5.16)

β ′ (g n ) G n ⇀ β ′ (g) G weakly in L 1 ((0, T ) × O R ), ∀R ≥ 0, (5.15) T 0 O |∇ v g n | 2 1 + g n dvdxdt ≤ C T ,
Λ En β(g n ) = β ′ (g n ) G n -ν β ′′ (g n ) |∇ v g n | 2 in D ′ ((0, T ) × O),
. Then g ∈ L ∞ (0, T ; L 1 (O)) is a solution of Λ E β(g) = β ′ (g) G -ν β ′′ (g) |∇ v g| 2 in D ′ ((0, T ) × O) (5.18)
for all β ∈ B 4 . Furthermore, the traces γg n and γg defined thanks to the Theorem 4.5 satisfy γg n r ⇀ γg in the renormalized sense, and γ + g n → γ + g a.e. (5.19) We shall need the following auxiliary results in the proof of Theorem 5.2.

Lemma 5.3 Let (u n ) be a bounded sequence of L 2 (Y ) such that u n ⇀ u weakly in L 2 (Y ). Then, there exists µ ∈ (C c (Y )) ′ , a non-negative measure, such that, up to the extraction of a subsequence,

|u n | 2 ⇀ |u| 2 + µ weakly in (C c (Y )) ′ .
Lemma 5.4 For any θ ∈ (0, 1) and M ∈ (0, ∞) we set

Φ(s) = Φ M,θ (s) := 1/θ (e θ s -1) if s ≤ M (s -M ) e θ M + 1/θ (e θ M -1) if s ≥ M, and β(s) := β 1 (s) = log(1 + s). Then      Φ ′ (s) ≥ 1, Φ • β(s) ր s when M ր ∞, θ ր 1,
and 0 ≤ -(Φ • β) ′′ (s) ≤ 1 -θ + e (θ-1) M 1 + s ∀s ≥ 0.
Lemma 5.5 Let g ∈ L ∞ (0, T ; L p loc (O)) be a solution to the Vlasov-Fokker-Planck equation

Λ E g = G + µ in D ′ ((0, T ) × O), (5.20) with E ∈ L 1 (0, T ; W 1,p ′ loc (O)), G ∈ L 1 loc ((0, T ) × O)
) and µ ∈ D ′ ((0, T ) × O), µ ≥ 0. For a given mollifer ρ k in R N , we set

g k := g * t ρ k * x ρ k * v ρ k and µ k := µ * t ρ k * x ρ k * v ρ k .
Then g k satisfies the Vlasov-Fokker-Planck equation

Λ E g k = G k + µ k in all compact set of (0, T ) × O, with G k → G strongly in L 1 loc ([0, T ] × O)).
The proof of Lemma 5.3 is classical, the one of Lemma 5.4 is elementary, and we refer to [START_REF] Diperna | On the Fokker-Planck-Boltzmann equation[END_REF] for the proof of Lemma 5.5. Proof of the Theorem 5.2.

Step 1: Proof of (5.18). This step is inspired from [START_REF] Diperna | On the Fokker-Planck-Boltzmann equation[END_REF] and it is clear from the theory of renormalized solution [START_REF] Diperna | Ordinary differential equations, transport theory and Sobolev spaces[END_REF] that it is enough to prove (5.18) only for β(s) := log(1 + s).

With the notation h n := β(g n ) and h = β(g) we have

∇ v h n = -β ′′ (g n ) ∇ v g n ⇀ -β ′′ (g) ∇ v g = ∇ v h weakly in L 2 ((0, T ) × O
) so that, thanks to Lemma 5.3, there is a bounded measure µ ≥ 0 such that, up to the extraction of a subsequence,

|∇ v h n | 2 ⇀ |∇ v h| 2 + µ weakly in D ′ ([0, T ] × Ō).
Passing to the limit n → ∞ in (5.17) we get

Λ E β(g) = β ′ (g) G -β ′′ (g) |∇ v g| 2 + µ in D ′ ((0, T ) × O).
We just point out that

E n β(g n ) ⇀ E β(g) weakly in L 1 ((0, T ) × O),
since β(g n ) → β(g) strongly in L 2 (0, T ; L p (O)) for all p < ∞ and E n ⇀ E weakly in L 2 (0, T ; L q (O)) for every q ∈ [1, N/(N -1)). We prove now that µ = 0 in (0, T ) × O.

With the notations introduced in Lemma 5.4 and Lemma 5.5 we have

Λ E Φ(h k ) = Φ ′ (h k ) β ′ (g) G -β ′′ (g) |∇ v g| 2 * t,x,v ρ k -Φ ′′ (h k ) |∇ v h k | 2 + Φ ′ (h k ) µ k .
Using that Φ ′ ≥ 1 (thanks to Lemma 5.4) and passing to the limit k → ∞ (thanks to Lemma 5.5), we get

Λ E (Φ • β)(g) ≥ Φ ′ (β(g)) β ′ (g) G -(Φ ′ (β(g)) β ′′ (g) + Φ ′′ (β(g)) (β ′ (g)) 2 ) |∇ v g| 2 + µ and then Λ E (Φ • β)(g) -(Φ • β) ′ (g) G ≥ (Φ • β) ′′ (g) |∇ v g| 2 + µ in D ′ ((0, T ) × O). (5.21)
In order to have an estimate of the left hand side we come back to equation (5.17), and we write

Λ En Φ • β(g n ) = (Φ • β) ′ (g n ) G n -(Φ • β) ′′ (g n ) |∇ v g n | 2 in D ′ ((0, T ) × O) since Φ•β ∈ B 4 .
Then, for all χ ∈ D((0, T )×O such that 0 ≤ χ ≤ 1 we have (thanks to Lemma 5.4)

T 0 O Φ • β(g n ) Λ En χ + (Φ • β) ′ (g n ) G n χ dvdxdt = = - T 0 O (Φ • β) ′′ (g n ) |∇ v g n | 2 χ dvdxdt ≤ [1 -θ + e (θ-1) M ] T 0 O |∇ v g n | 2 1 + g n dvdxdt.
Passing to the limit n → ∞ we get, thanks to (5.16),

T 0 O Φ • β(g) Λ E χ + (Φ • β) ′ (g) G χ dvdxdt ≤ [1 -θ + e (θ-1) M ] C T .
Then, coming back to (5.21), we have (thanks to Lemma 5.4 again)

T 0 O χ dµ ≤ - T 0 O Φ • β(g) Λ E χ + (Φ • β) ′ (g) G χ + (Φ • β) ′′ (g) |∇ v g| 2 dvdxdt ≤ 2 [1 -θ + e (θ-1) M ] C T ∀θ ∈ [0, 1], M > 0,
and letting M → ∞ and then θ → 1 we obtain µ = 0 on supp χ, which is precisely saying that µ = 0 on (0, T ) × O.

Step 2: Proof of (5.19). We fix φ ∈ D((0, T ) × Ō) such that 0 ≤ φ ≤ 1. By definition of γg n we have

T 0 Σ Φ • β(γ g n ) φ n(x) • v dvdσ x dt - T 0 O Φ • β(g n ) Λ En χ + (Φ • β) ′ (g n ) G n χ dvdxdt = = T 0 O (Φ • β) ′′ (g n ) |∇ v g n | 2 χ dvdxdt ≤ [1 -θ + e (θ-1) M ] T 0 O |∇ v g n | 2 1 + g n dvdxdt.
We note Φ • β the L 1 -weak limit of Φ • β(γ g n ). Passing to the limit n → ∞ we get

T 0 Σ Φ • β φ n(x) • v dvdσ x dt - T 0 O Φ • β(g) Λ E χ + (Φ • β) ′ (g) G χ dvdxdt ≤ ≤ [1 -θ + e (θ-1) M ] C T ,
and thus

T 0 Σ Φ • β φ n(x) • v dvdσ x dt - T 0 O Φ • β(g) Λ E χ + (Φ • β) ′ (g) G -(Φ • β) ′′ (g) |∇ v g| 2 χ dvdxdt ≤ 2 [1 -θ + e (θ-1) M ] C T .
Once again, by definition of γg, we obtain

T 0 Σ Φ • β -Φ • β(γ g) φ n(x) • v dvdσ x dt ≤ 2 [1 -θ + e (θ-1) M ] C T -→ M→∞,θ→1
0, and Φ • β ր r-lim γg n since Φ • β(s) ր s when M ր ∞, θ ց 1, so that γg = r-lim γg n .

In order to prove the a.e. convergence we only have to show, thanks to Proposition 3.5.3, that, up to the extraction of a subsequence, r-liminf β(γ + g n ) ≥ β(γ + g).

(5.22) Using Lemma 5.3 and the first step, we can pass to the limit in (5.17), up to the extraction of a subsequence, and we get

T 0 Σ β φ n(x) • v dvdσ x dt = T 0 O β(g) Λ E χ + (β ′ (g) G + β ′′ (g) |∇ v g| 2 + µ) χ dvdxdt = T 0 Σ (β(γg) n(x) • v + µ) φ dvdσ x dt,
where β = w-lim β(γg n ) is the weak limit in L 1 ((0, T ) × Σ) of β(γg n ). We deduce that β n(x) • v = β(γg) n(x) • v + µ on (0, T ) × Σ, and in particular

β ≥ β(γ + g) on (0, T ) × Σ + .
Since r-liminf β(γ + g n ) = β, that ends the proof of (5.22). ⊓ ⊔

6 Boltzmann, Vlasov-Poisson and Fokker-Planck equations

In this section we derive the a priori physical bound, then make precise the exact meaning of renormalized solution we deal with and finally state and present a proof of the corresponding stability results. In order not to repeat many times the exposition, we consider the full Vlasov-Poisson-Fokker-Planck-Boltzmann system (VPFPB in short)

∂f ∂t + v • ∇ x f -div v (∇ x V f + λ v) f -ν ∆ v f = Q(f, f ) in (0, ∞) × O, (6.1)
where ν ≥ 0, λ ∈ R, Q(f, f ) stands for the bilinear Boltzmann collision operator and V f is given by the mean of the Poisson equation

-∆V f = ρ f := R N f dv in (0, ∞) × Ω, V f = 0 on (0, ∞) × ∂Ω. (6.2)
We do not give the explicit expression for Q(f, f ) that we may find in [START_REF] Cercignani | The Boltzmann equation and its application[END_REF][START_REF] Diperna | On the Cauchy problem for Boltzmann equation: global existence and weak stability[END_REF] for example. The precise assumptions we make on the cross section are those introduced in [START_REF] Diperna | On the Cauchy problem for Boltzmann equation: global existence and weak stability[END_REF]. We only recall that the collision operator splits into a gain term and a loss term, Q

(f, f ) = Q + (f, f ) -Q -(f, f ), that it has the following collision invariants R 3 Q(f, f )   1 v |v| 2   dv = 0, (6.3)
and that the so-called entropy production term e(f ) ≥ 0 satisfies

R 3 e(f ) dv = - R 3 Q(f, f ) log f dv. (6.4)
Moreover, it has been established in [START_REF] Diperna | On the Cauchy problem for Boltzmann equation: global existence and weak stability[END_REF] the following estimate

∀ R > 0 ∃C R < ∞ BR Q ± (f, f ) 1 + f dv ≤ C R R N (1 + |v| 2 ) f + e(f ) dv, (6.5)
and in [START_REF] Villani | Square-root-renormalized solution of the Boltzmann equation[END_REF] (we also refer to [START_REF] Lions | Compactness in Boltzmann equation via Fourier integral operators and applications Part I[END_REF] for a related result) the more accurate estimate

∀ R > 0 ∃C R < ∞ BR |Q(f, f )| √ 1 + f dv ≤ C R R N (1 + |v| 2 ) f + e(f ) dv. (6.6)
We assume furthermore that f satisfies the boundary condition (1.1) and the initial condition (1.8), where f in is assumed to verify (1.9), as well as the following additional bound when ν > 0:

Ω |∇ x V fin | 2 dx < ∞ with -∆ x V fin = R 3
f in (x, v) dv on Ω, V fin = 0 on ∂Ω. (6.7) Lemma 6.1 For any non-negative initial datum f in such that (1.9)-(6.7) holds and any time T ∈ (0, ∞) there exists a constant C T ∈ (0, ∞) (only depending on T and on f in through the quantities C 0 and |∇ x V fin L 2 ) such that any solution f to the initial boundary value problem (6.1)-( 6

.2)-(1.1)-(1.8) satisfies (at least formally)

sup [0,T ] O f 1 + |v| 2 + | log f | dvdx + Ω |∇ x V f | 2 dx (6.8) + T 0 O e(f ) + ν |∇ v f | 2 f dvdxdt ≤ C T ,
as well as

T 0 ∂Ω E γ + f M + γ + f dσ x dt ≤ C T , (6.9)
where E is defined in (1.13). It is worth mentioning that the second estimate in (6.9) is an a posteriori estimate which we deduce from the interior estimate (6.8) and a Green formula.

Proof of (6.8) in Lemma 6.1. We claim that for f sufficiently regular and decreasing at the infinity all the integrations (by parts) that we shall perform are allowed. First, we simply integrate the equation (6.1) over all variables, and we get the conservation of mass

O f (t, .) dvdx = O f in dvdx ∀t ≥ 0. Next, setting h M (s) = s log(s/M ) and E = ∇ x V f , we compute ∂ ∂t h M (f ) + v • ∇ x h M (f ) + div v (E + λ v)h M (f ) -ν ∆ v h M (f ) = = h ′ M (f ) Q(f, f ) -ν h ′′ M (f ) |∇ v f | 2 -f (E + λ v) • ∇ v (log M ) +λ (h M (f ) -f h ′ M (f )) + 2 ν ∇ v f • ∇ v (log M ) + ν f ∆ v (log M ),
where h ′ M (s) = 1 + log (s/M ). We integrate this equation over the x, v variables using the collision invariants (6.3) and the entropy production identity (6.4), to obtain

d dt O h M (f ) dvdx + O e(f ) + ν |∇ v f | 2 f dvdx + Σ h M (γf ) v • n(x) dvdσ x = (6.10) = Ω E • j Θ dx + O λ |v| 2 Θ -1) + ν Θ f dvdx, where j(t, x) = R 3 v f (t, x, v) dv.
We first remark that integrating equation (6.1) in the velocity variable we have

∂ ∂t ρ + div x j = 0 on (0, ∞) × Ω,
and therefore

- Ω E • j Θ dx = Ω ∇V f • j Θ dx = Ω V f Θ ∂ρ ∂t dx = d dt Ω |∇ x V f | 2 2 Θ dx. (6.11)
Next, combining (6.10), (6.11) and the boundary estimate (2.5) we obtain

d dt O h M (f ) dvdx + Ω |∇ x V f | 2 2 Θ dx + O e(f ) + ν |∇ v f | 2 f dvdx + ᾱ ∂Ω E(γ + f ) dσ x ≤ C λ,ν O (1 + |v| 2 ) f dvdx.
Here and below, we set ᾱ = α in the case of the constant accommodation coefficient (1.2) and ᾱ is defined just after equation (1.6) in the case of mass flux dependent accommodation coefficient. Using the elementary estimate (2.6) and (2.7) we conclude that (6.8) holds, as well as the first estimate in (6.9).

In order to prove the second estimate in (6.9), we fix χ ∈ D(R N ) such that 0 ≤ χ ≤ 1, χ = 1 on B 1 and supp χ ⊂ B 2 and we apply the Green formula (4.9) written with φ = n(x) • v χ(v) and β(s) = √ 1 + s. We get

T 0 Σ 1 + γf (n(x) • v) 2 χ dvdσ x dt = O 1 + γf φ dvdx 0 T (6.12) + T 0 O 1 + f v • ∇ x + (∇ x V f + λ v) • ∇ v + ν ∆ v + N λ φ dvdxdt + T 0 O Q(f, f ) 2 (1 + f ) 1/2 + ν 4 |∇ v f | 2 (1 + f ) 3/2 φ dvdxdt.
Thanks to (6.8) and (6.6) and because ∇ x φ ∈ L ∞ , D 2 v φ ∈ L ∞ , we see that the right hand side term in (6.12) is bounded by a constant denoted by C ′ T and which only depends on C T defined in (6.8). On the other hand, from the boundary condition (1.1)-(1.2) or (1.1)-(1.6), we have

γ -f ≥ ᾱ M (v) γ + f on (0, T ) × Σ -. Therefore there is a constant C χ > 0 such that C χ T 0 ∂Ω γ + f dσ x dt ≤ T 0 Σ- γ + f ᾱ1/2 M 1/2 (v) χ (n(x) • v) 2 dvdσ x dt ≤ T 0 Σ- γ -f χ (n(x) • v) 2 dvdσ x dt ≤ C ′ T ,
which ends the proof of (6.9).

⊓ ⊔

We can now specify the sense of the solution we deal with. With DiPerna and Lions [START_REF] Diperna | On the Fokker-Planck-Boltzmann equation[END_REF], [START_REF] Diperna | On the Cauchy problem for Boltzmann equation: global existence and weak stability[END_REF][START_REF] Diperna | Global solutions of Boltzmann's equation and entropy inequality[END_REF], [START_REF] Lions | Compactness in Boltzmann equation via Fourier integral operators and applications Part I[END_REF] we say that 0 ≤ f ∈ C([0, ∞); L 1 (O)) is a renormalized solution of (6.1)-(6.2)-(1.1)-(1.8) if first f satisfies the a priori physical bound (6.8) and is a solution of

∂ ∂t β(f ) + v • ∇ x β(f ) + (∇ x V f + λ v) • ∇ v β(f ) -ν ∆ v β(f ) = (6.13) = β ′ (f ) (Q(f, f ) + λ N f ) -ν β ′′ (f ) |∇ v f | 2 in D ′ ((0, T ) × O),
for all time T > 0, and all β ∈ B 5 , the class of all functions β ∈ C

2 (R) such that |β ′′ (s)| ≤ C/(1+s), |β ′ (s)| ≤ C/ √ 1 + s, ∀s ≥ 0.
Thanks to (6.8) (and (6.6)) we see that each term in equation (6.13) makes sense. Next, the trace functions f (0, .) and γf defined by Theorem 4.5 through the Green formula (4.9) must satisfy (1.8) and (1.1), say almost everywhere. Finally, we will always assume that γf satisfies the additional bound (6.9). Our main result is the following stability or compactness result. Once again, in order not to repeat several times the proof, we establish our result for the full VPFPB system and the full VPB system, the same holds for the same equation with less terms. Theorem 6.2 Let (f n ) be a sequence of renormalized solutions to equation (6.1)-( 6.2) such that the associated trace functions γf n satisfy (1.1), with the linear reflection operator (1.2) when ν = 0 and a possibly mass flux depending accommodation coefficient (1.6) when ν > 0 (FP type models). Let us furthermore assume that both the sequence of solutions (f n ) and the trace sequence (γf n ) satisfy (uniformly in n) the natural physical a priori bounds

sup [0,T ] O f n 1 + |v| 2 + | log f n | dvdx + Ω |∇ x V fn | 2 dx (6.14) + T 0 O e(f n ) + ν |∇ v f n | 2 f n dvdxdt + T 0 ∂Ω E γ + f n M dσ x dt ≤ C T .
If f n (0, .) converges to f in weakly in L 1 (O) then, up to the extraction of a subsequence, f n converges weakly in L p (0, T ; L 1 (O)) for all T > 0 and p ∈ [1, ∞) (the convergence being strong when ν > 0) to a renormalized solution f to (6.1)-(6.2) with initial value f in and which satisfies the physical estimates (6.8). Furthermore, for any ε > 0 and T > 0, there exists a measurable set A ⊂ (0, T ) × ∂Ω such that meas ((0, T ) × ∂Ω \ A) < ε and γ + f n ⇀ γ + f weakly in L 1 (A × R N , dλ 1 ), (the convergence being strong when ν > 0). As a consequence we can pass to the limit in the boundary reflection condition (1.1) (and (1.6) when ν > 0), so that the trace condition is fulfilled and the trace estimate (6.9) holds.

Proof of the Theorem 6.2. From (6.14) we deduce, extracting a subsequence if necessary, that f n converges weakly in L p (0, T ; L 1 (O)) (∀p ∈ [1, ∞)) to a function f and that the local mass density ρ n = ρ fn satisfies (see [START_REF] Lions | Compactness in Boltzmann equation via Fourier integral operators and applications Part I[END_REF])

sup [0,T ] Ω ρ n (1 + | log ρ n |) dx ≤ C T .
In the case ν = 0, using the velocity averaging lemma of [START_REF] Golse | Regularity of the moments of solution of a transport equation[END_REF][START_REF] Diperna | L p regularity of velocity averages[END_REF] and the standard properties of the Poisson equation, we also show (see for instance [START_REF] Lions | Compactness in Boltzmann equation via Fourier integral operators and applications Part I[END_REF] and [START_REF] Stein | Singular integrals and differentiability properties of functions[END_REF])

ρ n -→ n→∞ ρ f in L p (0, T ; L 1 (Ω)) and ∇ x V fn -→ n→∞ ∇ x V f in L p (0, T ; W 1,1 ∩ L a (Ω))
for all T ∈ (0, ∞), p ∈ [1, ∞) and a ∈ [START_REF] Acerbi | Semi-continuity problems in the calculus of variations[END_REF][START_REF] Agoshkov | On the existence of traces of functions in spaces used in transport theory problems (in Russian)[END_REF]. It is also shown in [START_REF] Lions | Compactness in Boltzmann equation via Fourier integral operators and applications Part I[END_REF] that

Q ± (f n , f n ) 1 + δ f n ⇀ n→∞ Q±
δ weakly in L 1 ((0, T ) × O R ) and Q± δ ր Q ± (f, f ) a.e.

In the case ν > 0, since the term on the right hand side of equation (6.13) is bounded in L 1 , thanks to the uniform estimate (6.14), and since Λ E fn is an hypoelliptic operator (see [START_REF] Diperna | On the Fokker-Planck-Boltzmann equation[END_REF], [START_REF] Bouchut | Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system[END_REF], [START_REF] Hörmander | Hypoelliptic second order differential equations[END_REF]), we obtain that, say, log(1 + f n ) and next f n converge a.e. (see [START_REF] Bouchut | Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions[END_REF] and [START_REF] Diperna | On the Fokker-Planck-Boltzmann equation[END_REF]). We conclude that f n → f strongly in L p (0, T ; L 1 (O)), ∀p ∈ [1, ∞). It is also shown in [START_REF] Diperna | On the Fokker-Planck-Boltzmann equation[END_REF] that

Q(f n , f n ) 1 + f n → Q(f, f ) 1 + f strongly in L 1 ((0, T ) × O R ).
Therefore, using Theorem 5.1 or Theorem 5.2, we obtain that f satisfies the renormalized equation (6.13) (first for renormalizing function β ∈ B 4 and next for β ∈ B 5 ) and that γf n r ⇀ γf in the renormalized sense on (0, T ) × Σ, as well as γf n → γf a.e. on (0, T ) × Σ, when ν > 0. It is worth mentioning that f also satisfies the physical estimate (6.8), see [START_REF] Diperna | On the Fokker-Planck-Boltzmann equation[END_REF][START_REF] Diperna | Global solutions of Boltzmann's equation and entropy inequality[END_REF][START_REF] Lions | Compactness in Boltzmann equation via Fourier integral operators and applications Part I[END_REF] Next, from (1.1) we have

γ + f n ≤ ᾱ-1 M -1 (v) γ -f n on (0, T ) × Σ -, so that γ + f n r ⇀ ψ in (0, T ) × ∂Ω, with ψ ≤ ᾱ-1 M -1 (v) γ -f.
Furthermore, repeating the proof of Lemma 6.1 we get that ψ ∈ L 1/2 ((0, T ) × ∂Ω). Now, we can apply Theorem 3.9 (with m(v) = M (v), y = (t, x), d̟ y (v) = 1 Σ x + |n(x) • v| dv, φ n = γ + f n and dν(y) = dσ x dt), which says that for every ε > 0 there is A = A ε ⊂ (0, T ) × ∂Ω such that meas ((0, T ) × ∂Ω\A) < ε and γ + f n ⇀ γ + f weakly in L 1 (A × R N ).

In the case ν > 0, since we already know the a.e. convergence, this convergence is in fact strong in L 1 (A × R N ). There is no difficulty in passing to the limit in the boundary condition so that f satisfies (1.1) and f satisfies the same physical estimate (6.9) thanks to the convexity argument of Theorem 2.5.

⊓ ⊔ Remark 6.3 For the Boltzmann equation and the FPB equation, as well as for the VP system and the VPFP system when the Poisson equation (6.2) is provided with Neumann condition, we can prove the additional a priori estimate (2.3) on the trace function. As a consequence, we may also establish the a priori physical bound (6.8) for a time and position dependent wall temperature Θ = Θ(t, x) which satisfies 0 < Θ 0 ≤ Θ(t, x) ≤ Θ 1 < ∞.

Therefore, the stability result and the corresponding existence result can be generalized to these kind of boundary conditions. We refer to [START_REF] Arkeryd | On diffuse reflection at the boundary for the Boltzmann equation and related equations[END_REF] and [START_REF] Mischler | On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[END_REF] for more details. where M is the normalized Maxwellian (1.3). For that reflection operator (6.15), we can prove that a solution f to equations (6.1)-( 6.2)-(1.1) formally satisfies the a priori physical estimate (6.8)-(6.9) with E replaced by

E k (φ/M ) := v•n(x)>0 h φ M -h R φ M M v • n(x) dv.
By Jensen inequality one can prove that E k is non-negative, see [START_REF] Darrozès | Généralisation formelle du théorème H en présence de parois[END_REF], [START_REF] Cercignani | The mathematical theory of dilute gases[END_REF], [START_REF] Goudon | Sur quelques questions relatives à la théorie cinétique des gaz et à l'équation de Boltzmann[END_REF]. However, we do not know whether our analysis can be adapted to this general kernel. Nevertheless, considering a sequence (f n ) of solutions which satisfies the uniform interior estimate in (6.14), we can pass to the limit in (1.1) with the help of Theorem 5.1 or Theorem 5.2 and of Proposition 3.5.4, and we get that the limit function f is a solution which trace γf satisfies the boundary inequality condition (1.10). That extends and generalizes previous results known for the Boltzmann equation, see for instance [START_REF] Arkeryd | On diffuse reflection at the boundary for the Boltzmann equation and related equations[END_REF], [START_REF] Cercignani | Initial boundary value problems for the Boltzmann equation[END_REF], [START_REF] Mischler | On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system[END_REF].

A Appendix: More about the renormalized convergence

We come back to the notion of renormalized convergence and mainly discuss its relationship with the biting-L 1 weak convergence.

Remark A.1 1. Hypothesis ψ ∈ L 0 (Y ) in Theorem 3.7 (and (3.4)) is fundamental, since for example, the sequence (ψ n ) defined by ψ n = ψ ≡ +∞ ∀n does converge in the renormalized sense to ψ, but (ψ n ) does not converge (and none of its subsequence!) in the biting L 1 -weak sense.

2. The (asymptotically) boundedness of (ψ n ) in L 0 does not guarantee that (ψ n ) satisfies, up to the extraction of a subsequence, (3.3) or (3.4). An instructive example is the following: we define u(y) = 1/y on Y = [0, 1] that we extend by 1-periodicity to R, and we set ψ n (y) = u(n y) for y ∈ Y . Therefore, (ψ n ) is obviously bounded in L a (Y ) for all a ∈ [0, 1) and converges to ψ ≡ +∞ in the renormalized sense.

Proposition A.2 1. There exists (φ n ) which r-converges but does not b-converges.

2. There exists (φ n ) which b-converges but does not r-converges.

3. Given a sequence (φ n ), the property for any sub-sequence (φ n ′ ) there exists a sub-sequence (φ n ′′ ) of (φ n ′ ) such that φ n ′′ see [START_REF] Tartar | Compensated compactness and application to pde[END_REF]Theorem 5], [START_REF] Young | Lectures on the calculus of variations and optimal theory[END_REF]. Then define (φ n ) by setting φ 2 n = u n , φ 2 n+1 = v n . In such a way, we have exhibited a sequence (φ n ) which does not r-converge (for instance does not (T M )-renormalized converge) but converges to φ in the weak L 1 sense, and thus b-converges to φ. Moreover, for any sub-sequence (φ n ′ ), there exists a second sub-sequence (φ n ′′ ) which either converges to TM (if {n ′ } contain an infinity of even integer numbers) or to TM (if {n ′ } contain an infinity of odd integer numbers). Because TM ր φ and TM ր φ when M ր ∞, in both case φ n ′′ r-converges to φ, and (A.1) holds. ⊓ ⊔

  r-limsup φ n := sup ı∈I φ ı and r-liminf φ n := inf ı∈I φ ı .It is clear that if r-limsup φ n = r-liminf φ n then (φ n ) r-converges (up to the extraction of a subsequence).Proposition 3.5 1. If φ n r ⇀ φ, ψ n r ⇀ ψ and λ n → λ in R ⋆ + then φ n + λ ψ n r ⇀ φ + λ ψ. 2. Let φ n r⇀ φ and β be a non-negative and concave function then β(φ) ≥ r-limsup β(φ n ). 3. Let β be a strictly concave function, and (φ n ) be a sequence such that φ n r ⇀ φ and β(φ) ≤ r-liminf β(φ n ) then, up to the extraction a subsequence, φ n → φ a.e. in Y . 4. Let φ n r ⇀ φ and S be a bounded and non-negative operator of L 1 then S φ ≤ r-liminf S φ n . Proof of the Proposition 3.5. Step 1. From the elementary inequality

Theorem 4 . 4 (

 44 The case p ∈ [1, ∞)). Let g ∈ L ∞ (0, T ; L p loc ( Ō)) be a solution of equation(4.

O

  β(g(t 1 , .)) φ dxdv = O β(g(t 0 , .)) φ dxdv(4.13)

Remark 6 . 4

 64 Consider the general reflection operatorR φ = v ′ •n(x)>0 k(v, v ′ ) φ(v ′ ) v ′ • n(x) dv ′ (6.15)where the measurable function k satisfies the usual non-negative, normalization and reciprocity conditions k ≥ 0,v•n(x)<0 k(v, v ′ ) dv = 1, R M = M,(6.16)

⇀

  denotes either the b-convergence or the r-convergence As a consequence, the b-convergence and the r-convergence are not associated to any Hausdorff (separated) topological structure.Proof of Proposition A.2.Points 1 & 3. Let (φ n ) be the sequence defined byφ n = φ p,k = p 1 [k/p,(k+1)/p] where p ∈ N * , 0 ≤ k ≤ p -1 and n = 1 + 2 + ... + p + k. Then (φ n ) is bounded in L 1and clearly r-converges to 0, but does not b-converge. Moreover, for any subsequence (φ n ′ ) we can find a second subsequence (φ n ′′ ) such that φ n ′′ b-converges to 0.Points 2 & 3. Consider µ y = µ and ν y = ν two Young measures on Y = [0, 1] such that R z µ(dz) = R z ν(dz) =: φ ∈ L 1 (Y ), R T M (z) µ(dz) = R T M (z) ν(dz) ∀ M > 0,and define (u n ) (resp. (v n )) a sequence of L 1 functions tassociated to µ (resp. ν), such that for anyf ∈ C(R) f (u n ) ⇀ f := R f (z) µ y (dz) resp. f (v n ) ⇀ f := R f (z) ν y (dz) ,

Acknowledgments. I would like to thank T. Horsin and O. Kavian for many helpful discussions on weak-weak convergence. I express my gratitude to J. Soler for his kind hospitality while I was visiting at the Universidad de Granada where part of this work has been done. Financial support from TMR (ERB FMBX CT97 0157) for that visit is gratefully acknowledged. I also would like to thank F. Murat and F. Castaing for quoting additional references and for their useful comments.