Ugo Tricoli
email: ugo.tricoli@onera.fr

Leon Schlemmer

Using optical potentials with gain-loss to generate structural colors

We study the possibility to obtain structural colors through the use of supersymmetric transformations in optics such as the Darboux transform. Structural colors were originally discovered by studying the interference of light with natural photonics structures giving rise to vivid and spectacular tonalities. They differ fundamentally from ordinary colors based on light absorption at particular wavelengths, as they result from light interference only. To treat interference analytically, we make use of the Darboux transform to define materials with continuously varying spatial distributions of the refractive index that are exactly solvable for the electric field. Consequently, it is possible to calculate analytically the Transfer Matrix linked to the definition of the transmission and reflection coefficients. Interestingly, by using gain, anomalous transmission/reflection (i.e. larger than one) can be obtained, the physical system being open towards the external environment (the system is using external energy in order to increase both transmission and reflection). The generated active optical cavity can thus be used to amplify the incoming light in the desired spectral-angular region. The calculated refractive index distributions can be realized in practice as 1D multi-layered structures corresponding to optical filters in the visible.

Introduction

Structural colors were originally discovered by studying the interference of light with natural photonics structures giving rise to vivid and spectacular tonalities [START_REF] Kinoshita | Physics of structural colors[END_REF]. They differ fundamentally from ordinary colors based on light absorption at particular wavelengths as in pigments, dyes and metals as they are a consequence of light interference only. Many studies where dedicated to the understanding of natural photonic structures as they are found in animals and insects [START_REF] Anderson | An electron microscope study of some structural colors of insects[END_REF][START_REF] Kinoshita | Structural colors in nature: the role of regularity and irregularity in the structure[END_REF][START_REF] Noh | How noniridescent colors are generated by quasi-ordered structures of bird feathers[END_REF][START_REF] Vignolini | Pointillist structural color in pollia fruit[END_REF][START_REF] Burresi | Bright-white beetle scales optimise multiple scattering of light[END_REF]. On the other hand numerous other studies where dedicated to the attempt to produce structural colors through artificial structures based mainly on photonics [START_REF] Forster | Biomimetic isotropic nanostructures for structural coloration[END_REF][START_REF] Wiersma | Disordered photonics[END_REF] and plasmonics [START_REF] Yu | Transmissive/reflective structural color filters: theory and applications[END_REF]. Surprisingly, no study has been directed yet to explore the possibility to obtain structural colors through supersymmetry (SUSY) [START_REF] Cooper | Supersymmetry and quantum mechanics[END_REF]. We propose refractive index distributions that are used to produce iridescent structural colors i.e. strongly dependent on the incident direction. Supersymmetry, implemented through the Darboux transform (DT) [START_REF] Darboux | On a proposition relative to linear equations[END_REF], is explicitly broken truncating the refractive index distribution in space. Nevertheless the results remain analytical [START_REF] Krapez | Sequences of exact analytical solutions for plane waves in graded media[END_REF] being calculated through the Transfer Matrix [START_REF] Lekner | Light in periodically stratified media[END_REF][START_REF] Mostafazadeh | Transfer matrix in scattering theory: A survey of basic properties and recent developments[END_REF]. Due to the presence of loss and gain regions, the system is behaving as an active cavity being able to amplify the incoming light. We demonstrate how controlling the light incoming direction, many different intense colors can be generated by amplifying the radiation at the selected frequency. Only the spatial distribution of the refractive index is controlled while, for a given point in space, the refractive index is assumed to obey spectrally the Drude dispersion model, i.e. no absorption resonance is treated. As a consequence the observed colors result from geometric interference only and not from spectral absorption. Interestingly, almost the same dependence of colors on direction is predicted for the transmitted and reflected light. In this work we consider scattering of visible light from a sample with continuously varying (in space) refractive index with a large gain region and very small regions with absorption. We calculate transmission and reflection spectra through the Transfer Matrix theory [START_REF] Lekner | Light in periodically stratified media[END_REF][START_REF] Mostafazadeh | Transfer matrix in scattering theory: A survey of basic properties and recent developments[END_REF]. When SUSY is broken anomalous transmission/reflection is observed i.e. the transmission/reflection coefficients become larger than one. This can be seen as the manifestation of the fact that the system is open towards the external environment i.e. external (electromagnetic) energy is used to generate anomalous transmission/reflection. Indeed, a spectral-angular filter can be obtained for generating structural colors that are strongly direction dependent.

From the optical potential to the refractive index spatial distribution: Dispersion

We consider scattering of a monochromatic electromagnetic wave by an inhomogeneous dielectric slab of finite thickness along x but infinite extension in the yz-plane with a varying refractive index in the x direction n(x). The slab is surrounded by a non-absorbing background medium with refractive index n b . Only a scalar electric field corresponding to a TEpolarized wave is considered. The DT is used to define optical potentials V (x) that are continuously varying in space and are analytically solvable for the electric field in the corresponding Schrödinger-like equation [START_REF] Krapez | Sequences of exact analytical solutions for plane waves in graded media[END_REF]. Here the Darboux transformation energy is fixed to ε = 0.00005i (in units of the square of a reference wavenumber used for scaling) and n b = 1 i.e. the optical filter is immersed in air (see [START_REF] Tricoli | Susy designed broken pt-symmetric optical filters[END_REF] for details about the analytical DT). By using the optical potential definition we can calculate the spatial refractive index distribution by n(x) = n 2 b -V (x)/k 2 where k is the wavenumber in void of the incoming wave (and V (x) has the same units of ε). This assures a dispersion relation compatible with the Drude model for sufficiently small imaginary part of the refractive index. In order to break SUSY we truncate the refractive index distribution at some points in space (represented as dots in Fig. (1)). Selecting almost only the gain portion of the index (where the index imaginary part is negative) we define an active cavity. Importantly, the Kramers-Kronig relations [START_REF] Altarelli | Superconvergence and sum rules for the optical constants[END_REF] are not violated since the index imaginary part is sufficiently small. We note that for purely supersymmetric index profiles, we would expect reflectionless m aterials w ith u t ransmission f or a ll wavelengths a nd i ncoming d irections ( this i s the case for real and negative transformation energy) since the superpartner of the generated index is void. Indeed, when two index distributions are superpartners, they share the same scattering spectra [START_REF] Miri | Susy-inspired one-dimensional transformation optics[END_REF]. However, here a purely imaginary transformation energy is taken thus SUSY is broken by construction. In addition, we cut the spatial distribution at some points in order to get angular scattering spectra that are interesting for iridescent structural colors production.

Results: Iridescent structural colors in transmission and reflection

We analytically evaluate the transmission and reflection spectra in the visible for all angles of incidence. For the definition of the optical Transfer Matrix method for analytically solvable refractive index spatial distributions see [START_REF] Mostafazadeh | Transfer matrix in scattering theory: A survey of basic properties and recent developments[END_REF][START_REF] Born | Principles of optics: electromagnetic theory of propagation, interference and diffraction of light[END_REF]. Interestingly, for the refractive index distribution of Fig. [START_REF] Kinoshita | Physics of structural colors[END_REF] the net energy balance is positive indicating a gain effect (the absorption coefficient becomes negative). This can be seen as an evidence of the opening of the system toward the external environment (as it is known for quantum open systems [START_REF] Persson | Observation of resonance trapping in an open microwave cavity[END_REF][START_REF] Hatano | Some properties of the resonant state in quantum mechanics and its computation[END_REF][START_REF] Garmon | Bound states, scattering states, and resonant states in pt-symmetric open quantum systems[END_REF]). The dielectric system is using external energy in order to increase both transmission and reflection.

We show in Fig. (2) the transmission angular spectrum for incoming visible light (in the range [350, 700] nm) associated to the refractive index distribution of Fig. [START_REF] Kinoshita | Physics of structural colors[END_REF]. Notably, the optical potential is capable of operating spectral-angular filtering and is particularly efficient in selecting a narrow high intensity peak which is strongly dependent on the incoming direction. Remarkably, the spectral-angular filter has almost the same performance in the reflection configuration (not shown). We also calculate the RGB colors associated to different spectra for a particular incoming direction (we used a standard computer graphics RGB procedure as described in [START_REF]Javascript code to convert light wavelength to color[END_REF]). Interestingly, the proposed active optical device is able to generate structural colors from red to purple by changing the incoming direction (or alternatively for isotropic illumination, by changing the observation direction). However, for large angles the colors are less intense due to the absence of large radiation amplification.

Conclusions

We investigated the possibility to generate structural colors strongly dependent from direction (thus giving rise to iridescence) by using broken SUSY 1D-spatial distributions of the refractive index. Interestingly, the same device can be used for both transmission and reflection configurations. These results are only obtainable through the use gain i.e. an active cavity is defined which can efficiently accumulate the external electromagnetic energy in order to amplify the desired color channels. Due to the possibility to produce large surfaces with 1D multi-layered structure down to nanometer precision in the profile production, a direct application of the reported results could be the design of (strongly direction dependent) active color filters for the generation of colors with high intensity and bright tonalities.

Fig. 1 .

 1 Fig. 1. The refractive index distribution n(x). (a) Scattering geometry and a physical realization of n(x) a graded refractive index distribution with a large gain region and tiny absorption parts. (b) The refractive index distribution variation ∆n(x) = n(x)n b . The blue line represents Re[∆n(x)] while the yellow line represents Im[∆n(x)]. The region in between the red dots is used for spectra analytical calculation.

Fig. 2 .

 2 Fig. 2. Transmission angular spectrum of the refractive index distribution shown in Fig.(1). The corresponding structural colors (in RGB scheme) obtained for incidence angles from 0 • to 89 • with a step of 5 • are shown on the left side of the panel while on the right side is the transmitted normalized intensity (much greater than one due to amplification).


Leif Arkeryd
email: arkeryd@chalmers.se

The paper proves existence of stationary solutions to the Boltzmann equation in a bounded set of R 2 for given indata, hard forces and truncation in the collision kernel for small velocities and close to parallel colliding velocities. It does not use any averaging in velocity lemma and uses techniques from the discrete velocity stationary case recently developped in [7]-[8]-[9], where the averaging in velocity lemmas are not valid.

Stationary solutions to the Boltzmann equation in the plane.

1 Introduction.

Consider the stationary Boltzmann equation in Ω

⊂ R 2 , v • ∇ z f (z, v) = Q(f, f ), z ∈ Ω, v ∈ R 2 , (1.1)
where Ω is a strictly convex domain with C 1 boundary. The nonnegative function f represents the density of a rarefied gas with z the position and v the velocity. The operator Q is the nonlinear Boltzmann collision operator with angular cut-off and a truncation for small velocities and close to parallel colliding velocities, From now on denote χ η B by B. The inward and outward boundaries in phase space are

Q(f, f )(z, v) = R 2 S 1 χ η (v, v * , ω) B(v -v * , ω) f (z, v )f (z, v * ) -f (z, v)f (z, v * ) dv * dω. (1.2) S 1 is the unit circle in R 2 , v = v+v * 2 + |v-v * | 2 ω, v * = v+v * 2 -|v-v * | 2 ω. For η ∈]0,
∂Ω + = {(z, v) ∈ ∂Ω × R 2 ; v • n(z) > 0}, ∂Ω -= {(z, v) ∈ ∂Ω × R 2 ; v • n(z) < 0},
where n(z) denotes the inward normal on ∂Ω. Given a function f b defined on ∂Ω + , solutions f to (1.1) are sought with f (z, v) = f b (z, v), (z, v) ∈ ∂Ω + .

(1.5)

For any (z, v) ∈ Ω × R 2 , denote by

s + (z, v) = inf{s > 0; (z -sv, v) ∈ ∂Ω + }, s -(z, v) = inf{s > 0; (z + sv, v) ∈ ∂Ω -}, z + (z, v) = z -s + (z, v)v, z -(z, v) = z + s -(z, v)v. (1.6)
Solutions are understood in mild form, i.e.

f (z, v) = f b (z + (z, v), v) + s + (z,v) 0 Q(f, f )(z + (z, v) + sv, v)ds, a.a. (z, v) ∈ Ω × R 2 . (1.7)
The main result of the paper is the following.

Theorem 1.1 Let f b be a non negative measurable function such that

∂Ω + v • n(z) 1 + v 2 + ln f b f b (z, v)dσ(z)dv < ∞. (1.8)
Then equation (1.1) has a non negative solution satisfying the boundary condition (1.5).

For the stationary Boltzmann equation the control of mass and entropy is not straightforward, contray to the evolutionary case. Existence results in the slab, i.e. one-dimensional spatial and three-dimensional velocity frame, together with an invariance with respect to the two remaining space variables of the distribution function were first established for the nonlinear Boltzmann equation. Stationary integrable solutions to the Boltzmann equation in a slab have been proven in [START_REF] Arkeryd | L 1 solutions to the stationary Boltzmann equation in a slab[END_REF], [START_REF] Arkeryd | The stationary Boltzmann equation in a slab, with given weighted mass for hard and soft forces[END_REF] and [START_REF] Arkeryd | A compactness result related to the stationary Boltzmann equation in a slab, with applications to the existence theory[END_REF], for different boundary conditions, bounds on the entropy production term and a weighted moment of the distribution function giving control of the entropy. For higher space dimension, stationary unsigned solutions close to Maxwellians were constructed in convex domains [START_REF] Guiraud | Problème aux limites intérieur pour l'équation de Boltzmann linéaire[END_REF], [START_REF] Guiraud | Problème aux limites intérieur pour l'équation de Boltzmann en régime stationnaire, faiblement non linéaire[END_REF]. In [START_REF] Esposito | Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law[END_REF], the existence and uniqueness of the stationary solution to the Boltzmann equation close to a uniform Maxwellian have been proven in a bounded domain of R n , 1 ≤ n ≤ 3, for diffuse reflection boundary conditions. Its hydrodynamic limit to a solution to the steady incompressible Navier-Stokes-Fourier system has been performed in [START_REF] Esposito | Stationary solutions to the Boltzmann equation in the hydrodynamic limit[END_REF]. Existence of stationary solutions to the Boltzmann equation in a bounded domain of R n , n ≥ 1, and given indata has been proven in [START_REF] Arkeryd | The stationary Boltzmann equation in R n with given indata[END_REF]. There, scaling arguments from [START_REF] Arkeryd | On the stationary Boltzmann equation in R n[END_REF] were used. In this paper, we prove existence of solutions to the stationary Boltzmann equation in the plane with the help of the entropy production term and the construction of 'good' characteristics where the distribution function is bounded and 'bad' characteristics of arbitrarily small measure. This is inspired by recent results for discrete velocity models for the Boltzmann equation where averaging lemmas do not hold and new arguments are required. In [START_REF] Arkeryd | Stationary solutions to the two-dimensional Broadwell model[END_REF], [START_REF] Arkeryd | On stationary solutions to normal, coplanar discrete Boltzmann equation models[END_REF], [START_REF] Arkeryd | Discrete velocity Boltzmann equations in the plane: Stationary solutions[END_REF] a weaker property than L 1 compactness of averages in velocity, i.e. the L 1 compactness of the integrated collision frequencies of a sequence of approximations is proven. It strongly depends on the two-dimensional spatial dimension. In this paper we use the tools developped in [START_REF] Arkeryd | Stationary solutions to the two-dimensional Broadwell model[END_REF], [START_REF] Arkeryd | On stationary solutions to normal, coplanar discrete Boltzmann equation models[END_REF], [START_REF] Arkeryd | Discrete velocity Boltzmann equations in the plane: Stationary solutions[END_REF] and do not use any averaging lemma. Work is in progress to fill a gap in the proof of Lemma 4.1 of [START_REF] Arkeryd | On the evolutionary velocity-discrete Boltzmann equation[END_REF] that uses these techniques in the discrete velocity evolutionary case. The construction of a first sequence of approximations with damping and convolutions is performed in Section 2. In Section 3, the damping and convolutions are removed, leading to a more involved sequence of approximations. In Section 3, the phase space is split into 'good' characteristics where the approximations are uniformly bounded and 'bad' characteristics of arbitrarily small measure. In Section 4, the L 1 compactness of the integrated collision frequency sequence is proven. The passage to the limit in the mild form satisfied by the approximations is performed in Section 5.

2 First approximations.

In the paper we denote by c constants that do not depend on approximations.

To emphasize their dependence on the indatum f b , we sometimes denote them by c b . We use the following approximation scheme. Let (B α ) α∈]0,1[ , be a family of

C ∞ regularizations of B. Let (ϕ α ) α∈]0,1[ be mollifiers defined from ϕ ∈ C ∞ 0 (R 4 ) such that ϕ(z, v) = 0 for |z| ≥ 1 or |v| ≥ 1, ϕ(z, v)dzdv = 1, by ϕ α (z, v) = 1 α 4 ϕ( z α , v α ).
Outside the boundary the function to be convolved with µ α , is continued in the normal direction by its boundary value. Let μk be a smooth mollifier on ∂Ω × R 2 in a ball of radius 1 k . Denote by

f bk = min{f b , k} * μk . Lemma 2.1 For every (α, k) ∈]0, 1[×N * , there is a non negative solution F to αF + v • ∇ z F = B α F 1 + F k (z, v ) F * ϕ α 1 + F * ϕα k (z, v * ) - F 1 + F k (z, v) F * ϕ α 1 + F * ϕα k (z, v * ) dv * dω, (z, v) ∈ Ω × R 2 , (2.1)
F (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + . (2.2)
Proof of Lemma 2.1. It follows the lines of the proofs in Section 2 in [START_REF] Arkeryd | On stationary solutions to normal, coplanar discrete Boltzmann equation models[END_REF] that we refer to for details. Let (α, k) ∈]0, 1[×N * be given. Let K be the closed and convex subset of L 1 (Ω × R 2 ) defined by

K = {f ∈ L 1 + (Ω × R 2 ); f (z, v)dzdv ≤ 1 α ∂Ω + v • n(z)f b (z, v)dσ(z)dv}.
Define the map T from K into K by T (f ) = F , where F is the solution to

αF (z, v) + v • ∇ z F (z, v) = R 2 ×S 1 B α F 1 + F k (z, v ) f * ϕ α 1 + f * ϕα k (z, v * ) - F 1 + F k (z, v) f * ϕ α 1 + f * ϕα k (z, v * ) dv * dω, (z, v) ∈ Ω × R 2 , (2.3) F (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + .
(2.4) F = T (f ) can be obtained as the limit in L 1 + (Ω × R 2 ) of the sequence (F q ) q∈N defined by F 0 = 0 and

αF q+1 + v • ∇ z F q+1 = R 2 ×S 1 B α F q 1 + F q k (z, v ) f * µ α 1 + f * µα k (z, v * ) - F q+1 1 + F q k (z, v) f * µ α 1 + f * µα k (z, v * ) dv * dω , (2.5)
F q+1 (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + , q ∈ N. (2.6)
In exponential form F q+1 can be written as

F q+1 (z, v) = f bk (z + (z, v), v)e -αs + (z,v) e -0 -s + (z,v) 1 1+ F q k (z+sv,v) Bα f * µα 1+ f * µα k (z+sv,u * )du * dγ ds + 0 -s + (z,v) B α F q 1 + F q k (z + sv, v ) f * µ α 1 + f * µα k (z + sv, v * )e αs e s 0 1 1+ F q k (z+rv,v) Bα f * µα 1+ f * µα k (z+rv,u * )du * dγ dr . (2.7)
The sequence (F q ) q∈N is monotone. Indeed, F 0 ≤ F 1 , by the exponential form of F 1 . If F q ≤ F q+1 , then it follows from the exponential forms of F q+1 and F q+2 that F q+1 ≤ F q+2 . Moreover,

α F q+1 (z, v)dzdv ≤ ∂Ω + v • n(z)f bk (z, v)dσ(z)dv + B α F q -F q+1 1 + F q k (z, v) f * µ α 1 + f * µα k (z, v * )dzdvdv * dω, so that F q+1 (z, v)dzdv ≤ 1 α ∂Ω + v • n(z)f b (z, v)dσ(z)dv, q ∈ N. (2.8)
By the monotone convergence theorem, (F q ) q∈N converges in L 1 (Ω × R 2 ) to a solution F of (2.3)-(2.4). The solution of (2.3)-(2.4) is unique in the set of non negative functions. Indeed, let G be a non negative solution of (2.3)-(2.4). It follows by induction that

F q ≤ G, q ∈ N.
(2.9) Indeed, (2.9) holds for q = 0, since G ≥ 0. Assume (2.9) holds for q. Using the exponential form of F q+1 implies F q+1 ≤ G. Consequently,

F ≤ G. (2.10)
Moreover, subtracting the equation satisfied by G from the equation satisfied by F , and integrating the resulting equation on Ω × R 2 leads to

α Ω×R 2 (G -F )(z, v)dzdv + ∂Ω - |v • n(z)|(G -F )(z, v)dσ(z)dv = 0.
(2.11)

It results from (2.10)-(2.11) that G = F . The map T is continuous in the L 1 -norm topology (cf [6] pages 124-5). Namely, let a sequence (f q ) q∈N in K converge in L 1 (Ω × R 2 ) to f ∈ K. Set F q = T (f q ).
Because of the uniqueness of the solution to (2.3)-(2.4), it is enough to prove that there is a subsequence of (F q ) converging to F = T (f ). Now there is a subsequence of (f q ), still denoted (f q ), such that decreasingly (resp. increasingly) (G q ) = (sup r≥q f r ) (resp. (g q ) = (inf r≥q f r )) converges to f in L 1 . Let (S q ) (resp. (s q )) be the sequence of solutions to

αS q + v • ∇ z S q = B α S q 1 + S q k (z, v ) G q * µ α 1 + G q * µα k (z, v * ) - S q 1 + S q k (z, v) g q * µ α 1 + g q * µα k (z, v * ) dv * dω , (2.12
)

S q (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + , (2.13
)

αs q + v • ∇ z s q = B α s q 1 + s q k (z, v ) g q * µ α 1 + g q * µα k (z, v * ) - s q 1 + s q k (z, v) G q * µ α 1 + G q * µα k (z, v * ) dv * dω , (2.14)
s q (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + . (2.15) (S q
) is a non-increasing sequence, since that holds for the successive iterates defining the sequence. Then (S q ) decreasingly converges in L 1 to some S. Similarly (s q ) increasingly converges in L 1 to some s. The limits S and s satisfy (2.3)-(2.4). It follows by uniqueness that s = F = S, hence that (F q ) converges in L 1 to F . The map T is compact in the L 1 -norm topology. In [START_REF] Arkeryd | On stationary solutions to normal, coplanar discrete Boltzmann equation models[END_REF] an averaging lemma was used. Here we replace it by the following argument. Let (f l ) l∈N be a bounded sequence of L 1 and

F l = T (f l ). Denote by Fl = F l 1+ F l k , fl = f l * ϕ j 1+ f l * ϕ j k
, by B(0, V ) the open ball in R 2 centered at the origin and of radius V > 0, and by I(α, β) the interval with end points (α, β) ∈ R 2 . (F l ) l∈N is the sum of

f bk (z + (z, v), v) + 0 -s + (z,v) B α Fl (z + sv, v ) fl (z + sv, v * )dv * dωds l∈N * , (2.16) and - 0 -s + (z,v) B α Fl (z + sv, v) fl (z + sv, v * )dv * dωds l∈N * .
(2.17)

In order to prove that (2.16) is compact in L 1 , it is sufficient to prove that for any V > 0 and µ ∈]0, 1[,

0 -s + (z,v) B(0,V )×Aµ B α Fl (z + sv, v ) fl (z + sv, v * )dv * dωds l∈N * , (2.18)
is compact in L 1 (Ω × B(0, V ))
where

A µ = {ω ∈ S 1 ; | (v, ω)| ≥ µ and |π -(v, ω)| ≥ µ}. (2.19)
Indeed, noticing that v is parallel to v if and only if ω = ± v |v| , and expressing ω by its angle with the vector v |v| , the integral over Ω × B(0, V ) of

0 -s + (z,v) B(0,V )×A c µ B j Fl (z + sv, v ) fl (z + sv, v * )dv * dωdsdzdv is smaller than ck 2 V 2 µ. The sequence (2.18) is uniformly bounded in L 1 (Ω×B(0, V )).
Let us prove that it is uniformly equiintegrable with respect to the z variable. By the restriction to

A µ in (2.18), any (v, v ) ∈ R 2 × R 2
considered when integrating the absolute value of (2.18) over Ω × B(0, V ) forms a basis in R 2 . For any h ∈ R 2 , denote by (a(h), b(h)) its coordinates in this basis. Split the difference of (2.18) between (z, v) and (z + h, v) into the three following terms,

I(-s + (z,v),-s + (z+h,v)) Aµ B α Fl (z + h + sv, v ) fl (z + h + sv, v * )dv * dωds, 0 -s + (z,v) Aµ B α Fl (z + h + sv, v ) fl (z + h + sv, v * )dv * dω - Aµ B α Fl (z + b(h)v + sv, v ) fl (z + b(h)v + sv, v * )dv * dω ds = I(-s + (z,v)-a(h),-s + (z,v))∪I(0,a(h)) Aµ B α Fl (z + h + sv, v ) fl (z + h + sv, v * )dv * dω ds, 0 -s + (z,v) Aµ B α Fl (z + b(h)v + sv, v ) fl (z + b(h)v + sv, v * ) -fl (z + sv, v * ) dv * dω ds,
that tend to zero h → 0 when integrated over Ω × {v ∈ R 2 ; |v| < V }, and

0 -s + (z,v) Aµ B α Fl (z + b(h)v + sv, v ) -Fl (z + sv, v ) fl (z + sv, v * )dv * dωds. (2.20)
Notice that the integrand in the first line of (2.20) is a directional derivative in the direction v . Consequently, (2.20) is equal to

0 -s + (z,v) Aµ B α Fl (z + sv, v ) fl (z + sv, v * )e -αb(h) exp(- b(h) 0 B α fl (z + sv + rv , u * )du * dγ (1 + F l k )(z + sv + rv , v ) dr) -1 dv * dωds + 0 -s + (z,v) Aµ B α fl (z + sv, v * ) b(h) 0 B α Fl (z + sv + rv , V ) fl (z + sv + rv , V * )du * dγ e α(r-b(h)) exp - b(h) r B α fl (z + sv + tv , u * )du * dγ (1 + F l k )(z + sv + tv , v )
dt dr dv * dωds.

(2.21)

Here, V (resp V * ) denotes v -(v -u * , γ)γ (resp. u * + (v -u * , γ)γ).
(2.21) tends to zero when h → 0 when integrated over Ω × B(0, V ) in absolute value since all integrands are uniformly bounded and the domains of integration are of order h. This ends the proof of the uniform equiintegrability of (2.18) w.r.t. the z variable. The proof of its uniform equiintegrability w.r.t. the v variable is analogous. The L 1 compactness of (2.17) can be proven analogously. Hence the sequence (F l ) l∈N * is compact in L 1 . And so, the Schauder fixed point theorem applies to T , leading to a solution F of (2.1)-(2.2).

3 Removal of the damping and convolutions.

For any k ∈ N * , denote by

Q + k (resp. Q k , resp. ν k (F ), resp. D k
) the approximate gain term (resp. collision operator, resp. collision frequency, resp. entropy production term) defined by

Q + k (F, F )(v) = R 2 ×S 1 B F 1 + F k (v ) F 1 + F k (v * )dv * dω, (3.1)
ν k (F )(v) = 1 1 + F (v) k B F 1 + F k (v * )dv * dω, (3.2)
Q k (F, F ) = Q + k (F, F ) -F ν k (F ), D k (v) = R 2 ×S 1 B F k 1 + F k k (v ) F k 1 + F k k (v * ) (v * ) - F k 1 + F k k (v) F k 1 + F k k (v * ) ln F k 1 + F k k (v ) F k 1 + F k k (v * ) (v * ) 1 + F k k F k (v) 1 + F k k F k (v * ) dv * dω . (3.3) For any (α, k) ∈]0, 1[×N * , denote by F α,k the solution to (2.1)-(2.2) obtained in the previous section. (F α,k ) α∈]0,1[ is weakly compact in L 1 loc (Ω × R 2
) since it is bounded by a multiple of k 2 . Denote by F k the limit of a converging subsequence when α → 0. In the next lemma we prove that for a subsequence, the convergence is strong in L 1 (Ω × R 2 ).

Lemma 3.1

There is a sequence (α q ) q∈N tending to zero when q → +∞, such that when q → +∞, (F αq,k ) q∈N strongly converges in

L 1 (Ω × R 2 ) to F k . Moreover, F k is a solution to v • ∇ z F k = Q k (F k , F k ), (z, v) ∈ Ω × R 2 , (3.4)
F k (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + , (3.5)
and

(1 + v 2 )F k (z, v)dzdv ≤ c b , (3.6)
D k (z, v)dzdv ≤ c b , (3.7
)

∂Ω - F k (z, v) | v • n(z) | dσ(z)dv + ∂Ω -,F k ≤k F k ln F k (z, v) | v • n(z) | dσ(z)dv + ln k 2 ∂Ω -,F k ≥k F k | v • n(z) | dσ(z)dv ≤ c b , k ∈ N * . (3.8)
Proof of Lemma 3.1 Consider the approximation scheme (f α,ρ ) ρ∈N of F α,k , f α,0 = 0, (3.9)

αf α,ρ+1 (z, v) + v • ∇ z f α,ρ+1 (z, v) = B α F α,k 1 + F α,k k (z, v ) F α,k * µ α 1 + F α,k * µα k (z, v * ) - f α,ρ+1 1 + f α,ρ+1 k (z, v) f α,ρ * µ α 1 + f α,ρ * µα k (z, v * ) dv * dω, (3.10) f α,ρ+1 (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + , ρ ∈ N. (3.11)
f α,1 is explicitly given in terms of F α,k . It follows from the exponential forms of F α,k and f α,1 that

F α,k ≤ f α,1 , α ∈]0, 1[.
The sequence (f α,ρ ) ρ≥2 is constructed as follows. Denote by S the map from

(L 1 (Ω × R 2 )) 2 mapping (X, Z) into W = S(X, Z) ∈ L 1 (Ω × R 2 ) solution to αW + v • ∇ z W = B α F α,k 1 + F α,k k (z, v ) F α,k * µ α 1 + F α,k * µα k (z, v * ) - W 1 + X k (z, v) Z * µ α 1 + Z * µα k (z, v * ) dv * dω , W (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + .
Denote by f α,1,0 = S(0, f α,1 ), f α,1,r = S(f α,1,r-1 , f α,1 ),

F α,k,0 = S(0, F α,k ), F α,k,r = S(F α,k,r-1 , F α,k ), r ∈ N * .
First,

f α,1,0 ≤ F α,k,0 .
Then the sequence (f α,1,r ) r∈N (resp. (F α,k,r ) r∈N ) is increasing with limit

f α,2 (resp. F α,k ). It follows from f α,1,r ≤ F α,k,r , r ∈ N, that f α,2 ≤ F α,k . (3.12) Let f α,2,0 := S(0, f α,2 ), f α,2,r := S(f α,2,r-1 , f α,2 ), r ∈ N * .
It follows from (3.12) that

f α,2,0 ≥ F α,k,0 .
The sequence (f α,2,r ) r∈N is also increasing with limit f α,3 and with f α,2,r ≥ F α,k,r . Hence

f α,3 ≥ F α,k .
From here by induction on ρ, it holds that

f α,2ρ ≤ f α,2ρ+2 ≤ F α,k ≤ f α,2ρ+3 ≤ f α,2ρ+1 , α ∈]0, 1[, ρ ∈ N. (3.13)
By induction on r, for each r the sequence (f α,1,r ) α∈]0,1[ is translationally equicontinuous in α. The limit sequence (f α,2 ) α∈]0,1[ is also translationally equicontinuous. This is so, since given > 0, r and then h 0 can be taken so that

(f α,2 -f α,1,r )(z, v)dzdv < and |f α,1,r (z + h, v + h) -f α,1,r (z, v)|dzdv < , |h| < h 0 , | h| < h 0 .
It can analogously be proven that for each ρ ∈ N, (f α,ρ ) α∈]0,1[ is translationally equicontinuous in α. Let (α q ) q∈N be a sequence tending to zero. Take a subsequence in (α q ) q∈N , still denoted by (α q ) q∈N , such that (f αq,2 ) q∈N converges in L 1 to some f 0,2 when q → +∞.

Continuing by induction gives a sequence (f 0,ρ ) ρ∈N satisfying

f 0,2ρ ≤ f 0,2ρ+2 ≤ F k ≤ f 0,2ρ+3 ≤ f 0,2ρ+1 , ρ ∈ N, (3.14) v • ∇ z f 0,ρ+1 (z, v) = G(z, v) -B f 0,ρ+1 1 + f 0,ρ+1 k (z, v) f 0,ρ 1 + f 0ρ k (z, v * )dv * dω, f 0,ρ+1 (z, v) = f bk (z, v), (z, v) ∈ ∂Ω + .
Here, G is the weak L 1 limit when α → 0 of the gain term

B α F α,k 1 + F α,k k (z, v ) F α,k * µ α 1 + F α,k * µα k (z, v * )dv * dω .
In particular, (f 0,2ρ ) ρ∈N (resp. (f 0,2ρ+1 ) ρ∈N ) non decreasingly (resp. non increasingly) converges in L 1 to some g (resp. h) when ρ → +∞. The limits satisfy

0 ≤ g ≤ F k ≤ h, (3.15) v • ∇ z h = G -B h 1 + h k (z, v) g 1 + g k (z, v * )dv * dω, (3.16) v • ∇ z g = G -B g 1 + g k (z, v) h 1 + h k (z, v * )dv * dω, (3.17)
(h -g)(z, v) = 0, (z, v) ∈ ∂Ω + .
Subtracting (3.17) from (3.16) and integrating the resulting equation on Ω × R 2 gives that

∂Ω - | v • n(Z) | (h -g)(Z, v)dσ(Z)dv = 0, so that h -g = 0 also on ∂Ω -. Hence, s -(z,v) -s + (z,v) h(z + sv, v)e s 0
Bh(z+rv,u)dudγdr

B(h -g)(z + sv, v * )dv * dω ds = 0, (z, v) ∈ Ω × R 2 . (3.18)
(3.15) and (3.18) imply that g = h and is equal to

F k . (F αq,k ) q∈N converges to F k in L 1 (Ω × R 2
) when q → +∞. Indeed, given η > 0, choose ρ 0 big enough so that

f 0,2ρ 0 +1 -f 0,2ρ 0 L 1 < η and f 0,2ρ 0 -F k L 1 < η,
then q 0 big enough, so that

f αq,2ρ 0 +1 -f 0,2ρ 0 +1 L 1 ≤ η and f αq,2ρ 0 -f 0,2ρ 0 L 1 ≤ η, q ≥ q 0 .
Then split F αq,k -F k L 1 as follows,

F αq,k -F k L 1 ≤ F αq,k -f α,2ρ 0 L 1 + f α,2ρ 0 -f 0,2ρ 0 L 1 + f 0,2ρ 0 -F k L 1 ≤ f α,2ρ 0 +1 -f α,2ρ 0 L 1 +2η by (3.13) ≤ f α,2ρ 0 +1 -f 0,2ρ 0 +1 L 1 + f 0,2ρ 0 +1 -f 0,2ρ 0 L 1 + f 0,2ρ 0 -f α,2ρ 0 L 1 +2η ≤ 5η, q ≥ q 0 .
It remains to prove (3.6)-(3.8). Multiplying (3.4) by 1 + v 2 and integrating over Ω × R 2 , leads to

∂Ω - |v • n(z)|(1 + v 2 )F k (z, v)dσ(z)dv ≤ ∂Ω + v • n(z)(1 + v 2 )f b (z, v)dσ(z)dv. (3.19)
Denote by (v 1 , v 2 ) the components of v. Multiply (3.4) by v 1 and integrate it over Ω a × R 2 , where Ω a is the part of Ω with z 1 < a. Set S a = Ω ∩ {z 1 = a} and ∂Ω a = ∂Ω ∩ Ωa .

This gives

Sa×R 2 v 2 1 F k (a, z 2 , v)dz 2 dv = - ∂Ωa×R 2 v 1 v • n(z)F k (z, v)dzdv.
Ω×R 2 v 2 1 F k (z, v)dzdv ≤ c b . (3.21) ( v 2 2 F k (z, v)dzdv)
k∈N * is analogously bounded from above. Thus the boundedness of energy holds. Recalling the small velocity cut-off χ η defined in (1.3), this in turn implies the boundedness of mass. The boundedness of the mass outflow in (3.8) follows from an integration of (3.4) on Ω × R 2 . Finally, Green's formula for

F k ln F k 1+ F k k implies that for some c b > 0, ∂Ω - |v • n(z)| F k ln F k -(k + F k ) ln(1 + F k k ) (z, v)dσ(z)dv + Ω×R 2 D k (z, v)dzdv ≤ c b , k ∈ N * . Moreover, k ∂Ω - ln(1 + F k k )(z, v) | v • n(z) | dσ(z)dv ≤ ∂Ω - F k (z, v) | v • n(z) | dσ(z)dv ≤ c b .
Hence

∂Ω - F k ln F k 1 + F k k (z, v) | v • n(z) | dσ(z)dv + Ω×R 2 D k (z, v)dzdv ≤ c b . (3.22)
It holds that

∂Ω - F k ln(1 + F k k )(z, v) | v • n(z) | dσ(z)dv ≤ ∂Ω -,F k ≤k F k ln(1 + F k k )(z, v) | v • n(z) | dσ(z)dv + ∂Ω -,F k ≥k F k ln(1 + F k k )(z, v) | v • n(z) | dσ(z)dv ≤ ln 2 ∂Ω - F k (z, v) | v • n(z) | dσ(z)dv + ∂Ω -,F k ≥k F k ln 2F k k (z, v) | v • n(z) | dσ(z)dv ≤ c b + ∂Ω -,F k ≥k F k ln F k (z, v) | v • n(z) | dσ(z)dv -ln k 2 ∂Ω -,F k ≥k F k (z, v) | v • n(z) | dσ(z)dv.
Together with (3.22), this implies that

∂Ω -,F k ≤k F k ln F k (z, v) | v • n(z) | dσ(z)dv + ln k 2 ∂Ω -,F k ≥k F k | v • n(z) | dσ(z)dv ≤ c b .
This ends the proof of Lemma 3.1.

Compactness of the integrated collision frequency.

Denote by Q + (f, f ) (resp. ν(f )) the gain term (resp. the collision frequency) of the nonlinear Boltzmann collision operator,

Q + (f, f )(z, v) = Bf (z, v )f (z, v * )dv * dω, ν(f )(z, v) = Bf (z, v * )dv * dω, (4.1) so that Q(f, f ) = Q + (f, f ) -f ν(f ).
Lemma 4.1 For any V > 1, the sequence

s -(z,v) -s + (z,v) ν(F k )(z + sv, v)ds k∈N * is uniformly bounded by cV 2 on Ω × {v ∈ R 2 ; η < |v| < V }.
Proof of Lemma 4.1.

For any (Z, v) ∈ ∂Ω + with η < |v| < V , the truncation χ η in B implies that

ν(F k )(Z + sv, v) = |v * |>η B(v, v * , ω)F k (Z + sv, v * )dv * dω ≤ c χ η (v, v * , ω)(v 2 + v 2 * )F k (Z + sv, v * )dv * ≤ c V 2 + η 2 η 4 v 2 * sin 2 (v, v * )F k (Z + sv, v * )dv * . (4.2)
Let Ω Z,v be one of the two subsets of Ω split by the segment [Z, Z +s

-(Z, v)v] and ∂Ω Z,v = ∂Ω ∩ ΩZ,v . Let v ⊥ be
) ∈ Ω Z,v × R 2 . This gives s -(Z,v) 0 (v * • v ⊥ ) 2 |v| 2 F k (Z + sv, v * )dv * ds ≤ ∂Ω Z,v ×R 2 v * • v ⊥ |v| v * • n(z) F k (z, v * )dσ(z)dv * ≤ c b , (Z, v) ∈ ∂Ω + . (4.3)
Together with (4.2) this ends the proof of the lemma.

Lemma 4.2

The sequence (F k ) k∈N * is weakly compact in L 1 .

Proof of Lemma 4.2. By (3.6) it is sufficient to prove that for any V > 1 the sequence

(F k /Ω×{v∈R 2 ;|v|≤V } ) k∈N * is weakly compact in L 1 (Ω × {v ∈ R 2 ; η ≤ |v| ≤ V }).
It follows from the exponential form of F k (z, v) from the outgoing boundary ∂Ω -,

F k (z, v) = F k (z -(z, v), v) exp s -(z,v) 0 ν(F k )(z + sv, v)ds - s -(z,v) 0 Q + (F k , F k )(z + sv, v) exp s 0 ν(F k )(z + rv, v)dr ds,
and Lemma 4.1 that

F k (z, v) ≤ e cV 2 F k (z -(z, v), v), z ∈ Ω, η ≤ |v| ≤ V. (4.4)
By (3.8), (F k / ∂Ω -) k∈N * is weakly compact in L 1 |v•n(Z)| (∂Ω -).
This completes the proof of the lemma.

Lemma 4.3 For k ∈ N * and ∈]0, 1[, there is a subset Ω k, of characteristics of Ω × {v ∈ R 2 ; η < |v| < 1 },
with measure smaller than c , such that

F k (z, v) ≤ c 3 e c 2 , (z, v) ∈ Ω × {v ∈ R 2 ; η < |v| < 1 } \ Ω k, . (4.5)
Proof of Lemma 4.3.

Let ∈]0, 1[ be given. By the strict convexity of Ω and its C 1 regularity, the set

ω 1 := {(Z, v) ∈ ∂Ω × R 2 ; η < |v| < 1 and -2 < v • n(Z) < 0} (4.6)
is of measure smaller than c for some constant c > 0 and small enough. It follows from (3.8) that the measure of the subset of ∂Ω -where

η < |v| < 1 , v • n(Z) < -2 and F k (Z, v) > 1 3 , is smaller than c b . Denote this set by ω k, 2 . Define Ω k, as Ω k, = {(Z + sv, v); (Z, v) ∈ ω 1 ∪ ω k, 2 , s ∈ [-s + (Z, v), 0]}. (4.7)
Together with (4.4), this ends the proof of the lemma.

For any (k, ) ∈ N * ×]0, +∞[, denote by χ k, the characteristic function of (Ω k, ) c .

Lemma 4.4

For any V > 0 the sequence (

s -(z,v) -s + (z,v) ν(F k )(z + sv, v)ds) k∈N * is compact in L 1 (Ω × {v ∈ R 2 ; |v| < V }).
Proof of Lemma 4.4. Let V > 0 be given. By (3.6) and Lemma 4.2, it is sufficient to prove that for any > 0 and W > 0, the sequence

s -(z,v) -s + (z,v) |v * |<W Bχ k, F k (z + sv, v * )dv * ds k∈N * (4.8) is compact in L 1 (Ω × {v ∈ R 2 ; |v| < V }). By (3.6) this sequence is bounded in L 1 .
Let us prove that it is uniformly equiintegrable w.r.t. the z variable.

For any (α, β) ∈ R 2 , denote by I(α, β) the interval with end points α and β. For any h ∈ R 2 , split

s -(z+h,v) -s + (z+h,v) Bχ k, F k (z + h + sv, v * )dv * ds - s -(z,v) -s + (z,v) Bχ k, F k (z + sv, v * )dv * ds (4.9)
into

I(-s + (z,v),-s + (z+h,v))∪I(s -(z,v),s -(z+h,v)) Bχ k, F k (z + h + sv, v * )dv * ds, (4.10)
which absolute value tends to zero when integrated over Ω × R 2 and h → 0 by the continuity of (s + , s -) on Ω × {v; |v| ≤ V }, and

s -(z,v) -s + (z,v) Bχ k, F k (z + h + sv, v * ) -F k (z + sv, v * ) dv * ds. (4.11)
Almost every (v, v * ) ∈ R 2 × R 2 considered when integrating the absolute value of (4.11) over Ω × {v ∈ R 2 ; |v| < V } forms a basis in R 2 . Denote by (a(h), b(h)) the coordinates of h in this basis. Split (4.11) into

s -(z,v) -s + (z,v) Bχ k, F k (z + h + sv, v * ) -F k (z + b(h)v * + sv, v * ) dv * ds = s -(z,v) -s + (z,v) Bχ k, F k (z + h + sv, v * ) -F k (z + h + (s -a(h))v, v * ) dv * ds = I(-s + (z,v)-a(h),-s + (z,v))∪I(s -(z,v),s -(z,v)-a(h)) Bχ k, F k (z + h + sv, v * )dv * ds, (4.12)
and

s -(z,v) -s + (z,v) Bχ k, F k (z + b(h)v * + sv, v * ) -F k (z + sv, v * ) dv * ds = s -(z,v) -s + (z,v) Bχ k, f bk (z + (z + b(h)v * + sv, v * ), v * ) -f bk (z + (z + sv, v * ), v * ) dv * ds + s -(z,v) -s + (z,v) B I(0,b(h)) χ k, Q k (F k , F k )(z + sv + rv * , v * )drdv * ds. (4.13)
When integrated over Ω × {v ∈ R 2 ; |v| < V }, the limit when h → 0 of the first term of (4.13) is zero by the integrability of

f b . Notice that (χ k, Q k (F k , F k )) k∈N * is weakly compact in L 1 (Ω × {v ∈ R 2 ; |v| < V }), since χ k, Q k (F k , F k ) ≤ 1 ln Λ D k + cΛ 3 e c 2 ν(F k ), Λ > 1.
When integrated over Ω×{v ∈ R 2 ; |v| < V }, the limit when h → 0 of the second term of (4.13) is zero by the weak

L 1 compactness of (χ k, Q k (F k , F k )) k∈N * and lim h→0 b(h) = 0, (4.14)
uniformly on Ω × {v ∈ R 2 ; |v| < V }.

The uniform equiintegrability w.r.t. the v variable of

s -(z,v) -s + (z,v) |v * |<V Bχ k, F k (z + sv, v * )dv * ds k∈N *
follows from similar arguments.

Passage to the limit in the approximations

For each > 0, let F be the weak L 1 limit of a subsequence of (χ k, F k ) k∈N * . (F ) ∈]0,1[ is non increasing with respect to decreasing and bounded in L 1 . Let f be its strong L 1 limit when → 0. Notice that f is also the weak L 1 limit of (F k ) k∈N when k → +∞.

For proving that f is a mild solution of (1.1)-(1.5), it is sufficient to prove that for any β > 0, there is a set X β of characteristics with complementary set of measure smaller than cβ, such that if χ β denotes the corresponding characteristic function,

(χ β f )(z, v) = (χ β f b )(z + (z, v), v) + 0 -s + (z,v) (χ β Q(f, f ))(z + sv, v)ds, (z, v) ∈ Ω × R 2 . (5.1)
This in turn is satisfied if for any test function ϕ ∈ L ∞ ( Ω×R 2 ), continuously differentiable along characteristics, with v

• ∇ z ϕ ∈ L ∞ (Ω × R 2 )
, compact support and vanishing on ∂Ω -,

Ω×R 2 ϕχ β f (z, v)dzdv = Ω×R 2 ϕχ β f b (z + (z, v), v)dzdv + Ω×R 2 0 -s + (z,v) χ β f v • ∇ z ϕ + ϕχ β Q(f, f ) (z + sv, v)ds dzdv.
(5.2)

Let 0 > 0 be such that the support of ϕ is included in Ω×{v ∈ R 2 ; |v| ≤ 1 0 }. Define the set X β as follows. Using the weak L 1 (Ω × R 2 ) compactness of (F k ) k∈N * and the weak L 1 (∂Ω -) compactness of (F k /∂Ω -) k∈N * , pass to the limit when k → +∞ in

F k (z, v) ≤ e c 2 0 F k (z -(z, v), v), a.a. z ∈ Ω, η < |v| ≤ 1 0 , k ∈ N * . (5.3) It implies that f (z, v) ≤ e c 2 0 f (z -(z, v), v), a.a. z ∈ Ω, η < |v| ≤ 1 0 .
From here the proof follows the lines of the proof of Lemma 4.3, so that given β > 0, X β can be defined as a set of characteristics, with complementary set of measure smaller than cβ, such that

f (z, v) ≤ e c 2 0 β 3 , a.a. (z, v) ∈ X β . (5.4)
The following lemma is a preliminary step to pass to the limit when k → +∞ in quadratic terms along the 'good characteristics'

(z + sv, v), (z, v) / ∈ Ω k, . Lemma 5.1 For any test function ϕ ∈ L ∞ ( Ω × R 2 ), continuously differentiable along characteristics, with v • ∇ z ϕ ∈ L ∞ (Ω × R 2 ), compact support and vanishing on ∂Ω -, lim k→+∞ Ω×R 2 0 -s + (z,v) ϕχ β χ k, F k ν(F k )(z + sv, v)ds dzdv = Ω×R 2 0 -s + (z,v) ϕχ β F ν(f )(z + sv, v)ds dzdv. (5.5)
Proof of Lemma 5.1. Since ϕ has compact support, one can restrict to the passage to the limit when k → +∞ and V fixed of

Ω×{v∈R 2 ;|v|<V } 0 -s + (z,v) ϕχ β χ k, F k ν(F k )(z + sv, v)ds dzdv. (5.6)
By an integration by parts,

Ω×{v∈R 2 ;|v|<V } 0 -s + (z,v) ϕχ β χ k, F k ν(F k )(z + sv, v)ds dzdv = Ω×{v∈R 2 ;|v|<V } 0 -s + (z,v) ϕχ β χ k, f bk (z + (z, v), v) 0 -s + (z,v) ν(F k )(z + sv, v)ds(z + sv, v)ds dzdv + Ω×{v∈R 2 ;|v|<V } 0 -s + (z,v) χ β χ k, v • ∇ z ϕ F k + ϕQ k (F k , F k ) (z + sv, v) 0 s ν(F k )(z + rv, v)dr ds dzdv. (5.7)
Proof of Lemma 5.2.

For ∈]0, 0 [, write the mild form of ϕχ β χ k, F k and integrate it on Ω × R 2 . It results

Ω×R 2 ϕχ β χ k, F k (z, v)dzdv = Ω×R 2 ϕχ β χ k, f bk (z + (z, v), v)dzdv + Ω×R 2 0 -s + (z,v) χ β χ k, F k v • ∇ z ϕ(z + sv, v)ds dzdv + Ω×R 2 0 -s + (z,v) ϕχ β χ k, Q + (F k , F k ) -F k ν(F k ) (z + sv, v)ds dzdv.
(5.11)

By the weak L 1 compactness of (F k ) k∈N * and the linearity with respect to χ k, F k of the first two lines of (5.11), their passage to the limit when k → +∞ is straightforward. The passage to the limit when k → +∞ in the last term of (5.11) follows from Lemma 5.1. Finally, using monotonicity arguments together with the L ∞ boundedness of χ β f allows to pass to the limit when → 0 in

Ω×R 2 ϕχ β F (z, v)dzdv - Ω×R 2 ϕχ β f b (z + (z, v), v)dzdv - Ω×R 2 0 -s + (z,v) χ β F v • ∇ z ϕ(z + sv, v)ds dzdv + Ω×R 2 0 -s + (z,v)
ϕχ β F ν(f )(z + sv, v)ds dzdv, (5.12)

and obtain the limit ϕχ β χ k, (z + sv , v )B

Ω×R 2 ϕχ β f (z, v)dzdv - Ω×R 2 ϕχ β f b (z + (z, v), v)dzdv - Ω×R 2 0 -s + (z,v) χ β f v • ∇ϕ(z + sv, v)ds dzdv + Ω×R 2 0 -s + (z,v) ϕχ β f ν(f )(z + sv, v)ds dzdv. (5.13) Let us prove that lim →0 lim k→+∞ Ω×R 2 0 -s + (z,v) ϕχ β χ k, Q + (F k , F k )(z + sv, v)ds dzdv = Ω×R 2 0 -s + (z,v)
F k 1 + F k k (z + sv , v) F k 1 + F k k (z + sv , v * )dsdz dvdv * dω = Ω×R 2 ϕχ β χ k, s -(Z, v )B F k 1 + F k k (Z, v * )dv * dω F k 1 + F k k (Z, v)dZdv.
(5.16)

One can restrict to the study of the limit of (5.21)

Passing to the limit when (α, 1 , µ) → (0, 0, 0) in (5.21) leads to (5.14).

  1[ and fixed,χ η (v, v * , ω) = 0 if |v| ≤ η or |v * | ≤ η or |v | ≤ η or |v * | ≤ η or (v, v * ) < η or (v , v * ) < η, χ η (v, v * , ω) = 1, else. (1.3)Here (u, v) denotes the angle between vectors u and v. The function B is the kernel of the classical nonlinear Boltzmann operator for hard forces,|v -v * | β b(ω) with β ∈ [0, 2[, b ∈ L 1 + (S 1), b(ω) ≥ c > 0 a.e.(1.4)

  20) on [l, L], where l = inf{a; S a = ∅}, L = sup{a; S a = ∅}, and using (3.19) leads to

  one of the vectors orthogonal to v such that |v ⊥ | = |v|. Multiply (2.1) written in the variables (z, v * ) by v * • v ⊥ |v| and integrate the resulting equation over (z, v *

2 0

 2 ϕχ β Q + (f, f )(z + sv, v)ds dzdv.(5.14) For any (v, v * , ω), the change of variables (z, s) → (Z, s) = (z + sv , s), (5.15) moves the domain Ω × ] -s + (z, v ), 0[ into the domain {(Z, s) ; Z -sv ∈ Ω and s < 0}, i.e. (Z, -s) ∈ Ω×]0, s -(Z, v )[. Hence, Ω×R -s + (z,v)ϕχ β χ k, Q + (F k , F k )(z + sv, v)ds dzdv = Ω 0 -s + (z,v )

Ω×R 2 e c 2 F 2 s 2 s 2 s

 22222 {v * ∈R 2 ;|sin( v,v * )|>µ}×S 1ϕχ β χ k, s -(Z, v )B F k 1 + F k k (Z, v * )dv * dω F k 1 + F k k (Z, v)dZdv, µ ∈]0, 1[,(5.17)sinceΩ×R 2 {v * ∈R 2 ;|sin( v,v * )|<µ}×S 1 ϕχ β χ k, s -(Z, v )B F k v) → F k (Z, v) {v * ∈R 2 ;|sin( v,v * )|>µ}×S 1 ϕχ β χ k, s -(Z, v )BF k (Z, v * )dv * dω k∈N *is weakly compact in L 1 . Indeed, using the change of variables v * → v * for every (v, ω), which holds since|sin( v, v * )| > µ, F k (Z, v) ϕχ β χ k, s -(Z, v )BF k (Z, v * )dv * dω ≤ c ln Λ D k (Z, v) + c Λ 3 k (Z, v * )dv * , Λ > 1,where (D k ) k∈N * is defined in (3.1) and uniformly bounded in L 1 by (3.7).Consequently one can restrict to the passage to the limit when k → +∞ in(Z,v)∈Ω×R 2 ;s -(Z,v)>α {v * ∈R 2 ;|sin( v,v * )|>µ}×S 1 ϕχ β χ k, s -(Z, v )BF k (Z, v * )dv * dω χ k, 1 F k (Z, v)dZdv, (α, 1 ) ∈]0, 1[ 2 . (5.19) Moreover, (Z,v);s -(Z,v)>α {v * ∈R 2 ;|sin( v,v * )|<µ}×S 1 ϕχ β χ k, s -(Z, v )BF k (Z, v * )dv * dω χ k, 1 F k (Z, v)dZdv = {v * ∈R 2 ;|sin( v,v * )|<µ}×S 1 Z;s -(Z,v)>α s -(Z,v) 0 ϕχ β χ k, s -(Z, v ) χ k, 1 F k s -(Z, v) F k (Z, v * )dsdZ Bdvdv * dω = {v * ∈R 2 ;|sin( v,v * )|<µ}×S 1 Ω s -(z,v) max{0,α-s -(z,v)} ϕχ β χ k, s -(z + sv, v ) χ k, 1 F k s -(z + sv, v) F k (z + sv, v * )dsdz Bdvdv * dω = Ω×R -(z,v) max{0,α-s -(z,v)} χ k, 1 F k s -(z + sv, v) Bϕχ β χ k, s -(z + sv, v )F k (z + sv, v * )dv * ds dzdv. -(z,v) max{0,α-s -(z,v)} χ k, 1 F k s -(z + sv, v) Bϕχ β χ k, s -(z + sv, v )F k (z + sv, v * )dv * ds dzdv = Ω×R -(z,v) max{0,α-s -(z,v)} F 1 s -(z + sv, v){v * ∈R 2 ;|sin( v,v * )|<µ}×S 1 Bϕχ β s -(z + sv, v )f (z + sv, v * )dv * ds dzdv = (Z,v);s -(Z,v)>α {v * ∈R 2 ;|sin( v,v * )|<µ}×S 1 ϕχ β s -(Z, v )Bf (Z, v * )dv * dω F 1 (Z, v)dZdv.

The change of variables (z, s) → (Z, s) = (z + sv, s), (5.8) moves the domain Ω × ] -s + (z, v), 0[ into the domain

Hence,

Lemma 5.1 follows from the passage to the limit when k → +∞ in (5.9). It uses the weak L 1 compactness of (χ k, F k ) and (χ k, Q k (F k , F k )), the strong L 1 compactness of

dr . An integration by parts back in the s variable is finally performed, taking into account that

Lemma 5.2 f is a solution of (1.1)-(1.5), i.e. for test functions ϕ defined as in Lemma 5.1,

(5.10)