On stationary solutions to normal, coplanar discrete Boltzmann equation models

HAL Id: hal-02520761

https://hal.archives-ouvertes.fr/hal-02520761

Preprint submitted on 26 Mar 2020

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

On stationary solutions to normal, coplanar discrete Boltzmann equation models

Leif Arkeryd, Anne Nouri

To cite this version:

Leif Arkeryd, Anne Nouri. On stationary solutions to normal, coplanar discrete Boltzmann equation models. 2020. �hal-02520761�

On stationary solutions to normal, coplanar discrete Boltzmann equation models.

Leif ARKERYD and Anne NOURI

Mathematical Sciences, 41296 G¨oteborg, Sweden, [email protected]

Aix-Marseille University, CNRS, Centrale Marseille, I2M UMR 7373, 13453 Marseille, France, [email protected]

Abstract

The paper proves existence of renormalized solutions for a class of velocity-discrete coplanar station- ary Boltzmann equations with given indata. The proof is based on the construction of a sequence of approximations with L¹ compactness for the integrated collision frequency and gain term. L¹ compactness of a sequence of approximations is obtained using the Kolmogorov-Riesz theorem and replaces theL¹ compactness of velocity averages in the continuous velocity case, not available when the velocities are discrete.

.

1 Introduction.

The Boltzmann equation is the fundamental mathematical model in the kinetic theory of gases.

Replacing its continuum of velocities with a discrete set of velocites is a simplification preserving the essential features of free flow and quadratic collision term. Besides this fundamental aspect they can approximate the Boltzmann equation with any given accuracy [4], and are thereby useful for approximations and numerics. In the quantum realm they can also be more directly connected to microscopic quasi/particle models.

A discrete velocity model of a kinetic gas, is a system of partial differential equations having the form,

∂f_i

∂t(t, z) +v_i· ∇_zf_i(t, z) =Q_i(f)(t, z), t >0, z∈Ω, 1≤i≤p,

wheref_i, 1≤i≤p, are phase space densities at timet, positionz, velocityv_i, Ω⊂R^d, andv_i ∈R^d, 1 ≤i ≤p, are given discrete velocities. The collision operator Q = (Q_i)1≤i≤p with gain part Q⁺, loss part Q⁻, and collision frequency ν, is given by

Q_i(f) =

p

X

j,k,l=1

Γ^kl_ij(f_kf_l−f_if_j)

=Q⁺_i (f)−Q⁻_i (f), Q⁻_i (f) =fiνi(f), i= 1, ..., p.

12010 Mathematics Subject Classification; 60K35, 82C40, 82C99.

2Key words; stationary Boltzmann equation, discrete coplanar velocities, normal model, entropy.

The collision coefficients satisfy

Γ^kl_ij = Γ^kl_ji = Γ^ij_kl≥0. (1.1)

If a collision coefficient Γ^kl_ij is non-zero, then the conservation laws for momentum and energy, vi+vj =vk+vl, |v_i|²+|v_j|² =|v_k|²+|v_l|², (1.2) are satisfied. The discrete velocity model (DVM) is called normal (see [5]) if any solution of the equations

Ψ(vi) + Ψ(vj) = Ψ(v_k) + Ψ(v_l),

where the indices (i, j;k, l) take all possible values satisfying Γ^kl_ij >0, is given by Ψ(v) =a+b·v+c|v|²,

for some constantsa, c∈Rand b∈R^d. This paper studies stationary solutions to coplanar models, i.e. with vi ∈R², 1≤i≤p, in a strictly convex bounded open subset Ω ⊂R², with C¹ boundary

∂Ω and given indata. We consider

the generic situation of normal coplanar velocities with

no pair of velocities v_i, v_j,1≤i, j≤p, parallel, (1.3) and additionally that for some directionn0 ∈R², vi·n0 >0, 1≤i≤p. (1.4) For stationary solutions to the Broadwell model, that does not belong to this class, see [2], [6].

Denote by n(Z) the inward normal to Z ∈∂Ω. Denote the v_i-ingoing (resp. v_i-outgoing) part of the boundary by

∂Ω⁺_i ={Z ∈∂Ω;v_i·n(Z)>0}, (resp. ∂Ω⁻_i ={Z ∈∂Ω;v_i·n(Z)<0}).

Let

s⁺_i (z) = inf{s >0;z−svi ∈∂Ω⁺_i }, s⁻_i (z) = inf{s >0;z+svi ∈∂Ω⁻_i }, z∈Ω.

Write

z_i⁺(z) =z−s⁺_i (z)v_i (resp. z_i⁻(z) =z+s⁻_i (z)v_i) (1.5) for the ingoing (resp. outgoing) point on ∂Ω of the characteristics through z in direction vi. The boundary value problem

v_i· ∇f_i(z) =Q_i(f)(z), z∈Ω, (1.6)

f_i(z) =f_bi(z), z∈∂Ω⁺_i , 1≤i≤p, (1.7)

is considered in L¹ in one of the following equivalent forms;

the exponential multiplier form, fi(z) =f_bi(z⁺_i (z))e⁻

R^s

+ i(z)

0 νi(f)(z_i⁺(z)+svi)ds

+

Z s⁺_i(z) 0

Q⁺_i (f)(z⁺_i (z) +svi)e⁻

R^s

+ i(z)

s νi(f)(z⁺_i(z)+rvi)drds, a.a. z∈Ω, 1≤i≤p, (1.8)

the mild form,

f_i(z) =f_bi(z⁺_i (z)) +

Z s⁺_i(z) 0

Q_i(f)(z_i⁺(z) +sv_i)ds, a.a. z∈Ω, 1≤i≤p, (1.9) the renormalized form,

v_i· ∇ln(1 +f_i)(z) = Q_i(f) 1 +fi

(z), z∈Ω, f_i(z) =f_bi(z), z∈∂Ω⁺_i , 1≤i≤p, (1.10) in the sense of distributions.

Denote byL¹₊(Ω) the set of non-negative integrable functions on Ω. The main result of the present paper is

Theorem 1.1

Consider a coplanar collision operator in the generic case of (1.3)additionally satisfying (1.4), and non-negative ingoing boundary values f_bi, 1≤i≤p, with mass and entropy bounded,

Z

∂Ω⁺_i

vi·n(z)fbi(1 + lnfbi)(z)dσ(z)<+∞, 1≤i≤p.

There exists a stationary renormalized solution in L¹₊(Ω)p

to the boundary value problem (1.6)- (1.7) with finite entropy-dissipation.

Most mathematical results for stationary discrete velocity models of the Boltzmann equation have been obtained in one space dimension. An overview is given in [8]. In two dimensions, special classes of solutions to the Broadwell model are given in [6], [3], and [9]. The Broadwell model is a four-velocity model, with v1+v2=v3+v4= 0 andv1,v2 orthogonal. [6] contains a detailed study of the stationary Broadwell equation in a rectangle with comparison to a Carleman-like system, and a discussion of (in)compressibility aspects. A main result in [6] is the existence of continuous solutions to the two-dimensional stationary Broadwell model with continuous boundary data for a rectangle. The proof starts by solving the problem with a given gain term, and uses the compactness of the corresponding twice iterated solution operator to conclude by Schaeffer’s fixed point theo- rem. The paper [2] studies that problem in an L¹-setting, with the proof broadly within the frame of the present paper. In both those papers of ours, there is a priori control of mass and entropy dissipation. Denoting by f_i(t,·), 1 ≤i≤4, the density of the particles moving with velocity v_i at timet, the proof in [2] in an essential way uses the constancy of the sumsf₁+f₂ and f₃+f₄ along characteristics, which no longer holds in this paper. It is here replaced by a compactness property for the collision frequency and gain parts in the exponential form of the approximations employed.

The compactness is based on assumption (1.3) and the simultaneous presence of space integrals in two velocity directions.

The proof starts from bounded approximations with damping and convolution added, written in exponential multiplyer form, and solved by a fixed point argument. Then the damping and convo- lutions are removed by taking limits usingL¹-compactness of the integrated collision frequency and gain term. The compactness is proven by the Kolmogorov-Riesz theorem (see [10], [11]). The limit of the remaining approximations is obtained by using again the Kolmogorov-Riesz theorem.

2 Approximations.

The construction of the primary approximated boundary value problem with damping and convo- lutions is similar to the Broadwell case [2]. Denote a∧b the minimum of two real numbers aand

b. Take α >0 and set c_α = 1

α

p

X

i=1

Z

∂Ω⁺_i

(n(z)·v_i)f_bi(z)dσ(z), K_α={f ∈ L¹₊(Ω)p

;

p

X

i=1

Z

Ω

f_i(z)dz ≤c_α}. (2.1) Let µ_α be a smooth mollifier in R² with support in the ball centered at the origin of radius α.

Outside the boundary the function to be convolved with µ_α is continued in the normal direction by its boundary value. Let ˜µ_k be a smooth mollifier on∂Ω. Denote by

f_bi^k =

f_bi(·)∧k 2

∗µ˜_k, 1≤i≤p.

Let T be the map defined on K_α by T(f) =F, where F = (F_i)1≤i≤p is the solution of αF_i+v_i· ∇F_i =

p

X

j,l,m=1

Γ^lm_ij F_l 1 +^F_k^l

fm∗µα

1 +^fm_k^∗µα − Fi

1 +^F_kⁱ

fj∗µα

1 +^fj∗µ_k^α

, (2.2)

F_i(z⁺_i (z)) =f_bi^k(z_i⁺(z)). (2.3)

F =T(f) can be obtained as the limit in (L¹₊(Ω))^p of the sequence (F^q)q∈N defined byF⁰ = 0 and αF_i^q+1+vi· ∇F_i^q+1=

p

X

j,l,m=1

Γ^lm_ij F_l^q

1 +^F

q l

k

fm∗µα

1 +^fm_k^∗µα − F_i^q+1 1 +^F

q i

k

fj∗µα

1 +^fj∗µ_k^α

, (2.4)

F_i^q+1(z_i⁺(z)) =f_bi^k(z_i⁺(z)), q∈N. (2.5)

The sequence (F^q)q∈N is monotone. Indeed, F_i⁰ ≤F_i¹, 1≤i≤n,

by the exponential form of F_i¹. IfF_i^q ≤F_i^q+1, 1≤i≤p, then it follows from the exponential form that F_i^q+1≤F_i^q+2. Moreover,

α

p

X

i=1

F_i^q+1+

p

X

i=1

v_i· ∇F_i^q+1 =

p

X

i,j,l,m=1

Γ^lm_ij (F_l^q−F_l^q+1) 1 +^F

p l

k

fm∗µα

1 +^fm_k^∗µα ≤0, so that

p

X

i=1

Z

Ω

F_i^q+1(z)dz≤cα. (2.6)

By the monotone convergence theorem, (F^q)q∈N converges in L¹(Ω) to a solution F of (2.2)-(2.3).

The solution of (2.2)-(2.3) is unique in the set of non-negative functions. Indeed, letG= (Gi)1≤i≤p

be a non-negative solution of (2.2)-(2.3). It follows by induction that

∀q ∈N, F_i^q ≤G_i, 1≤i≤p. (2.7)

Indeed, (2.7) holds for q = 0, since G_i ≥ 0, 1 ≤ i ≤ p. Assume (2.7) holds for q. Using the exponential form ofF_i^q+1 impliesF_i^q+1 ≤G_i. Consequently,

F_i≤G_i, 1≤i≤p. (2.8)

Moreover, subtracting the partial differential equations satisfied by G_i from the partial differential equations satisfied by Fi, 1≤i≤p, and integrating the resulting equation on Ω, it results

α

p

X

i=1

Z

Ω

(Gi−Fi)(z)dz+

p

X

i=1

Z

∂Ω⁻_i

|n(z)·vi|(G_i−Fi)(z)dσ(z) = 0. (2.9) It results from (2.8)-(2.9) thatG=F.

The map T is continuous in the L¹-norm topology (cf [1] pages 124-5). Namely, let a sequence (f^q)q∈N in K_α converge in (L¹(Ω))^p to f ∈ K_α. Set F^q = T(f^q). Because of the uniqueness of the solution to (2.2)-(2.3), 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≥qf^r) (resp. (g^q) = (infr≥qf^r)) converges to f in L¹. Let (S^q) (resp.

(s^q)) be the sequence of solutions to αS_i^q+v_i· ∇S_i^q=

p

X

j,l,m=1

Γ^lm_ij S_l^q 1 +^S

q l

k

G^q_m∗µ_α 1 +^Gqm_k^∗µα

− S_i^q 1 +^S

q i

k

g^q_j ∗µα

1 +^g

q j∗µα

k

, (2.10)

S_i^q(z_i⁺(z)) =f_bi^k(z_i⁺(z)), (2.11)

αs^q_i +v_i· ∇_xs^q_i =

p

X

j,l,m=1

Γ^lm_ij s^q_l 1 +^s

q l

k

g_m^q ∗µ_α 1 +^gmq_k^∗µα

− s^q_i 1 +^s

q i

k

G^q_j ∗µα

1 +^G

q j∗µα

k

, (2.12)

s^q_i(z⁺_i (z)) =f_bi^k(z⁺_i (z)). (2.13)

(S^q) is a non-increasing sequence, since that holds for the successive iterates defining the sequence.

Then (S^q) decreasingly converges in L¹ to some S. Similarly (s^q) increasingly converges in L¹ to somes. The limitsS and ssatisfy (2.2)-(2.3). It follows by uniqueness thats=F =S, hence that (F^q) converges in L¹ toF.

The map T is also compact in the L¹-norm topology. Indeed, let (f^q)q∈N be a sequence in Kα

and (F^q)q∈N = (T(f^q))q∈N. The boundedness by k² of the terms in the collision operator, induces uniform L¹ equi-continuity of (F_i^q)q∈N with respect to the vi-direction, as follows from the mild form of the equations. For the uniform L¹ equi-continuity with respect to the vj-direction, j 6= i, consider for each q and withf :=f^q the sequence (G^q,r)r∈N defined byG^q,0= 0 and for r∈N^∗

αG^q,r_i +v_i· ∇G^q,r_i =

p

X

j,l,m=1

Γ^lm_ij G^q,r_l 1 +^G

q,r l

k

f_m∗µ_α

1 +^fm_k^∗µα − G^q,r_i 1 +^G

q,r−1 i

k

f_j∗µ_α 1 +^fj∗µ_k^α

, (2.14)

G^q,r_i (z⁺_i (z)) =f_bi^k(z⁺_i (z))), 1≤i≤p. (2.15) The existence of a unique solution for eachr follows as for the problem (2.2)-(2.3). By induction on r, prove that (G^q,r)q∈N is uniformly equicontinuous in thevj-direction. It holds forr = 0. Assume it holds forr−1∈N^∗and prove it for r. WritingG^q,r(z) in exponential form and using the uniform equicontinuity in thevj-direction of (G^q,r−1)q∈Nand the compactness of (f^q∗µα), it comes back to prove the uniform equicontinuity in the vj-direction of

z→

Z si(z⁻_i (z)) 0

G^q,r_l 1 +^G

q,r l

k

(z⁻_i (z) +sv_i)ds.

First,

z_i⁻(z+hvj) =z⁻_i (z) +avi+bv_l, with lim

h→0a(h) = lim

h→0b(h) = 0, uniformly with respect to z∈Ω. Consequently,

Z

|

Z si(z⁻_i (z)) 0

G^q,r_l 1 +^G

q,r l

k

(z⁻_i (z+hv_j) +sv_i)− G^q,r_l 1 +^G

q,r l

k

(z_i⁻(z) +sv_i)ds|dz

≤

Z Z si(z⁻_i (z)) 0

| G^q,r_l 1 +^G

q,r l

k

(z⁻_i (z) + (a+s)vi+bvl)− G^q,r_l 1 +^G

q,r l

k

(z⁻_i (z) + (a+s)vi)|dsdz (2.16) +

Z

|

Z _si(z⁻

i (z)) 0

G^q,r_l 1 +^G

q,r l

k

(z_i⁻(z) + (a+s)v_i)− G^q,r_l 1 +^G

q,r l

k

(z_i⁻(z) +sv_i)ds|dz. (2.17) The limit whenh→0 of (2.16) is zero, by the uniform L¹ equicontinuity of (G^q,r_l )q∈N with respect to thevl-direction. With the change of variabless→a+sin its first integral, (2.17) equals

Z

|

Z _si(z⁻

i (z))+a si(z_i⁻(z))

G^q,r_l 1 +^G

q,r l

k

(z_i⁻(z) +sv_i)ds− Z _a

0

G^q,r_l 1 +^G

q,r l

k

(z_i⁻(z) +sv_i)ds|dz, (2.18) which tends to zero when h tends to zero since both integrands are bounded by k. This proves the L¹ compactness of (G^q,r_i )q∈N. For q fixed, the sequence (G^q,r_i )r∈N is increasing, and its limit satisfies (2.2)-(2.3) withf =f^q, so the limit equalsF_i^q. Take a subsequence ofq still denoted byq, with (f^q∗µ_α) convergent inL¹ to some f^∞ whenq→ ∞, and a further subsequence so that (G^q,1) converges to some F^∞,1 in L¹. Continue by diagonalization to convergence of (G^q,r)_q toF^∞,r for all r∈N. The limits satisfy (2.14)-(2.15) withf∗µα replaced withf^∞, and G^q,r withF^∞,r giving an increasing sequence, with limit satisfying (2.2)-(2.3), where f∗µα is replaced withf^∞. So given a sequence in K_α, there is a subsequence with converging image underT. The compactness of T is thus proved.

Hence by the Schauder fixed point theorem,

Lemma 2.1 There is a fixed point in (L¹₊(Ω))^p to T i.e. a solution F ∈(L¹₊(Ω))^p to αFi+vi· ∇F_i =

p

X

j,l,m=1

Γ^lm_ij Fl

1 +^F_k^l

Fm∗µα

1 +^Fm_k^∗µα − Fi

1 +^F_kⁱ

F_j∗µ_α 1 +^Fj∗µ_k ^α

, (2.19)

Fi(z⁺_i (z)) =f_bi^k(z_i⁺(z)), 1≤i≤p. (2.20)

3 Removal of the damping and convolutions.

Let k > 1 be fixed. Denote by F^α the solution to (2.19)-(2.20) obtained in the previous section.

Each component ofF^αbeing bounded by a multiple ofk², (F^α)_α∈]0,1[is weakly compact in (L¹(Ω))^p. Denote byF^k the limit for the weak topology in (L¹(Ω))^p of a converging subsequence whenα→0.

Let us prove that for a subsequence, the convergence is strong in (L¹(Ω))^p.

Lemma 3.1 There is a sequence (α_q)q∈N tending to zero when q → +∞, such that (F^αq)q∈N

strongly converges to F^k in (L¹(Ω))^p when q→+∞.

Proof of Lemma 3.1

Consider the approximation scheme (f^α,κ)κ∈N ofF^α,

f_i^α,0= 0, (3.1)

αf_i^α,κ+1+v_i· ∇f_i^α,κ+1=

p

X

j,l,m=1

Γ^lm_ij F_l^α 1 +^F_k^lα

F_m^α ∗µ_α

1 +^Fmα_k^∗µα − f_i^α,κ+1 1 +^f

α,κ+1 i

k

f_j^α,κ∗µα

1 +^f

α,κ j ∗µα

k

, (3.2)

f_i^α,κ+1(z_i⁺(z)) =f_bi^k(z_i⁺(z)), 1≤i≤p, κ∈N. (3.3)

f^α,1is obviously given in terms ofF^α. It follows from the exponential form thatF_i^α ≤f_i^α,1,α ∈]0,1[.

Denote by S the map fromR^p×R^p mapping (X, Z) intoW =S(X, Z)∈R^p solution to αWi+vi· ∇W_i =

p

X

j,l,m=1

Γ^lm_ij F_l^α

1 +^F_k^lα

F_m^α∗µα

1 +^Fmα_k^∗µα − Wi

1 +^X_kⁱ

Zj∗µα

1 +^Zj∗µ_k ^α

, Wi(z⁺_i (z)) =f_bi^k(z_i⁺(z)), 1≤i≤p.

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,0=S(0, F^α), F^α,r =S(F^α,r−1, F^α), r ∈N^∗. First,

f_i^α,1,0≤F_i^α,0.

Then the sequence (f_i^α,1,r)r∈N (resp. (F_i^α,r)r∈N) is increasing with limitf_i^α,2 (resp. F_i^α). It follows from f_i^α,1,r ≤F_i^α,r,r ∈N, that

f_i^α,2≤F_i^α, 1≤i≤p. (3.4)

Let

f^α,2,0:=S(0, f^α,2), f^α,2,r :=S(f^α,r−1, f^α,2), r∈N^∗. It follows from (3.4) that

f_i^α,2,0≥F_i^α,0, 1≤i≤p.

The sequence (f_i^α,2,r)r∈N is also increasing with limitf_i^α,3 and withf_i^α,2,r ≥F_i^α,r. Hence f_i^α,3≥F_i^α.

From here by induction on κ, it holds that

f_i^α,2κ ≤f_i^α,2κ+2 ≤F_i^α≤f_i^α,2κ+3 ≤f_i^α,2κ+1, α∈]0,1[, κ∈N. (3.5) By induction onr, for eachr the sequence (f^α,1,r)_α∈]0,1[ is translationally equicontinuous inα. The limit sequence (f^α,2)_α∈]0,1[ is also translationally equicontinuous. This is so, since given > 0, r and thenh₀ can be taken so that

Z

(f^α,2−f^α,1,r)(z)dz < and Z

|f^α,1,r(z+h)−f^α,1,r(z)|dz < , |h|< h₀.

It can analogously be proven that for each κ∈N, (f^α,κ)_α∈]0,1[ is translationally equicontinuous in α. Let (αq)q∈N be a sequence tending to zero. Take a subsequence in (αq)q∈N, still denoted by (α_q)q∈N, such that (f^αq,2)q∈N converges inL¹ to somef^0,2 when q→+∞.

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

f_i^0,2κ ≤f_i^0,2κ+2 ≤F_i^k≤f_i^0,2κ+3≤f_i^0,2κ+1, κ∈N, (3.6) vi· ∇f_i^0,κ+1=Gi−

p

X

j,l,m=1

Γ^lm_ij f_i^0,κ+1 1 +^f

0,κ+1 i

k

f_j^0,κ 1 +^f

0κ j

k

, f_i^0,κ+1(z_i⁺(z)) =f_bi^k(z_i⁺(z)).

Here, G_i is the weakL¹ limit when α→0 of the gain term

p

X

j,l,m=1

Γ^lm_ij F_l^α 1 +^F

α l

k

F_m^α∗µ_α 1 +^Fmα_k^∗µα .

In particular, (f_i^0,2κ)κ∈N (resp. (f_i^0,2κ+1)κ∈N) non decreasingly (resp. non increasingly) converges inL¹ to somegi (resp. hi) when κ→+∞. The limits satisfy

0≤gi ≤F_i^k≤hi, v_i· ∇h_i =G_i−

p

X

j,l,m=1

Γ^lm_ij hi

1 +^h_kⁱ gj

1 +^g_k^j, (3.7)

v_i· ∇g_i =G_i−

p

X

j,l,m=1

Γ^lm_ij g_i 1 +^g_kⁱ

hj

1 +^h_k^j, (3.8)

(h_i−g_i)(z_i⁺(z)) = 0.

Integrating and summing gives that

p

X

i=1

Z

∂Ω⁻_i

|v_i·n(Z)|(h_i−g_i)(Z)dσ(Z) = 0,

i.e. that g_i = h_i also on ∂Ω⁻_i . Integrating the equation satisfied by h_i−g_i over the part of Ω on one side of a line orthogonal to n0, summing overiand using (1.4) implies thatg=h on that line, hence in all of Ω, and is equal toF_i^k. (F^αq)q∈N converges toF^k in (L¹(Ω))^pwhenq →+∞. Indeed, given η >0, chooseκ₀ big enough so that

kf_i^0,2κ0+1−f_i^0,2κ0 k_L1< η and kf_i^0,2κ0 −F_i^kk_L1< η, 1≤i≤p, then q0 big enough, so that

kf_i^αq,2κ0+1−f_i^0,2κ0+1k_L1≤η and kf_i^αq,2κ0−f_i^0,2κ0 k_L1≤η, q≥q₀. Then splitkF_i^αq −F_i^kk_L1 as follows,

kF_i^αq −F_i^kk_L1

≤kF_i^αq −f_i^α,2κ0 k_L1 +kf_i^α,2κ0 −f_i^0,2κ0 k_L1 +kf_i^0,2κ0−F_i^k k_L1

≤kf_i^α,2κ0+1−f_i^α,2κ0 k_L1 +2η by (3.5)

≤kf_i^α,2κ0+1−f_i^0,2κ0+1 k_L1 +kf_i^0,2κ0+1−f_i^0,2κ0 k_L1 +kf_i^0,2κ0 −f_i^α,2κ0 k_L1 +2η

≤5η, q≥q₀.

Lemma 3.2 For any k∈N^∗, there is a nonnegative continuous solution F^k to

v_i· ∇F_i^k =Q^+k_i −F_i^kν_i^k, (3.9)

F_i^k(z⁺_i (z)) =f_bi^k(z⁺_i (z)), 1≤i≤p, (3.10)

where Q^+k_i =

p

X

j,l,m=1

Γ^lm_ij F_l^k 1 +^F_k^lk

F_m^k 1 +^F_k^mk

, ν_i^k=

p

X

j,l,m=1

Γ^lm_ij F_j^k (1 +^F_k^ik)(1 +^F

k j

k ) . Proof of Lemma 3.2.

Passing to the limit whenq→+∞in (2.19)-(2.20) written forF^αq, implies thatF^kis a solution in (L¹₊(Ω))^p to (3.9)-(3.10). It remains to prove its continuity. Using twice its exponential form and the continuity off_b^k, it comes back to prove the continuity of

Z s⁺_i(z) 0

Z s⁺_j(z_i⁺(z)+svi) 0

G^k(z_j⁺(z_i⁺(z) +sv_i) +σv_j)dσds, i6=j, (3.11) for given measurable bounded functions G^k. The mapping

(s, σ)∈[0, s⁺_i (z)]×[0, s⁺_j (z_i⁺(z) +svi)]→Z =z_j⁺(z⁺_i (z) +svi) +σvj, (3.12) is a change of variables. Indeed, the strict convexity of Ω and the C¹ regularity of ∂Ω imply that z→z⁺_i (z) is well-defined andC¹ for anyi∈ {1,· · ·, p}. Hence the map (s, σ)→Z is one to one and C¹. Its Jacobian equals one sinceZ =Z_iv_i+Z_jv_j, withZ_i=s−s⁺_i (z) linear insand independent of σ, and Zj =σ−s⁺_j z+ (s−s⁺_i (z))vi

linear in σ.

Using this change of variable leads to the continuity of the map defined in (3.11).

Lemma 3.3 Solutions (F^k)k∈N^∗ to (3.9)-(3.10) have mass and entropy dissipation bounded from above uniformly with respect to k.

Proof of Lemma 3.3.

Choose an orthonormal basis (ex, ey) of R² so that neither the x-direction nor the y-direction is parallel to any ofv₁, ...,v_p. Observe that integrating (3.9)-(3.10) over Ω and summing overi, shows that outflow of mass equals inflow. We shall first obtain uniformly in k, an upper bound for the energy

p

X

i=1

v_i² Z

Ω

F_i^k(z)dz.

Recalling that the genericity condition (1.3) implies that all velocities are different from zero, the energy bound implies an upper estimate for the mass. Writev_i =ξ_ie_x+ζ_ie_y. Multiply the equation forF_i^k with ξi and integrate over Ωa= Ω∩ {(x, y);x≤a}. Set

Sa= Ω∩ {(x, y);x=a} and ∂Ωa =∂Ω∩Ω¯a. From (3.9)-(3.10) follows

p

X

i=1

ξ_i² Z

Sa

F_i^k(a, y)dy=

p

X

i=1

ξ_i Z

∂Ωa

(v_i·n(Z))F_i^k(Z)dσ(Z). (3.13)

For any (x, y)∈Ω let the line-segment through (x, y) in thex-direction (resp. y-direction) intersect the boundary ∂Ω atx⁻(y)< x⁺(y) (resp. y⁻(x)< y⁺(x)). Denote by

x⁻₀ := min

(x,y)∈Ω{x⁻(y)}, x⁺₀ := max

(x,y)∈Ω{x⁺(y)}. (3.14)

Integrating (3.13) on a∈[x⁻₀, x⁺₀] gives uniformly ink,

p

X

i=1

ξ_i² Z

Ω

F_i^k(z)dz =

p

X

i=1

ξ_i Z x⁺₀

x⁻₀

Z

∂Ωa

(v_i·n(Z))F_i^k(Z)dσ(Z)

da≤c_b, wherecbonly depends on the given inflow. AnalogouslyPp

i=1ζ_i²R

ΩF_i^k(z)dz≤cb. The boundedness of energy and with it mass, follows.

The entropy dissipation estimate is proved as follows. Denote by D^k the entropy production term for the approximation F^k,

D^k= X

ijlm

Γ^lm_ij Z

Ω

( F_i^k 1 +^F_k^ik

F_j^k 1 +^F

k j

k

− F_l^k 1 +^F_k^lk

F_m^k

1 +^F_k^mk ) lnF_i^kF_j^k(1 + ^F_k^lk)(1 + ^F_k^mk) (1 + ^F_k^ik)(1 + ^F

k j

k )F_l^kF_m^k (z)dz.

Multiply (3.9) by ln ^Fik

1+^{F k}_kⁱ

, add the equations in i, and integrate the resulting equation on Ω. It leads to

p

X

i=1

Z

∂Ω⁻_i

F_i^klnF_i^k−k(1 +F_i^k

k ) ln(1 +F_i^k k )

(Z)|v_i·n(Z)|dσ(Z) +D^k ≤c_b. Moreover,

k Z

∂Ω⁻_i

ln(1 +F_i^k

k )(Z)|vi·n(Z)|dσ(Z)≤ Z

∂Ω⁻_i

F_i^k(Z)|vi·n(Z)|dσ(Z)≤c_b. Hence

p

X

i=1

Z

∂Ω⁻_i

F_i^kln F_i^k 1 +^F_k^ik

(Z)|vi·n(Z)|dσ(Z) +D^k≤c_b. (3.15)

The uniform entropy dissipation bound holds, sincex→xln¹⁺

x k

x is bounded from above on ]0,+∞[.

The following lemma replaces an entropy control of (F^k)_k∈N^∗, under the condition (1.4).

Lemma 3.4 Assuming (1.4), it holds that

p

X

i=1

Z

z∈Ω;F_i^k(z)<k

F_i^klnF_i^k(z)dz+ lnk

p

X

i=1

Z

z∈Ω;F_i^k(z)≥k

F_i^k(z)dz < c_b, k∈N^∗, where c_b only depends on the given inflow.

Proof of Lemma 3.4.

The entropy flow of (F_i^k) is first controlled as follows. It holds that Z

∂Ω⁻_i

F_i^kln(1 +F_i^k

k )(Z)|vi·n(Z)|dσ(Z)

≤ Z

∂Ω⁻_i,F_i^k≤k

F_i^kln(1 +F_i^k

k )(Z)|vi·n(Z)|dσ(Z) +

Z

∂Ω⁻_i,F_i^k≥k

F_i^kln(1 +F_i^k

k )(Z)|vi·n(Z)|dσ(Z)

≤ln 2 Z

∂Ω⁻_i

F_i^k(Z)|vi·n(Z)|dσ(Z) + Z

∂Ω⁻_i,F_i^k≥k

F_i^kln2F_i^k

k (Z)|vi·n(Z)|dσ(Z)

≤c_b+ Z

∂Ω⁻_i,F_i^k≥k

F_i^klnF_i^k(Z)|v_i·n(Z)|dσ(Z)−lnk 2

Z

∂Ω⁻_i,F_i^k≥k

F_i^k(Z)|v_i·n(Z)|dσ(Z).

Together with (3.15), this implies that

p

X

i=1

Z

∂Ω⁻_i,F_i^k≤k

F_i^klnF_i^k(Z)|v_i·n(Z)|dσ(Z) + lnk 2

Z

∂Ω⁻,F_i^k≥k

F_i^k |v_i·n(Z)|dσ(Z)≤c_b. Set

e_x:=n₀, Ω_a = Ω∩ {(x, y);x≤a}, S_a= Ω∩ {(x, y);x=a}, ∂Ω_a=∂Ω∩Ω¯_a. Multiplying the equation for F_i^k by ln ^Fik

1+^{F k}_kⁱ

, 1 ≤ i ≤ p, summing the resulting equations and integrating over Ωa, implies that

p

X

i=1

v_i·n₀ Z

Sa

F_i^klnF_i^k−k(1 +F_i^k

k ) ln(1 +F_i^k k )

(a, y)dy

≤ −D_k+

p

X

i=1

Z

∂Ωa

F_i^klnF_i^k−k(1 +F_i^k

k ) ln(1 +F_i^k k )

(Z)(v_i·n(Z))dσ(Z)

≤c_b.

An integration on [x⁻₀, x⁺₀] defined in (3.14) implies that

p

X

i=1

v_i·n₀ Z

Ω

F_i^klnF_i^k−k(1 + F_i^k

k ) ln(1 +F_i^k k )

(z)dz ≤c_b.

Moreover, k

Z

Ω

ln(1 +F_i^k

k )(z)dz ≤ Z

Ω

F_i^k(z)dz ≤c_b, 1≤i≤p,

and Z

Ω

F_i^kln(1 +F_i^k k )(z)dz

≤ Z

z∈Ω;F_i^k(z)≤k

F_i^kln(1 +F_i^k

k )(z)dz+ Z

z∈Ω;F_i^k(z)≥k

F_i^kln(1 +F_i^k k )(z)dz

≤ln 2 Z

Ω

F_i^k(z)dz+ Z

z∈Ω,F_i^k(z)≥k

F_i^kln2F_i^k k (z)dz

≤cb+ Z

z∈∂Ω;F_i^k(z)≥k

F_i^klnF_i^k(z)dz−lnk 2

Z

z∈Ω;F_i^k(z)≥k

F_i^k(z)dz.

And so,

p

X

i=1

v_i·n₀Z

z∈Ω,F_i^k(z)<k

F_i^klnF_i^k(z)dz+ lnk 2

Z

z∈Ω;F_i^k(z)≥k

F_i^k(z)dz

< c_b. The use of assumption (1.4) gives

Z

z∈Ω,F_i^k(z)<k

F_i^klnF_i^k(z)dz+ lnk 2

Z

z∈Ω;F_i^k(z)≥k

F_i^k(z)dz < c_b, 1≤i≤p, k >2.

4 The passage to the limit in the approximations.

This section contains the proof of Theorem 1.1. The main part is a proof of strongL¹ compactness of (F^k)k∈N^∗, based on two compactness lemmas for integrated collision frequency and gain term.

Recall the exponential multiplier form for the approximations (F^k)k∈N^∗, F_i^k(z) =f_bi^k(z_i⁺(z))e⁻

R^s

+ i(z)

0 ν^k_i(z_i⁺(z)+svi)ds

+

Z s⁺_i(z) 0

Q^+k_i (z_i⁺(z) +sv_i)e^−R

s+ i(z)

s ν^k_i(F^k)(z⁺_i(z)+rvi)drds, a.a. z∈Ω, 1≤i≤p, (4.1) where ν_i^k and Q^+k_i are defined by

ν_i^k =X

jlm

Γ^lm_ij F_j^k (1 +^F_k^ik)(1 + ^F

k j

k )

, Q^+k_i =X

jlm

Γ^lm_ij F_l^k 1 +^F_k^lk

F_m^k 1 +^F_k^mk . An i-characteristics is a segment of points [Z−s⁺_i (Z)v_i, Z], where Z ∈∂Ω⁻_i .

By the strict convexity of Ω, there are for every i∈ {1,· · ·p} two points of∂Ω, denoted by ˜Z_i and Z¯i such that

z_i⁺( ˜Z_i) =z⁻_i ( ˜Z_i) and z⁺_i ( ¯Z_i) =z_i⁻( ¯Z_i).

Denote by Ω^k2_i (resp. Ω^k3_i) the set of points between ˜Zi (resp. ¯Zi) and the i-characteristics in Ω at distance from ˜Z_i (resp. ¯Z_i). Such subsets of Ω are introduced in order that all i-characteristics from Z ∈ Ω^k2_i ∪Ω^k3_ic

are segments of length uniformly bounded from below in terms of.