Discrete velocity Boltzmann equations in the plane: stationary solutions.

The paper proves existence of stationary mild solutions for normal discrete velocity Boltzmann equations in the plane with no pair of colinear interacting velocities and given ingoing boundary values. An important restriction of all velocities pointing into the same half-space in a previous paper is removed in this paper. A key property is L 1 compactness of integrated collision frequency for a sequence of approximations. This is proven using the Kolmogorov-Riesz theorem, which here replaces the L 1 compactness of velocity averages in the continuous velocity case, not available when the velocities are discrete.


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 velocities is a simplification, preserving the essential features of free flow and quadratic collision term.Besides this fundamental aspect, the discrete equations can approximate the Boltzmann equation with any given accuracy [10], [14], [18], 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, where f i (t, z), 1 ≤ i ≤ p, are phase space densities at time t, position z and velocities v i .The spatial domain is Ω.The given discrete velocities are v i , 1 ≤ i ≤ p.For f = (f i ) 1≤i≤p , the collision operator Q = (Q i ) 1≤i≤p with gain part Q + , loss part Q − , and collision frequency ν, is given by Γ lm ij f j , i = 1, ..., p.
The collision coefficients satisfy If a collision coefficient Γ lm ij is non-zero, then the conservation laws for momentum and energy, are satisfied.We call interacting velocities any couple of velocities (v i , v j ) such that for some (l, m) ∈ {1, • • •, p} 2 , Γ lm ij > 0. The discrete velocity model (DVM) is called normal (see [11]) if any solution of the equations where the indices (i, j; l, m) take all possible values satisfying Γ lm ij > 0, is given by for some constants a, c ∈ R and b ∈ R d .We consider the generic case of normal coplanar velocity sets with no pair of colinear interacting velocities (v i , v j ). (1. 3) The case is generic.Indeed, consider a normal velocity set such that for some interacting velocities (v i , v j ), v i and v j are colinear.Then there exists an arbitrary small vector v 0 such that the velocity set (v i + v 0 ) 1≤i≤p is normal and with no colinear interacting velocities.The paper considers stationary solutions to normal coplanar discrete velocity models satisfying (1.3), in a strictly convex bounded open subset Ω ⊂ R 2 , with C 2 boundary ∂Ω and given boundary inflow.Denote by n(Z) the inward normal to Z ∈ ∂Ω.Denote the v i -ingoing (resp.v i -outgoing) part of the boundary by for the ingoing (resp.outgoing) point on ∂Ω of the characteristics through z in direction v i .The stationary boundary value problem is considered in L 1 in one of the following equivalent forms ( [13]); the exponential multiplier form, the renormalized form, in the sense of distributions.Denote by L 1 + (Ω) the set of non-negative integrable functions on Ω.For a distribution function f = (f i ) 1≤i≤p , define its entropy (resp.entropy dissipation) by The main result of the paper is Theorem 1.1 Consider a coplanar normal discrete velocity model and a non-negative ingoing boundary value f b with mass and entropy inflows bounded, For the boundary value problem (1.5)-(1.6)satisfying (1.3), there exists a stationary mild solution in L 1 + (Ω) p with finite mass and entropy-dissipation.
0 for all j, l and m, then f i equals its ingoing boundary value, and the rest of the system can be solved separately.Such i's are not present in the following discussion.Most mathematical results for stationary discrete velocity models of the Boltzmann equation have been obtained in one space dimension.An overview is given in [15].Half-space problems [5] and weak shock waves [6] for discrete velocity models have also been studied.A discussion of normal discrete velocity models, i.e. conserving nothing but mass, momentum and energy, can be found in [8].In two dimensions, special classes of solutions to the Broadwell model are given in [7], [9] and [16].The Broadwell model, not included in the present results, is a four-velocity model, with v 1 + v 2 = v 3 + v 4 = 0 and v 1 , v 3 orthogonal.[9] 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 [12] is the existence of continuous solutions to the twodimensional stationary Broadwell model with continuous boundary data for a rectangle.The paper [1] solves that problem in an L 1 -setting.The proof uses in an essential way the constancy of the sums f 1 + f 2 and f 3 + f 4 along characteristics, which no longer holds in the present paper.For every normal model, there is a priori control of entropy dissipation, mass and entropy flows through the boundary.From there, main difficulties are to prove that for a sequence of approximations, weak L 1 compactness holds and the limit of the collision operator equals the collision operator of the limit.In [2], weak L 1 compactness of a sequence of approximations was obtained with assumption (1.3) together with the assumption that all velocities v i point out into the same half-plane.In this paper we keep assumption (1.3), remove the second assumption and provide a new proof of weak L 1 compactness of approximations using (1.3).Assumption (1.3) is also crucial for proving L 1 compactness of the integrated collision frequencies, that is important for the convergence procedure.Our paper also differs from [2] in the limit procedure.The frame of the limit procedure in [2] is the splitting into 'good' and 'bad' characteristics following the approach in our earlier stationary continuous velocity papers [3]- [4].Here we have instead recourse to sub-and super-solutions used in the classical evolutionary frame for renormalized solutions to the Boltzmann equation [13].
For the continuous velocity evolutionary Boltzmann equation [13], the compactness properties of the collision frequency use in an essential way the averaging lemma, which is not available for the discrete velocity Boltzmann model.In the present paper, the compactness properties are proven by the Kolmogorov-Riesz theorem.Also the argument used in the stationary paper [4] in the continuous velocity case for obtaining control of entropy, hence weak L 1 compactness of a sequence of approximations from the control of entropy dissipation, does not work in a discrete velocity case because the number of velocities is finite.
The proof starts in Section 2 from bounded approximations.In Section 3, L 1 compactness properties of the approximations are proven.Section 4 is devoted to the proof of Theorem 1.1.

Approximations.
Denote by N * = N \ {0} and by a ∧ b the minimum of two real numbers a and b.Let µ α be a smooth mollifier in R 2 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 μk be a smooth mollifier on ∂Ω in a ball of radius 1 k .Denote by The lemma introduces a primary approximated boundary value problem with damping and convolutions.

Lemma 2.1
For any α > 0 and k ∈ N * , there is a solution Proof of Lemma 2.1.For a proof of Lemma 2.1 we refer to the second section in [2].
Let k ∈ N * be given.Each component of F α,k is bounded by a multiple of k 2 .Therefore (F α,k ) α∈]0,1[ is weakly compact in (L 1 (Ω)) p .For a subsequence, the convergence is strong in (L 1 (Ω)) p as stated in the following lemma.

Lemma 2.2
There is a sequence (β(q)) q∈N tending to zero when q → +∞ and a function For a proof of Lemma 2.2 we refer to Lemma 3.1 in [2].
Denote by and by Dk the entropy production term of the approximations, All along the paper, c b denotes constants that may vary from line to line but is independent of parameters tending to +∞ or to zero. (2.6) ) have mass and entropy dissipation bounded from above uniformly with respect to k. Moreover their outgoing flows at the boundary are controlled as follows, (2.7) Proof of Lemma 2.3.Passing to the limit when q → +∞ in (2.1)-(2.2) written for ).For a proof of the rest of Lemma 2.3, we refer to Lemma 3.2 in [2].
3 On compactness of sequences of approximations.
This section is devoted to prove L 1 compactness properties of the approximations.In Proposition 3.1, weak L 1 compactness of (F k ) k∈N * is proven.Lemma 3.1 splits Ω into a set of i-characteristics with arbitrary small measure and its complement, where both the approximations and their integrated collision frequencies are bounded.In Lemma 3.2, the strong L 1 compactness of integrated collision frequency is proven.
the weak L 1 compactness of (F k ) k∈N * will follow from the uniform boundedness in L ∞ (∂Ω + i ) of where 3), there exists η > 0 such that for all interacting velocities (v i , v j ), and integrate it on one of the half domains defined by the segment [Z, Together with (3.3), this leads to the control of (3.2).
Recall the exponential multiplier form for the approximations (F k ) k∈N * , with ν k i and Proof of Lemma 3.1.By the strict convexity of Ω, there are for every i ∈ {1, • • •p} two points of ∂Ω, denoted by Zi and Zi such that Let li (resp.li ) be the largest boundary arc included in ∂Ω − i with one end point Zi (resp.Zi ) such that (3.7) It follows from the exponential form of F k i that The boundedness of the mass flow of ( It follows from (3.7)-(3.10)that the measure of the set is smaller than c b ǫ.The boundedness of the mass of (F k j ) k∈N * can be written Hence the measure of the set is bounded by c b ǫ.Together with (3.9), this implies that the measure of the complement of the set of Z ∈ ∂Ω − i , such that With it c b ǫ is a bound for the measure of the complement, denoted by Ω k,ǫ i , of the set of i-characteristics in Ω such that for all points z on the i-characteristics, (3.6) holds.
Given i ∈ {1, ..., p} and ǫ > 0, let χ k,ǫ i denote the characteristic function of the complement of Ω k,ǫ i .The following lemma proves the compactness in L 1 (Ω) of the k-sequence of integrated collision frequencies.

Lemma 3.2
The sequences 3), v i and v j span R 2 .Denote by (a, b) the corresponding coordinate system, (a − , a + ) defined by and by D the Jacobian of the change of variables z → (a, b).The uniform bound for the mass of (F k ) k∈N * proven in Lemma 2.3, implies that b − (a) By the Kolmogorov-Riesz theorem ( [17], [19]), the compactness of 0 −s + i (z) ν k i (z + sv i )ds k∈N * will follow from its translational equi-continuity in L 1 (Ω).Equicontinuity in the direction v i , and in the direction v j with the mild form (1.8) for F k j , come natural.Here the assumption (1.3) becomes crucial.The sequence is translationally equi-continuous in the v i -direction.Indeed, s + i (z + hv i ) = s + i (z) + h so that, denoting by I(0, h) the interval with endpoints 0 and h and using the uniform bound on the mass of (F k j ) k∈N * , Let us prove the translational equi-continuity of (3.11) in the v j -direction.By the weak L 1 compactness of (F k j ) k∈N * , it is sufficient to prove the translational equi-continuity in the v j -direction of where 3).Denote by (z + j (z + i (z)), z + j (z + i (z + hv j )) the boundary arc with end points z + j (z + i (z)) and z + j (z + i (z + hv j )) and of length tending to zero with h.Performing the change of variables s → Z = z + j (z + hv j + sv i ) (resp.s → Z = z + j (z + sv i )) in the first (resp.second) term of A k ij (z, h), and using that the sequence (f k bi ) k∈N * is bounded by f bi , it holds that uniformly with respect to k.Moreover, for some ω h (z) ⊂ Ω of measure or order | h | uniformly with respect to z ∈ Ω, The sequence (χ k,ǫ j Q k j ) k∈N * is weakly compact in L 1 .Indeed, 4 The passage to the limit in the approximations.
Let f be the weak L 1 limit of a subsequence of the solutions (F k ) k∈N * to (2.5)-(2.6),still denoted by (F k ) k∈N * .For proving that f is a mild solution of (1.5)-(1.6), it is sufficient to prove that for any η > 0 and i ∈ {1, • • •, p}, there is a set X η i of i-characteristics with complementary set of measure smaller than cη, such that where χ η i denotes the characteristic function of X η i .Define the set X η i as follows.For every ǫ > 0, pass to the limit when k → +∞ in and use the weak L 1 compactness of (χ k,ǫ i F k i ) k∈N * , the weak L 1 compactness and the uniform boundedness in L ∞ of (χ k,ǫ i F k i (z − i (z))) k∈N * , and the strong L 1 compactness of ( where F ǫ i is the limit of a subsequence of (χ k,ǫ i F k i ) k∈N * and ν i (f ) = p j,l,m=1 Γ lm ij f j .By the monotonicity in ǫ of (F ǫ ) ǫ∈]0,1[ (resp.F ǫ (z − i (z)) ǫ∈]0,1[ ) and the uniform boundedness of their masses, it holds that From here the proof follows the lines of the proof of Lemma 3.1, so that given η > 0, there is a set X η i of i-characteristics, with complementary set of measure smaller than cη, such that and By the weak L 1 compactness of (F k i ) k∈N * and the linearity with respect to χ k,ǫ i F k i of the first line of (4.5), its passage to the limit when k → +∞ is straightforward.Let us pass to the limit when k → +∞ in any term of the loss term of (4.5), denoted by Γ lm ij L k , where and J i is defined in (3.8).By integration by parts, L k equals Denote by (a, b) the coordinate system in the (v i , v j ) basis, (a − , a The first term in L k can be written as a − l k (a)da with l k defined as For each rational number a, the sequence of functions is by Lemma 3.2 strongly compact in L 1 , and by Lemma 3.1 uniformly bounded in L ∞ .The convergence follows for any rational number a.With a diagonal process, there is a subsequence of (l k ), still denoted by (l k ), converging for any rational a.Moreover, uniformly with respect to k and a, by the weak L 1 compactness of Thus (l k ) is a uniform converging sequence on [a − , a + ].The second term in L k can be treated analogously, (χ k,ǫ i f k bi ) k∈N * being uniformly bounded in L ∞ .The convergence follows.
In order to determine the limit of L k when k → +∞, remark that By a backwards integration by parts, In order to prove (4.4), let us prove that each term from Q +k i in (4.5) converges when k → +∞ to a limit smaller than Take Γ lm ij = 1, j ∈ J i , for simplicity.(µ 1 n ) n∈N * being the sequence of mollifiers defined at the beginning of Section 2 for α = 1 n , split (4.12) into , by (??), Denote by D the Jacobian of the change of variables z → (a, b).For some smooth function A, and any integrable function g, Hence, .15)For Λ large enough, pass to the limit when k → +∞ and n → +∞ in (4.14).Up to subsequences, the weak L 1 limits F ǫ i and Choose Λ large enough, ǫ small enough, Λ ′ large enough, ǫ ′ small enough, in this order.The passage to the limit when ǫ → 0 and ǫ ′ → 0 in (4.16) results from the monotone convergence theorem, the family (F ǫ ) ǫ∈]0,1[ being non decreasing, with mass uniformly bounded, together with the mass of (χ For β > 0, start from the equation for ϕχ η i F k i written in renormalized form, Hence in weak L 1 the sequence (β −1 ln 1 + βF k i ) k∈N * converges modulo subsequence to a function F β ≤ f when k → +∞.The mass of the limit increases to the mass of f , when β → 0. This gives in the final limit β → 0 for the l.h.s. of (4.18),

.19)
Using analogous arguments as for the limit of the loss term in Lemma 4.1, it holds that   Choose Λ large enough and split the domain of integration of every j ∈ J i term in (4.21) into It holds that  And so, (4.17) holds.Together with (4.4), this proves (4.1).