BOUNDARY LAYERS AND SHOCK PROFILES FOR THE DISCRETE BOLTZMANN EQUATION FOR MIXTURES

We consider the discrete Boltzmann equation for binary gas mixtures. Some known results for half-space problems and shock profile solutions of the discrete Boltzmann for single-component gases are extended to the case of two-component gases. These results include well-posedness results for halfspace problems for the linearized discrete Boltzmann equation, existence results for half-space problems for the weakly non-linear discrete Boltzmann equation, and existence results for shock profile solutions of the discrete Boltzmann equation. A characteristic number, corresponding to the speed of sound in the continuous case, is calculated for axially symmetric models. Some explicit calculations are also made for a simplified 6 + 4 -velocity model.


NICLAS BERNHOFF
In the planar stationary case, the DBE reduces to a system of ODEs. It is wellknown that the Boltzmann equation can be approximated up to any order by the DBE [13], [22], [27].
Half-space problems for the linearized Boltzmann equation are well investigated [2], and in the case of binary mixtures by Aoki, Bardos and Takata in [1]. For the linearized DBE a classification of well-posed half-space problems has been made in [4], based on results in [10] on the dimensions of the corresponding stable, unstable and center manifolds for singular points (Maxwellians for the DBE) to general systems of ODEs of the same type.
In [33] Ukai, Yang and Yu studied the non-linear case with inflow boundary conditions, assuming that the solutions tend to an assigned Maxwellian at infinity. The conditions on the data at the boundary needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. Similar problems have also been studied for the DBE in [32], [23], [24], and [5]. The quite general results in [5] include (for DVMs) the results obtained by Ukai, Yang and Yu in [33] for the continuous Boltzmann equation. In this connection, we also mention the recent paper by Yang [34], and the recent work by Liu and Yu in [26] on the center manifold theory of the half-space problem for the full Boltzmann equation, where also more references for the continuous case can be found.
The existence of shock profile solutions, c.f. [18] and [25], have been studied for the DBE in [16] and [9]. For the shock wave problem the DBE also becomes a system of ODEs. In [9] existence of shock profile solutions for the DBE is proved. The results concern weak shocks, i.e., when the shock speeds are close to a typical speed, corresponding to the sound speed in the continuous case. The shock-wave problem have also been studied for several explicit discrete velocity models for mixtures, see e.g. [19].
The case when one of the gases is a non-condensable gas (cf. [30]) is not included in this paper, but will be treated in a future paper [7].
The paper is organized as follows. In Section 2 we present the DBE for mixtures and some of its properties. We make an expansion around a bi-Maxwellian and obtain the linearized collision operator and the quadratic part and conclude that we actually obtain a system with the same structure as in the case of one species. We also remind a result in [10] on the dimensions of the corresponding stable, unstable and center manifolds for singular points (bi-Maxwellians for DVMs for binary mixtures) to general systems of ODEs of the same type. Then we present the extension of our results for boundary layers in [4] and [5], in Section 3, and for shock profiles in [9], in Section 4, to the case of binary mixtures. In Section 5, we calculate a number, corresponding to the speed of sound in the continuous case, for axially symmetric models. Finally, in Sections 6 and 7 we exemplify our theory for an explicit simplified model. We find exact shock profile solutions in Section 6 and consider non-linear boundary layers in the case of a moving wall with constant speed in Section 7, for a plane 6 + 4 -velocity model, where we have assumed that our flow is symmetric with respect to the x -axis.
2. Discrete velocity models for binary mixtures. The general discrete velocity model (DVM), or the discrete Boltzmann equation, for a binary mixture of the .., n α , and f α = f α (x, t, ξ) represents the microscopic density of particles (of the gas α) with velocity ξ at time t ∈ R + and position x ∈ R d . We denote by m α the mass of a molecule of the gas α. Here and below, α, β, γ ∈ {A, B}.
For a function g α = g α (ξ) (possibly depending on more variables than ξ), we will identify g α with its restriction to the set V α , but also when suitable consider it like a vector function where it is assumed that the collision coefficients Γ kl ij (β, α), with 1 ≤ i, k ≤ n α and 1 ≤ j, l ≤ n β , satisfy the relations with equality unless the conservation laws and Then the collision operator Q(f, f ) can be obtained from the bilinear expressions Denoting Q(f, g) = (Q 1 (f, g) , ..., Q n (f, g)), with n = n A + n B , we see that, for arbitrary f and g Q (f, g) = Q (g, f ) , (3) for all indices 1 ≤ i, k ≤ n α , 1 ≤ j, l ≤ n β and α, β ∈ {A, B}, such that Γ kl ij (β, α) = 0. By the relation (3) which is zero, independently of our choice of non-negative vector f (f α i ≥ 0 for all 1 ≤ i ≤ n α ), if and only if φ is a collision invariant.
We consider below (even if this restriction is not necessary in our general reasoning) only DVMs, such that any collision invariant is of the form for some constant a A , a B , c ∈ R and b ∈ R d . In this case the equation has the general solution (6). Discussions on constructions of DVMs for binary mixtures can be found in e.g. [11], [12], [20], [21], [14] and [15]. All bi-Maxwellians are of the form where φ = φ A , φ B is given by Eq. (6). Assuming that f is non-negative, we let φ = log f in Eq. (5) and obtain that in Eq.(1), the system ∂h ∂t . Furthermore, L is the linearized collision operator (n × n matrix, with n = n A + n B ) given by and the quadratic part S is given by By Eq.(3) and the relations M α i M β j = M α k M β l = 0, we obtain the equality
We consider below the case when D is non-singular, i.e. when all ξ α,1 i = 0 are non-zero. For the case of singular matrices D, see Remark 5 below.
We denote by n ± , where n + + n − = n, and m ± , with m + + m − = q, the numbers of positive and negative eigenvalues (counted with multiplicity) of the matrices D and D −1 L respectively, and by m 0 the number of zero eigenvalues of D −1 L. Moreover, we denote by k + , k − , and l, with k + + k − = k, where k + l = p, the numbers of positive, negative, and zero eigenvalues of the p × p matrix K (p = d + 3 for normal DVMs for binary mixtures), with entries k ij = y i , y j D = y i , Dy j , such that {y 1 , ..., y p } is a basis of the null-space of L, N (L). In our case, Here and below, we denote by ·, · the Euclidean scalar product on R n and denote ·, · D = ·, D· . In applications, the number p of collision invariants is usually relatively small compared to n (note that formally n = ∞ for the continuous Boltzmann equation whenas p ≤ 6). Also, the matrix D is diagonal and therefore all its eigenvalues are known. This explains the importance of the following result by Bobylev and Bernhoff [10] (see also [4]).
3. Applications to boundary layers. The main results for half-space problems for single species in [4] and [5] can now be applied in the case of binary mixtures. For the sake of completeness we present the results here. All proofs are similar to the ones for single species found in [4] and [5].
We consider the inhomogeneous (or homogeneous if g = 0) linearized problem where g = g(x) ∈ L 1 (R + , R n ), with one of the boundary conditions (O) the solution tends to zero at infinity, i.e.
|f (x)| e − x → 0 as x → ∞, for all > 0; at infinity. In the case of boundary condition (O) at infinity we additionally assume that We can (without loss of generality) assume that where
At x = 0 we assume the general boundary condition where C is a given n + × n − matrix and h 0 ∈ R n + . In applications, where C α are given n + α × n − α matrices. We introduce the operator C : R n → R n + , given by In order to be able to obtain existence and uniqueness of solutions of the linearized half-space problems we will assume that the matrix C fulfills the condition as we consider boundary condition (O) at infinity, the condition dim CX + = n + , with X + = span (u 1 , ..., u m + , y 1 , ..., y k + , z 1 , ..., z l ) , as we consider boundary condition (P) at infinity, and the condition (18) or the condition dim C X + = n + , with X + = span (u 1 , ..., u m + , y 1 , ..., y k + , z 1 + w 1 , ..., z l + w l ) , as we consider boundary condition (Q) at infinity.
We consider the non-linear system where the solution tends to zero at infinity. Furthermore, we fix a number σ, such that 0 < σ ≤ min {|λ| = 0; det(λD − L) = 0} and introduce the norm We have the following existence result.

Remark 2.
It was recently proved that one can get rid of the restrictive assumptions S (f (x), f (x)) , w j = 0 for j = 1, ..., l, on the quadratic part in the degenerate cases, in Theorem 3.3, for one-component as well as two-component gases [8], by a slight modification of the proof of Theorem 3.3, if one instead of condition (18) assume that dim C X + = n + , with X + = span (u 1 , ..., u m + , y 1 , ..., y k + , w 1 , ..., w l ) .
In the following theorem we present explicit conditions on h 0 , but then with restrictive conditions on the quadratic part.   We make the following assumptions on our DVMs.

NICLAS BERNHOFF
Then, and the degenerate values of c (the values of c for which l ≥ 1) are , where Here corresponds to the speed of sound in the continuous case.
6. Exact shock profiles for a plane 6+4-velocity model. We now consider the shock wave problem for a mixture, in which gas A is described by a 6-velocity model with velocities (±1, 0) and (±1, ±2m), and the gas B is described by the classical Broadwell model [17] in plane with velocities (±m, ±m).
, where we assume that m > 1. The case m < 1 can be studied in a similar way. If m = 1, then for this simplified model, the degenerate values would be c 0 = 0 and c ± = ±1. Especially, "the speed of sound" is 1, contradicting assumption 1[ii] in Section 4. However, in general we don't have to exclude the case m = 1.
Note that for the Broadwell model we have only two linearly independent collision invariants, as the mass vector and the energy vector are linearly dependent, even if mass, momentum, and energy all are preserved. However, the DVMs for gas A and the mixture will have the correct number of linearly independent collision invariants.

DISCRETE BOLTZMANN EQUATION FOR MIXTURES 13
For a flow symmetric around the x 1 -axis we obtain the reduced system We assume that D − cI is non-singular, i.e. that c / ∈ {±1, ±m}, and make the natural assumption σ 2 = σ 3 .
The set of collision invariants are generated by the collision invariants The density, momentum density, and energy density per unit volume are obtained by .