ON HALF-SPACE PROBLEMS FOR THE WEAKLY NON-LINEAR DISCRETE BOLTZMANN EQUATION

Existence of solutions of weakly non-linear half-space problems for the general discrete velocity (with arbitrarily finite number of velocities) model of the Boltzmann equation are studied. The solutions are assumed to tend to an assigned Maxwellian at infinity, and the data for the outgoing particles at the boundary are assigned, possibly linearly depending on the data for the incoming particles. 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. In the non-degenerate case (corresponding, in the continuous case, to the case when the Mach number at infinity is different of -1, 0 and 1) implicit conditions are found. Furthermore, under certain assumptions explicit conditions are found, both in the non-degenerate and degenerate cases. Applications to axially symmetric models are studied in more detail.

For a non-condensable gas (i.e. with no mass flux of the gas across the wall) we can put g 0 (ξ) ≡ 0. A particular case is the boundary conditions introduced by Maxwell in Ref. [30,Appendix], where T w is the temperature of the wall and α, with 0 ≤ α ≤ 1, is the accommodation coefficient. The case α = 1 is called diffuse reflection, and the case α = 0 specular reflection. The Maxwell boundary conditions can be obtained by taking with e 1 = (1, 0, 0), in Eq. (4). In this paper we study the corresponding problem for the general discrete velocity model in Refs. [15] and [23]. More general boundary conditions (see Eq. (30) below), corresponding to boundary condition (4) in the continuous case, are also considered. Discrete velocity models (DVMs) of the Boltzmann equation are models, where the velocity is discretized, i.e. the velocity is assumed to be able to take only a finite (or in general a discrete) number of different values. It is a well-known fact that the Boltzmann equation can be approximated the singular point (Maxwellian for DVMs) approached at infinity is fixed and small deviations of the solutions from the singular point is studied. The data for the outgoing particles at the boundary are assigned, possibly linearly depending on the data for the incoming particles. 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. In the non-degenerate case (corresponding, in the continuous case, to the case when the Mach number at infinity is different of -1, 0 and 1) implicit conditions have been found by using arguments by Ukai, Yang and Yu in Ref. [38] for the continuous Boltzmann equation. Furthermore, under certain assumptions explicit conditions are found, both in the non-degenerate and degenerate cases. The results extend, not only by more general boundary conditions, but also by more general assumptions, previous results for the discrete Boltzmann equation by Ukai in Ref. [37], and Kawashima and Nishibata in Refs. [28] and [29], and include also (for DVMs) the results obtained by Ukai, Yang and Yu in Ref. [38] for the continuous Boltzmann equation. Applications to axially symmetric models have also been studied, generalizing the results by Babovsky in Ref. [2].
All results are obtained for an arbitrary finite number of velocities. Similar results as in this paper can also be obtained for DVMs for mixtures. Existence of weak shock wave solutions for the discrete Boltzmann equation has also been proved based on the same ideas in Ref. [8].
This paper is organized as follows: In Section 2, we introduce the planar stationary discrete Boltzmann equation and review some of its properties. We make an expansion around an equilibrium Maxwellian, and review, Theorem 2.1 in Subsection 2.1, the results in Ref. [10] on the dimensions of the stable, unstable and center manifolds of the system of ODEs. The problem and the main results on existence and uniqueness are stated in Section 3 (Theorem 3.1 and Theorem 3.2). The boundary conditions at the "wall" are discussed in more detail in Section 4. In particular, inflow boundary conditions and Maxwell-type boundary conditions (Subsection 4.1) are considered. The results of [10] (stated in Theorem 2.1) are used to investigate the number of additional conditions needed to obtain well-posedness of the weakly non-linear problem in Section 5 and Section 6 respectively, and thereby to prove Theorem 3.1 (Section 5) and Theorem 3.2 (Section 6) in Section 3. Implicit conditions for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution in the non-degenerate case and also for the degenerate case, but then with some restrictions on the non-linear part of the collision operator, are obtained (Section 5). The results are in accordance with corresponding results for the continuous Boltzmann equation obtained in the non-degenerate case, with inflow boundary conditions in Ref. [38]. We also obtain explicit conditions for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution (Section 6), but with more restrictions, at least in the non-degenerate case, on the non-linear part. However, in some degenerate cases we obtain weaker restrictions on the non-linear part than in Theorem 3.1. The more general case when we allow velocities inducing a singular "velocity-matrix" (that is, if we allow velocities that have zero as first component) is discussed in Section 7. Applications to axially symmetric models is studied in Section 8. The degenerate cases for axially symmetric DVMs (in the "shock wave context"), if we have expanded around a non-drifting Maxwellian in Section 2, are discussed in Subsection 8.1. The results are in accordance with the results for the continuous Boltzmann equation in Ref. [21]. We also apply our results (Theorem 3.2) in Section 3 to a boundary layer problem of the type studied by Golse, Perthame and Sulem in Ref. [26] for the Boltzmann equation, and by Babovsky in Ref. [2] for DVMs (with quite restrictive conditions on the non-linear part of the collision operator). We first consider a plane 12-velocity DVM in Subsection 8.2, but also a more general axially symmetric DVM (cf. Ref. [2]) in Subsection 8.3.

2.
Discrete Boltzmann equation. The planar stationary system for the discrete Boltzmann equation (DBE) reads represents the microscopic density of particles with velocity ξ = (ξ 1 , ..., ξ d ) at position x = (x, x 2 , ..., x d ) ∈ R d . We also assume (except in Section 7) that ξ 1 i = 0, for i = 1, ..., n. For a function g = g(ξ) (possibly depending on more variables than ξ), we will identify g with its restriction to the set V, but also when suitable consider it like a vector function g = (g 1 , ..., g n ) , with g i = g (ξ i ) . Consistently, we say that g is non-negative (positive), g ≥ 0 (g > 0), if and only if Then Eq. (5) can be rewritten as Below we review some properties of the discrete Boltzmann equation. The collision operators Q i (F, F ) in Eq. (5) are given by the bilinear expressions where it is assumed that the collision coefficients Γ kl ij satisfy the relations Γ kl ij = Γ kl ji = Γ ij kl ≥ 0, with equality unless the conservation laws are satisfied (preservation of momentum and energy).

Remark 1.
Our main results, presented in Section 3, do not depend on the preservation of energy (even if we indeed use it in some of our applications), i.e., Eqs. (8) could be replaced by ξ i + ξ j = ξ k + ξ l , without affecting our main results. In fact, our main results do not depend on what set of collision invariants (cf. Eq. (9)) we have.

is a collision invariant if and only if
for all indices such that Γ kl ij = 0, or, equivalently, if and only if φ, Q (F, F ) = 0, for all non-negative functions F . We have the trivial collision invariants (also called the physical collision invariants) φ 0 = 1, φ 1 = ξ 1 , ..., φ d = ξ d , φ d+1 = |ξ| 2 (including all linear combinations of these). Here and below, we denote by ·, · the Euclidean scalar product on R n .
We consider below (even if this restriction is not necessary in our general context) only normal DVMs. That is, DVMs without spurious (or non-physical) collision invariants, i.e. any collision invariant is of the form for some constant a, c ∈ R and b ∈ R d (methods of their construction are described in Refs. [11], [13] and [14]). In this case the equation (10) has the general solution (11).
where φ is a collision invariant (11) (the latter equality is due to the assumption of normal DVMs). In general a, b and c can be functions of x, but since we assume that our solutions tend to a global, i.e. with absolute constant a, b and c, Maxwellian at infinity, our interest is in global Maxwellians, and so when we below refer to a Maxwellian, we will mean a global Maxwellian. Given a Maxwellian M we denote in Eq. (5), and obtain where L is the linearized collision operator (n × n matrix) given by and S is the quadratic part given by In more explicit forms, the operators (14) and (15) read Then the system (6) transforms into The diagonal matrix B (6) (under our assumptions) has no zero diagonal elements and is non-singular. If we denote f | x=0 = f 0 (the boundary conditions imposed by all ξ i ), then we can rewrite Eq. (16) as 2.1. Characteristic numbers. 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 B and B −1 L respectively, and by m 0 the number of zero eigenvalues of B −1 L. Moreover, we denote by k + , k − and l the numbers of positive, negative and zero eigenvalues of the p × p matrix K (p = d + 2 for normal DVMs), with entries k ij = y i , y j B = y i , By j , such that {y 1 , ..., y p } is a basis of the null-space of L, i.e. in our case span(y 1 , ..., Here and below, we denote ·, · B = ·, B· . We also recall the notation N (L) for the null-space of L.
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 ≤ 5). Also, the matrix B is diagonal and therefore all its eigenvalues are known. This explains the importance of the following result by Bobylev and Bernhoff in Ref. [10] (see also Ref. [7]).
[10] Theorem 2.1 is proved for any real symmetric matrices L and B, such that L is semi-positive and B is invertible.

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The Jordan normal form of B −1 L (with respect to the basis (17)- (19)) is where there are l blocks of the type 0 1 0 0 . For any h ∈ R n , we obtain 3. Statement of the problem and main results. We consider the non-linear system where the solution tends to zero at infinity, i.e. and The boundary conditions (21) correspond to the case when we have made the We can (without loss of generality) assume that We also define the projections R + : R n → R n + and R − : R n → R n − , n − = n − n + , by R + s = s + = (s 1 , ..., s n + ) and R − s = s − = (s n + +1 , ..., s n ) for s = (s 1 , ..., s n ).
At x = 0 we assume the general boundary conditions (cf. Eqs. (31) below) where C is a given n + × n − matrix and h 0 ∈ R n + . We introduce the operator C : R n → R n + , given by C = R + − CR − . We will also assume that the matrix C fulfills one of the conditions and dim CX + = n + , with X + = span (u 1 , ..., u m + , y 1 , ..., y k + , z 1 , ...., z l ) .
k + = 1, l = 3 and the collision invariants y 1 , y 2 , z 1 , z 2 and z 3 can be chosen as, cf. Ref. [21], Moreover, w j = L −1 ξ 1 z j , in the continuous case, and the continuous analogue of equation Lu = λBu is (see Ref. [16] for a discussion on the eigenvalue problem (27)). We also want to point out that, in the continuous case, the boundary conditions (before the expansion (13)), that correspond to conditions (24), are given by Eqs. (4).
We now state our main results.
Theorem 3.1. Let condition (26) be fulfilled and suppose that S (f (x), f (x)) , w j = 0 for j = 1, ..., l, and that h 0 , h 0 B+ is sufficiently small. Then with k + + l conditions on h 0 , the system (20) with the boundary conditions (21) and (24) has a locally unique solution.
Theorem 3.1 is proved in Section 5.
Theorem 3.2. Let condition (25) be fulfilled and assume that Then there is a positive number δ 0 , such that if then the system (20) with the boundary conditions (21) and (24) has a locally unique solution.
The proof of Theorem 3.2 is outlined in Section 6.

Remark 4. If condition (25) is fulfilled, then the condition
implies that we have k + + l conditions on h 0 .
Furthermore, if u = 0 and Cu = 0, then That is, dim CU + = dim U + = m + , and part ii) of the lemma is proved.
Part i) of the lemma is proved in a similar way (see also Ref. [7]).
Ukai considered the case with m + = n + and C = 0 in Ref. [37]. Then conditions (25), (26) and (28) are trivially fulfilled, since R + U + = R n + . This is the discrete correspondence of the case when the Mach number of the Maxwellian M ∞ is less for the full Boltzmann equation. This result was generalized by Kawashima and Nishibata in Ref. [28], where they still considered the case C = 0 (but allowed zero first components of the velocities, which was ruled out in Ref. [37]). Then conditions (25) and (26) are fulfilled by Corollary 1. They assumed that dim R + (BN (L)) ⊥ = m + (equivalent to assumption [A] in Ref. [28]), which implies that R + (BN (L)) ⊥ = R + U + , and hence, that l = 0 and R + U − ⊆ R + U + (Eq. (29) with C = 0). Therefore, condition (28) is fulfilled if the boundary data satisfies the consistency condition, equivalent to the condition h 0 ∈ R + (BN (L)) ⊥ , in Ref. [28]. In their subsequent paper [29], Kawashima and Nishibata assumed that dim R + (BN (L)) ⊥ = m + , ) ⊥ , and that u, u B < 0 if Cu = 0 and u = 0 (cf. Lemma 2.1 in Ref. [29]). Conditions (25) and (26) are fulfilled by the latter assumption and Lemma 3.3. By the first assumption, l = 0 and R + U − ⊆ R + U + . Therefore, the second assumption implies that condition (28) is fulfilled if the boundary data satisfies the consistency condition, equivalent to condition h 0 ∈ C (BN (L)) ⊥ , in Ref. [29]. Note that a 0 in Eq. (30) is assumed to be zero, a 0 = 0, in Ref. [29].
Remark 6. All our results can be extended in a natural way, to yield also for singular matrices B (see Section 7), if 4. Boundary conditions. If M = M ∞ = Ae b·ξ+c|ξ| 2 ∈ R n is the Maxwellian we have made the expansion (13) around, i.e., then the general boundary conditions (cf. boundary condition (4) in the continuous case) where C 0 is a given n + × n − matrix and a 0 ∈ R n + , at x = 0, lead to the following C and h 0 in Eq. (24), where I is the identity matrix and C 0d is the n + × n + matrix, with the elements for some Maxwellian M 0 , cf. Ref. [24]. The cases α = 0 and α = 1 correspond to specular and diffuse reflection, respectively. After the expansion (13), the Maxwell-type boundary conditions reads, cf. Ref. [7], where C d is the n + × n + matrix, with the elements

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We obtain that with equality if and only if u ∈ span( √ M − ).

4.1.
Maxwell-type boundary conditions. We now consider the Maxwell-type boundary conditions (32) .

5.
With damping term. We add (following the structure in Ref. [38] for the full Boltzmann equation) a damping term −γP + 0 f to the right-hand side of the system (20) and obtain where γ > 0 and First we consider the corresponding linearized inhomogeneous system where g = g(x) : R + → R n is a given function such that The system (37) with the boundary conditions (21) has (under the assumption that all necessary integrals exist) the general solution, using the notations in Eqs. (17)- (19), with η j (x) = g (x) , w j and β r (x) = g (x) , u r .
From the boundary conditions (24), we obtain the system For h 0 = 0 in (24), we have the trivial solution f (x) ≡ 0. Therefore, we consider only non-zero h 0 , h 0 = 0, below. The system (42) has (under the assumption that all necessary integrals exist) a unique solution if we assume that the condition (26) is fulfilled. We fix a number σ, such that 0 < σ ≤ min {|λ| = 0; det(λB − L) = 0} and σ ≤ γ and introduce the norm (cf. Ref. [29]) the Banach space and its closed convex subset where R is a, so far, undetermined positive constant. We assume that the condition (26) is fulfilled and introduce the operator Θ(f ) on X , defined by where with β 1 (f (0)) , ..., β m + (f (0)) , µ 1 (f (0)) , ..., µ k + (f (0)) and η 1 (f (0)) , ..., η l (f (0)) given by the system Lemma 5.2. Let f, h ∈ X and assume that the condition (26) is fulfilled. Then there is a positive constant K (independent of f and h), such that Proof. Let C −1 denote the inverse map of the linear map C = R + − CR − on X + = span (u 1 , ..., u m + , y 1 , ..., y k + , z 1 , ...., z l ) . The map C −1 is also linear and therefore bounded. We denote Then Hence, we obtain The quadratic function S(f, f ) is bilinear in its arguments and bounded, and hence, there is a positive constant K 2 (independent of f and h), such that Therefore, Proof. By estimates (44) and (45), there is a positive number K such that Let R = 2K and let δ 0 be a positive number, such that δ 0 < 1 R 2 . By estimates (45) and (46) The theorem follows by the contraction mapping theorem (see Ref. [33, p.2]). We denote by I γ the linear solution operator where f (x) is given by Similarly, we denote by I γ the nonlinear solution operator We assume that S (f, f ) , w j = 0 for j = 1, ..., l. By Theorem 5.4, the solution of Theorem 5.3 is a solution of the problem (20), (21) and (24) if and only if P + 0 I γ (h 0 ) ≡ 0.
Let g = g(x) : R + → R n be a given function, such that g(x) ∈ N (L) ⊥ for all x ∈ R + . The linearized inhomogeneous system with the boundary conditions (21) have (under the assumption that all necessary integrals exist) the general solution where and β 1 (x) , ..., β q (x) are given by Eq. (40). From the boundary conditions (24), we obtain the system The system (50) has (under the assumption that all necessary integrals exist) a solution if we assume that and a unique solution if and only if, additionally, condition (25) is fulfilled. 7. Extension to singular operators B. To study the case when the operator B is singular (i.e. the case when ξ 1 i = 0 for some i) we assume (cf. Refs. [29] and [7]) that and introduce the orthogonal projections P 0 : R n → N (B) and P 1 : R n → Im(B).
The assumption (52) ensures that the operator P 0 LP 0 is non-singular on N (B).
The system (47) is equivalent with the system (see Ref. [7]) where L = P 1 L(I − P 0 (P 0 LP 0 ) −1 P 0 L)P 1 , B = P 1 BP 1 and The restrictions, L Im and B Im , of L and B to Im(B), are linear operators ( n × n matrices, with n = n − dim(N (B))) on Im(B). Then the linear operators L Im and B Im on Im(B) have the following properties: L Im and B Im are real symmetric operators, L Im is semi-positive, B Im is non-singular, dim(N ( L Im )) = p, and the numbers k + , k − and l are the same for the system as for the original system (47). Furthermore, g(x) ∈ N ( L) ⊥ .
Proof. (cf. Ref. [7]) It is clear that the operators L and B are real and symmetric and that B is non-singular on Im(B). Hence, this is true also for the restrictions to Im(B). The linear operator L Im is semi-positive, since , and hence, for all h ∈ Im(B). By assumption (52), dim(N ( L)) = dim(N (L)) = p and N ( L) ⊆ P 1 N (L), since if Lh = L(P 0 + P 1 )h = 0, for h ∈ R n . Hence, Furthermore, the numbers k + , k − and l are the same for the system (53) as for the original system, since and so g(x), P 1 h = P 1 g(x), h + P 0 g(x), h = g(x), h = 0.

NON-LINEAR HALF-SPACE PROBLEMS 21
Denote where Θ is the operator (43) when L and S(f, f ) are replaced with L and S(f, f ) = and denote λ min = min |λ i | and λ max = max |λ i | , where λ 1 , ..., λ n−p are the non-zero eigenvalues of L. Then . We can now extend our main results in Section 3 to yield also for singular operators B.
8. Axially symmetric DVMs. In this section we consider only such symmetric sets of velocities V, such that for any combinations of signs (see also Ref. [7]). We can, without loss of generality, assume that .., ξ d i ) and ξ 1 i > 0, for i = 1, ..., N , with n = 2N .
Example 3. The plane 12-velocity model in Ref. [11], with velocities (±1, ±1) , (±1, ±3) and (±3, ±1) , and the infinitely many (obvious, from the constructions in Ref. [11] -"with three corners of a square in the model, add the fourth") symmetric normal extensions of this model are examples of (normal) such DVMs. These extensions include the plane square models, with (all combinations of) coordinates from the set of all odd integers with absolute values less or equal than a maximal odd integer (these models are called Nicodin p-th squares in Ref. [20], but are also, at least implicitly, constructed in Ref. [11]).
Under the assumptions (i)-(v) given above, the following theorem (see Ref. [26] for the case of the continuous Boltzmann Equation) follows by Theorem 3.2. Remark 9. The same problem, for d = 2, is also studied by Babovsky in Ref. [2], but then under the quite restrictive condition S(f, f ), w i = 0 for i = 1, 2 (in our notations).