Stationary solutions to the two-dimensional Broadwell model

Existence of renormalized solutions to the two-dimensional Broadwell model with given indata in L1 is proven. Averaging techniques from the continuous velocity case being unavailable when the velocities are discrete, the approach is based on direct L1-compactness arguments using the Kolmogorov-Riesz theorem.

The main result of the paper is the following.
Theorem 1.1 Given a non-negative boundary value f b with finite mass and entropy, there exists a stationary nonnegative renormalized solution in L 1 with finite entropy-dissipation to the Broadwell model (1.1).
Most mathematical results for discrete velocity models of the Boltzmann equation have been performed in one space dimension.An overview is given in [8].In two dimensions, special classes of solutions are given in [3] [4], and [9].[3] 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.
The existence of continuous solutions to the two-dimensional stationary Broadwell model with continuous boundary data for a rectangle, is proven in [6].That 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 theorem.
The present paper on the Broadwell model is set in a context of physically natural quantities.Mass and entropy flow at the boundary are given, and the solutions obtained, have finite mass and finite entropy dissipation.Averaging techniques from the continuous velocity case [7] being unavailable, a direct compactness approach is used, based on the Kolmogorov-Riesz theorem.The plan of the paper is the following.An approximation procedure for the construction of solutions to (1.1) is introduced in Section 2. The passage to the limita in the approximations is performed in Section 3.Here a compactness property of the approximated gain terms in mild form is carried over to the corresponding solutions themselves, using a particular sequence of successive alternating approximations and the Kolmogorov-Riesz theorem [10], [11].The approach also holds for domains which are strictly convex with C 1 boundary.A common approach to existence for stationary Boltzmann like equations is based on the regularizing properties of the gain term.In the continuous velocity case an averaging propery is available to keep this study of the gain term within a weak L 1 frame as in [2].However, in the discrete velocity case, averaging is not available.Instead strong convergence of an approximating sequence is here directly proved from the regularizing properties for the gain term (cf Lemma 3.5 below).But the technique in that proof is restricted to two dimensional velocities, whereas the averaging technique in the continuous velocity case is dimension independent.

Approximations.
Denote by L 1 + ([0, 1] 2 ) the set of non negative integrable functions on [0, 1] 2 , and by a∧b the minimum of two real numbers a and b.Approximations to (1.1) to be used in the proof of Theorem 1, are introduced in the following lemma.Lemma 2.1 For any k ∈ N * , there exists a solution The sequence of approximations (F k ) k∈N * is obtained in the limit of a further approximation with damping terms αF j and convolutions in the collision operator.
Step I. Approximations with damping and convolutions.Take α > 0 and set Let µ α be a smooth mollifier in (x, y) with support in the ball centered at the origin of radius α.
Let T be the map defined on K α by T (f ) = F , where F = (F i ) 1≤i≤4 is the solution of ) ) ) (2.11) (2.12) 2 ) of the sequence (F n ) n∈N defined by F 0 = 0 and The sequence (F n ) n∈N is monotone.Indeed, The same argument can be applied to prove that (2.17) It results from (2.16)-(2.17) that G = F .The map T is continuous in the L 1 -norm topology (cf [1] pages 124-5).Namely, let a sequence Because of the uniqueness of the solution to (2.8)-(2.13), it is enough to prove that there is a subsequence of (F l ) converging to F = T (f ).Now there is a subsequence of (f l ), still denoted (f l ), such that decreasingly (resp.increasingly) (G l ) = (sup m≥l f m ) (resp.(g l ) = (inf m≥l f m )) converges to f in L 1 .Let (S l ) (resp.(s l )) be the sequence of solutions to ) ) (2.20) (resp. (2.24) ) (2.26) (2.27) (2.28) ) is a non-increasing sequence, since that holds for the successive iterates defining the sequence.Then (S l ) decreasingly converges in L 1 to some S. Similarly (s l ) increasingly converges in L 1 to some s.The limits S and s satisfy (2.8)-(2.13).It follows by uniqueness that The boundedness by k 2 of the integrands in the r.h.s. of (2.8) and (2.10) induces uniform L 1equicontinuity of (F l1 ) l∈N (resp.(F l3 ) l∈N ) w.r.t. the x (resp.y) variable.Together with the L 1 -compactness of (f l * µ α ) l∈N , this implies uniform L 1 -equicontinuity w.r.t. the y variable of (H l1 ) l∈N , then of (F l1 ) l∈N .This proves the L 1 compactness of (F l1 ) l∈N .The L 1 compactness of (F li ) l∈N , 2 ≤ i ≤ 4 can be proven similarly.
Hence by the Schauder fixed point theorem there is a fixed point T (F ) = F , i.e. a solution F to (2.30) (2.31) (2.32) (2.33) (2.34) . Denote by F k a limit of a subsequence for the weak topology of By induction on l it holds that ) is translationnaly equicontinuous in the x-direction, since all integrands in its exponential form are bounded.It is translationnaly L 1 -equicontinuous in the y-direction by induction on l.Indeed, it is so for (F α 3 ) (resp.(F α 4 )) since ∂ y (e αy F α 3 ) ( resp.∂ y (e αy F α 4 )) is bounded by ek 2 , and ( is so for ( where G is the weak L 1 limit of ( ) α∈]0,1[ when α → 0. In particular, (g 2l 1 ) l∈N and (g 2l 2 ) l∈N (resp (g 2l+1 1 ) l∈N and (g 2l+1 2 ) l∈N ) non decreasingly (resp.non increasingly) converge in L 1 to some g 1 and g 2 (resp.h 1 and h 2 ) when l → +∞.The limits satisfy and The non negativity of h 1 − g 1 , g 1 , g 2 , h 1 and h 2 implies that h 1 − g 1 = 0.The same holds for h 2 − g 2 .Consequently , can be proven similarly.Passing to the limit when α → 0 in (2.30)-(2.35) is straightforward.And so, F k is a solution to (2.1)-(2.6).
3 Passage to the limit when k → +∞.
The study of the passage to the limit is split into six lemmas.In Lemma 3.1, uniform bounds are obtained for mass, entropy and entropy production term of the approximations.Lemma 3.2 splits [0, 1] 2 into 'large' sets of type 0 ≤ x ≤ 1 times a 'large' set in y for ( , where the approximations are uniformly bounded in L ∞ , and their complements where the mass of the approximations is small.Lemma 3.3 proves uniform equicontinuity with respect to the x (resp.y) variable of the two first (resp.last) components of the approximations.In Lemma 3.4, L 1 -compactness of a truncated gain term of the approximations is proven.Lemma 3.5 proves that the approximations form a Cauchy sequence in L 1 ([0, 1] 2 ).Their limit is proven to be a renormalized solution to the Broadwell model in Lemma 3.6 .In this section, c b denotes constant that only depend on the given boundary value f b .

Lemma 3.1
There are constants c b such that Proof of Lemma 3.1.Adding (2.1)-(2.4),integrating the resulting equation on [0, 1] 2 and taking (2.5)-(2.6)into account, implies that total outflow equals total inflow.Also using , resp.ln , resp.ln ), add the corresponding equations, and integrate the resulting equation on [0, 1] 2 .Denoting by D k the entropy production term for the approximation F k , Moreover, Hence Consequently, (0, y)dy And so, (3.3) holds.Moreover, for any Λ > 2 and k > 2, | (0, y)dy In particular, it holds that Consequently, for some subset ) and the boundedness of the f b2 entropy.

Lemma 3.3
There is c b > 0, and for ǫ > 0 given there is δ > 0 such that for |h| < δ, 1 in renormalized form (1.4) integrated on [x, x + h], where the integration from x + h > 1 tending to zero with h uniformly in k, is being omitted from the following computations; Denote by sgn the sign function, sgn(r) = 1 if r > 0, sgn(r) = −1 if r < 0. Multiply the previous equation by sgn ln(1 (X, y)dXdy (X, y)dXdy Recall that for any non negative real numbers x 1 > x 2 , there is θ ∈]0, 1[ such that And so the L 1 -norms of the translation differences of F k 1 and ln(1 There is also the small set with mass bounded by ǫ, where (x, y) → F k 1 (x + h, y) is not in Ω ǫΛ k1 .Together with (3.12) this proves the translational equicontinuity in the x-direction for k ≥ exp( 3c b ǫ ).The proof for as defined in Lemma 3.2, and take χ ǫΛ k1 as the corresponding cutoff function, Lemma 3.4 Let (α k ) k∈N be a non negative sequence bounded in L ∞ and compact in L 1 .The sequences and χ ǫΛ k1 (y) Proof of Lemma 3.4.For any γ > 1, using Lemmas 3.1-3.2, Choosing γ big enough, then h small enough, proves the translational L 1 equicontinuity in the x direction of χ ǫΛ k1 (y) x 0 the corresponding cutoff function, First, Moreover, , by the definition of ǫ 3 .
Proof of Lemma 3.6.
Start from a renormalized formulation of (2.1), for test functions ϕ ∈ (C 1 ([0, 1] 2 )) 4 .Using the strong L 1 convergence of the sequence (F k ) to pass to the limit when k → +∞ in the left hand side of (3.22), gives in the limit, For the passage to the limit when k → +∞ in the right hand side of (3.22), given η > 0 there is a subset A η of [0, 1] 2 with |A c η | < η, such that up to a subsequence, (F k ) k∈N * uniformly converges to F on A η and F ∈ L ∞ (A η ).Passing to the limit when k → +∞ on A η is straightforward.Moreover, lim η→0 A c η ϕ F 1 F 2 1 + F 1 (x, y)dxdy = 0 and lim (x, y)dxdy = 0, uniformly with respect to k, since ≤ 1, and lim The gain term can be estimated as follows.The uniform boundedness of the entropy production term of (F k ) is given in Lemma 3.1.A convexity argument together with the L 1 convergence of (F k ) to F (see [7]), imply that which tends to zero when η → 0. Similarly, using (3.3),This completes the proof of Theorem 2.1.

. 10 )
Proof of Lemma 3.3.The four cases F k 1 ,..., F k 4 are analogous.The detailed estimates are carried out for F k 1 .The translational L 1 equicontinuity in the x-direction for ln(1 + F k 1 ) is obtained as follows from the ∂ xterm in the renormalized equation.Consider h ∈ [0, 1[.Write the equation for F k

F 4 (F 2
x, y)dxdy ≤ c b .(3.23)It follows that, for any γ > 1, (x, y)dxdy, y)dxdy, which tends to zero when η → 0, uniformly in k.It follows that the right hand side of (3y)dxdy, when k → +∞.Consequently, F 1 satisfies the first equation of (1.1) in renormalized form.It can be similarly proven that (F j ) 2≤j≤4 is solution to the last equations of (1.1).
be reached in a similar way.Moreover,