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Statistical moments and integrability properties of monatomic gas mixtures with long range interactions

This document presents a priori estimates related to statistical moments and integrability properties for solutions of systems of monatomic gas mixtures modelled with the homogeneous Boltzmann equation with long range interactions for hard potentials. We detail the conditions for the generation and propagation of polynomial and exponential moments, and the integrability in Lebesgue spaces.


Introduction
In this paper we study a priori estimates for the homogeneous Boltzmann system describing multi-component monatomic gas mixtures for binary interactions in three dimension. We focus in long range and hard interactions using techniques that have been developed recently for the scalar equation combining recent result and notation presented in the works [18,11] for the case of cutoff angular kernels. The Cauchy problem will be addressed elsewhere.
The method of proof follows a classical approach where the generation and propagation of polynomial and exponential moments use, at its core, a Povzner inequality associated to the binary interaction of different monatomic species. The Povzner inequality for monatomic species is similar to the scalar version, yet less symmetric. This loss of symmetry poses important difficulties that modify the classical mathematical treatment. The mathematical analysis at the level of moments, polynomial and exponential, follows ideas from refined versions for the scalar case given in [18,22,15], however, it is worthwhile to mention that the moment analysis requires considering the system as a whole and not component by component independently. In other words, a global moment inequality has to be made for the system involving all species at once. This is particularly challenging for the case of exponential moment analysis which turns out to be subtle.
In a second stage, once the statistical moments are understood, a theory of generation and propagation of higher integrability norms is performed. We implement an approach given in [3] for the treatment of L p -norm generation in the range p ∈ (1, ∞]. Interestingly, for higher integrability generation and propagation, each species of the gas mixture can be considered independently by interpreting the rest of the components as given. This is related to the fact that the type of scattering considered in this document is only forward; in such a case only statistical moments are needed to prove higher Lebesgue integrability generation and propagation.
Let us mention here that regarding the homogeneous scalar equation, the study of moments have been addressed for the long range regime with hard and Maxwell potentials interactions in references such as [23,19,22,15,20]. The generation and propagation of higher integrability for the scalar problem have been addressed in [2,3,4,14] using methods initially presented in [1,2]. Boundedness of solutions for the homogeneous equation has been studied in [3,21,17] using different approaches and including polynomial and exponential weights, a probabilistic numerical method can be found in [16]. In addition, the gas mixture for monatomic components has been addressed in the so-called cutoff with hard interactions regime in references such as [9,18,11], the spatially inhomogeneous setting near thermal equilibrium is addressed in [10], the BGK approximation for multi-species can be found in [8], hydrodynamics expansions can be referred to [7], chemical reactions of multi-components is addressed in [13], fine mathematical properties of the linear multi-component model are addressed in [12].
The paper is organised in the following way: In section 2, we give some notations and preliminaries and state the main results of this manuscript. Then, in section 3, we prove Theorem 2.2, 2.3 and give some preliminary Lemma including Povzner inequality that we need to prove our first result. Section 4, is devoted to prove the uniform coercive estimate and our two last Theorem 2.5 and 2.6. Finally, in section 5, we give some general Lemma which help us to prove ours results. (2.2) [Q(F, F)] i := Here the collision Boltzmann operators Q ij (f i , f j ) measure the influence that the species f j exerts on the species f i . Consequently, for any i fixed the distribution function f i solves a Boltzmann equation where the collision operator takes into account the influence of all the species A i as 2.1.1. Collision process. Consider two interacting particles from the species A i with associated particle mass m i > 0 and A j with associated particle mass m j > 0. The shorthand ( ′ v ij , ′ v * ,ij ) denotes the particles velocities before collision. The dependence of such pre-collisional velocities with respect to the post-collisional velocities (v, v * ) is given by (2.4) Here σ ∈ S 2 is the scattering direction and u ± = u±|u|σ 2 . After collision, the species remain in the same species with the same associated mass. The collision laws (2.4) conserve momentum and kinetic energy, that is, the interactions are elastic. In this way one has that pre and post collisional velocities are interchangeable ( ′ v ij , ′ v * ,ij ) = (v ′ ij , v ′ * ,ij ) and the conservation laws m i |v ′ ij | 2 + m j |v ′ * ,ij | 2 = m i |v| 2 + m j |v * | 2 conservation of energy , (2.5) are valid. Now, note that in the collision laws (2.4) (or in the conservation laws (2.5)) one can divide the laws by I j=1 m j giving normalised component massesm i = mi I j=1 mj ∈ (0, 1), for each species 1 ≤ i ≤ I. For such reason, one can assume without loss of generality that the component masses add up to unity I j=1 m j = 1.
Remark 2.1. To simplify notation we eliminate the subindices from the shorthand (v ′ ij , v ′ * ,ij ) and simply write (v ′ , v ′ * ). Just keep in mind that the post and pre collisional relations will differ for each collision operators Q ij .

Collision operators.
The collision operator Q ij describing the binary interaction between the particles of species A i , denoted by f , and the particles of species A j , denoted by g, is explicitly defined by (2.6) Q where u = v − v * is the relative velocity between interacting particles, u = u |u| its direction, σ = u ′ |u ′ | the scattering direction, and B ij the collisional cross section. The B ij are positive functions satisfying the symmetric condition B ij = B ji and the following micro-reversibility assumption |u| . In this document we concentrate our efforts in hard potential interactions without angular cutoff. More precisely, the collision cross sections B ij , for 1 ≤ i, j ≤ I, are assumed to satisfy the explicit expression where b ij is the angular scattering kernel. Writing u · σ = cos θ with θ ∈ (0, π 2 ], the angular scattering satisfies the condition . The condition on the support of b implies that the regime we treat in this document is the so-called peaked forward scattering. Note that the symmetric condition implies that The following parameters, related to the minimum and the maximum value of the potential rates λ ij , will be important when considering the different properties of solutions, Additionally, we will consider the following parameters related to the angular singularities, Before continuing, let us mention that condition (2.9) is only needed in a vicinity of θ = 0. Away from this point b can be assumed just integrable. Additionally, let us remark that for the scalar Boltzmann equation the support of the angular scattering b(cos(θ)) can be assumed in the upper scattering sphere θ ∈ (0, π/2] with no loss of generality since v ′ ↔ v ′ * under the change σ ↔ −σ. This is not the case for monatomic gas mixtures due to lack of symmetry in the collision laws. We point out, however, that the arguments and results of Section 3 are all valid, without modification, with a kernel b(cos(θ)) defined in (0, π]. Morevover, from a physical perspective, the most generic assumption for the backscattering is b( e · σ)1 { e·σ≤0} ∈ L 1 σ (S 2 ). In other words, terms associated to the backscattering are of lower order in comparison to the forward and can be included in the analysis of Section 4 at a technical price, which may slightly impact some of the arguments in this section. For the interested reader, we refer to [11] for the nuances of the cutoff scattering analysis.
2.1.3. Functional spaces. First, we recall some notations and definitions that will be important along the manuscript. · The mixture's bracket form is defined as · The scalar k th -polynomial moment, with k ≥ 0, is defined for any 0 ≤ f ∈ L 1 k as For the gas mixture F = [f i ] 1≤i≤I we use the shorthand that is, the subindex i in the shorthand m k,i (t) prescribes the bracket and the associate component in F. The cumulative k th moment of the gas mixture is defined as The Banach L p n spaces associated to the mixture are defined as follow, refer to [11], For the special case p = +∞, the space is defined as The L log L space is defined as We also work with the Sobolev spaces endowed with the norm Thus, splitting the sum in the sets {i ≥ j} and {i < j} we can interchange indexes i ↔ j to obtain that Recalling that B ji = B ij and the micro reversibility condition (2.7), we can interchange variables (2.10) The weak form of Boltzmann collision operator imply the conservation properties of the system. In particular, which leads to conservation of mass for each single component and for the system as a whole Moreover, when we choose the test functions ψ i = m i v and ψ i = m i |v| 2 in the weak form (2.10) we obtain the conservation of momentum and energy for the system as a whole (but not for each single species) More precisely, if F is a solution of Boltzmann gas mixture system then conservation of mass per species ,

This latter yields
Also, we recall the gas mixture entropy and the gas mixture entropy production, given by It is possible to deduce from the weak form (2.10) that D[F] ≤ 0, refer to [13], which yields that Consequently, Note that . Therefore, by conservation of total mass, energy, and dissipation of entropy, it holds that . (2.12) Thus, we consider initial data satisfying that so that F L log L ≤ log(2)m 0 + H 0 + C m 3 4 2 < ∞ as argued in (2.12). In this way, solutions to the gas mixture will formally lie in the space In particular, we take initial data F 0 ∈ U (D 0 , E 0 ).

Main Results.
Recall that we address a priori estimates related to statistical moments and higher integrability for a system of spatially homogeneous Boltzmann equations describing a gas mixture of monatomic components A 1 , ..., A I , where the component A i has associated particle mass m i > 0 The i th entry of the vector F : was defined in (2.2). Also recall in the following that λ ij ∈ (0, 2] and s ij ∈ (0, 2) with 1 ≤ i, j ≤ I. Keep in mind that the following are a priori estimates, so we consider solutions F(t) to (4.12) with sufficient regularity so that all computations performed are allowed. Finally, in the results is explicit that constants depend of the parameters of the model, that is, on the λ ij , s ij , κ 1 ij , κ 2 ij , m i , and θ 2 b ij L 1 (S 2 ) . Such constant will depend on the mass, enegy and entropy D 0 and E 0 of the gas mixture initial configuration as well.
Proposition 2.2. Take λ ♮ > 0 and F 0 ∈ U (D 0 , E 0 ). For any integer n ≥ 2, the statistical moments satisfy the equation where, for each 1 ≤ i, j ≤ I, Furthermore, the following estimates hold for all t > 0.
(i) For any fixed 1 ≤ i, j ≤ I there exist positive constants C 1 ij and C 2 ij depending on the parameters of the model, D 0 and E 0 , such that the pair (f i (t), f j (t)) satisfies (ii) In addition, the system F(t) satisfies the inequality The positive constants C 1 and C 2 have the aforementioned dependence.
Based on Proposition 2.2 item (ii) one can prove the following results about higher algebraic/exponential tail integrability. Theorem 2.3 (Polynomial Moments). Take λ ♮ > 0 and F 0 ∈ U (D 0 , E 0 ). Then, for any r > 2 there exist positive constants C 1 and C 2 depending on r, the parameters of the model, D 0 and E 0 , such that Furthermore, in the particular case of even integers r = 2n ≥ 4 if m 2n (0) < ∞ then Using the two aforementioned results, we deduce the following theorem.
There exist constants T > 0 and σ > 0 depending only on the parameters of the model, D 0 and E 0 , such that (ii) For any A > 0, σ 0 > 0 and ρ ∈ (0, 2], there exists a (computable) constant σ > 0 depending on the parameters of the model, D 0 , E 0 , ρ, σ 0 , and A, such that if Once statistical moments are understood for the system, it is possible to obtain higher integrability generation and propagation as stated in the following results. We remark that the results now refer to individualised components. Theorem 2.5 (Higher integrability). Fix 1 ≤ i ≤ I, and let λ ♮ > 0 and s i > 0. Assume also that F 0 ∈ U (D 0 , E 0 ). Then, it holds for any p ∈ (1, ∞) and t 0 > 0 that The constant C t0 t −β 0 + 1 (for some β > 0) depends only on t 0 , the parameters of the model, D 0 and E 0 . Furthermore, if f i,0 ∈ L 1 2n ∩ L p with sufficiently large number of moments n ≥ 2, then (2. 16) sup The positive constant C depends, in addition, on the L 1 2n norm of f i,0 . The number of moments can be taken as 2n ≥ r s := 3λi si . In light of Theorem 2.5, one has the following result as a consequence.
where the constant C t0 t −β 0 + 1 (for some β > 0) depends only on t 0 , the parameters of the model, D 0 and E 0 .
Furthermore, if f i,0 ∈ L 1 2n ∩ L 2 with sufficiently large number of moments 2n ≥ 3λi si , then where C > 0 depends, in addition, on the L 1 2n norm of f i,0 .
The case p = +∞ is treated in our last result.
Theorem 2.7 (Boundedness). Fix 1 ≤ i ≤ I, and let λ ♮ > 0 and s i > 0. Assume also that F 0 ∈ U (D 0 , E 0 ). Then, given t 0 > 0 there exists a constant C t0 > 0 depending on the parameters of the model, D 0 , E 0 , and t 0 , such that The constant is controlled as C t0 for a positive constant C depending additionally on the L 1 2n norm of f i,0 .
Let us remark that the fact that we can infer properties of higher integrability/regularity emergence and propagation for individualised components in Theorem 2.5, Corollary 2.6, and Theorem 2.7 is a consequence of the forward scattering hypothesis in the b ij . As previously mentioned, including the backscattering in the interactions adds lower order terms that can be managed at a technical price. We refer to [11] for a detailed analysis on the handling of "cutoff" terms.

Generation and propagation of statistical moments
In this section, the proof are inspired from classical approaches, in particular we extend the Povzner lemma given in [15] to this setting of system of equations.
We proceed in two steps.
Step 1. We show that there exists a constant λ 1 ij > 0 such that We remark that the integral in a n is nonnegative because for X = Then, using (2.9) we get that Furthermore, for all θ ∈ [0, n − 1 2 ] we have that which yields whence, recalling that n ≥ 2, Consequently, Step 2. We estimate b n . We start by recalling that Let C n = {(p, q, k, l) ∈ A n ; l ∈ 2N}, we compute that This can be rewritten as where, setting A n,a = (p, q, k, l) ∈ C n ; p + k 2 + l 2 = a (whence q + k 2 + l 2 = n − a) , one has that K n,a = (p,q,k,l)∈An,a Fix a ∈ [1, n − 1], then (p, q, k, l) ∈ A n,a if and only if there exist r, s ∈ N such that k = 2r, l = 2s, p = a − r − s, and q = n − a − r − s. Consequently, Using the fact that Thus, From the definition of J n,a one has that Let us show that L sij ,n,a is estimated in terms of K 0,n,n−a . Indeed, using the substitution θ → π−θ we see that Consequently, we deduce that Now, we estimate K sij ,n,a . For θ ∈ (0, π/2] we have that θ ≤ 2 sin θ and where we used the change of variables x = m 2 i +m 2 j +2mimj cos θ (mi+mj ) 2 in the latter. Hence where Γ(·) is Euler's Gamma function. Consequently, This concludes the proof.

3.2.
Generation and Propagation of polynomial moments. The purpose of Proposition 2.2 is to derive from the Boltzmann system (2.14) an ordinary differential inequality for polynomial moment of order 2n, with n ≥ 2, based on the Povzner estimate from Lemma 3.1 to implement statistical moment analysis. In particular, the key idea in regard of exponential tails generation and propagation is to realise that the mixture system's control inequality is obtained by individualising the ij-scalar components satisfying the Povzner lemma, Lemma 3.1, specifically described in Proposition 2.2 item (i).
Proof of Proposition 2.2. Let n ≥ 2 and 1 ≤ i, j ≤ I, and recall that Consequently, a direct consequence of the weak formulation of the Boltzmann mixture equation n a Let us begin proving the first result by proceeding in two steps.
Step 1. Finding a lower bound for A n . Since F ∈ U (D 0 , E 0 ), one can use a slight modification of the proof of [6, Lemma 9] to obtain an explicit constant c := c(D 0 , E 0 ) > 0 such that

Consequently,
Step 2. Finding a convenient upper bound for B ij n . Using that, see for example [18], we get that where we performed the change of index a → n − a in the intermediate step, and the fact that 1 a ≤ n s ij /2 a s ij /2+1 for a ≤ n. Furthermore, the terms in parenthesis are symmetric with respect to a, that is m 2a+λij ,i (t)m 2(n−a),j (t) for a = 1, 2, 3, · · · equals m 2a,j (t)m 2(n−a)+λij ,i (t) for a = n − 1, n − 2, n − 3, · · · .
And similarly for the second group of moments. Note, in addition, that the first half of weights controls the second half because a → n a is decreasing. Hence, a=1 n a n sij /2 a sij /2+1 m 2a+λij ,i (t)m 2(n−a),j (t) Now, since n − a ≥ a for 1 ≤ a ≤ ⌊n/2⌋ we can invoke the Lemma 5.1 with b = 2(n − a) and β = λ ij , which is a generalisation of the classical lemma [15,Lemma 7], to conclude that resulting in the estimation Consequently, where the constants can be taken as This is precisely statement (i).
Now, statement (ii) follow from (i) after some observations. First, for the coercive term it readily follows that The upper estimate related to S ij n follows after some moment interpolations. Note that (since n/a ≥ 1) a=1 n a n s/2 a s/2+1 m 2a (t)m 2(n−a)+λ (t) , Furthermore, 2 < 2(n − a) + λ < 2n (recalling that n ≥ 2), therefore, the following interpolations hold Consequently, where in the latter we used Young's inequality. Consequently, which proves item (ii) of the proposition.

Proof of Theorem 2.3. Use the interpolation
Thus, a comparison principle for order, see [6,Lemma 18], gives that the super solution m * 2n (t) controls the 2n th -moment . For a general r > 2, consider the minimal integer n ≥ 2 such that 2 < r ≤ 2n, then Thus, we conclude that This concludes the proof of the result. To this end we use a methodology reminiscent of [15,5]. The proof relies on generation and propagation of polynomial moments stated in Theorem 2.3 and the following lemma. (i) (System) For σ 0 ∈ (0, ∞), α ∈ [1, ∞) and K ∈ [1, ∞), we have

Exponential moments. Let us now focus in the generation and propagation of exponential moments result of Theorem 2.4. Our goal is to show that the partial sums
(ii) (Component) Let ρ ∈ (0, 2], σ 0 ∈ (0, 1] and K ≥ 1, there exists σ 1 := σ 1 (ρ, σ 0 , K) such that Proof. We start with (i) noticing that by Holder's inequality (recall that α ≥ 1), Then, Regarding to (ii), note that For n ≥ 1 we set k n = ⌈ 2n ρ ⌉ ∈ [ 2n ρ , 2n ρ + 1). Using that ρ k n ≥ 2n we obtain that and consequently, Invoking Stirling's formula n! ∼ √ 2πn(n/e) n and since k n = ⌈ 2n ρ ⌉ we conclude that for some constant A ∈ (0, ∞) depending only on ρ it holds that We refer to [15] for additional details. Moreover, since σ 0 ∈ (0, 1] then σ kn 0 ≥ σ 2n ρ +1 0 , and consequently after choosing, for example, 3.3.1. Generation of exponential moments. First we introduce for σ ∈ (0, 1] and α > 0, to be chosen later sufficiently small and large respectively, p ≥ 2 and t > 0 Additionally, we define the objects, for 1 ≤ i, j ≤ I, We start by time differentiation of E p [F](t) and then using Proposition 2.2 item (i). Since (3.6) The following lemma is crucial as it expresses the convolution structure of the moments in the summation. This is where the nonlinear part is estimated.
For 1 ≤ a ≤ ⌊n/2⌋ one has that n sij /2 ≤ 2(n − a) sij /2 . Therefore, with obvious definitions for each term. Now, note that interchanging the summation it hols that where Setting N ǫ = ⌈2 ǫ −1 ⌉, it holds that for a ≥ N ǫ implies 2 a −1 ≤ ǫ. Hence, Using the first result of Theorem 2.3, we deduce that there are explicit constants C and C ǫ such that Thus, Similarly, we also have that In this way, adding (3.7) and (3.8), it holds that Furthermore, for t ∈ (0, 1] it holds that where B ǫ := 2A ǫ + 2 ǫ C , and D ǫ : This concludes the proof.
, and F 0 ∈ U (D 0 , E 0 ). There are explicit constants K and L, depending on the parameters of the model, E 0 , and D 0 , such that for any σ ∈ (0, 1] and p ≥ 2, it holds that Proof. First observe that for κ ≥ 1, to be chosen later sufficiently large, Thus, overall we get that Interpolation gives that
We can finally proceed with the proof of exponential moment generation.
Proof of the generation result of Theorem 2.4. We devise the proof in two steps.
Step 1. We show first that for λ ♮ > 0, s ♮ > 0 and α = max 1, 2−s ♮ λ ♮ , there exist σ ∈ (0, 1] and T ∈ (0, 1], depending only on the parameters of the model such that Invoking Theorem 2.3 it holds that for some constant C p ∈ (0, ∞), Now, using identity (3.6) and lemmas 3.3 and 3.4, we get that λ ♮ IL, and Choosing ǫ := min 1≤i,j≤I , and σ 0 := min 1≤i,j≤I we get for any σ ≤ min{1, σ 0 } Similar, for σ ≤ σ 1 := min 1≤i,j≤I , we obtain that where the latter inequality is a direct consequence of the definitions of F i Now, let us introduce the time This time is indeed positive by continuity of the polynomial moments (we pointed out previously that lim t→0 E p [F](t) = m 0 ). Thus, for all t ∈ (0, T p ], we have that Then, we comclude that for all t ∈ (0, T p ] it holds that Recalling that E p [F](0) = m 0 , we deduce directly from the definition of T p that Thus, for σ ≤ min{1, σ 0 , σ 1 }, Step 2. Let us proceed with the concluding argument. To this end, fix the rate and observe that α = max 1, 2−s ♮ λ ♮ = 2 ρ . Using the result of Step 1, we obtain that Then, Lemma 3.2 item (i) with σ 0 = (σt) 2/λ ♮ and K = 2m 0 implies that which is precisely the statement of the exponential moment generation.

3.3.2.
Propagation of exponential moment. The argument to propagate exponential moments follows a similar path to that of propagation; we present the main steps for completeness. Let us introduce for σ ∈ (0, 1], to be chosen sufficiently small, and p ≥ 2, Analogously, for all 1 ≤ i, j ≤ I, we denote Note that the analogous to (3.6) in this context is given by As before, we have the key estimate on the convolution structure of moments.

Lebesgue integrability generation and propagation
After some technical generalization of a coercivity estimate for the collision operator originated in [1] for Maxwell Molecules and refined in [2] for other potentials, it is possible to implement an energy estimate method, see [14,3], to prove generation and propagation of higher Lebesgue integrability.

4.1.
A uniform coercive estimate. Recall that the dynamic of the gas mixture satisfies F(t) ∈ U (D 0 , E 0 ). Let us make an important observation related to such space U It follows from the compactness of U (D 0 , E 0 ) in L 1 that there exist positive constantsR andr depending only on D 0 and E 0 (independent of v 0 ) such that where χ A denotes a characteristic function of the set A ⊂ R 3 . Of course one can takeR > 1 >r with no loss of generality.
Proposition 4.1. Fix 1 ≤ i ≤ I and assume that s ij ∈ (0, 2) and λ ij ∈ (0, 2] for all 1 ≤ j ≤ I. Assume also that D 0 , E 0 > 0, and G = [g j ] 1≤j≤I ∈ U (D 0 , E 0 ). Then there exist positive constants c i , C i depending only on the parameters of the model, D 0 and E 0 , such that Proof. Compute with a suitable f Using the identity where the terms are given by and Step 1. This step is based on the scalar case given in [2]. Let us estimate I 1 (g j , f ) in a region of high velocities |v * | ≤R ≪ |v|. ChooseR andr such that (4.1) holds. Let φR be a non-negative smooth function not greater than one, which is 1 for {|v| ≥ 4R} and 0 for {|v| ≤ 2R}. In view of we have that Observe that where the latter follow after using that u + · u − = 0 and |u| 2 = |u + | 2 + |u − | 2 in the definition of v ′ . Using the mean value theorem there exists τ ∈ (0, 1) such that the inequalities hold In the second inequality we used the uniform equivalences in (4.2). Since , Now we estimate I 1 in a region where v * and v have comparable sizes, yet, separated. For the set B(4R) we take the finite covering For each A k choose a non-negative smooth function φ A k which is 1 on A k and 0 on |v − v k | ≥r 2 . Note thatr Then, recalling that m i < 1 andR ≥ 1, Again, using the mean value theorem it holds for |v * | ≤R that Consequently, we obtain in a similar fashion as before that 3 is the number of sets in the covering. Consequently, where we used (4.3) for the first term and (4.4) for the N remaining.
Step 2. Let us estimate I 2 (g j , f ). To this purpose use the Cancellation Lemma 5.1, specifically Remark 5.3, that gives In addition, recall that This proves the proposition with c i := The following corollary of Proposition 4.1 extends the L 2 frame to L p .
Proof. This arguments is based on [3, Lemma 1]. Using the weak formulation, it holds for any Using that, with notation p ′ = p p−1 , we conclude that For the term J 1 (g j , f ) we use the Cancellation Lemma 5.1 to obtain that where recall that the scattering b i was defined as Consequently, Given that λ ij ≤ 2, we can take Therefore, it holds that where Using the coercive estimate given in the proof of Proposition 4.1 one deduces that Furthermore, using the Cancellation Lemma 5.1 for the terms J 1 and J 2 , it follows that The result follows adding the estimates.
Proof of Theorem 2.7. We follow the spirit of the argument used in the proof of [3, Theorem 2] after fixing 1 ≤ i ≤ I. For solution of the Boltzmann system we apply Lemma 4.3 with G = F and f = f i to conclude that (4.12) 1 2 Let us introduce the levels and times (K > 0 and t * > 0) Here K > 0 will be chosen later sufficiently large. Define also the energy functional (with notation for a constantC depending on (E 0 , ξ) (so the limit ξ → 2 is not allowed) and α 0 := The result follows from here associating the common powers and applying Young's inequality.

Cancellation Lemma.
We show in this section the cancellation Lemma of Boltzmann equation for monatomic gas mixtures. All results are proven assuming a integrable approximation of the scattering kernel, so that the collision operator can be separated, and then arguing by density. For the classical Boltzmann equation the reader can consult [1] where technical details are filled.
In all the results below we assume the scattering is forward, that is, B ij has support in the set { u · σ ≥ 0}.

5.3.
Coercive estimate of the Dirichlet form. In this subsection we explore a coercive estimate for Dirichlet form of the collision operator for monatomic gas mixtures. The argument is inspired in the classical case developed in [1] and is based in a series of lemmata. Recall the weak formulation for a suitable functions f , g, and ϕ(v), Lemma 5.4 (Bobylev identity). Let f ∈ L 2 (R 3 ) and g ∈ L 1 (R 3 ). For it holds that where ξ = ξ + ij + ξ − ij . More explicitly, Proof. Plugging ϕ(v) = e −iv·ξ in the weak formulation (5.3), we get that Note that, a key remark by Bobylev, Thus, The result follows from here.
Proof. Expanding the quadratic term gives three terms, namely, We begin with the middle term. By the pre-post collisional change of variables and Parseval's identity, Furthermore, using Bobylev's identity Lemma 5.4 in the right-side it holds that The right-side is equal to its complex conjugate since the left-side is real valued. Now, we note that S 2 b ij ( u · σ)dσ does not depend on the unit vector u ∈ S 2 , consequently it follows that where we have applied the usual Plancherel identity. Finally, for the term involving f (v ′ ) 2 we first make the change of variables (v, v * ) → (v − v * , v * ), and then v → v + ij (and rename as v) as in Cancellation lemma to obtain that which achieve the proof of Lemma 5.5 after adding the three respective terms.
Corollary 5.6. Let f ∈ L 2 (R 3 ) and g ∈ L 1 (R 3 ) with g ≥ 0. Then, Proof. In Lemma 5.5 use Cauchy-Schwarz inequality to control the latter two terms as The result follows after observing that for g ≥ 0 one has that F (g)(0) − F (g)(ξ − ij ) ≥ 0. Lemma 5.7. Suppose that b ij satisfies assumption (2.8) and (2.9). Then, there exists a constant K ij > 0 such that The constant can be taken as K ij := K ij (g) = mj mi+mj 2 1 2−sij κ 1 ij C g |S| with the constant C g > 0 depending only on g L 1 1 and g L log L . Proof. Using [1, Lemma 3], there exists a constant C g depending on g L 1 1 and g L log L such that F (g)(0) − |F (g)(ξ − ij )| ≥ C g |ξ − ij | 2 ∧ 1 ≥ 0 . Then,