ON THE MULTI-SPECIES BOLTZMANN EQUATION WITH UNCERTAINTY AND ITS STOCHASTIC GALERKIN APPROXIMATION

In this paper the nonlinear multi-species Boltzmann equation with random uncertainty coming from the initial data and collision kernel is studied. Well-posedness and long-time behavior – exponential decay to the global equilibrium – of the analytical solution, and spectral gap estimate for the corresponding linearized gPC-based stochastic Galerkin system are obtained, by using and extending the analytical tools provided in [M. Briant and E.S. Daus, Arch. Ration. Mech. Anal. 3 (2016) 1367–1443] for the deterministic problem in the perturbative regime, and in [E.S. Daus, S. Jin and L. Liu, Kinet. Relat. Models 12 (2019) 909–922] for the single-species problem with uncertainty. The well-posedness result of the sensitivity system presented here has not been obtained so far neither in the single species case nor in the multi-species case. Mathematics Subject Classification. 35Q20 and 65M70. Received April 6, 2020. Accepted May 6, 2021.

the solution in the long time [31]. To our knowledge, uncertainty quantification (UQ) for any nonlinear multispecies kinetic model has not been studied so far, while general single-species linear and non-linear collisional kinetic problems with multiple scales and uncertainty were studied in [27].
Research on uncertainty quantification for kinetic equations has not started until recently, and the reason for the growing interest in these problems is the following. Kinetic equations, derived from -body Newton's equations via the mean-field limit [3], typically contain an integral operator modeling interactions between particles. Since calculating the collision kernel from first principles is impossible for complex particle systems, only empirical formulas are used for general particles [9]. Consequently, this inevitably brings modeling errors, so the collision kernel contains some uncertainty. Other sources of uncertainties may come from inaccurate measurements of the initial or boundary data, forcing or source terms. We refer to the book [24] and the recent articles and reviews [13,15,[21][22][23][25][26][27][28] for more detailed studies in this direction.
The main goal of this paper is to study the well-posedness and long-time behavior of the nonlinear multispecies Boltzmann equation under the impact of random uncertainty and its stochastic Galerkin approximation in the perturbative regime. The first part of our paper (Sect. 3) studies the well-posedness and exponential decay of the solution with random initial data and collision kernel in suitable Sobolev spaces in the perturbative setting, in which the initial data is assumed to be close to the global equilibrium. Our proof is based on the analysis of the Cauchy theory of the multi-species Boltzmann equation with uncertainty in the weighted Lebesgue space 1 ∞ (⟨ ⟩ ) ∞ (see (3.7) for the precise definitions) with a polynomial weight of order > 0 (where 0 is the threshold derived in Section 6 of [8], which recovers in the particular case of a multi-species hard spheres mixture (with equal molar masses) the optimal threshold of finite energy 0 = 2 obtained in the single-species setting in [19]).
The additional difficulty in our framework with uncertainty compared to the deterministic setting is to handle the extra high-order derivatives in the random parameter , which naturally appear from the fact that we introduce uncertainty into the model. We refer to the equations obtained by taking the -derivatives of the th component of the density functions governed by the multispecies Boltzmann equation as the sensitivity equations. We manage to control these new terms containing high-order -derivatives by designing a new decomposition built upon the factorization of Gualdani et al. in [19], with a mathematical induction in the order of -derivatives. This factorization technique was established by Gualdani et al. in [19], later adapted to the nonlinear perturbative setting in [7], and generalized to the multi-species deterministic framework with different molar masses in [8]. For more details on the factorization method see Section 3.1. We want to emphasize that there has not been established any rigorous existence analysis for uncertain kinetic equations in any previous work [10,11,23,26,27] yet, even not for the single-species case.
Concerning the task of numerically solving kinetic equations with uncertainties, one of the standard and efficient numerical methods is the generalized polynomial chaos approach in the stochastic Galerkin (referred to as gPC-SG) framework [17,20,32]. Compared to the classical Monte Carlo method, the gPC-SG approach enjoys a spectral accuracy in the random space-if the solution is sufficiently smooth-while the Monte Carlo method converges with the rate of (1/ √ ), where is the number of simulations. Note that the smoothness of the solution in the random space is one motivation for us to use the SG method. However, other types of non-intrusive methods, such as the stochastic collocation method, could also work well especially for highdimensional problems, but for us it seemed to be mathematically more interesting to study the sensitiveness of the Galerkin system and its convergence.
The second part of our paper (Sect. 4) obtains the spectral gap estimate for the linearized gPC-Galerkin system. Compared to [11] on the single-species gPC-SG Boltzmann system, the generalization to the multispecies case here can be done by adapting techniques from the proof for the multi-species H-theorem, see for instance [12,14]. Establishing this spectral estimate is essential in order to understanding the long-time behavior of the gPC-SG approximation.
We remark that our work relies on several existing literature on UQ for general kinetic models [24], sensitivity analysis [27], spectral convergence of the gPC-Galerkin method [11] and multi-species Boltzmann equations [8]. Readers may refer to those work for a more detailed overview.
The paper is organized as the following. In Section 2, we introduce the multispecies Boltzmann equation with uncertainty and present the assumptions for the two main results of this paper. In Section 3, we show the existence and uniqueness of the sensitivity equations in the perturbative setting and establish the exponential decay of each order -derivative of the solution. In Section 4, we extend the previous work [11,27] to the multi-species setting and obtain the spectral gap for the linearized gPC-SG system. Finally, we formulate our conclusions in Section 5.

The multispecies Boltzmann equations with uncertainty
The evolution of a dilute ideal gas composed of ≥ 2 different species of chemically non-interacting monoatomic particles with same molar particle masses can be modeled by the following system of Boltzmann equations (see [6,8,12] for the deterministic case), with some uncertainty characterized by a random variable ∈ , coming from both the initial data and the collision kernels, ) is the distribution function of the system, with (1 ≤ ≤ ) describing the distribution function of the th species. The spatial domain T 3 is the three-dimensional torus. For the sake of simplicity of the presentation, compared to [8], we set all the molar masses to be equal, e.g., = 1, for = 1, · · · , . The right-hand side of the kinetic Equation (2.1) is the th component of the nonlinear collision operator Q(F) = ( 1 (F), · · · , (F)), and is defined by where models interactions between particles of species and (1 ≤ , ≤ ), where we used the shorthands ′ = ( ′ ), = ( ), * = ( ′ * ) and * = ( * ). The velocities before and after the collisions are described by the following relation: which follows from the fact that we assume the collisions to be elastic, i.e., the momentum and kinetic energy are conserved on the microscopic level: Here the collision kernel depends on the relative velocity | − * |, the cosine of the deviation angle , and the random variable ∈ ⊆ R. For simplicity, we consider a one-dimensional random space, but our analysis can be easily extended to higher dimensional cases as well.
The global equilibrium, which is the unique stationary solution to (2.1), is given by ∞ = ( ∞ 1 , · · · , ∞ ), with where for 1 ≤ ≤ , By translating and scaling the coordinate system, one can assume ∞ = 0 and B ∞ = 1, and then the global equilibrium becomes

Main assumptions on the random collision kernel
We summarize here the assumptions on the random collision kernel that are needed throughout the whole paper: (H1) The following symmetry holds for each ∈ ⊆ R: The collision kernels for each ∈ ⊆ R are decomposed into the product where the functions Φ ≥ 0 are called the kinetic part and the angular part (cos , ) > 0 is assumed to be uncertain. (H3) We consider the case of hard potentials ∈ (0, 1] or Maxwellian molecules ( = 0), and thus the kinetic part takes the form: (H4) For the angular part, for each ∈ ⊆ R we assume a strong form of Grad's angular cutoff, i.e., ∃ , (H5) In addition, we assume the following condition on | | for all : where ∈ N is determined by the regularity of the random initial data, and is the same upper bound as in (2.6).
In (H1)-(H4), for each fixed the same conditions are assumed as in the deterministic problem [8]. The new assumption appears in (H5). We mention that our analysis in this work also applies to the case when the kinetic part Φ of the collision kernel is assumed uncertain, i.e., takes the form: (| − * |, cos , ) = Φ (| − * |, ) (cos ).

State of the art on the multi-species deterministic Boltzmann equation
As already mentioned above, the main difficulty of the deterministic multi-species Boltzmann equation compared to the single-species Boltzmann equation lies in the different conserved quantities: namely, the mass of each species is conserved, while for the momentum and kinetic energy only the sum of all the species is conserved, see [14,18]. Because of this, the proof of an explicit spectral-gap estimate of the linearized single-species operator [30] had to be changed significantly in the multi-species framework in [12] by carefully exploiting these new collision invariants. The stability of this spectral-gap estimate around non-equilibrium Maxwellian distributions was studied in [2]. The full Cauchy theory for the inhomogeneous Boltzmann equation for mixtures in the perturbative regime was formulated without going to any higher order Sobolev regularity [8], by using the factorization method of [19]. Besides this, in [8] a new multi-species Carleman's representation and a new Povzner-type inequality was proved, due to the loss of symmetry arisen from different masses. In [4,5], compactness of one part of the linearized multi-species operator was studied, moreover, in [3] it was shown that in the diffusive limit, the multi-species Boltzmann equation converges to the Maxwell-Stefan system. In [1], the Chapman-Enskog asymptotics for a mixture of gases was presented.
Finally, we also want to mention the very recent work [16] on the homogeneous multi-species Boltzmann system, for which it seems to be rather hard to conduct the sensitivity analysis and study the long-time behavior in the UQ setting, since the logarithmic entropy functional cannot be evaluated for the -derivatives of the distribution function, due to their lack of positivity.

Existence and exponential decay of the solution to the sensitivity system
This section will discuss the existence of a solution and the exponential decay to global equilibrium of the multi-species Boltzmann equation in the perturbative setting with random initial data and collision kernel. In the following, we will introduce the same notation and we will use similar techniques as in [8], where the Cauchy theory for the (deterministic) multi-species Boltzmann system was studied. Using the ansatz ( , , , ) = ( ) + ( , , , ), (3.1) the equation for f = ( 1 , · · · , ) satisfying the perturbed multi-species Boltzmann equation reads as where L = ( 1 , · · · , ) is the linearized Boltzmann collision operator with its th (1 ≤ ≤ ) component given by 3), and the nonlinear Boltzmann collision operator Q = ( 1 , · · · , ) is defined in (2.2) and (2.3).

Presentation and discussion of the main result
The proof of the main result of Section 3 uses techniques of Section 6 from [8] which rely on the idea of a nonlinear version of the factorization method of [19] presented in [7].
We first briefly recall some propositions in [8] to prepare us for the analysis. Define the truncation function and its support included in the set where ∈ (0, 1) is to be chosen. Define the splitting of the linear operator G = ( 1 , · · · , , · · · , ) as is a multiplicative operator called collision frequency, which also depends on the random variable : The results in [8] have shown that A ( ) has some regularizing effects and that is hypodissipative. Notice that The notation Π G is the orthogonal projection onto Ker(G) in 2 , (M −1/2 ). Recall the shorthand notation and the function spaces that we will use: The following theorem, which is our main result of Section 3, gives the existence, Sobolev regularity and long-time behavior of the solution in the random space.

8)
such that for all , As a consequence, f satisfies for all , where the constant depends on the initial data of f 0 for = 0, · · · , .
Since we need the following Lemmas given in [8] in the proof for the main Theorem 3.1, we paraphrase them below. For each fixed ∈ , Lemmas 3.2, 3.3 and Lemma 3.4 are the same as Lemma 6.2, 6.3 and 6.6 of [8], respectively. .
Then for all f , g such that̃︀ (f , g) is well-defined, the latter belongs to [Ker(L)] ⊥ , and ∃ > 0 such that ∀1 ≤ ≤ and each f and g, The strategy of the proof is to introduce a new adaptation of the factorization method of Gualdani et al. [19] to our probabilistic setting studied in this paper. The core idea is to decompose the full linear operator G (defined in (3.3)) into the hypodissipative operator A ( ) (see (3.6)) and the regularizing operator G 1 ( ) (see (3.6)), and to decompose the sensitivity system (3.8) into a system of equations, such that the hypodissipative and regularizing effects of the operators can be used to obtain the result of Theorem 3.1.
The additional challenge here in our framework with uncertainty compared to the deterministic results in [7,8,19] is to find a way of handling the extra high-order derivatives in the random parameter , which naturally appear from the fact that we introduce uncertainty into the model. Thus, the main difference and new challenge in our work compared to all the previous works on the deterministic problem is that a new decomposition, denoted by g = g 1 + g 2 , for each order -derivative of the distribution function has to be introduced. One needs to carefully design this new decomposition into the coupled system for g 1 , g 2 (see Eqs. (3.15)-(3.16)) such that the hypodissipative and regularising properties for the new operators (see the definitions for ( ) and ( ) in Eq. (3.11)) can be proved and used in a similar way as in the deterministic problems. Finally, a suitable induction in the order of -derivatives needs to be applied. Compared to the previous work on the sensitivity analysis for a class of (single-species) collisional kinetic equations with multiple scales and random inputs [27], we want to highlight the following differences in this work: First, here we conduct the sensitivity analysis for the multi-species Boltzmann system, while [27] studied a class of single-species kinetic equations, including the Boltzmann equation with random initial data and collision kernel. Second, here we rigorously prove the existence of solutions to the sensitivity equations, and its exponential decay to the equilibrium in the norm || · || 1 ∞ (⟨ ⟩ ) ∞ .

The proof of Theorem 3.1
We shall prove Theorem 3.1 by induction. The deterministic case of = 0 is shown in [8]. Now assume that Proposition 3.1 holds for all 0 ≤ ≤ − 1 with ≥ 1, we shall prove that the result holds for = .
First, one needs to calculate (f ) and (f ). Denote ( ) Compared with ( ) , ( ) shown in (3.4), the only difference in ( ) , ( ) is that one replaces the angular part of the kernel to be here instead of . The -order -derivative of the G operator is given by Then the -order -derivative of the collision operator is Combine (3.8), (3.12) and (3.13), then g := f satisfies for each the equation (3.14) Decomposition: In the form of g = g 1 + g 2 with g 1 ∈ 1 ∞ (⟨ ⟩ ) and g 2 ∈ ∞ , (⟨ ⟩ −1/2 ), then (g 1 , g 2 ) satisfy the following system of equations The above decomposition of the solution g = g 1 + g 2 follows [8], which also adopted the idea in [19] for the single-species Boltzmann equation. Compared to the deterministic case studied in [8], the differences here are the last three terms on the right-hand-side of (3.15), which appear due to the uncertainty dependence, and the last term on the right-hand-side of (3.16). They need to be grouped properly in the equation for g 1 or g 2 . First, we show a simple Lemma: Also, we have the estimate for 0 ≤ ℓ ≤ , The proof is given in Appendix A. By Lemma 3.4, (3.17) implies that In "Term ○ ⋆", the second term is exactly the left-hand-side of (3.17). By using the assumption (2.7) and Lemma 3.5, the first term is estimated by Thus "Term ○ ⋆ " can be bounded by Another thing we would like to mention before starting the main steps of the proof: Proposition 6.1 and at the end of Section 6.1.2 from ref. [8] shows that the solution f is small in the following sense, and one can assume that ∫︁ where 1 , 2 are constants depending on the initial data ||f 0 || 1 ∞ (⟨ ⟩ ) and an exponential decay factor − .
Step (ii): existence. Let g 1 (0) = 0 and consider the following iteration on equation (3.23) with ∈ N: with the initial data g 1 ( +1) (0, , , ) = g 0 . We omit including the superscript in B ( ) here. Note that in (3.25), the last two terms on the right-hand-side do not involve the time iteration index of the scheme. Our goal is to show that (︀ g 1 ( ) )︀ ∈N is a Cauchy sequence in ∞ 1 ∞ (⟨ ⟩ ).
By the Duhamel formula along the characteristics for all , 26) where 0, ( , , ) is the th component of the initial data g 0 . Similarly we write Since we are in the case of hard potentials and Maxwellian molecules, we know that ( ) ≥ 0 > 0. Subtract (3.27) from (3.26), take the 1 ∞ (⟨ ⟩ )-norm of (g 1 ( +1) − g 1 ( ) ) and sum over , by using the relatioñ︀ one gets for each , , (3.28) where Lemma 3.3 and Lemma 3.4 on estimates of the operator B and̃︀ Q are used. On the other hand, where we used the fact that the integral in is bounded by 1; exchanged the integration domains in and 1 , and used Lemma 3.3 and Lemma 3.4 again. Adding up (3.28) and (3.29), by using (3.21), one has Assumption (3.22) indicates that

3.2.2.
Step 2: discussion for g 2 As for g 2 , it satisfies the linear Equation (3.16), which is in a similar form as Equation (6.3) from [8] except for the last term involving lower order -derivatives of f . We thereby mimic Proposition 6.8 from [8] and get the following: Moreover, ∃ some constants 2 > 0, 2 > 0 such that for all , where 2 depends on the initial data of f 0 for 1 ≤ ≤ .
The proof is similar to [8], so we omit most details. Theorem 5.4 of [8] implies that there is a unique solution g 2 to the differential system (3.30), given by where G ( ) is the semigroup generated by G in ∞ , (⟨ ⟩ M −1/2 ); A ( ) and A b k ( ) are vector operators with the th component ( ) and ( ) , (1 ≤ ≤ ). We use the regularising property of A ( ) operator given in Lemma 3.2, and similarly for A b k ( ) due to that follows the same assumption as . The exponential decay of h and all the lower order -derivatives of f , i.e., || f | 1 ∞ (⟨ ⟩ ) (0 ≤ ≤ − 1) is used, by the assumption for h in this proposition and by induction.
We showed that Proposition 3.1 is true for = (1 ≤ ≤ ) by induction, one concludes that the result in Proposition 3.1 holds for all = 0, · · ·, , where is associated to the regularity of the initial data f 0 in the random space.

Spectral gap of the linearized gPC Galerkin system
In this part, we generalize the single-species gPC-SG system to the multi-species gPC-SG system by adapting the idea from the proof of the multi-species H-theorem [14] and in particular for the Boltzmann model [12], combined with the previous work considering the uncertainty [11,27]. We consider in this Section the case of random initial data and random collision kernel, where the distribution of the one-dimensional random variable is given by ( ). The same notation and perturbative setting are followed as that in [11,12]. Denote Assume that the distribution function is close to the global equilibrium such that we can write and :: It has been shown in [12] that the linearized Boltzmann system (4.2) satisfies the H-theorem with the linearized entropy ( ) = 1 where (·, ·) 2 is the scalar product on 2 = 2 (R 3 ; R ). is that we will extend the spectral gap analysis from the single-species case studied in [11] to the multi-species Boltzmann system, thus it is better to follow the same perturbative setting as in [11].
One can approximate the distribution for the th species (or ℎ ) by using the ansatz and conducting a standard Galerkin projection, one obtains the following gPC-SG system for , (with 1 ≤ ≤ , 1 ≤ ≤ ): , + · ∇ , = ⟨ ( ), ⟩ 2 ( ( )) . (4.5) In this part of the study for the gPC-Galerkin system, besides (H1)-(H5), we need the following additional assumptions (recall that is the angular part of the collision kernel in (2.5)): (B1) Assume that is linear in , This assumption is reasonable and a common practice, see the Karhunen-Loève expansion [29]. (B2) Assume the leading part (0) and the perturbative part (1) in (4.6) satisfy the condition where is associated to the energy defined in [11]. (B3) The random variable has a compact support, that is, Remark 4.2. We want to mention that due to (B1), our global assumption (H5) has the particular form: The assumptions (B1)-(B3) are the same as that in [11] except now we are in the multi-species framework.
The main result of Section 4 is the following theorem: , and additionally, assume that for all 1 ≤ , ≤ , (cos ) in (4.7) satisfy the same assumptions as (cos ) in the deterministic case in [8], then we obtain an explicit spectral gap estimate for the linearized operator in the gPC stochastic Galerkin system, in a proper weighted norm, where is a positive constant independent of , || · || Λ is some weighted 2 norm.

The proof of Theorem 4.3
We denote the right-hand-side of (4.5) by Term a ○, then where the subscript in :: ( ) is omitted, and we use (4.3) and approximate ℎ (and ℎ ) by ℎ (and ℎ ) given in (4.4); the term Θ [ℎ ] above is denoted by For the readers' convenience, we use indices ( , ) to denote different species, while ( , ) stand for the index of the gPC coefficients.
Step 3: Change ( , * ) to ( * , ) on (4.10), then exchange and , one has where we used * = * followed by * = * , and = . Adding up Equations The each index pair ( , ), the above formulation (4.12) is exactly the same as Equation (39) from [11] except now we are in the multispecies setting. A similar analysis follows here, and we put it in Appendix B. Then in analogous to Equation (44) from [11], one finally obtains that where we define ℎ = Integrating on of (4.13), we finally get The proof of Theorem 4.3 is done. We generalized the spectral gap proof for the linearized numerical collision operator of the single-species Boltzmann equation studied in [11] to the multi-species setting, which will be prepared for studying the long-time behavior and spectral convergence for the numerical solution (and numerical error) for the gPC Galerkin system, as done for the analytical solution in Section 3. We mention that in [27], hypocoercivity of the SG system and regularity of its solution in a weighted Sobolev norm, as well as spectral accuracy and exponential decay in time of the numerical error of the gPC-SG method has been established. In [12], the authors have studied the convergence to equilibrium in 1 , space for the linearized multi-species Boltzmann equations, nevertheless the study of convergence to equilibrium in higher Sobolev space , for the nonlinear deterministic equations is not yet developed, so a complete above-mentioned study in the uncertainty framework for the gPC Galerkin system remains a future work.

Conclusion
In this paper, we consider the nonlinear multi-species Boltzmann equation with uncertainty coming from both the initial data and collision kernels. Well-posedness and regularity in the random space of the solution to the sensitivity system -the PDE obtained from taking derivatives in the random space, long-time behavior (exponential decay to the global equilibrium) of the analytic solution, spectral gap of the linearized corresponding gPC-based stochastic Galerkin system are established. Combine the two cases, then (3.17) is proved.
Proof. (3.18): We recall Proof of Lemma 6.6 from [8], the difference is that here ℓ involves the -derivatives of the collision kernel :