LOCAL SENSITIVITY ANALYSIS AND SPECTRAL CONVERGENCE OF THE STOCHASTIC GALERKIN METHOD FOR DISCRETE-VELOCITY BOLTZMANN EQUATIONS WITH MULTI-SCALES AND RANDOM INPUTS

. In this paper we study the general discrete-velocity models of Boltzmann equation with uncertainties from collision kernel and random in- puts. We follow the framework of Kawashima and extend it to the case of diﬀusive scaling in a random setting. First, we provide a uniform regularity analysis in the random space with the help of a Lyapunov-type functional, and prove a uniformly (in the Knudsen number) exponential decay towards the global equilibrium, under certain smallness assumption on the random perturbation of the collision kernel, for suitably small initial data. Then we consider the generalized polynomial chaos based stochastic Galerkin approximation (gPC-SG) of the model, and prove the spectral convergence and the exponential time decay of the gPC-SG error uniformly in the Knudsen number.

1. Introduction. In this paper, we are interested in the discrete-velocity models (DVMs) of the Boltzmann equations with multi-scales and random inputs. The study of DVMs of the Boltzmann equations is of considerable interest in the kinetic theory of gases, which describes the time evolution of particles of gases in the case where the particles are allowed to move in the space with finitely many velocities.
Starting from 1970s, there have been plenty of works that studied the discretevelocity Boltzmann kinetic models. The diffusive limit for the Carleman-type kinetic models were investigated in [36,33,38,37], while the L 1 -stability was established in [9]. The decay of solutions of the Carleman model was given in [13,14] without any scaling. For another example of DVMs, the Broadwell model, Inoue and Nishida showed the decay of solution in one dimension and the hydrodynamical limit in the compressible Euler scaling, as the mean free path goes to zero [15]. For general DVMs, we would like to mention the framework that Kawashima constructed [28,24,29,25,42,39,26]. In particular, he proved the global existence and long-time the so-called sensitivity analysis [41], since it shows the insensitivity of the solution to the random perturbation under the assumed conditions. Then we consider the gPC-SG approximation for the same model, and prove the spectral convergence of the method and the exponential time decay of the gPC-SG error uniformly in the Knudsen number.
The study of discrete velocity kinetic models is not just of theoretical interest. It will also have an impact for numerical computations, since any numerical kinetic model needs to discretize velocity, thus becomes de facto discrete-velocity models. Moreover, the lattice Boltzmann methods [6], popular in numerical simulations of incompressible flows, are also discrete-velocity models.
This paper is organized as follows. In Section 2, we introduce the generalized form of DVMs of the Boltzmann equations with randomness and describe the notations used in this paper. In Section 3, we show the regularity of the solutions of DVMs, which results in the decay toward global equilibrium. Section 4 proves the spectral convergence and error estimates of the gPC-SG method.
2. DVM of Boltzmann equations with random inputs.
2.1. The basic setup. In this article, we consider the initial value problem for discrete-velocity Boltzmann equation in dimensionless form as following where f i = f i (t, x, z) represents the mass density of particles with velocity v i ∈ R d at time t and position x, depending on a random variable z with π(z) as its probability density function. d ≥ 1 is the dimension of space and velocity. The random variable z lies in I z . f is a vector function with component f i . ε is the Knudsen number, the ratio of the mean free path over a typical length scale of the problem. Each B i is a binary collision operator given by where α i are positive constants, and A ij kl are non-negative constants. A kl ij are so-called transition rates related to the collisions The transition rates are positive constants which, according to the indistinguishability property of the gas particles and the reversibility of the collision, satisfy Remark 1. For discrete velocity Boltzmann equations, it is easy to deduce the high dimensional problems to one dimension [26].

2.2.
Examples of DVMs. One famous example of discrete-velocity model of the Boltzmann equation is the Carleman Model [4], The other example is the Broadwell model [3], Here v is a positive constant and σ(z) = 1.

Notations.
In this paper, we will work on vector functions If f and g are two complex-valued vectors, then the standard dot product (multiplication) in C is defined as Denote the inner product as For functions f = f (x, z), the above norm is actually a function of z, we define the expect value of sum of square of Sobolev norm (including both z and x derivatives): In particular, f H s SENSITIVITY ANALYSIS FOR DISCRETE-VELOCITY KINETIC MODELS ...

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Besides, for functions f = f (z), we define the Sobolev norm in the random space as 3. Uniformly exponential decay to the global equilibrium. In this section, we extend the deterministic framework of Kawashima [23,26,27] about convergence toward the global equilibrium for the DVMs of Boltzmann equation to the case with uncertainty.
In particular, we will consider a solution which is a small perturbation of the global equilibrium. To this aim, we shall introduce basic concepts concerning (1) and summarize their properties [23,26,2] which will be used later.
We denote by M the set of summational invariants. Then 0 < dim M < m because (α 1 , . . . , α m ) T ∈ M and M = R m . Denote . . , m. Let d = dim M. and ψ (j) , j = 1, . . . , d and φ (k) , k = d + 1, . . . , m, be constant vectors such that where M ⊥ denotes the orthogonal complement of M in R m . For f ∈ R m , we define w = (w 1 , . . . , w d ), w j = (f, ψ (j) ), j = 1, . . . , d.  In particular, f > 0 is called a global equilibrium if it is a locally equilibrium and is independent of t and x, which means it is a constant vector.
Let B(f, g) = (B 1 (f, g), . . . , B m (f, g)) T and B(f, g) = σ(z)B(f, g).   All the definitions and the lemmas above can be found in [23,26]. Let M be the global equilibrium, which can be uniquely determined by the initial data. We shall seek the solution of (1)- (2) in the form where M = (M 1 , . . . , M m ) T > 0 and The fluctuation g satisfies The operators L and B have the following properties [23]. Lemma 3.6.
1. L is real symmetric and positive semi-definite; its null space is given by 2. B is bi-linear and satisfies B(f, g) ∈ Null(L) ⊥ for any f, g ∈ R m , where Null(L) ⊥ is the orthogonal complement of Null(L) in R m . 3. There exist λ 0 and λ 1 such that and where P ⊥ denotes the orthogonal projection onto Null(L) ⊥ .
Denote P as the projection operator onto Null(L), then it is not hard to find: Proof. Note that g is the perturbation around the global equilibrium M . Since f and M share the same moments, this directly yields this lemma.

3.2.
The estimate for the linearized equation. Let's first consider linearized equation of (8) with the same initial data, where L = σ(z)L is the bounded linear collision operator defined by (9). We also assume that (13) is "dissipative" in the following sense (see [23,24,26]): Assumption. For any complex-valued vector function f, there exists a bounded real anti-symmetric matrix M such that the symmetric part of M V + L is positive definite. That is, there exist a constant λ 2 > 0 such that where [M V ] denotes the symmetric part of M V .
Next we want to show the decay estimate following the framework of Kawashima [29,26] using the Fourier transform. Suppose g can be written as where the Fourier coefficient We first state a technical lemma with more assumptions for the collision kernel.
and for n = 1, . . . , r, 0 < σ min ≤ σ(z) ≤ σ max , and |∂ n z σ| ≤ σ max for some constant σ min and σ max . Besides, assume that the Fourier coefficients g k of the initial data is in H r z . Then for all integer 0 ≤ n ≤ r and for all k ∈ Z/{0}, there exist positive constants c rj , a rj and c n such that Remark 3. The Lyapunov functional is positive and equivalent to the Sobolev norm for α j small.
Proof. We will use mathematical induction in this proof. For n = 0, taking the Fourier transform in x, for k = 0 one gets Taking inner product with g k (in C m ), and since V and L are real symmetric, then the real part of (19) reads

SHI JIN AND YINGDA LI
Multiply (−εikM ) and take inner product with g k . Since iM is Hermitian, then the real part is Thus we have Here we use the boundedness of the matrix M. If one choose δ 0 = ελ2 2 , then it follows (23) One multiplies the first and the second inequalities by (1 + k 2 ) and α 0 , respectively, and then adds them up. It follows Choosing α 0 such that gives Assume that the inequality (17) holds true for all n ≤ r. After adding all those inequalities, one arrives at where c = min{c 0 , . . . , c n }. If we rewrite the equation, it follows where andα r = min{α 0 , . . . , α r }. For n = r + 1, take derivative with respect to z variable to the equation.
After taking inner product with ∂ r+1 z g k , the real part becomes which yields Multiplying (29) by (−εikM ) to both sides and taking inner product with ∂ r+1 z g k , then the real part gives Thus If one chooses δ 1 = λ2ε 4 , it follows (34) Similar to the case of the zero-th derivative in z, one times inequality (34) by αr+1 1+k 2 and adds it with inequality (31) to get As what was done before, choose α r+1 such that It follows Multiplying (26) with β, which will be determined later, and adding it to (36) yields If we choose β such that Finally, it follows Then we finish proving our lemma.
Next we obtain the time decay of g for the linearized equation (13). Proof. We expand g(t, x, z) by the Fourier transform in x direction Then by Parseval's identity, one gets Take k = 0 in (19), we have Applying P ⊥ to (40) and multiplying it by the complex conjugate of P ⊥ g 0 , it follows 1 2 which implies (42) For each component j = 1, . . . , m, using (12) one has Thus |P g 0 (t)| 2 = m j=1 (P g 0 (t)) 2 j = 0. For k = 0, from Lemma 3.8, one can have As long as α j j = 1, . . . , r are small enough, we can construct If one chooses α = min{α 0 , . . . α r }, then (E α1,...,αr,r ) t + λ 2 α 4 k 2 1 + k 2 E α1,...,αr,r ≤ 0. (45) This inequality implies where δ = λ2α 4 and Note we defined a weighted Sobolev norm that is equivalent to the standard Sobolev norm f H r in the random space. Then where C is a constant independent of ε. And where α = min{α 0 , . . . , α r } for α i ≤ min{ λ0λ2σmin It is easy to generate to higher regularity in x space, since we can have Remark 4. Note that although δ depends on ε (through α), it depends on ε in a good way. For example, without loss of generality one can assume ε ≤ 1, then (50) yields a uniform exponential decay.
The next result states that one can still obtain exponential decay of the solution to (8) even with the bilinear operator B, if the initial data is small enough.
Proof. Consider a semi-group generated by Therefore, one gets the formula which implies Set G(t) = sup 0≤τ ≤t e δτ g(τ ) 2 H s x H r z . By the definition of G(t), the last term on the right-hand side of (53) is dominated by G(t) 2 t 0 e −δt e −δτ dτ . Therefore we arrive at the inequality The results of Theorem 3.10 prove that the random fluctuation g decays exponentially, thus f of the solution to (8) converges to the deterministic global equilibrium M. In other words, the solution is insensitive to the random inputs in initial data and collision kernel σ(z) under the assumption (16).
4. The spectral convergence of the gPC-SG method. In this section, we will first give a brief review of the generalized polynomial chaos approach in the stochastic Galerkin (SG) framework, state some properties of the SG solution and then prove the spectral convergence of the SG method and the exponential decay in time toward the global equilibrium of its solution.
4.1. The gPC-SG approximation. Let {φ k (z)} ∞ k=1 be the series of orthonormal polynomial basis in the Hilbert space W ∞ µ (I z ) corresponding to a random measure dµ, where Here δ ij is the Kronecker delta function. One can expand f as is the coefficient of the gPC expansion. For any fixed integer K, define the projection operator P K : W ∞ µ (I z ) → W K µ where W K µ is the subspace spanned by {φ k (z)} K k=1 . Then We seek the solution in W K µ , that is in the form of Correspondingly, where g K = 1 . Insert this ansatz into equation (8), one obtains the gPC-SG system for g k : for each 1 ≤ k ≤ K and the initial condition is given by The collision operators are given by wherẽ

4.2.
Estimate for the gPC coefficients. To get the spectral convergence of the gPC method, we follow the argument in [40]. We shall get an estimate on the solutions first. Assume that for some positive constant p. Then it follows that Here are some examples satisfying (61). For the case I z = [−1, 1] with uniform distribution, φ k is the normalized Legendre polynomials, and (61) holds for p = 1 2 . For the case I z = [−1, 1] with the distribution π(z) = 2 π √ 1−z 2 , φ k are the normalized Chebyshev polynomials, and (61) holds with p = 0. Since φ k is a (k − 1) th degree polynomial, orthogonal to all lower order polynomials and if we are assuming that σ(z) is linearly depending on z, S ijk = 0 if (i − 1) + (j − 1) + 1 < k − 1. Thus S ijk may be nonzero only when i + j ≥ k (62) holds. Note that there is symmetry for i, j, k in S ijk , and S ijk may be nonzero also when j + k ≥ i, k + i ≥ j (63) hold. One can derive from (61) that Define the energy by and we want to estimate this energy. To this aim, after multiplying k q to system (58), one arrives Then we have the following lemma: Lemma 4.1. Assume condition (60). Let q > p + 2 and suppose the collision kernel linearly depends on z, i.e. σ(z) = σ 0 + σ 1 z. Then Proof. We begin by rewriting the left side of (67) into Consider the case of i ≥ j. Since i q ≥ k 2 q and (64), then Thus the i ≥ j term in RHS of (68) can be estimated by where in the second inequality we use (69), and χ ijk is the indicator function for index (i, j, k). If fixing i, one can rewrite the RHS of (70) as By (62) and (63), χ ijk = 0 only when i − j ≤ k ≤ i + j. It means that there are at most 2j terms in I i above. With assumption q > p + 2, it holds that For the case of i ≤ j, one can exchange the indexes i and j to have the same estimate. Then we finish the proof.

Remark 6.
The assumption on the linearity in z is a common practice in UQ research. It is known that uncertainties are usually modelled by stochastic process, and according to the Karhunen-Loeve theory, any stochastic process can be approximated by a linear combination of uncorrelated random variables (z in this paper). Our analysis could be extended to more general function of z but the algebra will become messy and lose the clarity of the analysis, so we do not carry it out here.
Next we obtain the exponential decay of E K (t) for σ(z) with a smaller random perturbation.
Theorem 4.2. Assume condition (60). Let q > p + 2 and suppose the collision kernel linearly depends on z in the following way, σ(z) = σ 0 + εσ 1 z with 0 < σ min ≤ M λ12 2q+9 , then the energy defined by (65) can be estimate as Proof. The proof is similar to Theorem 3.10, the interested reader will find them in Appendix 5.
Once we obtain the estimate of energy, we can get exponential decay of the gPC solutions.
Corollary 1. With the assumption above, there exist constants C and C which are independent of ε and K so that and Proof.
due to q > p + 2. In addition, 4.3. The gPC error estimate. In order to estimate the gPC error g − g K , we denote where R K and E K refer to truncation error and projection error, respectively. Then using the strategy of [34], we can have the following theorem.
Theorem 4.3. Assume condition (60). Let q > p + 2 and suppose the collision kernel linearly depending on z, i.e. σ(z) = σ 0 + ε 2 σ 1 z with σ 0 , σ 1 independent of z (thus 0 < σ min ≤ σ(z) ≤ σ max ). If initially g e in 2 H s x H r z ≤ C I , g 0 2 H s x H r z ≤ C 0 , and ifC 0 = C 0 max{C π C 0 , 1} such that where δ is defined in Theorem 4.2 and C π is a constant independent of K and , then the gPC error has following estimate where C (linearly depending on T) and δ are constants independent of K and ε.
Proof. By Theorem 3.10 and standard estimate for truncation error of orthogonal polynomial approximations where C π is a constant independent on K. Let the projection error be whereg k = Iz gφ k dµ and we denote e = [e 1 , . . . , e K ]. Let Since g K is the gPC solution, then for all k = 1, . . . , K, Due to T (g), φ k µ = 0, one can have For the first term inside of T , it follows Similarly, one can show ∂ x R K , φ k µ = 0. Then (82) becomes Then (83) becomes an equation for e k (t, x), ∂ t e k + 1 ε V e k x + 1 ε 2 L k (e k ) = B k (g − g K , g) + B k (g K , g − g K ) − 1 ε 2 L k (R K ) Since is indeed One can follow the proof in [11] to treat the non-linear term in (87) as' whereC 0 is a constant from initial data ( g K (0) and g(0) ) and independent of K and ε. In the above estimate, we use the Cauchy-Schwartz inequality in the first and fourth inequalities, and the fifth one is due to (51) and (74). In the last inequality, we use (79). Similarly, Then (87) becomes Set S(t) = sup 0≤τ ≤t K 2r−1 e δτ K k=1 e k 2 Lx . Multiplying K 2r−1 e δt to both sides of (90), one has One can usually choose E K (0) = 0. Hence, one may obtain from (77) that that is where C(T ) (linearly depending on T ) and δ are constants independent of K and ε. For higher derivatives in x, one can take H s x norm on equation (86) and sum those K equations. Then by similar analysis, one will have Then combining with (79), we finish the proof.