Spectral Convergence of the Stochastic Galerkin Approximation to the Boltzmann Equation with Multiple Scales and Large Random Perturbation in the Collision Kernel

In [L. Liu and S. Jin, Multiscale Model. Simult., 16, 1085-1114, 2018], spectral convergence and long-time decay of the numerical solution towards the global equilibrium of the stochastic Galerkin approximation for the Boltzmann equation with random inputs in the initial data and collision kernel for hard potentials and Maxwellian molecules under Grad's angular cutoff were established using the hypocoercive properties of the collisional kinetic model. One assumption for the random perturbation of the collision kernel is that the perturbation is in the order of the Knudsen number, which can be very small in the fluid dynamical regime. In this article, we remove this smallness assumption, and establish the same results but now for random perturbations of the collision kernel that can be of order one. The new analysis relies on the establishment of a spectral gap for the numerical collision operator.


2.
Introduction of the Boltzmann equation with uncertainties. We first give a review of some of the results in [11] that will be useful in our proof. Consider the initial value problem for the Boltzmann equation where f = f (t, x, v, z) is the particle density distribution that depends on time t, particle position x ∈ T dx (periodic box of d x dimension), velocity v ∈ R dv and a random variable z. The numbers d x , d v ≥ 1 denote the dimension of the spatial and velocity spaces, and z is a random variable that lies in the domain I z ⊂ R with compact support, which is used to account for the random uncertainties or inputs. The operator Q is quadratic and models the binary collisional interactions between particles. The parameter ǫ is the dimensionless Knudsen number, the ratio of particle mean free path over the domain size. The choice α = 1 refers to the incompressible Navier-Stokes scaling, and α = 0 corresponds to the Euler (or acoustic in this article) scaling. Moreover, we assume periodic boundary conditions on the torus T dx . 3 For notational simplicity, we set d v = 3 in the following. Since we consider random collision kernels, the operator Q is defined by where we used the abbreviations f ′ = f (v ′ ), g * = g(v * ) and g ′ * = g(v ′ * ), and S 2 is the three-dimensional unit sphere. Note that v ′ and v ′ * are the post-collisional velocities of particles depending on the pre-collisional velocities v and v * . During elastic collisions, the momentum and kinetic energy of the involved particles are conserved, namely, where σ ∈ S 2 is a parameter on the 2-dimensional unit sphere. The collision kernel B = B(|v − v * |, cos θ, z) is a non-negative function depending on the modulus of the relative velocity |v − v * |, the cosinus of the deviation angle θ with and the random variable z ∈ I z .
Properties of the collision operators: First, conservation of mass, momentum and energy is satisfied, i.e.
Next, we have the dissipation of entropy which is known as the celebrated Boltzmann's H-theorem. Moreover, where M loc is the local equilibrium state given by a Maxwellian distribution The global equilibrium is the unique stationary solution to (1) and is given by where by translating and scaling the coordinate system, we assumed ρ = 1, u = 0 and T = 1 in (5). For further properties of the Boltzmann equation, see [3].
One of the central questions in kinetic theory is to understand the long-time behavior of the solution, and for this, the hypocoercive effects of the kinetic equations play a pivotal role, see [5,7,8,15,16]. A hypocoercivity framework for generic nonlinear collisional kinetic equations was established in [13,2]. In [11], this framework was extended to nonlinear collisional kinetic equations with random initial data and/or random collision kernels, which allow to study the long-time sensitivity, regularity, and exponential decay of the (random) solution towards the (deterministic) global equilibrium, for both the random kinetic equations and their stochastic Galerkin approximations. Note that these studies have been carried out only for solutions near the global equilibrium, i.e. in a perturbative setting, such that the solution can be defined under suitable Sobolev norms. Let Now inserting this ansatz into the model (1), the fluctuation h satisfies where the linearized collision operator is defined by while the nonlinear operator has the form The linearized operator L is acting on L 2 v = {f | R 3 f 2 dv < ∞}, with a finite dimensional kernel N (L) = Span{ϕ 1 , · · · , ϕ n }, where {ϕ i } 1≤i≤n is an orthonormal family of polynomials in v corresponding to the manifold of local equilibria for the linearized kinetic models. The orthogonal projection on N (L) in L 2 v is defined by In the classical case of hard potentials and Maxwellian molecules under Grad's angular cutoff, meaning that the collision kernel B has the form with the kinetic part Φ satisfying and the angular part b being locally integrable, assumptions (1.1), (1.2), (1.3) in [12] are satisfied. Thus, [12, Theorem 1.1] holds, and consequently L has the following local coercivity property with an explicitly computable coercivity constant λ > 0: where 5 stands for the microscopic part of h, and the coercivity norm is (11). For all z, we make the same assumption for b(cos θ, z) as in [12, assumption (1.3)], namely: 3. A gPC based Stochastic Galerkin Method. We first review the gPC-SG method for solving kinetic equations with uncertainties [9]. For more general differential equations, see for examples [6,17]. We want to approximate f (or h) in the following way Here where π(z) is the probability distribution function of z, which is given a priori in our problem. Note that f can be expanded by Finally, we define the projection operator P K as We will consider the one-dimensional random variable z in the sequel. By inserting the ansatz (14) into (7) and conducting a standard Galerkin projection, one obtains the following gPC-SG system for h k for each 1 ≤ k ≤ K: with periodic boundary conditions and the initial data

ESTHER S. DAUS, SHI JIN AND LIU LIU
In (16), the operator L k (h K ) is given by where we denote S K×K by the K × K matrix with (k, j)-th component Note that S is symmetric. The nonlinear term is with the tensor T K×K×K defined by We first review [11, Theorem 5.1], which is the main result on the SG systems of that paper.
Theorem 3.1. Assume that the collision kernel B satisfies the assumptions where b is linear in z, and has the particular form where z ∈ I z has a compact support, that is, and We also assume the technical condition [14] ||ψ with a parameter p ≥ 0. Let q > p + 2, and define the energy E K by with the initial data satisfying E K (0) ≤ η.
(25) Finally, let for all s ≥ s 0 , 0 ≤ ǫ d ≤ 1 and for 0 ≤ ǫ ≤ ǫ d , let h K be a gPC solution of (16) with (26) Then we get that where η,η, τ ,τ and C are all positive constants independent of K and z.
Note that the theorems on the estimate of E K , the gPC error and spectral convergence of the SG method in [11] require the same set of assumptions on B and gPC polynomial basis ψ, given by (19)-(23). One may also put the random dependence on φ in the collision kernel B, which brings a similar analysis and the same conclusion. 4. The main result. The main purpose of this section is to obtain the same results as in Theorem 3.1, whereas relaxing the condition (22) We will focus on the linearized operator L k below. We denote and we introduce the operator Θ by To remove the O(ε) assumption on the collision kernels, we need to reinvestigate the estimate for the term involving the linearized operator L k , namely By (8), by the definition (29).

ESTHER S. DAUS, SHI JIN AND LIU LIU
Remark 1. In [11], we assume that B satisfies (19)-(22), that is, with |∂ z b| = |b 1 | ≤ O(ε). Thus |S kk | ≤ C b , using (21), for k = j we have The procedure below makes it possible to remove the assumption that the random perturbation in the collision kernels require a bound of order O(ε), and avoids using the assumption that |S kj | ≤ Cε for k = j as it was shown in [11].
Consider the term Thus (for simplicity, we omit writing the integral domain below), Step 1: We make the change of variables (v, v * , σ) → (v ′ , v ′ * , k) with k = (v − v * )/|v −v * | on the right hand side of (33), which has unit Jacobian and is involutive. Note that S kj also depends on |v − v * | through B, with the relation |v − v * | = |v ′ − v ′ * |, thus in the following steps S kj will not be affected by the coordinate transformation. Then one obtains where we used M ′ M ′ * = MM * , due to the conservation of kinetic energy, i.e., |v| 2 + |v * | 2 = |v ′ | 2 + |v ′ * | 2 .
Step 2: Then one makes the change of variables (v, v * ) → (v * , v) on the right hand side of (33) and obtains Step 3: Now we use again the change of variables (v, v * , σ) → (v ′ , v ′ * , k) with k = (v − v * )/|v − v * | on the right hand side of (35), thus where M ′ M ′ * = MM * is used.
We now consider the following term: where in the second inequality we used k k−1 ≤ 2 and k k+1 < 1; in the third inequality the Cauchy-Schwarz inequality is used: and |xy| ≤ 1 2 |x| 2 + |y| 2 is used in the fourth inequality. Under the assumption on b given in (41), then Since M > 0, M * > 0, φ(|v − v * |) ≥ 0, by using (43), one has Term I controlled by: where the equivalence between (38) and (33) is used again in the second last row, and (32) is used in the last equality, except that now we haveD instead of S kj in the integral, with LD defined as replacing B byD = φ(|v − v * |)D(cos θ) in (29), with D(cos θ) sharing the same assumption as b(cos θ) in [12, assumption (1.3)], so that the coercivity property (12) still holds, namely: Integrating on x, one finally has where C λ is a constant independent of K and z.
The estimate on the nonlinear term F k (k = 1, · · · , n) will be the same as in [11]. The reason is that the upper bound for the triple index coefficient matrix that we need stays the same as before (see the inequality (5.9) in [11]): will dominateC. Another difference from [11] is that we no longer need to "absorb" the non-diagonal part of the linearized term into the non-linear term, which has the same order of coefficients (O( 1 ε ) in the incompressible Navier-Stokes scaling). The rest of the proof is the same as that for Theorem 5.1 in [11]. We omit the details here.
The following Theorem has the same conclusion as Theorem 3.1 (i.e., Theorem 5.1 in [11]), namely, the gPC based Stochastic Galerkin method for the Boltzmann equation with random inputs and both scalings of α is of spectral accuracy, and the total gPC error decays exponentially in time. The significant difference here is that we are now able to remove the "bad" assumption on the small O(ε) random perturbation of the collision kernel shown in (22). Theorem 4.1. We first give a summary of the assumptions needed on the collision kernel B: and instead of (22) in [11], assume that for some constant R > 0. Let the gPC polynomial basis satisfy with a parameter p ≥ 0, and q in (45) satisfies q > p + 2. The initial conditions for energy E K (defined in (24)) and solution h shown in (25)-(26) are also satisfied. Then we have Here η,η, τ ,τ and C are all positive constants and independent of K and z. 13 Remark. The idea of using change of variables and writing Term I in the form of (38) is inspired by [4], where an explicit spectral-gap estimate and a convergence to equilibrium of the linearized multi-species Boltzmann equation was studied. In [4], they consider a system of Boltzmann equations that models the evolution of a dilute ideal gas composed of n ≥ 2 different species, Q i is the i-th component of the nonlinear collision operator, defined by The linearized equation of (47) is given by One important step relevant to our analysis is the following idea. Split the operator L = L m + L b with L m = (L m 1 , · · · , L m n ) and L b = (L b 1 , · · · L b n ), given by then (f, L m (f )) L 2 v and (f, L b (f )) L 2 v can be written in bilinear forms with squares: