Compactness property of the linearized Boltzmann collision operator for a multicomponent polyatomic gas

The linearized Boltzmann collision operator is fundamental in many studies of the Boltzmann equation and its main properties are of substantial importance. The decomposition into a sum of a positive multiplication operator, the collision frequency, and an integral operator is trivial. Compactness of the integral operator for monatomic single species is a classical result, while corresponding results for monatomic mixtures and polyatomic single species are more recently obtained. This work concerns the compactness of the operator for a multicomponent mixture of polyatomic species, where the polyatomicity is modeled by a discrete internal energy variable. With a probabilistic formulation of the collision operator as a starting point, compactness is obtained by proving that the integral operator is a sum of Hilbert-Schmidt integral operators and operators, which are uniform limits of Hilbert-Schmidt integral operators, under some assumptions on the collision kernel. The assumptions are essentially generalizations of the Grad's assumptions for monatomic single species. Self-adjointness of the linearized collision operator follows. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere like model. Then it follows that the linearized collision operator is a Fredholm operator, and its domain is also obtained.


Introduction
The Boltzmann equation is a fundamental equation of kinetic theory of gases, e.g., for computations of the flow around a space shuttle in the upper atmosphere during reentry [1].Studies of the main properties of the linearized collision operator are of great importance in gaining increased knowledge about related problems, see, e.g., [10] and references therein, and for related half-space problems [2,4,3,7].The linearized collision operator is obtained, by considering deviations of an equilibrium, or Maxwellian, distribution.It can in a natural way be written as a sum of a positive multiplication operator, the collision frequency, and an integral operator −K.Compact properties of the integral operator K (for angular cut-off kernels) are extensively studied for monatomic single species, see, e.g., [14,11,10,19], and more recently for monatomic multi-component mixtures [9,5] and polyatomic single species, where the polyatomicity is modeled by either a discrete or a continuous internal energy variable [5,6].See also [8] for the case of molecules undergoing resonant collisions, i.e., collisions where internal energy is transferred to internal energy, and correspondingly, translational energy to translational energy, during the collisions.The integral operator can be written as the sum of a Hilbert-Schmidt integral operator and an approximately Hilbert-Schmidt integral operator (cf.Lemma 4 in Section 4) [13], and so compactness of the integral operator K can be obtained.This work extends the results of [5] for monatomic multicomponent mixtures and polyatomic single species, where the polyatomicity is modeled by a discrete internal energy variable [12,15], to the case of polyatomic multicomponent mixtures, where the polyatomicity is modeled by discrete internal energy variables.To consider mixtures of monatomic and polyatomic molecules are of highest relevance in, e.g., the upper atmosphere [1].
Following the lines of [5,6], motivated by an approach by Kogan in [18, Sect.2.8] for the monatomic single species case, a probabilistic formulation of the collision operator is considered as the starting point.With this approach, it is shown, based on slightly modified arguments from the ones in [5], that the integral operator K can be written as a sum of compact operators in the form of Hilbert-Schmidt integral operators and approximately Hilbert-Schmidt integral operators -which are uniform limits of Hilbert-Schmidt integral operators -and so compactness of the integral operator K follows.The operator K is self-adjoint, as well as, the collision frequency, why the linearized collision operator, as the sum of two self-adjoint operators of which one is bounded, is also self-adjoint.
For models corresponding to hard sphere models in the monatomic case, bounds on the collision frequency are obtained.Then the collision frequency is coercive and becomes a Fredholm operator.The set of Fredholm operators is closed under addition with compact operators.Therefore also the linearized collision operator becomes a Fredholm operator by the compactness of the integral operator K.For hard sphere like models the linearized collision operator satisfies all the properties of the general linear operator in the abstract half-space problem considered in [4].
The rest of the paper is organized as follows.In Section 2, the model considered is presented.The probabilistic formulation of the collision operators considered and its relations to more classical formulations [12,15] are accounted for in Section 2.1.Some classical results for the collision operators in Section 2.2 and the linearized collision operator in Section 2.3 are reviewed.Section 3 is devoted to the main results of this paper, while the main proofs are addressed in Sections 4 − 5; a proof of compactness of the integral operator K is presented in Section 4, while a proof of the bounds on the collision frequency appears in Section 5. Finally, the appendix concerns a proof of a crucial -for the compactness -lemma, which is an extension of a corresponding lemma for the monatomic mixture case [9,5].

Model
This section concerns the model considered.A probabilistic formulation of the collision operator is considered, whose relation to a more classical formulation is accounted for.Known properties of the model and corresponding linearized collision operator are also reviewed.
Moreover, denote r = s α=1 rα and consider the real Hilbert space The evolution of the distribution functions is (in the absence of external forces) described by the (vector) Boltzmann equation where the (vector) collision operator .., Q s rs is a quadratic bilinear operator that accounts for the change of velocities and internal energies of particles due to binary collisions (assuming that the gas is rarefied, such that other collisions are negligible), where the component Q α i is the collision operator for particles of species aα with internal energy Ii for i ∈ {1, ..., rα} and α ∈ {1, ..., s}.

Collision operator
The (vector) collision operator .., Q s rs has components that can be written in the following form for some constant ϕ 1 1 , ..., ϕ 1 r 1 , ..., ϕ s 1 , ..., ϕ s rs ∈ R r .Here and below the abbreviations are used.In the collision operator (3) the gain term -the term containing the product f ′ α,k f ′ β,l * -accounts for the gain of particles of species aα with microscopic velocity ξ and internal energy I α i (at time t and position x)here (ξ, I α i ) and ξ * , I β j represent the post-collisional particles, while the loss term -the term containing the product fα,if β,j * -accounts for the loss of particles of species aα with microscopic velocity ξ and internal energy I α i -here (ξ, I α i ) and ξ * , I β j represent the pre-collisional particles.The corresponding (signed) internal energy gap is The transition probabilities are of the form, cf.[5], where δ3 and δ1 denote the Dirac's delta function in R 3 and R, respectively; taking the conservation of momentum and total energy (2) into account.Here and below we use the (inconsistent) shorthanded expressions for given scattering cross-section σ αβ : R The scattering cross sections σ αβ kl,ij , with (α, β, i, j, k, l) ∈ Ω, are assumed to satisfy the microreversibility conditions Furthermore, to obtain invariance of change of particles in a collision, it is assumed that the scattering cross sections σ αβ kl,ij , with (α, β, i, j, k, l) ∈ Ω, satisfy the symmetry relations (fixing the pairs (ξ, while The invariance under change of particles in a collision, which follows directly by the definition of the transition probability (5) and the symmetry relations (7) , (8) for the collision frequency, and the microreversibility of the collisions (6), implies that the transition probabilities (5) for {α, β} ⊂ {1, ..., s} satisfy the relations Applying known properties of Dirac's delta function, the transition probabilities -aiming to obtain expressions for |g ′ | in the arguments of the delta-functions -may be transformed to Remark 1 Note that, cf.[16],

By a change of variables g
lowed by one to spherical coordinates, noting that the observation that can be made, resulting in a more familiar form of the Boltzmann collision operator for mixtures with polyatomic molecules modeled with a discrete energy variable, cf.e.g.[12,15].
Remark 2 Note that, when considering spherical coordinates, we, maybe unconventionally, often represent the direction by a vector in S 2 , rather than with azimuthal and polar angels, still referring to it as spherical coordinates.By representing the direction by a unit vector, the sine of the polar angle will not appear as a factor in the Jacobian, resulting in the Jacobian to be the square of the radial length.

Collision invariants and Maxwellian distributions
The following lemma follows directly by the relations (9).
The multiplication operator Λ defined by The following lemma follows immediately by Lemma 1.
Remark 4 A property of the nonlinear term, although of no relevance to the studies here, is that it is orthogonal to the kernel of L, i.e.Γ (h, h) ∈ (ker L) ⊥ h .This follows, since any element in ker L is of the form M 1/2 g for some collision invariant g, while for any collision invariant g 3

Main Results
This section is devoted to the main results, concerning a compactness property in Theorem 1 and bounds of the collision frequencies in Theorem 2.
The following result may be obtained.
Theorem 1 will be proven in Section 4.

Corollary 1
The linearized collision operator L, with scattering cross sections satisfying (20), is a closed, densely defined, self-adjoint operator on L 2 (dξ) r .
Proof.By Theorem 1, the linear operator L = Λ − K is closed as the sum of a closed and a bounded operator, and densely defined, since the domains of the linear operators L and Λ are equal; D(L) = D(Λ).Furthermore, it is a self-adjoint operator, since the set of self-adjoint operators is closed under addition of bounded self-adjoint operators, see Theorem 4.3 of Chapter V in [17].Now consider the scattering cross sections -cf.hard sphere models - for some positive constants C αβ > 0 for all (α, β, i, j, k, l) ∈ Ω.Note that assumption (21) reduces to the hard sphere model for monatomic multicomponent mixtures [5] in the case of vanishing internal energy gap and unit weigths ϕ 1 1 = ... = ϕ 1 r 1 = ... = ϕ s 1 = ... = ϕ s rs = 1.In fact, it would be enough with the bounds for some positive constants C± > 0, on the scattering cross sections.

Corollary 2
The linearized collision operator L, with scattering cross sections (21) (or (22)), is a Fredholm operator, with domain Proof.By Theorem 2 the multiplication operator Λ is coercive and, hence, a Fredholm operator.The set of Fredholm operators is closed under addition of compact operators, see Theorem 5.26 of Chapter IV in [17] and its proof, so, by Theorem 2, L is a Fredholm operator.
Moreover, by Theorem 2, Corollary 3 For the linearized collision operator L, with scattering cross sections (21) (or (22)), there exists a positive number λ, ImL.As a Fredholm operator, L is closed with a closed range, and as a compact operator, K is bounded, and so there are positive constants ν0 > 0 and cK > 0, such that (h, Lh) ≥ ν0(h, h) and (h, Kh) ≤ cK (h, h).
Remark 5 By Proposition 3 and Corollary 1 − 3, for hard sphere like models the linearized operator L fulfills the properties assumed on the linear operators in [4], and hence, the results therein can be applied for hard sphere like models.

Compactness
This section concerns the proof of Theorem 1.
Note that in the proof the kernels are rewritten in such a way that ξ * -and not ξ ′ and ξ ′ * -always will be an argument of the distribution functions.As for single species, either ξ * is an argument in the loss term (like ξ) or in the gain term (unlike ξ) of the collision operator.However, in the latter case, unlike for single species, for mixtures one have to differ between two different cases (considering interspecies collision operators); either ξ * is the velocity of particles of the same species as the particles with velocity ξ, or not.The kernels of the terms from the loss part of the collision operator will be shown to be Hilbert-Schmidt in a quite direct way.Some of the terms -for which ξ * is the velocity of particles of the same species as the particles with velocity ξ -of the gain parts of the collision operators will be shown to be uniform limits of Hilbert-Schmidt integral operators, i.e. approximately Hilbert-Schmidt integral operators in the sense of Lemma 4. By applying the following lemma, Lemma 5, (for disparate masses), which is a generalization of corresponding lemma for monatomic mixtures by Boudin et al in [9], see also [5], it will be shown that the kernels of the remaining terms -for which ξ * is the velocity of particles of a species different to the species of the particles with velocity ξ -from the gain parts of the collision operators, are Hilbert-Schmidt.
Denote, for any (non-zero) natural number N , Then we have the following lemma from [13], that will be of practical use for us to obtain compactness in this section.
Lemma 4 (Glassey [13, Lemma 3.5.1],Drange [11]) Assume that b(ξ, ξ * ) ≥ 0 and let T f Then the operator T is the uniform limit of Hilbert-Schmidt integral operators, and we say that the kernel b(ξ, ξ * ) is approximately Hilbert-Schmidt, while T is an approximately Hilbert-Schmidt integral operator.The reader is referred to Lemma 3.5.1 in [13] for a proof.
Lemma 5 [9] For (α, β, i, j, k, l) ∈ Ω, assume that mα = m β , and Then there exists a positive number ρ, 0 < ρ < 1, such that A proof of Lemma 5, based on the proof of the corresponding lemma [9] for monatomic mixtures in [5], is accounted for in the appendix.The proof is constructive, in the way that an explicit value of such a number ρ, namely is produced in the proof.Now we turn to the proof of Theorem 1.Note that throughout the proof C will denote a generic positive constant.
We now continue by proving the compactness for the three different types of collision kernel separately.Note that, if α = β, by applying the last relation in ( 27), k αβ,ij (ξ, ξ * ), and we will remain with only two cases -the first two below.Even if mα = m β , the kernels k αβ,ij (ξ, ξ * ) are structurally equal, why we (in principle) remain with (first) two cases (the second one twice).
Assume the internal energy gap ∆I αβ ik,jl = I α i + I β k − I α j − I β l , as well as, the velocities ξ and ξ * , to be given.Then a collision will be uniquely determined by a vector w orthogonal to g = ξ −ξ * .This follows, since, by conservation of momentum and total energy (2) (reminding the relabeling of the velocities and internal energies), the relation between |ξ − ξ ′ | and |ξ ′ * − ξ * | can be obtained, while also 2. Indeed, note that -aiming to obtain expressions for g ′ and χ+ in the arguments of the delta-functions, the expression (28) of k αβ,ij may be rewritten in the following way Here, see Figure 2, implying that the kinetic energy part of the exponent of the product where Hence, by assumption (20) and the Cauchy-Schwarz inequality, . Indeed, by changing variables ξ * → g, with g = ξ − ξ * , and then to spherical coordinates, Next we aim for proving that the integral of k αβ,ij (ξ, ξ * ) with respect to ξ over R 3 is bounded in ξ * .Indeed, directly by the bound (33) on k (34) Hence, due to the symmetry k αβ,ji (ξ * , ξ) (29), by a change of variables ξ → g = ξ − ξ * , followed by one to spherical coordinates, Finally, heading for proving the uniform convergence of the integral of k (α) αβ,ij with respect to ξ * over the truncated domain hN to the one over all of R 3 , the following bound on the integral over R 3 can be obtained for |ξ| = 0. Indeed, by bound (34), by changing variables ξ * → g = ξ − ξ * , then to (conventional) spherical coordinates, with ξ as zenithal direction, and hence, ϕ as polar angle, followed by the change of variables ϕ → η = R + 2 |ξ| cos ϕ + 2χ jl ik (R), with dη = −2 |ξ| sin ϕ dϕ, Then, by the bounds (34) and (35), sup Hence, by Lemma 4 the operators αβ,ij (ξ, ξ * ) hα,j * dξ are compact on L 2 (dξ) for {i, j} ⊆ {1, ..., rα} and {α, β} ⊆ {1, ..., s}.
Firstly, assume that mα = m β .Assume that the internal energy gap ∆ αβ il,kj = I α i + I β l − I α k − I β j and the velocities ξ and ξ * are given.Then a collision will be uniquely determined by a unit vector η Then, by a change of variables {ξ mα − m β and then to spherical coordinates, where αβ,ij may be transformed to Here, see Figure 3, Then, by Lemma 3, since relation (26) follows by energy conservation, we have the following relation between the kinetic parts of the exponents of the products for some positive number ρ, where 0 < ρ < 1.However, also and  (20), and bound (37), one may obtain the following bound and hence, by expression (39), assumption (20), and inequality (37), one may obtain the following bound and hence, by expression (38), assumption (20), and inequality (37), the bound
6 Appendix: Proof of Lemma 5 This appendix concerns a proof of Lemma 5.

Figure 2 :
Figure 2: Typical collision of K