Linearized Boltzmann collision operator for a mixture of monatomic and polyatomic chemically reacting species

At higher altitudes, for high temperature gases, for instance near space shuttles moving at hypersonic speed, not only mechanical collisions are affecting the gas flow, but also chemical reactions have an impact on such hypersonic flows. In this work we insert chemical reactions, in form of dissociations and recombinations (associations), in an existing model for a mixture of mono- and polyatomic (non-reacting) species. More general chemical reactions, e.g., bimolecular ones, can be obtained by instant combinations of associations and dissociations. The polyatomicity is modelled by a continuous internal energy variable and the evolution of the gas is described by a Boltzmann equation. In, e.g., the Chapman-Enskog process - and related half-space problems - the linearized Boltzmann collision operator plays a central role. Here we extend some important properties of the linearized Boltzmann collision operator to the introduced model with chemical reactions. A compactness result, stating that the linearized operator can be decomposed into a sum of a positive multiplication operator - the collision frequency - and a compact integral operator, is proven under reasonable assumptions on the collision kernel. The strategy is to show that the terms of the integral operator are (at least) uniform limits of Hilbert-Schmidt integral operators and therefore also compact operators. Self-adjointness of the linearized operator is a direct consequences. Moreover, bounds on - including coercivity of - the collision frequency are obtained for hard sphere like, as well as hard potentials with cutoff like models. As consequence, Fredholmness, as well as, the domain, of the linearized collision operator are obtained


Introduction
At atmospheric reentry of space shuttles, or, higher altitudes flights at hypersonic speed the vehicles excite the air around them to high temperatures.At high altitudes the pressure is lower and therefore the air is excited at lower temperatures than at the sea level [20].In high temperature gases, not only mechanical collisions affect the flow, but also chemical reactions have an impact on such flows [20,14].Typical chemical reactions in air at high temperatures are dissociation of oxygen, O 2 ⇄ 2O, and nitrogen, N 2 ⇄ 2N, but at higher temperatures even ionization of oxygen and nitrogen atoms, O ⇄ O + + e − and N ⇄ N + + e − , respectively [20].
At high altitude the gas is rarefied and the Boltzmann equation is used to describe the evolution of the gas flow, e.g., around a space shuttle in the upper atmosphere during reentry [2,14].Studies of the main properties of the linearized Boltzmann collision operator are of great importance in gaining increased knowledge about related problems, see, e.g., [13,14] and references therein.
The linearized collision operator, obtained by considering deviations of an equilibrium, or Maxwellian, distribution, 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., [19,17,18,13,23], and more recently for monatomic multi-component mixtures [10,4].Extensions to polyatomic gases, where the polyatomicity is modeled by either a discrete, or, a continuous internal energy variable for single species [4,5,12], or, mixtures [6,7] have very recently been conducted.Compactness results are also recently obtained for models of polyatomic single gases, with a continuous internal energy variable, where the molecules undergo resonant collisions (for which internal energy and kinetic energy, respectively, are conserved) [9].In this work 1 , we extend the results for mixtures of mono-and polyatomic (non-reacting) species in [6], where the polyatomicity is modeled by a continuous internal energy variable, cf., [15,2,1], to include chemical reactions, in form of dissociations and recombinations (associations) [14].More general chemical reactions, e.g., bimolecular ones [15], can be obtained by instant combinations of associations and dissociations.
Following the lines of [4,5,6,7], motivated by an approach by Kogan in [22, 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 that the integral operator K can be written as a sum of Hilbert-Schmidt integral operators and operators, which are uniform limits of Hilbert-Schmidt integral operators (cf.Lemma 3 in Section 4) -and so compactness of the integral operator K follows.Self-adjointness of the operator K and the collision frequency, imply that the linearized collision operator, as the sum of two self-adjoint operators whereof (at least) one is bounded, is also self-adjoint.
For models corresponding to hard sphere models, as well as hard potentials with cut off models, in the monatomic case, bounds on the collision frequency are obtained.Then the collision frequency is coercive and, hence, a Fredholm operator.The resultant Fredholmness -vital in the Chapman-Enskog process -of the linearized operator, is due to that the set of Fredholm operators is closed under addition with compact operators.Unlike for hard potential models, the linearized operator is not Fredholm for soft potential models, even in the monatomic case.The domain of collision frequency -and, hence, of the linearized collision operator as well -follows directly by the obtained bounds.
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 [3], and, hence, the general existence results in [3] apply.
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 a more classical parameterized expression are accounted for in Section 2.1.Some results for the collision operator in Section 2.2 and the linearized collision operator in Section 2.3 are presented.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 operators K is presented in Section 4, while a proof of the bounds on the collision frequency appears in Section 5.

Model
This section concerns the model considered.Probabilistic formulations of the collision operators are considered, whose relations to more classical formulations are accounted for.Properties of the models and corresponding linearized collision operators are also discussed.
The distribution functions are of the form f = (f 1 , ..., f s ), where {t, I} ⊂ R + , x = (x, y, z) ∈ R 3 , and ξ = (ξ x , ξ y , ξ z ) ∈ R 3 , is the distribution function for species a α .Moreover, 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 mech = (Q m1 , ..., Q ms ) is a quadratic bilinear operator, see [1,6], 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) and the transition operator Q chem = (Q c1 , ..., Q cs ) accounts for the change of velocities and internal energies under possible chemical reactions -processes where two particles associate into one, or, vice versa, one particle dissociates into two.
A chemical process (dissociation or association/recombination) {a γ , a ς } ←→ {a β }; one given particle of species a β for β ∈ I poly dissociating in two particles of species a γ and a ς for (γ, ς) ∈ I 2 , or, vice versa, two given particles of species a γ and a ς for (γ, ς) ∈ I 2 associating in a particle of species a β for β ∈ I poly , can be represented by two post-/preprocess elements, each element consisting of a microscopic velocity and possibly also an internal energy, Z ′ and Z ′ * , and one corresponding post-/pre-process element, Z * , consisting of a microscopic velocity and an internal energy.Denote the set of all indices (β, γ, ς) ∈ I poly ×I 2 of valid considered chemical processes {a γ , a ς } ←→ {a β } by C. Assume below that (β, γ, ς) ∈ C.
Due to mass, momentum, and total energy conservation, the following relations have to be satisfied by the elements It follows that the center of mass velocity is Furthermore, for any possible chemical process {a γ , a ς } ←→ {a β }, the following relation on the numbers of internal degrees of freedom is assumed: On the other hand, a mechanical collision can, given two particles of species a α and a β , {α, β} ⊂ {1, ..., s}, respectively, be represented by two pre-collisional elements Z and Z * , and two corresponding post-collisional elements Z ′ and Z ′ * [6].The notation for pre-and post-collisional pairs may be interchanged as well.Due to momentum and total energy conservation, the following relations have to be satisfied by the elements It follows that the center of mass velocity and the total energy in the center of mass frame where g = ξ − ξ * and g ′ = ξ ′ − ξ ′ * , are conserved.
Note that in the literature it is usual to use a slightly different setting [11,15,2], where already renormalized distribution functions are considered, opting to consider a weighted measure -where the renormalization weights appear as weights -with respect to I.However, this is merely due to a different scaling of the distribution functions considered.
The transition probabilities W αβ are of the form [6] where σ αβ = σ αβ (|g| , cos θ, I, I * , I ′ , I ′ * ) > 0 a.e., with Applying known properties of Dirac's delta function, the transition probabilities may be transformed to [6] For more details about the mechanical collision operator, including more familiar formulations, we refer to [6].
The chemical process operator Q chem = (Q c1 , ..., Q cs ) has components that can be written in the following form and Here δ c denotes the Dirac's delta function in R 3 or R 4 if α ∈ I mono or α ∈ I poly , respectively.For (β, γ, ς) ∈ C introduce a positive number -the energy of the transition state of the process -K β γζ , such that That is, for a chemical process {a γ , a ς } ←→ {a β } to take place has to be satisfied.Note that by condition ( 7) The transition probabilities W β γζ are for (β, γ, ς) ∈ C of the form where with , and ∆I = I * − I ′ 1 γ∈I poly − I ′ * 1 ζ∈I poly .Remind that δ 3 and δ 1 denote the Dirac's delta function in R 3 and R, respectively -δ 1 and δ 3 taking the conservation of momentum and total energy into account, while m γ + m ζ = m β .Note that for γ ∈ I mono the scattering cross sections σ γζ β are independent of I ′ , while correspondingly, for ζ ∈ I mono the scattering cross sections σ γζ β are independent of I ′ * .Furthermore, it is assumed that the scattering cross sections σ ζγ β and σ β γζ for (β, γ, ς) ∈ C satisfy the microreversibility conditions Moreover, to obtain invariance of change of particles in a collision, it is assumed that the scattering cross sections σ ζγ β for (β, γ, ς) ∈ C satisfy the symmetry relations Applying known properties of Dirac's delta function, the transition probabilities may for (β, γ, ς) ∈ C be transformed to Then, also, for (β, γ, ς) ∈ C , with where By a series of change of variables: , followed by a change to spher- and, finally, if also Similarly, where Then, with ϕ α (I) = I δ (α) /2−1 for α ∈ I, for two monatomic constituents, i.e., with (γ, ζ) ∈ I 2 mono , (mono/mono-case) , or, for one monatomic and one polyatomic constituent, respectively, i.e., either with (γ, ζ) ∈ I mono × I poly (mono/poly-case), , or, correspondingly, with (γ, ζ) ∈ I poly × I mono (poly/mono-case), and, finally, for two polyatomic constituents, i.e., with (γ, ζ) ∈ I 2 poly , (poly/polycase) , resulting in more explicit forms of the chemical process operators.For the mono/mono-case, note that Explicitly, the internal energy gaps are given by ∆I = I * in the mono/monocase, ∆I = I * − I ′ * in the mono/poly-case, ∆I = I * − I ′ in the poly/mono-case, while ∆I = I * − I ′ − I ′ * in the poly/poly-case.
We have the following proposition.
Definition 3 A function g = (g 1 , ..., g s ), with g α = g α (Z), is a collision invariant if it is a chemical process invariant as well as a mechanical collision invariant.
Introduce the vector space and denote by U 0 := {u 1 , ..., u s } , where s a basis of U. We have the following proposition.
For example, if (β, γ, ς) ∈ C, such that the species a β , a γ , and a ς do not take part in any other chemical process, then if γ = ς one can replace {e β , e γ , e ς } in the set of generating mechanical collision invariants by {e β + e γ , e β + e ς } to describe generators for the set of collision invariants, while if γ = ς one can correspondingly replace {e β , e γ } in the set of generating mechanical collision invariants by {2e β + e γ }.
Remark 2 Introducing the H-functional an H-theorem can be obtained.

Linearized collision operator
Consider (without loss of generality) a deviation of a non-drifting, i.e. with u = 0 in expression (15), Maxwellian distribution M = (M 1 , ..., M s ) ( 15) , (17) of the form Insertion in the Boltzmann equation ( 2) results in the system with the linearized collision operator L = L mech +L chem , where the components of the linearized collision operator L chem = (L c1 , ..., L cs ) are given by while S = S mech + S chem , where the components of the non-linear operator S chem = (S c1 , ..., S cs ) are given by Here, for with and Moreover, the components of the linearized collision operator L mech = (L m1 , ..., L ms ) [6] are given by with while the components of the quadratic term S mech = (S m1 , ..., S ms ) [6] are given by for α ∈ I.The multiplication operator Λ defined by Λ(f ) = νf , where ν = diag (ν 1 , ..., ν s ) , with ν α = ν mα + ν cα , is a closed, densely defined, self-adjoint operator on h.It is Fredholm as well if and only if Λ is coercive.We remind the following properties of L mech [6].
We have the following lemma.
Proof.By Lemma 1 and Proposition 3, (Lh, g) = (h, Lg) is immediate, and Furthermore, h ∈ ker L if and only if (Lh, h) = 0, which will be fulfilled if and for all (β, γ, ς) ∈ C, i.e., if and only if M −1/2 h is a chemical process invariant.
The last part of the lemma now follows by Proposition 2 and 3.
Remark 3 Note also that it trivially follows that the quadratic term is orthogonal to the kernel of L, i.e., S (h) ∈ (ker L) ⊥ h .

Main Results
This section is devoted to the main results, concerning compact properties in Theorem 2 and bounds of collision frequencies in Theorem 3. Below we consider the particular case ϕ α (I) = I δ (α) /2−1 for α ∈ I.
Here and below E αβ = 1 if α ∈ I mono and E * αβ = 1 if β ∈ I mono , while, otherwise, We remind the following compactness result for K mech in [6].
Then the operator Assume that for some positive number χ, such that 0 < χ < 1, there is for all (β, γ, ς) for E β ≥ K β γζ , on the scattering cross sections, or, equivalently, the bound for E β ≥ K β γζ , on the collision kernels.Here and below E γζ = 1 if γ ∈ I mono and E * γζ = 1 if ζ ∈ I mono , while, otherwise, Then the following result may be obtained.
Then the operator K chem = (K c1 , ..., K cs ), with the components K cα given by ( 23) The proof of Lemma 2 will be addressed in Section 4. The following theorem follows by Theorem 1 and Lemma 2.
Theorem 2 Assume that for all (β, γ, ς) ∈ C the scattering cross sections σ β γζ satisfy the bound (28) for some positive number χ, such that 0 < χ < 1, and that for all (α, β) ∈ I 2 poly the scattering cross sections σ αβ satisfy the bound (27) for some positive number τ , such that 0 < τ < 1.Then the operator By Theorem 2, 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 selfadjoint 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 [21].

Corollary 1
The linearized collision operator L, with scattering cross sections satisfying (28), is a closed, densely defined, self-adjoint operator on h.Now consider, for some nonnegative number η less than 1, 0 ≤ η < 1, -cf.hard sphere models for η = 0 -the scattering cross sections for some positive constants C β γζ > 0 for (β, γ, ς) ∈ C, and with In fact, it would be enough with the bounds for some nonnegative number η, such that 0 ≤ η < 1, and some positive constants C ± > 0, on the scattering cross sections -cf.hard potential with cut-off models.
By Theorem 3 the multiplication operator Λ is coercive, and thus it is a Fredholm operator.Furthermore, the set of Fredholm operators is closed under addition of compact operators, see Theorem 5.26 of Chapter IV in [21] and its proof, so, by Theorem 3, L is a Fredholm operator.

Corollary 2
The linearized collision operator L, with scattering cross sections (29) and (30) (or (31) and (32)), is a Fredholm operator with domain For hard sphere like models we obtain the following result.
Corollary 3 For the linearized collision operator L, with scattering cross sections (29) and (30) (or (31) and (32)), where η = 0, there exists a positive number λ, 0 < λ < 1, such that The proof is the same as in the non-reactive case, cf., e.g., [ Remark 4 By Proposition 4 and Corollary 1−3 the linearized operator L fulfills the properties assumed on the linear operators in [3], and hence, the results for half-space problems with general boundary conditions therein can be applied to hard sphere like models.

Compactness
This section concerns the proof of Lemma 2. Note that in Section 2.3 the kernels are written in such a way that Z * always will be an argument of the distribution function.Either Z and Z * are both arguments in the dissociated term of the collision operator, or, either of Z and Z * is the argument in the associated term.
In the former case, the kernel will be shown to be Hilbert-Schmidt , while, in the latter case, the terms will be shown to be uniform limits of Hilbert-Schmidt integral operators, i.e., approximately Hilbert-Schmidt in the sense of Lemma 3.
Then T Lemma 3.5.1] is the uniform limit of Hilbert-Schmidt integral operators and we say that the kernel b(Z, Z * ) is approximately Hilbert-Schmidt, while T is an approximately Hilbert-Schmidt integral operator.The reader is referred to Glassey [18, Lemma 3.5.1]for a proof of Lemma 3. Now we turn to the proof of Theorem 2. Note that throughout the proof C will denote a generic positive constant.Moreover, remind that ϕ α (I) = I δ (α) /2−1 for α ∈ I below.
Proof.For α ∈ I write expression (25) as dZ ′ , and Under assumption (28) the following bound for the transition probabilities W α βγ = W α βγ (Z, Z * , Z ′ ), see also Remark 1, may be obtained It follows immediately, by expression (28) for k 1α βγ = k 1α βγ (Z, Z * ), that Firstly, assume that γ ∈ I poly .By relation (6), since δ (γ) ≥ 2, Then, since Furthermore, it follows, for h N given by notation (34), that Hence, k 1α βγ (Z, Z * ) 1 hN ∈ L 2 (dZ dZ * ) for any (nonzero) natural number N , since Then It follows that, sup Now, assume that (β, γ) ∈ I 2 mono .Then Furthermore, it follows, for h N given by notation (34), that Furthermore, It follows, for h N given by notation (34), that Furthermore, Then by Lemma 3, Under assumption (28) the following bound for the transition probabilities W γ αβ = W γ αβ (Z ′ , Z, Z * ), see also Remark 1, may be obtained Here the first inequality in the following bounds was applied to obtain the last bound,

Bounds on the collision frequency
This section concerns the proof of Theorem 3. Note that throughout the proof, C > 0 will denote a generic positive constant.Moreover, remind that ϕ α (I) = I δ (α) /2−1 for α ∈ I below.