On a diffusion-kinetic equation arising in extended kinetic theory

Jacek Banasiak^{∗} Department of Mathematics and Applied Mathematics University of Natal, Durban 4041, SOUTH AFRICA

Abstract

This paper is devoted to the study of solvability of an evolution equation obtained as a hydrodynamic limit of a linear Boltzmann equation with inelastic scattering term. An interesting feature of this equation is that it combines the diffusion operator and a singular operator of a kinetic type. The study is carried out in the L_{1} space which is natural from physical point of view and allows to apply modifications of methods of the kinetic theory. Key words: evolution equations, positive semigroups, inelastic scattering, singular perturbations, diffusion approximation

AMS Subject Classification 34G10, 82C40, 47D06, 35Q35

1 Introduction

In recent years there has been a renewed interest in kinetic models describing processes involving inelastic scattering of particles, i.e., scattering involving an exchange of quanta of energy between the test particles and the background (see e.g. [10]). Such models appear in the semiconductor theory where they describe scattering of electrons on impurities and/or on crystalline lattice (e.g. [12, 13, 14]); they also describe phenomena occurring in the neutron and electron transport in gases and are relevant to scattering of high energy neutrons on nuclei.

Linear Boltzmann equation describing such phenomena has the following form

∂_{t}f = ϵ^{−p}Sf + ϵ^{−q}C^{i}f + ϵ^{−r}C^{e}f,  (1.1)

where (t, x, v) → f(t, x, v) is the distribution function of particles in the 6-dimensional phase space ℝ_{x,v}^{6}, S is the streaming operator (Sf)(x, v) = −v · ∂_{x}f(x, v), and C^{i} and C^{e} are, respectively, inelastic and elastic scattering operators defined below by Eqs. (2.3) and (2.4). The number ϵ is a small positive parameter, p, q, r ∈ ℕ_{0}, and the resulting coefficients ϵ^{−p}, ϵ^{−q}, ϵ^{−r} reflect possibly different time-scales of the processes modeled by S, C^{i} and C^{e}.

∗Research supported by the Foundation of Research and Development of South Africa

In this paper we shall not discuss all the possible hydrodynamic limits corresponding to various values of p, q and r. An exhaustive treatment of these can be found in [4, 7, 8]. Here we shall concentrate only on the case with p = 1, q = 0 and r = −2 which, in our opinion, is the most interesting from the mathematical point of view.

Standard asymptotic procedure, outlined in the Appendix, shows that in the asymptotic limit, as ϵ → 0, the suitably defined hydrodynamic quantity ρ should satisfy the following equation

∂_{t}ρ = Dρ + Kρ,  (1.2)

where D is a diffusion operator and K is an unbounded operator structurally similar to C^{i}.

The main aim of this paper is to prove that a suitable realization of the operator D + K in the space X = L_{1}(ℝ^{6}) is the generator of a positive semigroup which is conservative on the positive cone of X.

In Section 2 we collect all the basic definitions and prove that the operator D obtained via the compressed asymptotic procedure is indeed a diffusion operator in spatial variables.

In Section 3 we shoe that a modification of the Miyadera-Voigt perturbation theory, developed in [1] for the transport equation, can be used also for (1.2). This method can be used in many cases when other generalizations of the Miyadera-Voigt theory (see e.g. [17]) fail, but unfortunately it enables to prove only that there exists a semigroup (G(t))_{t≥0} generated by an extension T_{e} of D + K without providing a constructive characterization of T_{e}. In particular, it doesn’t say whether T_{e} coincides with D + K, or even with \overline{D + K}, and these are the only physically relevant cases, as only then one can prove that the total number of particles is preserved throughout the evolution.

We deal with the problem of characterization of T_{e} in Section 4. In Subsection 4.1 we prove that for the class of scattering potentials including Maxwell and soft potentials we have T_{e} = D + K, and in Subsection 4.2 we show that for all other potentials T_{e} = D + K. The proofs of both these facts utilize suitably modified techniques developed in [1, 2].

In the Appendix we collect mathematical properties of the elastic scattering operator C^{e}, which allow the asymptotic analysis of Eq. (1.1) and briefly sketch the asymptotic procedure leading to Eq. (1.2).

Acknowledgement. The author is grateful to Prof. G. Frosali and Dr L. Demeio for pointing out the interesting phenomena which occur for different scalings of Eq. (1.1).

2 Formal asymptotic analysis

Let us start with definitions of scattering operators appearing in Eq. (1.1). The operator describing elastic collisions is given by

(C^{e}f)(x, v) = −f(x, v) ∫_{S^{2}} λ(x, v, ω · ω′)dω′ + ∫_{S^{2}} λ(x, v, ω · ω′)f(x, vω′)dω′  (2.3)

where S^{2} denotes the unit sphere in ℝ_{v}^{3} and v = vω with v ∈ [0, ∞[ and ω ∈ S^{2}. Since in the presented considerations λ does not play any significant role, for simplicity we assume that λ is a measurable, bounded function:

λ_{min} ≤ λ(x, v, z) ≤ λ_{max}

for all (x, v, z) ∈ ℝ^{3} × [0, ∞[×[−1, 1] and some 0 < λ_{min} ≤ λ_{max} < +∞.

The inelastic scattering operator is of the form

(C^{i}f)(x, v) = −f(x, v)\frac{1}{v} ∫_{S^{2}} (n_{2}v_{+}^{2}ν(v_{+}, ω · ω′) + n_{1}H(v_{−}^{2})v^{2}ν(v, ω · ω′)) dω′

+\frac{n_{1}}{v} ∫_{S^{2}} v_{+}^{2}ν(v_{+}, ω · ω′)f(x, v_{+}ω′)dω′ + \frac{n_{2}}{v} ∫_{S^{2}} H(v_{−}^{2})v^{2}ν(v, ω · ω′)f(x, v_{−}ω′)dω′.(2.4)

where H denotes the Heavyside function and v_{±} = \sqrt{v^{2} ± 1}. We assume that ν is a measurable function satisfying the following requirements:

ν_{min}v^{s−1} ≤ ν(v, ω · ω′) ≤ ν_{max}v^{s−1} for v ∈ [1, ∞[,  (2.5)

where s ≤ 1. For technical reasons we had to assume that ν is independent of the spatial variable x.

Remark 2.1 From physical point of view the assumptions mean that the elastic scattering is governed by Maxwell type potential whereas the inelastic scattering is quite arbitrary: the case of hard spheres corresponds to s = 1, for hard potentials we take 0 < s < 1, soft potentials are obtain with s < 0 and the Maxwell potential is recovered with s = 0.

As stated in the Introduction, we are interested in the case with p = 1, q = 0 and r = −2. For these values of parameters we obtain the following version of Eq. (1.1)

∂_{t}f = C^{i}f + \frac{1}{ϵ}Sf + \frac{1}{ϵ^{2}} C^{e}f.  (2.6)

Remark 2.2 With this scaling the elastic collisions are the dominant force driving the system towards the equilibrium with inelastic collisions acting on much slower time-scale. Yet, interesting enough, in the hydrodynamic limit the inelastic scattering contribution is of the same order of magnitude as the diffusion.

It can be proved (see e.g. [7, 8]) that the proper hydrodynamic quantity for Eq. (2.6) (which in the functional-analytic terminology is the spectral projection of the solution f to (2.6) onto the null-space of the operator C^{e}) is given by

\overline{ρ}(t, x, v) = (ℙf)(t, x, v) = \frac{1}{4π} ∫_{S^{2}} f(t, x, vω)dω.  (2.7)

The compressed (abstract Chapman-Enskog) asymptotic method [16] gives the following abstract ”drift-diffusion” equation, the solution of which, ρ, is expected to approximate \overline{ρ}

∂_{t}ρ = ℙC^{i}ℙρ + Dρ,  (2.8)

where the abstract ”diffusion operator” is given by

D = −ℙSℚ(ℚC^{e}ℚ)^{−1}ℚSℙ  (2.9)

with the transport operator given by S = −vω · ∂_{x} and ℚ = I − ℙ. The derivation of (2.8) is briefly sketched in Appendix. Here we shall prove that, for each fixed v > 0, D is indeed a diffusion operator in x variable.

Proposition 2.1 We have

(Dρ)(x, v) = ∂_{x}(d(x, v)∂_{x}ρ(x, v))  (2.10)

with d > 0 for almost all x, v.

Proof. Firstly, since ℙf is independent of ω and S is linear in ω, we have ℙSℙf = 0. Next

ℙSf = \frac{1}{4π} v∂_{x} ∫_{S^{2}} ωfdω

and ℙSℚ = ℙS, ℚSℙ = Sℙ. By Theorem 5.1 ℚC^{e}ℚ is invertible on W. It is also homogeneous in v, and since ω · ∂_{x}ρ ∈ W, we obtain

Dρ = \frac{v^{2}}{4π} ∂_{x} · (∫_{S^{2}} (ω(ℚC^{e}ℚ)^{−1}ω · ∂_{x}ρ)dω) = v^{2} ∑_{i,j=1}^{3} ∂_{x_{i}} (d_{ij}∂_{x_{j}}ρ) ,  (2.11)

where d_{ij} = − ∫_{S^{2}} ω_{i}F_{j}dω and F_{j} is the unique in W solution to

F_{j}(x, v) ∫_{S^{2}} λ(x, v, ω · ω′)dω′ − ∫_{S^{2}} λ(x, v, ω · ω′)F_{j}(x, vω · ω′)dω′ = ω_{j}.  (2.12)

Due to the rotational invariance, we can always select the coordinate axes in such a way that ω = ω_{j}e_{j} so that ω · ω′ = ω_{j} and Eq. (2.12) becomes only ω_{j} dependent. Since the structure of the equation remains unchanged, the considerations of Theorem 5.1 applies giving the solution of this equation depending only on ω_{j}: F_{j}(x, vω) = F̄_{j}(x, v, ω_{j}). Since the original equation is uniquely solvable in W, F̄_{j} is the sought solution of Eq. (2.12). Hence,

d_{ij}(x, v) = − ∫_{S^{2}} ω_{i}F̄_{j}(x, v, ω_{j})dω = \{ \begin{matrix}−d_{i} & if & j = i \\ 0 & if & j ≠ i \end{matrix}

Using again the rotational invariance, we reach the conclusion that d_{i}(x, v) = d(x, v) for i = 1, 2, 3. Next we observe that, under given assumptions on λ, the operator C^{e} is a bounded operator in L_{2}(S^{2}) (for almost every x, v). Moreover, for f belonging to

W_{2} = {f ∈ L_{2}(S_{2}); ∫_{S^{2}} fdω = 0}

we obtain (dropping for a moment an unessential dependence on x and v)

∫_{S^{2}} (fC^{e}f)(ω)dω = −\frac{1}{2} ∫_{S^{2}} ∫_{S^{2}} λ(ω · ω′)(f(ω) − f(ω′))(f(ω) − f(ω′))dωdω′

≤ −\frac{1}{2}λ_{min} (∫_{S^{2}} ∫_{S^{2}} f^{2}(ω)dωdω′ + ∫_{S^{2}} ∫_{S^{2}} f^{2}(ω′)dωdω′

− ∫_{S^{2}} ∫_{S^{2}} f(ω)f(ω′)dωdω′ − ∫_{S^{2}} ∫_{S^{2}} f(ω′)f(ω)dωdω′)

≤ −4πλ_{min}∥f∥_{L_{2}(S^{2})}.

This shows that ℚC^{e}ℚ is invertible in W_{2}. Let f = (ℚC^{e}ℚ)^{−1}(ω_{i}); then

d = − ∫_{S^{2}} ω_{i}F̄_{i}(ω_{i})dω = − ∫_{S^{2}} f(ℚC^{e}ℚf)dω ≥ 4πλ_{min}∥f∥_{L_{2}(S^{2})}.

Consequently, the operator D defined by (2.9) is given by (2.10).

Next we turn our attention to the term ℙC^{i}ℙ. For v ≥ 1 we denote

m(v) = v ∫_{S^{2}} ∫_{S^{2}} ν(v, ω · ω′)dω′dω.

Because ρ = ℙf is independent of ω, we obtain

ℙC^{i}ℙf = −\frac{1}{4π} (\frac{n_{2}v_{+}}{v} m(v_{+}) + n_{1}H(v_{−}^{2})m(v)) ρ + \frac{1}{4π} (\frac{n_{1}v_{+}}{v} m(v_{+})ρ_{+} + n_{2}H(v_{−}^{2})m(v)ρ_{−}) ,

(2.13)

where ρ_{±}(x, v) = ρ(x, \sqrt{v^{2} ± 1}).

3 Solvability of the drift-diffusion equation

According to the previous section we have to prove the solvability of the following Cauchy problem

∂_{t}ρ = v^{2}∂_{x} (d∂_{x}ρ) − \frac{1}{4π} (\frac{n_{2}v_{+}}{v} m(v_{+}) + n_{1}H(v_{−}^{2})m(v)) ρ

+\frac{1}{4π} (\frac{n_{1}v_{+}}{v} m(v_{+})ρ_{+} + n_{2}H(v_{−}^{2})m(v)ρ_{−}) ,

ρ(0) = ρ^{∘}  (3.14)

in X = X_{x} ⊗ X_{v} = L_{1}(ℝ_{x}^{3}) ⊗ L_{1}(ℝ_{v}^{3}). Since in the velocity space X_{v} all the terms have been averaged with respect to ω, this space is reduced to L_{1}(ℝ_{+}, v^{2}dv).

Let us consider the operators appearing in Eq. (2.13). Firstly, in the operator D the variable v is a parameter, hence we can consider operators D_{v} (D with fixed v), acting in X_{x}. From [4] it follows that for each fixed v ≥ 0 the operator D_{v} generates a positive semigroup of contractions in X_{x} with domain L_{1,2}(ℝ^{3}) (Bessel potential space, see e.g. [11]) for v > 0 and L_{1}(ℝ^{3}) for v = 0. We denote these semigroups by (G_{D_{v}}(t))_{t≥0} and proceeding as in [4] we obtain that a semigroup of contractions in X is generated by D defined on the natural domain

D(D) = {f ∈ X; f(·, v) ∈ D(D_{v}), (x, v) → (D_{v}f)(x, v) ∈ X}.

In a standard way we can prove that this semigroup is conservative for positive initial data. In fact, let us first consider the operators D_{v} with fixed v > 0. It follows that the domains D_{v}^{k} coincide with the Bessel potential space L_{1,2k}(ℝ^{3}). Hence, for nonnegative ϕ^{∘}∈ C_{0}^{∞} we have f(t) = G_{D_{v}}(t)ϕ ∈ W_{1}^{k}(ℝ^{3}) for t ≥ 0 and for arbitrary k. Therefore ∫_{ℝ^{3}}(D_{v}f)(t, x)dx = 0

and ∥f(t)∥_{X_{x}} = ∥ ϕ^{∘} ∥_{X_{x}}. Since any nonnegative L_{1}(ℝ^{3})-function can be approximated in norm by nonnegative C_{0}^{∞} function, we obtain

∥G_{D_{v}}(t) f^{∘} ∥_{X_{x}} = ∥ f^{∘} ∥_{X_{x}}.  (3.15)

Integration with respect to v yields the statement for the semigroup (G_{D}(t))_{t≥0}.

Let us define

(Aρ)(v) = α(v)ρ(v) = \frac{1}{4π} (\frac{n_{2}v_{+}}{v} m(v_{+}) + n_{1}H(v_{−}^{2})m(v)) ρ(v).

We see that A is unbounded in X_{v} due to Eq. (2.5), and c_{1}v^{−1} ≤ α(v) ≤ c_{2}v^{−1} as v → 0^{+} and C_{1}v^{s} ≤ α(v) = C_{2}v^{s}, as v → ∞, where s ≤ 1.

By K we denote the sum of the other two operators

(Kρ)(v) = (K_{−}ρ)(v) + (K_{+}ρ)(v) = \frac{n_{1}v_{+}}{v} m(v_{+})ρ_{+}(v) + n_{2}H(v_{−}^{2})m(v)ρ_{−}(v).

It follows (see e.g. (3.22)) that D(K) = D(A) (the detailed analysis of domains of these operators is given in [3]). We shall prove the following theorem.

Theorem 3.1 There exists a semigroup of contractions, say (G(t))_{t≥0}, generated by an extension T_{e} of the operator (T , D(T )) = (D − A + K, D(D) ∩ D(A))

Proof. The proof follows the lines of Arlotti’s generalization of the Miyadera-Voigt perturbation theorem ([1]).

Firstly, we define D_{α} = D − A and consider the Cauchy problem

∂_{t}ρ = D_{α}ρ, ρ(0) =ρ^{∘}  (3.16)

in the space X.

Since A and D commute, D_{α} generates a semigroup (of contractions) in X such that D(D)∩ D(A) is a core of D(D_{α}), [18]. Since v here is a parameter, we consider the family of operators D_{α,v} = D_{v} − α(v)I in X_{x}. Clearly, also each D_{α,v}, defined on D(D_{α,v}) = D(D_{v}) generates a semigroup of contractions, say (G_{D_{α,v}}(t))_{t≥0}, which satisfies

∥G_{D_{α,v}}(t)ρ(·, v)∥_{X_{x}} ≤ exp(−α(v)t)∥ρ(·, v∥_{X_{x}},  (3.17)

with the equality sign if ρ is positive. As the first step, we shall identify D(D_{α}). Using (3.17) and the semigroup representation of the resolvent we obtain for λ > 0

∥(R(λ, D_{α,v})ρ)(·, v)∥_{X_{x}} ≤ \frac{1}{λ + α(v)}∥f(·, v)∥_{X_{v}}.  (3.18)

By (2.5) and (3.18) we obtain for any f ∈ X

∥α(v)(R(λ, D_{α,v})f)(·, v)∥_{X_{x}} ≤ \frac{α(v)}{λ + α(v)}∥f(·, v)∥_{X_{x}} ≤ ∥f(·, v)∥_{X_{x}},  (3.19)

hence

∥AR(λ, D_{α})∥_{X} ≤ ∥f∥_{X}  (3.20)

and D(D_{α}) ⊂ D(A).

Using the fact that for a fixed v the operator A_{v} is bounded, we obtain the following relation for resolvents in X_{x}:

R(λ, D_{α,v}) = R(λ, D_{v})[I − α(v)R(λ, D_{α,v})].

Since the operator AR(λ, D_{α}) is bounded in X by Eq. (3.20), the above equation shows that D(D_{α}) ⊂ D(D) and consequently D(D_{α}) ⊂ D(D) ∩ D(A) which yields

D(D_{α}) = D(D) ∩ D(A).)  (3.21)

Next we note that for any ρ ∈ X we have

∫_{0}^{+∞}K|ρ(x, v)|v^{2}dv = ∫_{0}^{+∞}\frac{n_{1}v_{+}}{v} m(v_{+})|ρ_{+}(x, v)|v^{2}dv + n_{2} ∫_{1}^{∞}m(v)ρ_{−}(x, vv^{2}dv

= n_{1} ∫_{0}^{+∞}H(z_{−}^{2})m(z)|ρ(z)|z^{2}dz + n_{2} ∫_{0}^{+∞}\frac{u_{+}}{u} m(u)|ρ(u)|u^{2}du

= ∫_{0}^{+∞}α(v)|ρ(x, v)|v^{2}dv,  (3.22)

and, using the fact that for positive operators we have |Kρ| ≤ K|ρ|, we obtain

∥Kρ∥_{X} ≤ ∥αρ∥_{X}.  (3.23)

If ρ is the solution to (3.16), then (3.23) gives

∫_{0}^{t} ∥Kρ(s)∥_{X}ds ≤ ∫_{ℝ_{v}^{3}} (α(v)∥ ρ^{∘} (·, v)∥_{X_{x}} (∫_{0}^{t} e^{−α(v)s}ds)) dv

≤ ∫_{ℝ_{v}^{3}} (1 − e^{−α(v)t})∥ ρ^{∘} (·, v)∥_{X_{x}}dv ≤ ∥ ρ^{∘} ∥_{X} − ∫_{ℝ_{v}^{3}} e^{−α(v)t}∥ ρ^{∘} (·, v)∥_{X_{x}}dv
≤ ∥ ρ^{∘} ∥_{X} − ∥G_{D_{α}}(t) ρ^{∘} ∥_{X},  (3.24)

where we used (3.17) in the reverse direction: −e^{−α(v)t}∥ ρ^{∘} ∥_{X_{x}} ≤ −∥G_{D_{α,v}}(t) ρ^{∘} ∥_{X_{x}}.

This estimate allows to use verbatim the approach of [1]. Namely, we define iterates: S_{0}(t) = G_{D_{α,v}}(t) and

S_{n+1}(t)f = ∫_{0}^{t} S_{n}(t − s)KG_{D_{α,v}}(s)fds.  (3.25)

It is easy to see that each S_{n} is a strongly continuous function on [0, ∞[ and it follows from (3.24) that for any f ∈ X and t ≥ 0

∥S_{n}(t)f∥_{X} ≤ ∥f∥_{X} − ∑_{k=0}^{n−1} ∥S_{k}(t)f∥_{X}.  (3.26)

From this estimate we see that the series ∑_{k=0}^{∞} S_{k}(t) converges to a strongly continuous function

of t ∈ [0, ∞[ with values in the space of bounded linear operators in X

G(t) = ∑_{k=0}^{∞} S_{k}(t).

It follows [1] that G is a strongly continuous semigroup whose generator is an extension T_{e} of T .

4 Identification of the domain of T_{e}

The problem of identifying the actual domain of T_{e} is not trivial (see [1, 2]). Below we shall present a modification of techniques used in [1, 2]) which can be used in the discussed case. We shall prove that if s ≤ 0, then D(T_{e}) = D(T ) and if 0 < s ≤ 1, then T_{e} = T̄.

4.1 The case s ≤ 0

Before we formulate the main result of this subsection, we note that the same arguments as in [1] can be used to prove that the iterates S_{n} can be found also from the following recurrence relation:

S_{n+1}(t)f = ∫_{0}^{t} G_{D_{α}}(t − s)KS_{n}(s)fds,  (4.1)

and for each f ∈ D(D_{α}) the function t → AS_{n}(t)f is continuous.

Theorem 4.1 If s ≤ 0, then the domain of the generator T_{e} coincides with D(T ).

Proof. We split ℙC^{i}ℙ = C_{+}^{i} + C_{−}^{i}, where

C_{+}^{i}ρ = −α_{+}ρ + K_{+}ρ = −\frac{n̄_{2}v_{+}}{v} m(v_{+}) + n̄_{2}H(v_{−}^{2})m(v)ρ_{−}
C_{−}^{i}ρ = −α_{−}ρ + K_{−}ρ = −n̄_{1}H(v_{−}^{2})m(v)ρ + \frac{n̄_{1}v_{+}}{v} m(v_{+})ρ_{+}  (4.2)

where n̄_{i} = n_{i}/4π, i = 1, 2. Since s ≤ 0, the operator C_{−}^{i} is bounded. Hence, due to the Bounded Perturbation Theorem, it is enough to consider the generation of the semigroup by D+C_{+}^{i}. The operator C_{+}^{i} has all the necessary properties to prove Theorem 3.1 so that there exists a semigroup, say (G_{+}(t))_{t≥0}, generated by an extension T_{e,+} 0f of D + C_{+}^{i}. We denote by (Z_{+}(t))_{t≥0} the auxiliary semigroup generated by D − α_{+}I and by S_{n,+} the corresponding iterates defined as in Eq. (4.1). For f ∈ D(C^{i}) we have

∥α_{+}Z_{+}(t)f∥_{X} ≤ ∫_{ℝ_{v}^{3}} α_{+}(v)e^{−α_{+}(v)t}∥(G_{D}(t)f)(·, v)∥_{X_{x}}dv ≤ ∥α_{+}f∥.  (4.3)

In particular, Z_{+}(t)f ∈ D(K).

Due to assumptions (2.5) we can write

c_{1}vi(v) ≤ m(v) ≤ c_{2}vi(v)  (4.4)

for all v ≥ 1, where i(v) = min(1, v^{s−1}). Using this notation we have

∥α_{+}K_{+}f∥ ≤ M′ ∫_{S^{2}} ( ∫_{1}^{+∞}v_{+}^{2}i(v_{+})i(v)∥f(·, v_{−})∥_{X_{x}}v^{2}dv) dω′

= M′ ∫_{S^{2}} (∫_{0}^{∞}(z^{2} + 2)i(\sqrt{z^{2} + 2})i(z_{+})z_{+}z^{−1}∥f(·, z)∥_{X_{x}}z^{2}dz) dω′
≤ M∥α_{+}f∥_{X},  (4.5)

for some constant M, where we used \sqrt{z^{2} + 2} ≤ \sqrt{2}\sqrt{z^{2} + 1} and \sqrt{z^{2} + 2}i(\sqrt{z^{2} + 2}) ≤ 1 for z ≥ 0, and the first part of estimate (4.4).

Now, by (4.1), (4.3) and (4.5) we see that for f ∈ D(C^{i})

∥α_{+}S_{n,+}(t)f∥ ≤ ∫_{0}^{t} ∥α_{+}Z_{+}(t − s)K_{+}S_{n−1,+}(s)f∥ds ≤ ∫_{0}^{t} ∥α_{+}K_{+}S_{n−1}(s)fds∥

≤ M ∫_{0}^{t} ∥α_{+}S_{n−1}(s)f∥ds  (4.6)

Since, as in Eq. (3.24), we have

∥α_{+}S_{1,+}(t)f∥ ≤ M ∫_{0}^{t} ∥α_{+}Z_{+}(s)f∥ds ≤ M∥f∥,

by induction and density we get

∥α_{+}S_{n,+}(t)f∥ ≤ M^{n} \frac{t^{n−1}}{(n − 1)!}∥f∥  (4.7)

for any f ∈ X. From (4.7) we obtain for λ > M

∫_{0}^{∞}e^{−λs}∥α_{+}G_{+}(t)f∥ds ≤ ∫_{0}^{∞}e^{−λs}∥α_{+}Z_{+}(t)f∥ds + ∥f∥ ∫_{0}^{∞}e^{−λs}M ∑_{n=0}^{∞} \frac{M^{n}s^{n}}{n!} ds

≤ ∥f∥ + \frac{M}{λ − M} ∥f∥ = \frac{λ}{λ − M} ∥f∥  (4.8)

where we used again the estimate (3.24). Since we know that each function t → α_{+}S_{n,+}(t)f is continuous whenever f ∈ D(D_{α}), we obtain that t → α_{+}G_{+}(t)f is continuous for such f as the sum of an almost uniformly converging series. Together with (4.8) this shows that

α_{+} ∫_{0}^{∞}e^{−λs}G_{+}(s)fds = ∫_{0}^{∞}e^{−λs}α_{+}G_{+}(s)fds.

for f ∈ D(D_{α}) and this can be extended onto X by density due to closedness of α_{+}I and (4.8). Next, since continuity of t → α_{+}G_{+}(t)f yields the continuity of t → K_{+}G_{+}(t)f for f ∈ D(D_{α}), we get similarly

K_{+} ∫_{0}^{∞}e^{−λs}G_{+}(s)fds = ∫_{0}^{∞}e^{−λs}K_{+}G_{+}(s)fds  (4.9)

for such f. By (4.8) the left hand side can be extended onto X, and since by (3.23) the convergence of α_{+}f_{n} to α_{+}f implies that K_{+}f_{n} converges to K_{+}f, we obtain that Eq. (4.9) holds on X. Now we have by (4.7)

G_{+}(t)f = ∑_{n=0}^{∞} S_{n,+}(t)f = Z_{+}(t)f + lim_{N→∞} ∫_{0}^{t} Z_{+}(t − s) (∑_{n=1}^{N} K_{+}S_{n−1,+}f) ds

= Z_{+}(t)f + ∫_{0}^{t} Z_{+}(t − s)K_{+}G_{+}(s)fds  (4.10)

and this holds for any f ∈ X. Operating with the Laplace transform on both sides of (4.10) we obtain by [19], p. 8 and the abstract convolution formula

(λI − T_{e,+})^{−1} = (λI − (D − α_{+}I))^{−1}f + (λI − (D − α_{+}I))^{−1}K_{+}(λI − T_{e,+})^{−1}f

which shows that D(T_{e,+}) ⊂ D(D − α_{+}I) and consequently T_{e,+} = D + C_{+}^{i}. Since C_{−}^{i} is a bounded operator, we obtain T_{e} = D + C_{+}^{i} + C_{−}^{i} = D + C^{i}.

4.2 The case 0 < s ≤ 1

In this case the multiplying function α is unbounded also as v tends to infinity and consequently both C_{+}^{i} and C_{−}^{i} are unbounded. To deal with this case we adapt another method, also developed by Arlotti [2]. As this method is not well-established, we shall briefly sketch the main ideas specified to our case.

We consider the measure space (ℝ_{x,v}^{6}, dxdv) where dxdv is the Lebesgue measure, and the Banach space X = L_{1}(ℝ_{x,v}^{6}, dxdv) endowed with the usual norm. We have proved that the operators D, A and K acting in X have following properties:

(a) D and D_{α} = D − A are infinitesimal generators of two positive contraction semigroups (G_{D}(t))_{t≥0} and (G_{D_{α}}(t))_{t≥0}; K is a positive operator.

(b) D(A) = D(K) = D and

∫_{ℝ^{6}} (−Af + Kf)dxdv = 0,

for all f ∈ D_{+} (where D_{+} is the intersection of D with the positive cone X_{+} of X.

From Theorem 3.1 we know that there exists a positive contraction semigroup (G(t))_{t≥0} such that its generator T_{e} is an extension of D − A + K. To provide a better characterization of D(T_{e}) we extend all the discussed operators in the following way.

Let E denote the set of all the extended real-valued measurable functions defined on ℝ_{x,v}^{6}; clearly X ⊂ E. We define the subset F ⊂ E by the following condition: f ∈ F if and only if for every nonnegative and nondecreasing sequence of functions (f_{n})_{n∈ℕ} satisfying sup_{n} f_{n} = |f| we have sup_{n}(I − D_{α})^{−1}f_{n} ∈ X.

We also assume that f ∈ D(K) if and only if f^{+} = max{f, 0}, f^{−} = max{−f, 0} and |f| all belong to D(K). Through K we construct another subset of E, say G, defined as the set of all functions f ∈ X such that for any nonnegative, nondecreasing sequence (f_{n})_{n∈ℕ} of elements of D(K) such that sup_{n} f_{n} = |f| we have sup ess_{n}Kf_{n} < +∞. We have D(D_{α}) ⊆ D ⊆ G ⊆ X ⊆ F ⊆ E. By D(D_{α})_{+}, D_{+}, G_{+}, X_{+}, F_{+} and E_{+} we denote the corresponding positive cones.

It can be proved that we can correctly define the mapping L : F_{+} → X_{+} by

Lf = sup_{n}(I − D_{α})^{−1}f_{n}

where 0 ≤ f_{n} ≤ f_{n+1} for any n, and sup_{n} f_{n} = f.

To proceed, in [2] two more assumptions were adopted.

(d) The mapping L is injective.

(e) If f ∈ G_{+} and (f′_{n})_{n∈ℕ}, (f′′_{n})_{n∈ℕ} are two nondecreasing sequences of elements of D_{+} satisfying f = sup_{n} f′_{n} = sup_{n} f′′_{n}, then sup_{n} Kf′_{n} = sup_{n} Kf′′_{n}. The common value will be denoted by Kf.

We extend the mapping L onto F by

Lf = Lf^{+} − Lf^{−} ∈ X,

and similarly we define K on G by

Kf = Kf_{+} − K_{−}f,

where the values of K on positive elements were defined in assumption (e).

Denote g = (I − T_{e})h ∈ X. It is known that h can be expressed by

h = ∑_{k=0}^{∞} L(KL)^{k}g.

Let us denote, for any g ∈ X

f_{n} = ∑_{k=0}^{n}(KL)^{k}g

and

h_{n} = Lf_{n}

We note that for nonnegative g we can consider limits of both sequences in the sense of monotonic convergence almost everywhere, as L and K are positive operators. We denote by f and h the respective limits provided they exist. In this case we obtain Lf = h.

The result of [2] which we shall apply here reads as follows.

Theorem 4.2 Let all the above assumptions are satisfied. If for any g ∈ X_{+} the limit of the following real convergent sequence satisfy

lim_{n→∞} ∫_{ℝ^{6}} Dh_{n}dxdv ≤ lim_{n→∞} ∫_{ℝ^{6}} (h − f + Kh)dxdv,

then T_{e} = \overline{D − A + K}.

We have for f ∈ X

(I − D_{α})^{−1}f = ∫_{0}^{∞}e^{−s(1+α(v))}G_{D}(s)fds.

The definition of L given in [2] is rather awkward. We shall interpret it from the point of view of the theory of Sobolev associated semigroups (see e.g. [18], p. 19-20) showing that the assumption (d) is superfluous.

Let us define the space X_{−1} as the completion of X with respect to norm

∥f∥_{−1} = ∥(I − D_{α})^{−1}f∥_{X}.

Then the semigroup (G_{D_{α}}(t))_{t≥0} extends continuously to a semigroup (G_{D_{α,−1}}(t))_{t≥0} in X_{−1} generated by the extension of D_{α} to the domain D(D_{α,−1}) = X ⊂ X_{−1}. The resolvent also extends by density to a bounded one-to-one operator R(1, D_{α,−1}) : X_{−1} → X_{−1} with the range exactly equal to X. We have the following lemma.

Lemma 4.1 The operator L is a restriction of R(1, D_{α,−1}). As a consequence, the assumption (d) is satisfied.

Proof. Let g ∈ X_{+} satisfies g = Lf. This means that g = sup_{n}(I − D_{α})^{−1}f_{n} for a non-decreasing sequence of nonnegative functions f_{n} ∈ X_{+} such that sup_{n} f_{n} = f. Since (I − D_{α})^{−1} is a positive operator, the sequence ((I − D_{α})^{−1}f_{n})_{n=0}^{∞} is also non-decreasing and g ≥ (I − D_{α})^{−1}f_{n} for any n ∈ ℕ. Because g is integrable, we obtain

lim_{n→∞} ∫_{ℝ^{6}} (I − D_{α})^{−1}f_{n}dxdv = ∫_{ℝ^{6}} gdxdv

and

lim_{n→∞} ∫_{ℝ^{6}} |g − (I − D_{α})^{−1}f_{n}|dxdv = ∫_{ℝ^{6}} gdxdv − lim_{n→∞} ∫_{ℝ^{6}} (I − D_{α})^{−1}f_{n}dxdv = 0

This shows that ((I − D_{α})^{−1}f_{n})_{n∈ℕ} converges in X and therefore g = Lf = R(1, D_{α,−1})f. The extension for arbitrary f is done by linearity.

Returning to our problem, we recall that

Kf = \frac{n_{1}v_{+}}{v} m(v_{+})f_{+} + n_{2}H(v_{−}^{2})m(v)f_{−},

where f_{±}(v) = f(\sqrt{v^{2} ± 1}) and it is clear that the assumption (e) is satisfied.

Hence, all the assumptions of Theorem 4.2 are satisfied.

Using the conservativity of the semigroup, Eq. (3.15), we obtain for any f ∈ X_{+}

∫_{ℝ_{x}^{3}} R(1, D_{α,v})fdx = ∫_{0}^{∞}(e^{−s(1+α(v))} ∫_{ℝ_{x}^{3}} G_{D}(t)fdx)ds = (1 + α(v))^{−1} ∫_{ℝ_{x}^{3}} fdx

Let now f ∈ F_{+} and let (f_{n})_{n∈ℕ} be a nondecreasing sequence of nonnegative functions converging to f, which defines Lf. Using the above equation for f_{n} we see that the left hand side converges and hence for almost every v, f(·, v) ∈ L_{1}(ℝ_{x}^{3}) and the above equality can be written as

(1 + α(v)) ∫_{ℝ_{x}^{3}} Lfdx = ∫_{ℝ_{x}^{3}} fdx.

Using the conservativity of G_{D} we obtain that ∫_{ℝ^{6}} Dh_{n}dxdv = 0 for nonnegative h_{n} in the

domain of D, and with the above equality we see that the condition of Theorem 4.2 is reduced to the following statement.

Proposition 4.1 If, for any nonnegative h ∈ L_{1}(ℝ_{v}^{3}) which is such that αh − Kh ∈ L_{1}(ℝ_{v}^{3}), the following holds

∫_{ℝ_{v}^{3}} (α(v)h(v) − (Kh)(v))dv ≤ 0,  (4.11)

then T_{e} = \overline{D − A + K}.

After these preliminaries we can finally prove the main theorem of this subsection.

Theorem 4.3 Let 0 < s ≤ 1. If n_{2}ν_{min}/n_{1}ν_{max} < 1, then T_{e} = \overline{D − A + K}.

Proof. It is enough to prove (4.11) for functions h independent of ω. In such a case it is convenient to re-write this inequality in terms of the energy variable ξ = v^{2} so that it reads

∫_{0}^{∞}(n_{2}\sqrt{ξ_{1}}m(ξ_{1})h(ξ) + n_{1}H(ξ_{−}1)\sqrt{ξ}m(ξ)h(ξ)  (4.12)
−n_{1}\sqrt{ξ_{1}}m(ξ_{1})h(ξ_{1}) − n_{2}H(ξ_{−1})\sqrt{ξ}m(ξ)h(ξ_{−1})) dξ ≤ 0,

where ξ_{±1} = ξ ± 1. This integral is the limit of integrals over the intervals [1/N, N], N ∈ ℕ

b_{N} = ∫_{1/N}^{N} (n_{2}\sqrt{ξ_{1}}m(ξ_{1})h(ξ) + n_{1}H(ξ_{−1})\sqrt{ξ}m(ξ)h(ξ)

−n_{1}\sqrt{ξ_{1}}m(ξ_{1})h(ξ_{1}) − n_{2}H(ξ_{−1})\sqrt{ξ}m(ξ)h(ξ_{−1})) dξ.

Since h is integrable and all the coefficients are regular enough, we have integrability of the first three terms over [1/N, N], N ∈ ℕ, and since b_{N} exists, also

∫_{1/N}^{N} H(ξ_{−1})\sqrt{ξ}m(ξ)h(ξ_{−1})dξ = ∫_{1}^{N} \sqrt{ξ}m(ξ)h(ξ_{−1})dξ < ∞.  (4.13)

Therefore we can integrate each term separately and having changed the variables we get

b_{N} = n_{1} ( ∫_{1}^{1+1/N}\sqrt{ξ}m(ξ)h(ξ)dξ − ∫_{N}^{N+1}\sqrt{ξ}m(ξ)h(ξ)dξ)

+n_{2} ( ∫_{N}^{N+1}\sqrt{ξ}m(ξ)h(ξ_{−1})dξ − ∫_{1}^{1+1/N}\sqrt{ξ}m(ξ)h(ξ_{−1})dξ)

The sequence (b_{N})_{N∈ℕ} is convergent by assumption; we must prove that lim_{N→∞} b_{N} ≤ 0. Let us analyse the terms of the above expression. It is clear that for b′_{N} = ∫_{1}^{1+1/N} \sqrt{ξ}m(ξ)h(ξ)dξ we have lim_{N→∞} b′_{N} = 0. Also, from (4.13) we infer that if b′′_{N} = ∫_{1}^{1+1/N} \sqrt{ξ}m(ξ)h(ξ_{−1})dξ, then lim_{N→∞} b′′_{N} = 0. Let us denote c_{N} = b_{N} − n_{1}b′_{N} + n_{2}b′′_{N}, N ≥ 1. Then we have

c_{N} = −n_{1} ∫_{N}^{N+1}\sqrt{ξ}m(ξ)h(ξ)dξ + n_{2} ∫_{N}^{N+1}\sqrt{ξ}m(ξ)h(ξ_{−1})dξ
≥ −n_{1}ν_{min}N^{S}a_{N} + n_{2}ν_{max}(N + 1)^{S}a_{N−1}  (4.14)

where S = (s+1)/2, a_{N} = ∫_{N}^{N+1}h(ξ)dξ for N ≥ 1 and a_{0} = ∫_{0}^{1} h(ξ)\sqrt{ξ}dξ. Clearly ∑_{N=1}^{∞} \sqrt{N}a_{N} <

∞. From (4.14) we obtain the following inequality

n_{1}ν_{min}a_{N} ≤ −(N + 1)^{S}α^{N} ∑_{i=1}^{N} \frac{α^{−i}c_{i}}{i^{S}(i + 1)^{S}} + α^{N}(N + 1)^{S}n_{1}ν_{min}a_{0}  (4.15)

where α = n_{2}ν_{max}/n_{1}ν_{min}. According to the assumption, α < 1. Assume that lim_{N→∞} b_{N} > 0, then also lim_{N→∞} c_{N} > 0 and for some N_{0} we must have c_{N} ≥ c > 0 for N > N_{0}. From (4.15) we get

n_{1}ν_{min}N^{S}a_{N} ≤ −N^{S}(N + 1)^{S}α^{N} ∑_{i=1}^{N_{0}} \frac{α^{−i}c_{i}}{i^{S}(i + 1)^{S}} − cN^{S}(N + 1)^{S}α^{N} ∑_{i=N_{0}+1}^{N} \frac{α^{−i}}{i^{S}(i + 1)^{S}}
+α^{N}N^{S}(N + 1)^{S}n_{1}ν_{min}a_{0}

and the first and third terms tend to zero. For the middle term we observe that

∑_{i=N_{0}+1}^{N} \frac{α^{−i}}{i^{S}(i + 1)^{S}} ≥ ∫_{N_{0}}^{N−1}\frac{α^{−t}dt}{t^{S}(t + 1)^{S}}

and

lim_{N→∞} N^{S}(N + 1)^{S}α^{N} ∫_{N_{0}}^{N−1}\frac{α^{−t}dt}{t^{S}(t + 1)^{S}} _{N} = −\frac{α}{ln α} > 0.

Hence, for sufficiently large N we have n_{1}ν_{min}N^{S}a_{N} < 0 which is a contradiction, as h ≥ 0. Consequently, lim_{N→∞} b_{N} ≤ 0.

5 Appendix

5.1 Mathematical properties of the elastic scattering operator for

Lorentz gas

Here we shall prove a number of properties of the elastic scattering operator which are relevant to the asymptotic analysis of the Boltzmann equation (2.6).

Theorem 5.1 Let all the assumptions of Section 2 are satisfied. Then the operator C^{e} is a bounded operator in X = L_{1}(ℝ_{x,v}^{6}) with the following properties

(i) For any f ∈ X and any non-decreasing function κ we have

∫_{ℝ^{6}} κ(f)C^{e}fdxdv ≤ 0.  (5.1)

(ii) The null-space N(C^{e}), the range R(C^{e}) and the spectral projection ℙf onto N(C^{e}) are given by

N(C^{e}) = {f ∈ L_{1}(ℝ_{x,v}^{6}); f is independent on ω}.  (5.2)
R(C^{e}) = W = {f ∈ L_{1}(ℝ_{x,v}^{6}); ∫_{S^{2}} fdω = 0}.  (5.3)

ℙf = \frac{1}{4π} ∫_{S^{2}} fdω.  (5.4)

(iii) For f ∈ W we have

∫_{ℝ^{6}} sign(f)C^{e}fdxdv ≤ −4πλ_{min}∥f∥_{X}  (5.5)

and hence the spectral bound s(C^{e}) of C^{e} satisfies s(C^{e}) < −4πλ_{min}.

Proof. Since in this theorem x plays a role of a parameter, we shall drop it from the notation. Using the symmetry of λ in ω and ω′ we get

2 ∫_{S^{2}} (gC^{e}f)(vω)dω = − ∫_{S^{2}} ∫_{S^{2}} λ(v, ω · ω′)(g(vω) − g(vω′))(f(vω) − f(vω′))dωdω′

Taking any bounded non-decreasing function κ : ℝ → ℝ we obtain

2 ∫_{S^{2}} (κ(f)C^{e}f)(vω)dω = − ∫_{S^{2}} ∫_{S^{2}} λ(v, ω·ω′)(κ(f(vω))−κ(f(vω′)))(f(vω)−f(vω′))dωdω′ ≤ 0

Integrating over the remaining variables we get the H-theorem.

Let C^{e}f = 0; then the left hand side of the above equation is zero, and if we take a strictly monotonic κ, then this is possible only if

f(x, vω) = f(x, vω′),  (5.6)

for almost all x, v, that is, to the kernel of C^{e} may belong only functions independent of ω. On the other hand, such functions clearly belong to N(C^{e}), therefore the kernel of C^{e} is given by (5.2).

Denote W = {f ∈ L_{1}(ℝ_{v}^{3}); ∫_{S^{2}} fdω = 0}; for f ∈ W we get

∫_{S^{2}} (sign(f)C^{e}f(vω)dω

= −\frac{1}{2} ∫_{S^{2}} ∫_{S^{2}} λ(v, ω · ω′)(sign(f(vω)) − sign(f(vω′)))(f(vω) − f(vω′))dωdω′

≤ −\frac{1}{2}λ_{min} (∫_{S^{2}} ∫_{S^{2}} sign(f(vω))f(vω)dωdω′ + ∫_{S^{2}} ∫_{S^{2}} sign(f(vω′))f(vω′)dωdω′

− ∫_{S^{2}} ∫_{S^{2}} sign(f(vω))f(vω′)dωdω′ − ∫_{S^{2}} ∫_{S^{2}} sign(f(vω′))f(vω)dωdω′)

≤ −4πλ_{min}∥f∥_{L_{1}(S^{2})}

where, upon integration with respect to the remaining variables, we obtain that C^{e}|_{W} − αI is dissipative for α > −4πλ_{min}. Since C^{e}|_{W} is bounded, C^{e}|_{W} − αI must be m-dissipative, therefore if α > −4πλ_{min}, then α ∈ ρ(C^{e}|_{W}).

It is also clear that the spectral projection onto N(C^{e}) is given by (5.4) which ends the proof of the remaining part of (ii).

5.2 Formal derivation of the drift-diffusion equation

Following the compressed asymptotic method [16] we write the solution f to (2.6) in the following form: f = ℙf + ℚf = ρ + w and accordingly we split the equation getting

∂_{t}ρ = ℙC^{i}ℙρ + ℙC^{i}ℚw + \frac{1}{ϵ}ℙSℚw
∂_{t}w = ℚC^{i}ℙρ + ℚC^{i}ℚw + \frac{1}{ϵ}ℚSℙρ + \frac{1}{ϵ}ℚSℚw + \frac{1}{ϵ^{2}} ℚC^{e}ℚw,  (5.7)

where we used ℙC^{e} = C^{e}ℙ = 0 and ℙSℙ = 0. We expand w = w_{0} + ϵw_{1} + … and leave ρ unexpanded. Substituting this expansion into the second equation and equating coefficients at the same powers of ϵ we get immediately w_{0} = 0 and w_{1} = −(ℚC^{e}ℚ)^{−1}ℚSℙρ. The existence of w_{1} is ensured by Theorem 5.1 (iii). Inserting this expansion for w into the first equation of (5.7) and discarding higher order terms in ϵ we obtain

∂_{t}ρ = ℙC^{i}ℙρ − ℙSℚ(ℚC^{e}ℚ)^{−1}ℚSℙρ.  (5.8)

Remark 5.1 The outlined procedure has proved to yield correct approximations of the solutions to linear kinetic equations for large class of problems [16, 4, 9] and the rigorous analysis of the order of the error for the kinetic equation discussed in this paper can be carried out along the similar lines. A more exhaustive discussion of asymptotic properties for various scalings (1.1) will appear in a separate paper.

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