1.Introduction
The unknown quantity modeled by equations in kinetic theory is the probability distribution function f(t,x,v) of a population of particles which is a function of time, position and velocity. The linear Boltzmann equation describes the evolution of the distribution function modeling the collision of a population of particles with a background medium:
∂tf(t,x,v)+v·∇xf(t,x,v)=σ(x)(∫Vf(t,x,w)dμ(w)−f(t,x,v)),
where
μ is a Borel probability measure on the space of velocities
V. The coefficient
σ(x) is the scattering coefficient of the background material. Our objective is to perform an asymptotic analysis when the background medium is inhomogeneous – say, a composite material with microstructure. We consider two asymptotically small parameters associated with the above kinetic model: (a) mean free path
ε of the particles between two interactions; (b) the scale of heterogeneity
η of the background medium – it can be the average distance between two neighboring inhomogeneities or the average size of the inhomogeneities.
In the above kinetic model, the scattering coefficient σ(x) represents the background medium. The inhomogeneous nature of the background medium implies that smaller the parameter η is, more rapid the oscillations are in σ(x). As is standard in the theory of homogenization, we consider a family of scattering coefficients indexed by η, i.e. ση(x) and study an associated family of solutions to the kinetic model. We also wish to study the evolution of the local equilibria for the above kinetic model. This corresponds to scaling the above kinetic model using parabolic scaling with the parameter ε. The objective of this article is to study the following scaled linear Boltzmann equation
ε∂tfε,η(t,x,v)+v·∇xfε,η(t,x,v)=ση(x)ε(∫Vfε,η(t,x,w)dμ(w)−fε,η(t,x,v))
in the simultaneous limit as both the scaling parameters
ε,
η vanish. The diffusion approximation of the linear Boltzmann equation corresponds to the
ε→0 limit in the above scaled equation (see [
6] and references therein for the state-of-the-art on the techniques used in the diffusion approximation of linear transport equations). Therefore in the regime
ε≪η, one can first perform the diffusion asymptotic for a fixed
η yielding a parabolic equation with heterogeneous coefficient
ση(x). The
η→0 limit corresponds to deriving an homogenized equation for the thus obtained heterogeneous parabolic equation. This can either be performed using the asymptotic expansions method [
11] or the two-scale convergence method [
1,
21] when
ση(x)=σ(x/η) is
η-periodic. The method of H-convergence [
19] (see also [
20]) can be used when the family
ση(x) is a general family of
L∞ coefficients (see [
2] for a pedagogical exposition of the method of H-convergence). In the regime
η≪ε, we need to homogenize the linear Boltzmann equation in the limit
η→0 for a fixed
ε (see [
12] on the homogenization of linear transport equations). This shall be followed by a diffusion asymptotic in the
ε→0 limit. These points have already been observed in an expository article by F. Golse [
13].
The present article addresses the issue of simultaneous limit procedure when both the small parameters ε and η vanish. This problem has been addressed in [5,9,10,22] when the heterogeneous scattering coefficient is periodic. A first work in this direction goes back to the work of R. Sentis [22] where the heterogeneity length scale η is related to the mean free path as η=εβ with β<1. Recently, there has been a revival of this problem. The works [5,9] address this problem in the periodic setting and in the regime ε≪η. The approach in [5,9] is to introduce a new parameter εη and study some cell problems involving the new small parameter εη. They extensively use the method of two-scale convergence. We also cite [3] where the spectral problem associated with the scaled linear Boltzmann equation is studied when η=ε. For the simultaneous limit procedures in the case η=ε, we cite [16,17] (see also Chapter 7 in the lecture notes [4]).
Our work, essentially, goes beyond the periodic setting in the spirit of the compensated compactness theory [18,19] developed by F. Murat and L. Tartar in the context of homogenization of elliptic and parabolic problems in the late 1970s. All the results and computations in this article are presented for a special case of the stationary linear Boltzmann, i.e. the probability density function is supposed to be time independent. All the results can be straightaway generalized to the time-dependent setting (see Remark 4).
Similar to the work of R. Sentis [22], we assume that the two scaling parameters are related as η=εβ where β∈(0,∞). Note, however, that the results in [22] hold only when β<1 and under periodicity assumption on the heterogeneous scattering coefficient. Our main result in one dimensional setting (Theorem 1) is that the solutions to the linear Boltzmann equation converge to the solutions of an elliptic problem whenever β⩽2. We also obtain an explicit form of the effective diffusion coefficient in the limit equation – given in terms of some velocity averages and the weak-* limit of the heterogeneous scattering coefficient. The one-dimensional setting is very special as the divergence operator coincides with the gradient operator resulting in uniform H1 estimates for certain family of second moments – refer to Section 5 for further details.
Our main result in any arbitrary dimension is that if the heterogeneous coefficient satisfies
‖σ¯(x)−σε(x)σ¯(x)‖H−12(Ω)=O(ε1+)
for some
σ¯∈L∞(Ω) and where the family
σε(x) is nothing but the family
ση(x) with
η=εβ, then the solutions to the linear Boltzmann equation converge to the solutions of a diffusion equation in physical space. More precisely, we show that the family
fε,η(x,v)⇀ρ(x)weakly in L2(Ω×V;dxdμ) as ε,η→0,
where
ρ(x) is the
L2-weak limit of the family of local densities and that it is the unique solution to a diffusion equation. Our result in arbitrary dimension (Theorem
2) essentially employs the moments approach inspired by [
7] and uses the regularity of velocity averages guaranteed by the now well-known velocity averaging lemma [
14,
15].
Plan of the paper. In Section 2, we present the linear kinetic model, the scaling parameters and the scaling considered in this article. Section 3 gives some uniform (with respect to the scaling parameters ε and η) estimates on the solutions to the linear transport equation and associated velocity averages. In Section 4, we briefly explain the moments method as given in [7] and apply this method to the linear Boltzmann equation. Section 5 is devoted to deriving the limit equation (in the ε→0 limit) in one dimensional setting – Theorem 1. In Section 6, we give a result in any arbitrary dimension under a certain assumption on the scattering coefficient. Finally, in Section 7 we give some concluding remarks and perspectives.
2.Stationary linear Boltzmann equation
Let f(x,v) be the distribution function which depends on x∈Ω⊂Rd (space position) and v∈V (velocity). The distribution function models the probability density of mono-kinetic particles interacting with the background medium. The velocity space can be either of the following:
V=Rd;V=Sd−1;V=B(0,l):={v∈Rd s.t. |v|⩽l}.
We denote by
μ a Borel probability measure on
V. We further suppose that
(1)∫Vvdμ(v)=0.
In order to define the boundary conditions, taking
n(x) to be the unit exterior normal to Ω at the point
x∈∂Ω, we introduce the following notations:
Σ:={(x,v)∈∂Ω×V}Phase space Boundary,Σ+:={(x,v)∈∂Ω×V s.t. v·n(x)>0}Outgoing Boundary,Σ−:={(x,v)∈∂Ω×V s.t. v·n(x)<0}Incoming Boundary,Σ0:={(x,v)∈∂Ω×V s.t. v·n(x)=0}Grazing set.
Denote by
γ+f (respectively
γ−f), the trace of
f on
Σ+ (respectively on
Σ−).
The goal is to perform the simultaneous limiting procedure for the scaled problem:
(2a)fε,η(x,v)+1εv·∇xfε,η(x,v)+1ε2ση(x)Lfε,η(x,v)=g(x)for (x,v)∈Ω×V,(2b)γ−fε,η(x,v)=0for (x,v)∈Σ−,
where the linear Boltzmann operator is the following integral operator:
(3)Lg(v):=g(v)−∫Vg(w)dμ(w)for any g∈L1(V;dμ).
The family of heterogeneous scattering coefficients indexed by
η, i.e.
ση(x) is assumed to be a family of differentiable function such that there exist uniform (with respect to
η) constants
a,b,c such that
0<a⩽ση(x)⩽b<+∞∀x∈Ω
and
‖∇ση‖L∞(Ω)⩽cη−1.
The source term in (
2a) is assumed to be square-integrable in the space variable, i.e.
‖g‖L2(Ω)⩽C.
We suppose that the heterogeneity length scale
η and the mean free path
ε are related as
η=εβfor β>0.
Hence we drop the superscript
η in (
2a) and index the family of heterogeneous scattering coefficients by
ε, i.e.
σε(x) which inherits the following bounds from
ση given above.
(4)0<a⩽σε(x)⩽b<+∞∀x∈Ωand‖∇σε‖L∞(Ω)⩽cε−β,
where the constants
a,b,c are uniform with respect to
ε.
Remark 1.
Consider σ∈W1,∞(Rd) such that for a.e. x∈Rd,
0<a⩽σ(x)⩽b<∞and‖∇σ‖L∞(Rd)⩽c<∞.
Now consider the family
σε(x):=σ(xεβ).
Then, the assumptions given in (
4) on the heterogeneous coefficient are satisfied by the above defined family of coefficients.
The main objective of this article is to study the following stationary problem in the ε→0 limit:
(5a)fε(x,v)+1εv·∇xfε(x,v)+1ε2σε(x)Lfε(x,v)=g(x)for (x,v)∈Ω×V,(5b)γ−fε(x,v)=0for (x,v)∈Σ−.
We shall prove that the entire family
fε(x,v) of solutions to the above kinetic equation exhibit the following compactness property:
fε(x,v)⇀ρ(x)weakly in L2(Ω×V;dxdμ(v)),
where
ρ(x) is the
L2-weak limit of the associated local densities, i.e.
∫Vfε(x,v)dμ(v)⇀ρ(x)weakly in L2(Ω)
and the above weak limit uniquely solves a second order elliptic equation:
Aρ(x)=g(x)in Ω.
Further details on the elliptic operator
A shall be given in Sections
5 and
6.
3.Uniform a priori bounds
In order to perform the asymptotic analysis in the ε→0 limit for (5a)–(5b), we derive uniform (with respect to ε) L2-estimates on the solution family fε(x,v) and some associated velocity averages.
We first show that the Dirichlet form associated with the integral operator L in L2(V;dμ) is positive semi-definite.
Lemma 1.
For any ϕ∈L2(V;dμ), we have
∫Vϕ(v)Lϕ(v)dμ(v)=12∬V×V(ϕ(v)−ϕ(w))2dμ(w)dμ(v)⩾0.
Proof.
By the definition of the Boltzmann operator (3), we have:
∫Vϕ(v)Lϕ(v)dμ(v)=∫V|ϕ(v)|2dμ(v)−∬V×Vϕ(v)ϕ(w)dμ(w)dμ(v).
Split the first integral on the right hand side of the above expression into two, thus yielding
12∫V|ϕ(v)|2dμ(v)+12∫V|ϕ(w)|2dμ(w)−∬V×Vϕ(v)ϕ(w)dμ(w)dμ(v).
Thanks to
μ being a probability density on
V, we can rewrite the above expression as
12∬V×V(|ϕ(v)|2+|ϕ(w)|2−2ϕ(v)ϕ(w))dμ(w)dμ(v)⩾0.
Hence the result. □
Next, by using the energy approach (i.e. by choosing appropriate multiplier), we prove an entropy inequality associated with the stationary model (5a)–(5b).
Lemma 2.
The solution fε(x,v) to the linear Boltzmann equation (5a)–(5b) satisfies the following entropy inequality:
∫Ω∫V|fε(x,v)|2dμ(v)dx+1ε2∫Ω∬V×Vσε(x)(fε(x,v)−fε(x,w))2dμ(w)dμ(v)dx⩽∫Ω|g(x)|2dx.
Proof.
Multiply (5a) by fε(x,v) and integrate over V yielding
∫V|fε(x,v)|2dμ(v)+12ε∫Vv·∇x|fε(x,v)|2dμ(v)+σε(x)ε2∫Vfε(x,v)Lfε(x,v)dμ(v)=∫Vg(x)fε(x,v)dμ(v).
Using Lemma
1 for the Dirichlet form and integrating over Ω yields:
∫Ω∫V|fε(x,v)|2dμ(v)dx+12ε∫Ω∫Vv·∇x|fε(x,v)|2dμ(v)dx+∫Ωσε(x)2ε2∬V×V(fε(x,v)−fε(x,w))2dμ(w)dμ(v)dx=∫Ω∫Vg(x)fε(x,v)dμ(v)dx.
Consider the transport term in the previous expression and perform an integration by parts in the space variable yielding (with
dΓ(x) as the surface measure on
∂Ω):
∫Σ(v·n(x))|γfε(x,v)|2dμ(v)dΓ(x)=∫Σ+(v·n(x))|γ+fε(x,v)|2dμ(v)dΓ(x)
because of the zero absorption boundary condition (
5b) on the incoming phase-space boundary. In the right hand side of the above expression, make the change of variables:
v→v∗:=Rxv for each
x∈∂Ω, where
Rxv=v−2(v·n(x))n(x) is the reflection operator, yielding:
∫Σ+(v·n(x))|γ+fε(x,v)|2dvdΓ(x)=−∫Σ−(v∗·n(x))|γ−fε(x,v∗)|2dμ(v∗)dΓ(x)=0
again by the absorption boundary condition (
5b). Hence the transport term does not contribute. We are left with the following expression:
∫Ω∫V|fε(x,v)|2dμ(v)dx+∫Ωσε(x)2ε2∬V×V(fε(x,v)−fε(x,w))2dμ(w)dμ(v)dx=∫Ω∫Vg(x)fε(x,v)dμ(v)dx.
Apply Cauchy–Schwarz inequality and Young’s inequality to the term involving the source term:
∫Ω∫Vg(x)fε(x,v)dμ(v)dx⩽(∫Ω∫V|g(x)|2dμ(v)dx)1/2(∫Ω∫V|fε(x,v)|2dμ(v)dx)1/2⩽12∫Ω|g(x)|2dx+12∫Ω∫V|fε(x,v)|2dμ(v)dx.
Using this inequality in the previous equality yields the entropy inequality. □
Our next result is to derive some uniform L2-estimates using the entropy inequality. We use the following notation:
⟨h⟩:=∫Vh(v)dμ(v)for all h∈L1(V;dμ).
Lemma 3.
Let fε(x,v) be the solution to (5a)–(5b). We have the following estimates:
(6a)‖fε‖L2(Ω×V;dxdμ)⩽‖g‖L2(Ω),(6b)‖(fε−⟨fε⟩)‖L2(Ω×V;dxdμ)⩽εa‖g‖L2(Ω),(6c)‖⟨fε⟩‖L2(Ω)⩽‖g‖L2(Ω).
Proof.
The uniform estimate (6a) follows directly from the entropy inequality (Lemma 2).
Next, we focus on the uniform estimate (6b). We have
‖fε(x,·)−⟨fε⟩(x)‖L2(V;dμ)2=∫V(∫V(fε(x,v)−fε(x,w))dμ(w))2dμ(v).
Applying Cauchy–Schwarz we get:
∫V(fε(x,v)−fε(x,w))dμ(w)⩽(∫V(fε(x,v)−fε(x,w))2dμ(w))1/2
because
μ is a probability measure on
V. Hence we have
∫Ω‖fε(x,·)−⟨fε⟩(x)‖L2(V;dμ)2dx⩽∫Ω∬V×V(fε(x,v)−fε(x,w))2dμ(w)dμ(v)dx⩽ε2a‖g‖L2(Ω)2,
where the last inequality is because of the entropy inequality (Lemma
2).
Our next goal is to prove the uniform estimate (6c). Consider
‖⟨fε⟩‖L2(Ω)2=∫Ω(∫Vfε(x,v)dμ(v))2dx.
By Cauchy–Schwarz inequality, we have:
(∫Vfε(x,v)dμ(v))2⩽(∫V|fε(x,v)|2dμ(v))(∫V12dμ(v))=∫V|fε(x,v)|2dμ(v)
because
μ is a probability measure on
V. Thus we have:
‖⟨fε⟩‖L2(Ω)⩽‖fε‖L2(Ω×V;dxdμ)⩽‖g‖L2(Ω).
Hence the result. □
Next, we prove a crucial estimate on the velocity average ⟨vfε⟩. To begin with, we observe that the integral operator L is self-adjoint in L2(V;dμ), i.e.
∫Vψ(v)Lϕ(v)dμ(v)=∫Vϕ(v)Lψ(v)dμ(v)for all ϕ,ψ∈L2(V;dμ).
This observation follows from the following successive equalities for the inner product in
L2(V;dμ):
∫Vψ(v)Lϕ(v)dμ(v)=∫Vψ(v)ϕ(v)dμ(v)−∫Vψ(v)∫Vϕ(w)dμ(w)dμ(v)=∫Vϕ(v)ψ(v)dμ(v)−∫Vϕ(v)∫Vψ(w)dμ(w)dμ(v)=∫Vϕ(v)Lψ(v)dμ(v).
Next, we give a very important representation for the velocity variable in terms of the integral operator
L.
Lemma 4.
For each ith component of the velocity variable, we have:
(7)vi=Lvi.
Proof.
Take ϕ(v)=vi and apply the integral operator L on to ϕ, i.e.
Lvi=vi−∫Vwidμ(w)=vi
thanks to our assumption (
1). □
Lemma 5.
Let fε(x,v) be the solution to (5a)–(5b). We have the following estimate:
(8)‖⟨vfε⟩‖[L2(Ω)]d⩽ε⟨|v|2⟩‖g‖L2(Ω).
Proof.
Consider the velocity average:
1ε⟨vfε⟩=1ε∫Vvfε(x,v)dμ(v).
Crucial argument is to substitute for the velocity variable in the above expression using Lemma
4 which yields:
1ε⟨vfε⟩=1ε∫Vfε(x,v)Lvdμ(v)=1ε∫VvLfε(x,v)dμ(v)
because
L is self-adjoint. Substituting for the Boltzmann operator in the above expression, we get
1ε⟨vfε⟩=∬V×Vv1ε(fε(x,v)−fε(x,w))dμ(w)dμ(v)⩽(∬V×V|v|2dμ(w)dμ(v))1/2(∬V×V1ε2(fε(x,v)−fε(x,w))2dμ(w)dμ(v))1/2,
where Cauchy–Schwarz inequality is used. Squaring the above inequality, we get
|1ε⟨vfε⟩|2⩽⟨|v|2⟩∬V×V1ε2(fε(x,v)−fε(x,w))2dμ(w)dμ(v).
Integrate the above inequality on Ω yielding:
∫Ω|1ε⟨vfε⟩|2dx⩽⟨|v|2⟩∫Ω∬V×V1ε2(fε(x,v)−fε(x,w))2dμ(w)dμ(v)dx⩽⟨|v|2⟩‖g‖L2(Ω)2,
where we have used the entropy inequality (Lemma
2), thus proving the crucial estimate. □
Next, we prove uniform estimates on ⟨(v⊗v)fε⟩ and its divergence.
Lemma 6.
Let fε(x,v) be the solution to (5a)–(5b). We have the following estimates:
(9a)‖⟨(v⊗v)fε⟩‖[L2(Ω)]d×d⩽⟨|v⊗v|2⟩‖g‖L2(Ω),(9b)‖∇x·⟨(v⊗v)fε⟩‖[L2(Ω)]d⩽b‖g‖L2(Ω).
Proof.
Consider
‖⟨(v⊗v)fε⟩‖[L2(Ω)]d×d2=∫Ω(∫V(v⊗v)fε(x,v)dμ(v))2dx.
Cauchy–Schwarz inequality yields:
(∫V(v⊗v)fε(x,v)dμ(v))2⩽⟨|v⊗v|2⟩(∫V|fε(x,v)|2dμ(v)).
Integrating the above inequality over Ω yields:
‖⟨(v⊗v)fε⟩‖[L2(Ω)]d×d2⩽⟨|v⊗v|2⟩∫Ω∫V|fε(x,v)|2dμ(v)dx.
Using the uniform estimate (
6a) from Lemma
3 yields the estimate (
9a).
Next, we focus on the estimate (9b). To that end, multiply the stationary problem (5a) by the ith component of the velocity variable and integrate over V yielding:
ε⟨vifε⟩+∑j=1d⟨vivj∂fε∂xj⟩+σε(x)ε⟨vifε⟩=0.
The scattering coefficient is bounded in
L∞(Ω). Hence the crucial estimate (
8) of Lemma
5 would straightaway imply the following:
for each i∈{1,…,d}‖∑j=1d⟨vivj∂fε∂xj⟩‖L2(Ω)⩽b‖g‖L2(Ω),
thus proving the estimate (
9b). □
Remark 2.
The result of Lemma 6 says that the matrix valued function ⟨(v⊗v)fε⟩ is in the Hilbert space [H(div;Ω)]d, i.e. each row vector of ⟨(v⊗v)fε⟩ belongs to
(10)H(div;Ω):={u∈[L2(Ω)]d such that ∇x·u∈L2(Ω)}.
4.Moments method
We present a moments based approach to derive the limit behavior for (5a)–(5b) as ε→0. This method is essentially borrowed from [7]. For readers’ convenience, we shall present this approach step-by-step.
Step I.
Integrate (5a) over V:
(11)⟨fε⟩+1ε∇·⟨vfε⟩=g(x).
Step II.
Multiply (5a) by the velocity variable v and integrate over V:
(12)εσε(x)⟨vfε⟩+1σε(x)∇·⟨(v⊗v)fε⟩+1ε⟨vfε⟩=0.
Step III.
Multiply (11) by a test function φ(x)∈H01(Ω) and integrate over Ω:
(13)1ε∫Ω⟨vfε⟩·∇φ(x)dx=∫Ω⟨fε⟩(x)φ(x)dx−∫Ωg(x)φ(x)dx.
Step IV.
Take dot product of the vector equation (12) with ∇φ(x) and integrate over Ω:
(14)∫Ω(1σε(x)∇·⟨(v⊗v)fε⟩)·∇φ(x)dx+1ε∫Ω⟨vfε⟩·∇φ(x)dx+ε∫Ω1σε(x)⟨vfε⟩(x)·∇φ(x)dx=0.
Using (13) in (14) yields the following expression in which we need to pass to the limit.
(15)∫Ω(1σε(x)∇·⟨(v⊗v)fε⟩)·∇φ(x)dx=∫Ωg(x)φ(x)dx−∫Ω⟨fε⟩(x)φ(x)dx−ε∫Ω1σε(x)⟨vfε⟩(x)·∇φ(x)dx.
The
moments method culminates in passing to the limit as
ε→0 in (
15) using some compactness properties of the family
fε(x,v). In this article, this final step of the moments method is achieved in Section
5 for the one-dimensional case and in Section
6 for any arbitrary dimension. Observe that the expression (
15) can be treated as a weak formulation for the following second-order differential equation in the
x variable:
(16)−∇x·(1σε(x)∇·⟨(v⊗v)fε⟩)=Gε(x),
where
Gε:Ω→R is defined as
(17)Gε(x):=g(x)−⟨fε⟩(x)+ε∇x·(1σε(x)⟨vfε⟩(x)).
Next, we give a result showing that the family
Gε(x) is uniformly bounded in
L2(Ω) provided the exponent
β takes values in certain interval of
[0,∞).
Lemma 7.
Let fε(x,v) be the solution to (5a)–(5b) and suppose the exponent β⩽2. Let Gε(x) be the family of scalar functions defined by (17). There exists a constant C, independent of ε, such that
‖Gε‖L2(Ω)⩽C.
Proof.
By chain rule, we have
(18)Gε(x)=g(x)−⟨fε⟩(x)+εσε(x)∇x·⟨vfε⟩(x)−ε∇xσε(x)[σε(x)]2·⟨vfε⟩(x).
Substitute for
∇x·⟨vfε⟩ using the continuity equation (
11) in the above expression yielding:
Gε(x)=g(x)−⟨fε⟩(x)+ε2σε(x)g(x)−ε2σε(x)⟨fε⟩(x)−ε∇xσε(x)[σε(x)]2·⟨vfε⟩(x)=(1+ε2σε(x))(g(x)−⟨fε⟩(x))−ε∇xσε(x)[σε(x)]2·⟨vfε⟩(x).
Computing the
L2-norm for
Gε we have:
‖Gε‖L2(Ω)⩽‖(1+ε2σε(x))‖L∞(‖g‖L2(Ω)+‖⟨fε⟩‖L2(Ω))+ε‖∇σε[σε]2‖L∞‖⟨vfε⟩‖[L2(Ω)]d⩽C(1+1a)(‖g‖L2(Ω)+‖⟨fε⟩‖L2(Ω))+C(ca2)ε1−β‖⟨vfε⟩‖[L2(Ω)]d,
where we have used the growth assumption (
4) on the heterogeneous coefficient
σε. The assumption on the exponent
β (i.e.
β⩽2) and the uniform estimates (
6c) on
⟨fε⟩ help us arrive at the uniform
L2-bound for
Gε(x). □
5.One dimensional setting
In this section, we treat a special setting: both the spatial and velocity domains are one-dimensional, i.e. x∈(−ℓ,+ℓ) and v∈V where V is either R or (−1,+1). We consider the transport equation for the one particle distribution function fε(x,v):
(19a)fε+1εvdfεdx+σε(x)ε2(fε−⟨fε⟩)=g(x)for (x,v)∈(−ℓ,+ℓ)×V,(19b)fε(x,v)=0 for x=±ℓ and v∈V,
where the heterogeneous coefficient
σε(x) are assumed to satisfy the same regularity assumptions as before, i.e. (
4).
Theorem 1.
The solution family fε(x,v) to the one-dimensional stationary linear Boltzmann equation (19a)–(19b) exhibits the following compactness property:
fε(x,v)⇀ρ(x)weakly in L2((−ℓ,+ℓ)×V;dxdμ),
where ρ(x) is the unique solution to the stationary diffusion equation:
(20a)ρ(x)−ddx(Dσ∗dρdx)=g(x)for x∈(−ℓ,+ℓ),(20b)ρ(x)=0for x=±ℓ,
where D is a constant equal to ⟨v2⟩ and σ∗ is the L∞ weak-∗ limit of the sequence σε.Proof.
The a priori estimates of Lemma 6 in the one-dimensional setting imply that the sequence ⟨v2fε⟩ is uniformly bounded in H1(−ℓ,+ℓ). Hence we can extract a sub-sequence such that
(21)⟨v2fε⟩→Dρstrongly in L2(−ℓ,+ℓ),(22)ddx⟨v2fε⟩⇀Ddρdxweakly in L2(−ℓ,+ℓ),
where
D is a constant given by
⟨v2⟩ and
ρ(x) is the
L2 weak limit of the local densities, i.e.
∫Vfε(x,v)dμ(v)⇀ρ(x)weakly in L2(−ℓ,+ℓ).
The second order differential equation (i.e. the one similar to (
16)) that we get via the moments method (see Section
4) in this one-dimensional setting is the following:
(23)−ddx(1σε(x)d⟨v2fε⟩dx)=Gε(x),
where
Gε(x) is given by
Gε(x):=g(x)−⟨fε⟩(x)+εddx(1σε(x)⟨vfε⟩(x)).
Define
(24)ζε(x):=1σε(x)ddx⟨v2fε⟩.
We have the following uniform
L2-bound:
(25)‖ζε‖L2(−ℓ,+ℓ)⩽1a‖ddx⟨v2fε⟩‖L2(−ℓ,+ℓ)⩽ba‖g‖L2(−ℓ,+ℓ),
where we have used the assumption (
4) that
σε is bounded from below and the uniform a priori bound (
9b) from Lemma
6. From (
23), it follows that
ζε(x) solves
(26)−ddxζε(x)=Gε(x).
The uniform
L2-estimate on
Gε (Lemma
7) yields a uniform bound on the derivative of
ζε:
(27)‖dζεdx‖L2(−ℓ,+ℓ)=‖Gε‖L2(−ℓ,+ℓ)⩽C.
The estimates (
25) and (
27) together imply that the sequence
ζε is uniformly bounded in
H1(Ω). The compact embedding
H1(−ℓ,+ℓ)↪L2(−ℓ,+ℓ) implies that we can extract a sub-sequence and there exists a limit
ζ0 such that
(28)ζε→ζ0strongly in L2(−ℓ,+ℓ).
By the definition (
24) of
ζε, we have:
σε(x)ζε(x)=ddx⟨v2fε⟩.
Because of the strong convergence in (
28), we have the following:
σεζε⇀σ∗ζ0weakly in L2(−ℓ,+ℓ),
where
σ∗ is the
L∞ weak-
∗ limit of the sequence
σε. Identifying the above weak limit with the weak limit (
22), we get:
ζ0(x)=Dσ∗dρdx.
Upon passing to the limit (as
ε→0) in (
26), we have the following equation:
−dζ0dx=g(x)−ρ(x).
Substituting for
ζ0 in the above equation, we get:
ρ(x)−ddx(Dσ∗dρdx)=g(x)for x∈(−ℓ,+ℓ).
The unique solvability of the limit equation (
20a)–(
20b) follows from the Lax–Milgram theorem. □
6.Arbitrary dimensions
In the previous section, using the moments method, we managed to prove the ε→0 limit in the one-dimensional setting under the assumption that the exponent β⩽2. In this section, we prove that the ε→0 limit for the linear Boltzmann equation (5a)–(5b) in arbitrary dimensions can be obtained under some smallness criterion on the H−12 norm of a quotient involving the heterogeneous coefficient σε(x).
Theorem 2.
Suppose there exists σ¯(x)∈L∞(Ω), bounded away from zero, such that
(29)‖σ¯(x)−σε(x)σ¯(x)‖H−1/2(Ω)=O(ε1+).
Then the family of local densities ⟨fε(x,·)⟩ exhibit the following compactness property:
∫Vfε(x,v)dμ(v)⇀ρ(x)weakly in L2(Ω),
where the limit local density ρ(x) satisfies the following diffusion equation:
(30)ρ(x)−∇·(Dσ¯(x)∇ρ(x))=g(x)for x∈Ω,(31)ρ(x)=0for x∈∂Ω
with D a constant matrix equal to ⟨v⊗v⟩.Proof.
Let us rewrite our steady state model problem (5a) as follows:
εfε+v·∇fε+σ¯(x)ε(fε−⟨fε⟩)=(σ¯(x)−σε(x))(fε−⟨fε⟩ε)+εg(x).
Applying the moments approach outlined in Section
4 to the above problem, we arrive at the following weak formulation with a smooth test function
Ψ(x) vanishing on
∂Ω:
(32)ε∫Ω1σ¯(x)⟨vfε⟩·∇Ψ(x)dx+∫Ω(1σ¯(x)∇·⟨(v⊗v)fε⟩)·∇Ψ(x)dx+∫Ω⟨fε⟩Ψ(x)dx=∫Ω(σ¯(x)−σε(x)σ¯(x))⟨vfε⟩εdx+∫Ωg(x)Ψ(x)dx.
We have for the first term on the left hand side of the above equality:
|ε∫Ω1σ¯(x)⟨vfε⟩·∇Ψ(x)dx|⩽ε‖σ¯−1‖L∞(Ω)‖⟨vfε⟩‖[L2(Ω)]d‖∇Ψ‖[L2(Ω)]d⩽Cε2,
thanks to Lemma
5. Next, for the first term on the right hand side of (
32) we have
∫Ω(σ¯(x)−σε(x)σ¯(x))⟨vfε⟩εdx=1ε⟨⟨(σ¯−σεσ¯)Id,⟨vfε⟩⟩⟩[H−1/2(Ω)]d,[H1/2(Ω)]d
which implies:
|∫Ω(σ¯(x)−σε(x)σ¯(x))⟨vfε⟩εdx|⩽1ε‖(σ¯−σεσ¯)Id‖[H−1/2(Ω)]d‖⟨vfε⟩‖[H1/2(Ω)]d.
The hypothesis (
29) and the velocity-averaging Lemma [
14,
15] together imply that the above term is of
O(ε+). With all the above observation, we are left to pass to the limit in the following expression:
O(ε2)+∫Ω(1σ¯(x)∇·⟨(v⊗v)fε⟩)·∇Ψ(x)dx+∫Ω⟨fε⟩Ψ(x)dx=O(ε+)+∫Ωg(x)Ψ(x)dx.
Observe that
limε→0∫Ω(1σ¯(x)∇·⟨(v⊗v)fε⟩)·∇Ψ(x)dx=−limε→0∑i,j=1d∫Ω⟨vivjfε⟩∂xj(1σ¯(x)∂xiΨ(x))dx=−∑i,j=1d∫ΩDijρ(x)∂xj(1σ¯(x)∂xiΨ(x))dx,
where
Dij=⟨vivj⟩. Thus, in the limit, we arrive at the following expression:
∫ΩDσ¯(x)∇ρ(x)·∇Ψ(x)dx+∫Ωρ(x)Ψ(x)dx=∫Ωg(x)Ψ(x)dx,
which is nothing but the weak formulation of the limit problem (
30)–(
31). The unique solvability of (
30)–(
31) follows again by the application of Lax–Milgram theorem. Hence the entire family converges to the limit local density. □
Next, we make an interesting observation on the smallness of the H−12-condition (29) with regard to a rapidly oscillating periodic function. Remark that the smallness assumption (29) of the H−12-norm in the periodic setting corresponds to having the exponent β>2.
Lemma 8.
Take σε(x)=2+sin(xεβ) with β>2 and take σ¯=2. Then
1ε‖σ¯−σεσ¯‖H−12(0,2π)=O(ε+).
Proof.
As the denominator in the quotient is a constant, we shall ignore it for the calculations to follow. We shall compute the H−1-norm by testing against a function φ∈H1:
∫02πsin(xεβ)φ(x)dx=−εβ∫02π∂x(cos(xεβ))φ(x)dx=εβ∫02πcos(xεβ)∂xφ(x)dx−εβ(cos(ε−β2π)φ(2π)−cos(0)φ(0))
which implies that
‖sin(xεβ)‖H−1(0,2π)∼εβ.
As we have
‖sin(xεβ)‖L2(0,2π)=O(1),
interpolating between
H−1 and
L2 implies that
‖sin(xεβ)‖H−12(0,2π)∼εβ2.
Hence we have
1ε‖σ¯−σε‖H−12(0,2π)∼εβ2−1.
The choice of
β>2 indeed implies that the
H−12-norm is of
O(ε+). □
Even though the result of Lemma 8 is given in one dimension, the proof carries over to any arbitrary dimension.
Remark 3.
Note that the smallness condition on the H−1/2-norm in Theorem 2 is quite strong as suggested by Lemma 8 which essentially says that any smooth function depending on the argument xεβ would satisfy the smallness assumption (29) provided β>2. Do note that we have treated the case β⩽2 for the one-dimensional case in Theorem 1. If we were to suppose that the heterogeneous coefficient has the following structure
σε(x)=σ¯(x)+α(ε)hε(x),
i.e. the highly heterogeneous oscillations are of small amplitude, of size
α(ε), which vanish in the
ε→0 limit then the limit procedure can be carried out without reverting to the condition (
29). However, this is trivial as the heterogeneities are dying out in the
ε→0 limit. The smallness assumption (
29) in Theorem
2 and the velocity averaging lemma (see Proof of Theorem
2) does allow us to have genuine heterogeneity in the coefficients
σε(x).
Remark 4.
Even though the results presented so far – Theorem 1 and Theorem 2 – concern the stationary transport model, analogous results for the associated non-stationary model follow straightaway. Consider
∂tfε=Tεfε,fε(0)=fin(x,v)
with the operator
Tεfε:=−1εv·∇xfε+σε(x)ε2(∫Vfε(t,x,w)dμ(w)−fε(t,x,v)).
The following holds for the semigroup associated with
Tε:
(p−Tε)−1=∫0∞e−ptetTεdt.
Inverting the Laplace transform, we get
etTεfin=12πilimℓ→∞∫γ−iℓγ+iℓept(p−Tε)−1findp.
Taking the
ε→0 limit in the previous expression, we get
limε→0etTεfin=12πilimℓ→∞∫γ−iℓγ+iℓeptlimε→0(p−Tε)−1findp.
The asymptotic limit obtained in Theorem
2 for the resolvent helps us get
limε→0etTεfin=12πilimℓ→∞∫γ−iℓγ+iℓept(p−D)−1ρindp=etDρin,
where we have used the following notations:
Du:=∇x·(Dσ¯(x)∇xu)andρin(x):=∫Vfin(x,v)dμ(v).
7.Concluding remarks
We have seen in Theorem 1 that we can get an explicit expression for the effective diffusion coefficient. This is analogous to the H-limits in one dimensional setting in the theory of H-convergence [20]. The theory of H-convergence, however, goes beyond the one-dimensional setting in getting explicit expressions while dealing with laminated materials. Our computations show that this is indeed the case in our setting. Results in this flavor will be in a later publication of the authors [8]. The main handicap of our result in the β⩽2 case is that we are unable to handle dimensions higher than one. As noted in Remark 2, the estimates are in H(div;Ω) for the matrix-valued function ⟨(v⊗v)fε⟩. This is in stark contrast to the classical estimates used in the H-convergence theory with regard to elliptic problems. In a work which is in progress [8], the authors have made some progress in getting around the available less regularity information via constructing suitable class of test functions which emphasizes the importance of transport behavior at the scale of the microstructure. This approach employs the famous div-curl lemma (see [12] for a kinetic analogue of the div-curl lemma). Finally, it would be interesting to address this simultaneous limit in the case of linear Fokker–Planck equation. The authors shall address this problem in the near future.
