# Simultaneous diffusion and homogenization asymptotic for the linear Boltzmann equation

#### Abstract

This article is on the simultaneous diffusion approximation and homogenization of the linear Boltzmann equation when both the mean free path *ε* and the heterogeneity length scale *η* vanish. No periodicity assumption is made on the scattering coefficient of the background material. There is an assumption made on the heterogeneity length scale *η* that it scales as

## 1.Introduction

The unknown quantity modeled by equations in kinetic theory is the probability distribution function

*μ*is a Borel probability measure on the space of velocities

*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 *η* is, more rapid the oscillations are in *η*, i.e. *ε*. The objective of this article is to study the following scaled linear Boltzmann equation

*ε*,

*η*vanish. The diffusion approximation of the linear Boltzmann equation corresponds to the

*η*yielding a parabolic equation with heterogeneous coefficient

*η*-periodic. The method of H-convergence [19] (see also [20]) can be used when the family

*ε*(see [12] on the homogenization of linear transport equations). This shall be followed by a diffusion asymptotic in the

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

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

Our main result in any arbitrary dimension is that if the heterogeneous coefficient satisfies

*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

## 2.Stationary linear Boltzmann equation

Let

*μ*a Borel probability measure on

*f*on

The goal is to perform the simultaneous limiting procedure for the scaled problem:

*η*, i.e.

*η*) constants

*η*and the mean free path

*ε*are related as

*η*in (2a) and index the family of heterogeneous scattering coefficients by

*ε*, i.e.

*ε*.

##### Remark 1.

Consider

The main objective of this article is to study the following stationary problem in the

## 3.Uniform a priori bounds

In order to perform the asymptotic analysis in the *ε*)

We first show that the Dirichlet form associated with the integral operator

##### Lemma 1.

*For any* *, we have*

##### Proof.

By the definition of the Boltzmann operator (3), we have:

*μ*being a probability density on

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* *to the linear Boltzmann equation (**5a**)–(**5b**) satisfies the following entropy inequality:*

##### Proof.

Multiply (5a) by

Our next result is to derive some uniform

##### Lemma 3.

*Let* *be the solution to (**5a**)–(**5b**). We have the following estimates:*

##### Proof.

The uniform estimate (6a) follows directly from the entropy inequality (Lemma 2).

Next, we focus on the uniform estimate (6b). We have

*μ*is a probability measure on

Our next goal is to prove the uniform estimate (6c). Consider

*μ*is a probability measure on

Next, we prove a crucial estimate on the velocity average

##### Lemma 4.

*For each ith component of the velocity variable, we have:*

##### Proof.

Take *ϕ*, i.e.

##### Lemma 5.

*Let* *be the solution to (**5a**)–(**5b**). We have the following estimate:*

##### Proof.

Consider the velocity average:

Next, we prove uniform estimates on

##### Lemma 6.

*Let* *be the solution to (**5a**)–(**5b**). We have the following estimates:*

##### Proof.

Consider

Next, we focus on the estimate (9b). To that end, multiply the stationary problem (5a) by the *i*th component of the velocity variable and integrate over

##### Remark 2.

The result of Lemma 6 says that the matrix valued function

## 4.Moments method

We present a moments based approach to derive the limit behavior for (5a)–(5b) as

##### Step II.

Multiply (5a) by the velocity variable *v* and integrate over

##### Step III.

Multiply (11) by a test function

##### Step IV.

Take dot product of the vector equation (12) with

Using (13) in (14) yields the following expression in which we need to pass to the limit.

*moments method*culminates in passing to the limit as

*x*variable:

*β*takes values in certain interval of

##### Lemma 7.

*Let* *be the solution to (**5a**)–(**5b**) and suppose the exponent* *. Let* *be the family of scalar functions defined by (**17**). There exists a constant C, independent of ε, such that*

##### Proof.

By chain rule, we have

*β*(i.e.

## 5.One dimensional setting

In this section, we treat a special setting: both the spatial and velocity domains are one-dimensional, i.e.

##### Theorem 1.

*The solution family* *to the one-dimensional stationary linear Boltzmann equation (**19a**)–(**19b**) exhibits the following compactness property:*

*where*

*is the unique solution to the stationary diffusion equation:*

*where*

*is a constant equal to*

*and*

*is the*

*weak-*

^{∗}

*limit of the sequence*

*.*

##### Proof.

The a priori estimates of Lemma 6 in the one-dimensional setting imply that the sequence

^{∗}limit of the sequence

## 6.Arbitrary dimensions

In the previous section, using the moments method, we managed to prove the

##### Theorem 2.

*Suppose there exists* *, bounded away from zero, such that*

*Then the family of local densities*

*exhibit the following compactness property:*

*where the limit local density*

*satisfies the following diffusion equation:*

*with*

*a constant matrix equal to*

*.*

##### Proof.

Let us rewrite our steady state model problem (5a) as follows:

Next, we make an interesting observation on the smallness of the

##### Lemma 8.

*Take* *with* *and take* *. Then*

##### Proof.

As the denominator in the quotient is a constant, we shall ignore it for the calculations to follow. We shall compute the

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

##### 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

## 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

## Acknowledgements

This work was initiated during C.B.’s visit to the University of Cambridge. It was supported by the ERC grant matkit. The authors warmly thank Rémi Sentis for his constructive suggestions during the preparation of this article. The authors would also like to thank Thomas Holding for his valuable remarks on a preliminary version of this manuscript and Clément Mouhot for helpful suggestions. H.H. acknowledges the support of the ERC grant MATKIT and the EPSRC programme grant “Mathematical fundamentals of Metamaterials for multiscale Physics and Mechanic” (EP/L024926/1).

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