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 for . In one space dimension, we prove that the solutions to the kinetic model converge to the solutions of an effective diffusion equation for any in the limit. In any arbitrary phase space dimension, under a smallness assumption of a certain quotient involving the scattering coefficient in the norm, we again prove that the solutions to the kinetic model converge to the solutions of an effective diffusion equation in the limit.
The unknown quantity modeled by equations in kinetic theory is the probability distribution function 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:
In the above kinetic model, the scattering coefficient represents the background medium. The inhomogeneous nature of the background medium implies that smaller the parameter η is, more rapid the oscillations are in . As is standard in the theory of homogenization, we consider a family of scattering coefficients indexed by η, i.e. 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
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  where the heterogeneity length scale η is related to the mean free path as with . Recently, there has been a revival of this problem. The works [5,9] address this problem in the periodic setting and in the regime . The approach in [5,9] is to introduce a new parameter and study some cell problems involving the new small parameter . They extensively use the method of two-scale convergence. We also cite  where the spectral problem associated with the scaled linear Boltzmann equation is studied when . For the simultaneous limit procedures in the case , we cite [16,17] (see also Chapter 7 in the lecture notes ).
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 , we assume that the two scaling parameters are related as where . Note, however, that the results in  hold only when 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 . 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 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
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  and apply this method to the linear Boltzmann equation. Section 5 is devoted to deriving the limit equation (in the 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 be the distribution function which depends on (space position) and (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:
The goal is to perform the simultaneous limiting procedure for the scaled problem:
Consider such that for a.e. ,
The main objective of this article is to study the following stationary problem in the limit:
3.Uniform a priori bounds
We first show that the Dirichlet form associated with the integral operator in is positive semi-definite.
For any , we have
By the definition of the Boltzmann operator (3), we have:
Multiply (5a) by and integrate over yielding
Our next result is to derive some uniform -estimates using the entropy inequality. We use the following notation:
Next, we focus on the uniform estimate (6b). We have
Our next goal is to prove the uniform estimate (6c). Consider
Next, we prove a crucial estimate on the velocity average . To begin with, we observe that the integral operator is self-adjoint in , i.e.
For each ith component of the velocity variable, we have:
Take and apply the integral operator on to ϕ, i.e.
Consider the velocity average:
Next, we prove uniform estimates on and its divergence.
The result of Lemma 6 says that the matrix valued function is in the Hilbert space , i.e. each row vector of belongs to
We present a moments based approach to derive the limit behavior for (5a)–(5b) as . This method is essentially borrowed from . For readers’ convenience, we shall present this approach step-by-step.
Integrate (5a) over :
Multiply (5a) by the velocity variable v and integrate over :
Multiply (11) by a test function and integrate over Ω:
Take dot product of the vector equation (12) with and integrate over Ω:
By chain rule, we have
5.One dimensional setting
In this section, we treat a special setting: both the spatial and velocity domains are one-dimensional, i.e. and where is either or . We consider the transport equation for the one particle distribution function :
The a priori estimates of Lemma 6 in the one-dimensional setting imply that the sequence is uniformly bounded in . Hence we can extract a sub-sequence such that
In the previous section, using the moments method, we managed to prove the limit in the one-dimensional setting under the assumption that the exponent . In this section, we prove that the limit for the linear Boltzmann equation (5a)–(5b) in arbitrary dimensions can be obtained under some smallness criterion on the norm of a quotient involving the heterogeneous coefficient .
Suppose there exists , bounded away from zero, such that
Let us rewrite our steady state model problem (5a) as follows:
Next, we make an interesting observation on the smallness of the -condition (29) with regard to a rapidly oscillating periodic function. Remark that the smallness assumption (29) of the -norm in the periodic setting corresponds to having the exponent .
Take with and take . Then
As the denominator in the quotient is a constant, we shall ignore it for the calculations to follow. We shall compute the -norm by testing against a function :
Even though the result of Lemma 8 is given in one dimension, the proof carries over to any arbitrary dimension.
Note that the smallness condition on the -norm in Theorem 2 is quite strong as suggested by Lemma 8 which essentially says that any smooth function depending on the argument would satisfy the smallness assumption (29) provided . Do note that we have treated the case for the one-dimensional case in Theorem 1. If we were to suppose that the heterogeneous coefficient has the following structure
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 . 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 . The main handicap of our result in the case is that we are unable to handle dimensions higher than one. As noted in Remark 2, the estimates are in for the matrix-valued function . 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 , 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  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.
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).
G. Allaire, Homogenization and two-scale convergence, SIAM Journal on Mathematical Analysis 23: (6) ((1992) ), 1482–1518. doi:10.1137/0523084.
G. Allaire, Shape Optimization by the Homogenization Method, Vol. 146: , Springer Science & Business Media, (2002) .
G. Allaire and G. Bal, Homogenization of the criticality spectral equation in neutron transport, ESAIM: Modélisation Mathématique et Analyse Numérique 33: (4) ((1999) ), 721–746. doi:10.1051/m2an:1999160.
G. Allaire and F. Golse, Transport et Diffusion, Lecture Notes, Ecole Polytechnique, (2013) .
G. Bal, N. Ben Abdallah and M. Puel, A corrector theory for diffusion-homogenization limits of linear transport equations, SIAM Journal on Mathematical Analysis 44: (6) ((2012) ), 3848–3873. doi:10.1137/110857301.
C. Bardos, E. Bernard, F. Golse and R. Sentis, The diffusion approximation for the linear Boltzmann equation with vanishing scattering coefficient, Commun. Math. Sci. 13: (3) ((2015) ), 641–671. doi:10.4310/CMS.2015.v13.n3.a3.
C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, Journal of Statistical Physics 63: (1–2) ((1991) ), 323–344. doi:10.1007/BF01026608.
C. Bardos and H. Hutridurga, Div-curl approach for diffusion-homogenization limits for linear transport equations, 2016, In preparation.
N. Ben Abdallah, M. Puel and M.-S. Vogelius, Diffusion and homogenization limits with separate scales, Multiscale Modeling & Simulation 10: (4) ((2012) ), 1148–1179. doi:10.1137/110828964.
N. Ben Abdallah and M.-L. Tayeb, Diffusion approximation and homogenization of the semiconductor Boltzmann equation, Multiscale Modeling & Simulation 4: (3) ((2005) ), 896–914. doi:10.1137/040611227.
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, (1978) .
L. Dumas and F. Golse, Homogenization of transport equations, SIAM Journal on Applied Mathematics 60: (4) ((2000) ), 1447–1470. doi:10.1137/S0036139997332087.
F. Golse, Particle transport in nonhomogeneous media, in: Mathematical Aspects of Fluid and Plasma Dynamics, Springer, (1991) , pp. 152–170. doi:10.1007/BFb0091366.
F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, Journal of Functional Analysis 76: (1) ((1988) ), 110–125. doi:10.1016/0022-1236(88)90051-1.
F. Golse, B. Perthame and R. Sentis, Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale de l’un opérateur de transport, C. R. Acad. Sci. Paris Sér. I Math 301: (7) ((1985) ), 341–344.
T. Goudon and A. Mellet, Homogenization and diffusion asymptotics of the linear Boltzmann equation, ESAIM: Control, Optimisation and Calculus of Variations 9: ((2003) ), 371–398. doi:10.1051/cocv:2003018.
T. Goudon and F. Poupaud, Approximation by homogenization and diffusion of kinetic equations, 2001.
F. Murat, Compacité par compensation, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 5: (3) ((1978) ), 489–507.
F. Murat and L. Tartar, H-convergence, in: Séminaire d’Analyse Fonctionnelle et Numérique de L’Université D’Alger, Mimeographed Notes, (1978) .
F. Murat and L. Tartar, in: H-Convergence, Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl., Vol. 31: , (1997) , pp. 21–43. doi:10.1007/978-1-4612-2032-9_3.
G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM Journal on Mathematical Analysis 20: (3) ((1989) ), 608–623. doi:10.1137/0520043.
R. Sentis, Approximation and homogenization of a transport process, SIAM Journal on Applied Mathematics 39: (1) ((1980) ), 134–141. doi:10.1137/0139011.