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## Homogenization of transport equations (1999)

Citations: | 12 - 0 self |

### Citations

692 |
Geometrical methods in the theory of ordinary differential equations volume 250 of Grundlehren,
- Arnol’d
- 1983
(Show Context)
Citation Context ...ontexts. For example, the classical averaging method for perturbations of ordinary differential equations, initiated by Bogoliubov–Mitropolski and studied in detail by several authors since then (see =-=[3]-=-, [4], [27], and the references therein), can be recast in terms of linear partial differential equations of order 1; see, for example, [21] and [22] for a general discussion, and [17], where a proof ... |

390 |
Ordinary differential equations, transport theory and Sobolov spaces,” Inventiones mathematicae
- Perna, Lions
- 1989
(Show Context)
Citation Context ...l averaging result is obtained in the case of vector fields having only Sobolev—and not Lipschitz—regularity, with the help of the DiPerna–Lions theory of generalized flows and renormalized solutions =-=[13]-=-. Here we shall not concentrate on the type of linear transport equations associated to vector fields (Liouville equations), but instead on linear transport equations with absorption and scattering te... |

315 |
Analyse Math'ematique et Calcul Num'erique pour les Sciences et Techniques, Tome 1
- Dautray
- 1985
(Show Context)
Citation Context ...ficient σ ≡ σ(x, v) is the absorption cross section of the material, while the integral kernel k ≡ k(x, v, v′) is the local scattering cross section of the material. We refer the interested reader to =-=[16]-=- for a more thorough description of the physical situations described by this type of equation. As we said in the introduction, one should think of (3.1) as a model describing the advection of neutron... |

140 |
Microlocal defect measures
- Gérard
- 1991
(Show Context)
Citation Context ...y-averages (velocity averages) of an do not contain oscillations in the variable x. This situation is clearly analogous to the Murat–Tartar div-curl lemma. We refer to the formalism due to P. Gérard =-=[18]-=-, encompassing both the compensated compactness results as well as the analogue of Proposition 2.2 obtained by replacing L1 or L∞ spaces by L2 spaces; the main new tool in this work is the notion of m... |

116 |
Regularity of the moments of the solution of a transport equation
- Golse, Lions, et al.
- 1988
(Show Context)
Citation Context ...odels, the condition (2.2) on the sequence (an) is usually obtained by applying the method of velocity averaging. We recall it for the sake of being complete. Proposition 2.3 (velocity averaging; see =-=[24]-=-). Let µ be a positive regular Borel measure on Rd such that, for all hypercubes P ⊂ Rd, 0 < µ(P ) < +∞, and (2.17) ∀(τ, ξ) ∈ R×Rd \ {0} , µ({y ∈ Rd | τ + y · ξ = 0}) = 0 . Let (an) be a sequence of L... |

85 |
Chernov, Statistical properties of two-dimensional hyperbolic billiards
- Sinai, I
- 1991
(Show Context)
Citation Context ...port equation (besides their intrinsic interest in statistical physics); see [23], [5], [15], [9], as well as the fundamental contributions by Bunimovich–Sinai [10] and Bunimovich, Sinai, and Chernov =-=[11]-=-. The analogous problem in the case of diffusion equations has been studied by Cioranescu–Murat [12]. 2. Compensated compactness for kinetic models. In the homogenization problem for diffusion equatio... |

74 |
Diffusion approximation and computation of the critical size
- Bardos, Santos, et al.
- 1984
(Show Context)
Citation Context ...< L. (Here, the notation a << b means that a/b → 0 asymptotically.) In such a case, one can approximate first the solution of the transport equation by that of a diffusion equation (see, for example, =-=[6]-=- for a very simple account of this theory) and then apply the existing homogenization results for diffusion equations (see, for example, [7] to the approximating equation). This strategy has been deve... |

71 |
Compacité par compensation, Ann. Scuola Norm
- Murat
- 1978
(Show Context)
Citation Context ...r is as follows: • In section 2, we present a mathematical result which can be regarded as an analogue, in the case of kinetic models, of the compensated compactness theorems of Murat and Tartar (see =-=[29]-=-, [33], [34]). • In section 3, we show how this result can be applied to the most general result possible concerning the homogenization problem for the transport equation; this section is analogous to... |

70 |
H-Measures, a New Approach for Studying Homogenization. Oscillations and Concentration Effects
- Tartar
- 1990
(Show Context)
Citation Context ...ell as the analogue of Proposition 2.2 obtained by replacing L1 or L∞ spaces by L2 spaces; the main new tool in this work is the notion of microlocal defect measure independently introduced by Tartar =-=[35]-=-. 3. The general homogenization result. We consider in this section the general homogenization problem for the linear transport equation (3.1) ∂tf + v · ∇xf + σ(x, v)f − ∫ Rd k(x, v, v′)f(t, x, v′)dv′... |

61 |
Boundary layers and homogenization of transport processes
- Bensoussan, Lions, et al.
- 1979
(Show Context)
Citation Context ...e account of this theory) and then apply the existing homogenization results for diffusion equations (see, for example, [7] to the approximating equation). This strategy has been developed notably in =-=[8]-=-, [32] (we also refer to the literature indicated there). The homogenization problem for the transport equation itself has received much less attention; however, there are very interesting contributio... |

55 |
Multiphase Averaging for Classical Systems
- Lochak, Meunier
- 1988
(Show Context)
Citation Context ...or example, the classical averaging method for perturbations of ordinary differential equations, initiated by Bogoliubov–Mitropolski and studied in detail by several authors since then (see [3], [4], =-=[27]-=-, and the references therein), can be recast in terms of linear partial differential equations of order 1; see, for example, [21] and [22] for a general discussion, and [17], where a proof of Anosov’s... |

55 |
H-convergence, Topics in the mathematical modelling of composite materials
- Murat, Tartar
- 1997
(Show Context)
Citation Context ...result possible concerning the homogenization problem for the transport equation; this section is analogous to the study of H-convergence by Murat and Tartar on diffusion equations (see, for example, =-=[28]-=-). • In section 4, we discuss the case of periodic inhomogeneities; we show in detail why the method used on diffusion equations does not apply here because of small divisor problems analogous to thos... |

32 |
Statistical properties of Lorentz gas with periodic configuration of scatterers
- Sinai
- 1981
(Show Context)
Citation Context ...of homogenization problems for the transport equation (besides their intrinsic interest in statistical physics); see [23], [5], [15], [9], as well as the fundamental contributions by Bunimovich–Sinai =-=[10]-=- and Bunimovich, Sinai, and Chernov [11]. The analogous problem in the case of diffusion equations has been studied by Cioranescu–Murat [12]. 2. Compensated compactness for kinetic models. In the homo... |

31 |
Diffusions with random coefficients
- Papanicolaou, Varadhan
- 1982
(Show Context)
Citation Context ...divisor problems and how they are usually dealt with). • In section 5, we consider the homogenization problem for the transport equation in random media such as those studied by Papanicolaou–Varadhan =-=[31]-=- in the case of diffusion equations. HOMOGENIZATION OF TRANSPORT EQUATIONS 1449 The results in sections 2 and 3 were announced in [19]; those in section 4 were described without proof in [20]. The hom... |

29 |
Estimations fines des coefficients homogénéisés, in Ennio De Giorgi’s Colloquium
- Tartar
- 1983
(Show Context)
Citation Context ...ows: • In section 2, we present a mathematical result which can be regarded as an analogue, in the case of kinetic models, of the compensated compactness theorems of Murat and Tartar (see [29], [33], =-=[34]-=-). • In section 3, we show how this result can be applied to the most general result possible concerning the homogenization problem for the transport equation; this section is analogous to the study o... |

17 |
Analyse asymptotique spectrale d’un problème de di↵usion neutronique
- Allaire, Malige
- 1997
(Show Context)
Citation Context ...ne with any component of the mixture are exponentially distributed; see also [36]. Finally, we mention two recent contributions on the spectral problem in the context of neutron transport theory [1], =-=[2]-=-. In the present paper, we shall restrict our attention to the case where l << λ and we shall consider L and λ as not asymptotically small. In other words, the only small scale here will be l. Therefo... |

14 |
Diffusion approximation for billiards with totally accommodating scatterers
- Bardos, Dumas, et al.
- 1997
(Show Context)
Citation Context ...o known as the Lorentz gas model) which can be viewed as particular examples of homogenization problems for the transport equation (besides their intrinsic interest in statistical physics); see [23], =-=[5]-=-, [15], [9], as well as the fundamental contributions by Bunimovich–Sinai [10] and Bunimovich, Sinai, and Chernov [11]. The analogous problem in the case of diffusion equations has been studied by Cio... |

14 |
Un terme étrange venu d’ailleurs (i
- Cioranescu, Murat
- 1982
(Show Context)
Citation Context ...s well as the fundamental contributions by Bunimovich–Sinai [10] and Bunimovich, Sinai, and Chernov [11]. The analogous problem in the case of diffusion equations has been studied by Cioranescu–Murat =-=[12]-=-. 2. Compensated compactness for kinetic models. In the homogenization problem for diffusion equations, a fundamental role is played by a series of results known as the compensated compactness method,... |

10 | On the mean free path for a periodic array of spherical obstacles
- Dumas, Dumas, et al.
- 1996
(Show Context)
Citation Context ...wn as the Lorentz gas model) which can be viewed as particular examples of homogenization problems for the transport equation (besides their intrinsic interest in statistical physics); see [23], [5], =-=[15]-=-, [9], as well as the fundamental contributions by Bunimovich–Sinai [10] and Bunimovich, Sinai, and Chernov [11]. The analogous problem in the case of diffusion equations has been studied by Cioranesc... |

10 |
Compensated compactness and applications to p.d.e. in nonlinear analysis and mechanics
- Tartar
- 1979
(Show Context)
Citation Context ...s follows: • In section 2, we present a mathematical result which can be regarded as an analogue, in the case of kinetic models, of the compensated compactness theorems of Murat and Tartar (see [29], =-=[33]-=-, [34]). • In section 3, we show how this result can be applied to the most general result possible concerning the homogenization problem for the transport equation; this section is analogous to the s... |

8 |
Approximation and homogenization of a transport process
- Sentis
- 1980
(Show Context)
Citation Context ...ount of this theory) and then apply the existing homogenization results for diffusion equations (see, for example, [7] to the approximating equation). This strategy has been developed notably in [8], =-=[32]-=- (we also refer to the literature indicated there). The homogenization problem for the transport equation itself has received much less attention; however, there are very interesting contributions by ... |

7 |
Linear transport theory in a random medium
- Levermore, Pomraning, et al.
- 1986
(Show Context)
Citation Context ...o the literature indicated there). The homogenization problem for the transport equation itself has received much less attention; however, there are very interesting contributions by Levermore et al. =-=[25]-=- and Levermore, Pomraning, and Wong [26] in the case of some particular random media—they study mainly the case of random binary mixtures such that the lengths of the intersections of any given straig... |

5 |
Transport dans les milieux composites fortement contrast,s I: Le module du billard
- Golse
- 1994
(Show Context)
Citation Context ...s (also known as the Lorentz gas model) which can be viewed as particular examples of homogenization problems for the transport equation (besides their intrinsic interest in statistical physics); see =-=[23]-=-, [5], [15], [9], as well as the fundamental contributions by Bunimovich–Sinai [10] and Bunimovich, Sinai, and Chernov [11]. The analogous problem in the case of diffusion equations has been studied b... |

4 |
Homogénéisation d’une équation spectrale du transport neutronique
- Allaire, Bal
- 1997
(Show Context)
Citation Context ...ht line with any component of the mixture are exponentially distributed; see also [36]. Finally, we mention two recent contributions on the spectral problem in the context of neutron transport theory =-=[1]-=-, [2]. In the present paper, we shall restrict our attention to the case where l << λ and we shall consider L and λ as not asymptotically small. In other words, the only small scale here will be l. Th... |

4 |
Renewal Theory for Transport Processes in Binary Statistical Mixtures
- Levermore, Wong, et al.
- 1988
(Show Context)
Citation Context ...omogenization problem for the transport equation itself has received much less attention; however, there are very interesting contributions by Levermore et al. [25] and Levermore, Pomraning, and Wong =-=[26]-=- in the case of some particular random media—they study mainly the case of random binary mixtures such that the lengths of the intersections of any given straight line with any component of the mixtur... |

3 |
Multiphase averaging for generalized flows on manifolds. Ergodic Theory Dynam. Systems 14
- Dumas, Golse, et al.
- 1994
(Show Context)
Citation Context ... since then (see [3], [4], [27], and the references therein), can be recast in terms of linear partial differential equations of order 1; see, for example, [21] and [22] for a general discussion, and =-=[17]-=-, where a proof of Anosov’s very general averaging result is obtained in the case of vector fields having only Sobolev—and not Lipschitz—regularity, with the help of the DiPerna–Lions theory of genera... |

2 |
Particle Transport in Nonhomogeneous Media, in Mathematical Aspects of Fluid and Plasma
- Golse
- 1991
(Show Context)
Citation Context ...Varadhan [31] in the case of diffusion equations. HOMOGENIZATION OF TRANSPORT EQUATIONS 1449 The results in sections 2 and 3 were announced in [19]; those in section 4 were described without proof in =-=[20]-=-. The homogenization problem for random media was announced in [14]. Finally, we quote some references on billiards dynamics (also known as the Lorentz gas model) which can be viewed as particular exa... |

2 |
Moyennisation des champs de vecteurs et EDP; Journ'ees EDP de St Jean de
- Golse
- 1990
(Show Context)
Citation Context ...ski and studied in detail by several authors since then (see [3], [4], [27], and the references therein), can be recast in terms of linear partial differential equations of order 1; see, for example, =-=[21]-=- and [22] for a general discussion, and [17], where a proof of Anosov’s very general averaging result is obtained in the case of vector fields having only Sobolev—and not Lipschitz—regularity, with th... |

2 |
Perturbations de systèmes dynamiques et moyennisation en vitesse des EDP
- Golse
- 1992
(Show Context)
Citation Context ...tudied in detail by several authors since then (see [3], [4], [27], and the references therein), can be recast in terms of linear partial differential equations of order 1; see, for example, [21] and =-=[22]-=- for a general discussion, and [17], where a proof of Anosov’s very general averaging result is obtained in the case of vector fields having only Sobolev—and not Lipschitz—regularity, with the help of... |

1 |
On the distribution of ther free path lengths for the periodic Lorentz gas
- Bourgain, Golse, et al.
- 1998
(Show Context)
Citation Context ...se three scales would not suffice. For example, the average size of the inhomogeneities may be of a very different order than the average distance between two neighboring inhomogeneities; we refer to =-=[9]-=- for a very simple yet fundamental example where such considerations indeed play a very important role. It may also be that the geometric structure of the inhomogeneities requires more than one length... |

1 |
Méthodes Mathématiques et Numériques pour les Modelés Cinétiques, Thèse de Doctorat, Universite
- Dumas
- 1995
(Show Context)
Citation Context ... TRANSPORT EQUATIONS 1449 The results in sections 2 and 3 were announced in [19]; those in section 4 were described without proof in [20]. The homogenization problem for random media was announced in =-=[14]-=-. Finally, we quote some references on billiards dynamics (also known as the Lorentz gas model) which can be viewed as particular examples of homogenization problems for the transport equation (beside... |

1 |
Remarques sur l’homogénéisation des équations de transport
- Golse
- 1987
(Show Context)
Citation Context ...on in random media such as those studied by Papanicolaou–Varadhan [31] in the case of diffusion equations. HOMOGENIZATION OF TRANSPORT EQUATIONS 1449 The results in sections 2 and 3 were announced in =-=[19]-=-; those in section 4 were described without proof in [20]. The homogenization problem for random media was announced in [14]. Finally, we quote some references on billiards dynamics (also known as the... |

1 |
Papanicolaou,Macroscopic properties of composites, bubbly fluids, suspensions and related problems
- unknown authors
- 1985
(Show Context)
Citation Context ...rojections of H such that (5.7) Tx = ∫ Rd eiλ·xU(dλ) . The example of a Poisson distribution of inhomogeneities. This particular case is usually considered as important for physical applications; see =-=[30]-=-. It is formulated as follows: let Ω = {countable subsets of Rd}. The Poisson probability P on Ω is defined by the relation ∀A ⊂ Rd , P ({ω ∈ Ω | 8(ω ∩A) = n}) = e−λ|A|λ n|A|n n! . The action of Rd on... |

1 |
Radiative transfer in statistically inhomogeneous mixtures
- Vanderhaegen
- 1986
(Show Context)
Citation Context ...ia—they study mainly the case of random binary mixtures such that the lengths of the intersections of any given straight line with any component of the mixture are exponentially distributed; see also =-=[36]-=-. Finally, we mention two recent contributions on the spectral problem in the context of neutron transport theory [1], [2]. In the present paper, we shall restrict our attention to the case where l <<... |