Homogenization of linear transport equations. A new approach
[Homogénéisation d’équations de transport linéaires. Une nouvelle approche]
Journal de l'École polytechnique — Mathématiques, Tome 7 (2020) , pp. 479-495.

Cet article propose une nouvelle approche de l’homogénéisation des équations de transport linéaires induites par une suite uniformément bornée de champs de vecteurs b ε (x) et dont les solutions u ε (t,x) coïncident en t=0 avec une suite bornée de L loc p ( N ) pour un certain p(1,). En supposant que la suite b ε ·w ε 1 est compacte dans L loc q ( N ) (q exposant conjugué de p) pour un champ de gradients w ε 1 borné dans L loc N ( N ) N et qu’il existe une suite uniformément bornée σ ε >0 telle que σ ε b ε est à divergence nulle si N=2 ou est un produit vectoriel de (N-1) gradients bornés dans L loc N ( N ) N si N3, on montre que la suite σ ε u ε converge faiblement vers une solution d’une équation de transport. Il s’avère que la compacité de b ε ·w ε 1 remplace la condition d’ergodicité du cas périodique bidimensionnel classique et permet de traiter des champs de vecteurs non périodiques en toute dimension. Le résultat d’homogénéisation est illustré par différents exemples généraux.

The paper is devoted to a new approach of the homogenization of linear transport equations induced by a uniformly bounded sequence of vector fields b ε (x), the solutions of which u ε (t,x) agree at t=0 with a bounded sequence of L loc p ( N ) for some p(1,). Assuming that the sequence b ε ·w ε 1 is compact in L loc q ( N ) (q conjugate of p) for some gradient field w ε 1 bounded in L loc N ( N ) N , and that there exists a uniformly bounded sequence σ ε >0 such that σ ε b ε is divergence free if N=2 or is a cross product of (N-1) bounded gradients in L loc N ( N ) N if N3, we prove that the sequence σ ε u ε converges weakly to a solution to a linear transport equation. It turns out that the compactness of b ε ·w ε 1 is a substitute to the ergodic assumption of the classical two-dimensional periodic case, and allows us to deal with non-periodic vector fields in any dimension. The homogenization result is illustrated by various and general examples.

Reçu le : 2019-05-20
Accepté le : 2020-02-24
Publié le : 2020-03-06
DOI : https://doi.org/10.5802/jep.122
Classification : 35B27,  35F05,  37C10
Mots clés: Homogénéisation, équation de transport, flot dynamique, redressement
@article{JEP_2020__7__479_0,
     author = {Marc Briane},
     title = {Homogenization of linear transport equations. A new approach},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     pages = {479-495},
     doi = {10.5802/jep.122},
     zbl = {07179026},
     language = {en},
     url = {jep.centre-mersenne.org/item/JEP_2020__7__479_0/}
}
Marc Briane. Homogenization of linear transport equations. A new approach. Journal de l'École polytechnique — Mathématiques, Tome 7 (2020) , pp. 479-495. doi : 10.5802/jep.122. https://jep.centre-mersenne.org/item/JEP_2020__7__479_0/

[1] Y. Amirat, K. Hamdache & A. Ziani - “Homogénéisation d’équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux”, Ann. Inst. H. Poincaré Anal. Non Linéaire 6 (1989) no. 5, p. 397-417 | Article | Zbl 0699.35170

[2] Y. Amirat, K. Hamdache & A. Ziani - “Homogénéisation par décomposition en fréquences d’une équation de transport dans R n ”, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991) no. 1, p. 37-40

[3] Y. Amirat, K. Hamdache & A. Ziani - “Homogenisation of parametrised families of hyperbolic problems”, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992) no. 3-4, p. 199-221 | Article | MR 1159181

[4] Y. Brenier - “Remarks on some linear hyperbolic equations with oscillatory coefficients”, in Third International Conference on Hyperbolic Problems, Vol. I, II (Uppsala, 1990), Studentlitteratur, Lund, 1991, p. 119-130 | Zbl 0768.35057

[5] M. Briane - “Isotropic realizability of fields and reconstruction of invariant measures under positivity properties. Asymptotics of the flow by a non-ergodic approach”, SIAM J. Appl. Dyn. Syst. 18 (2019) no. 4, p. 1846-1866 | Article | MR 4019185 | Zbl 07147349

[6] R. Caccioppoli - “Sugli elementi uniti delle trasformazioni funzionali: Un teorema di esistenza e di unicità ed alcune sue applicazioni”, Rend. Sem. Mat. Univ. Padova 3 (1932), p. 1-15 | Numdam | Zbl 58.1116.02

[7] B. Dacorogna - Direct methods in the calculus of variations, Applied Math. Sci., vol. 78, Springer, New York, 2008 | MR 2361288 | Zbl 1140.49001

[8] R. J. DiPerna & P.-L. Lions - “Ordinary differential equations, transport theory and Sobolev spaces”, Invent. math. 98 (1989) no. 3, p. 511-547 | Article | MR 1022305 | Zbl 0696.34049

[9] H. S. Dumas, J. A. Ellison & F. Golse - “A mathematical theory of planar particle channeling in crystals”, Phys. D 146 (2000) no. 1-4, p. 341-366 | Article | MR 1787420 | Zbl 0976.82048

[10] H. S. Dumas & F. Golse - “The averaging method for perturbations of mixing flows”, Ergodic Theory Dynam. Systems 17 (1997) no. 6, p. 1339-1358 | Article | MR 1488321 | Zbl 0902.58026

[11] H. S. Dumas, F. Golse & P. Lochak - “Multiphase averaging for generalized flows on manifolds”, Ergodic Theory Dynam. Systems 14 (1994) no. 1, p. 53-67 | Article | MR 1268709 | Zbl 0799.58066

[12] F. Golse - “Moyennisation des champs de vecteurs et EDP”, in Journées “Équations aux Dérivées Partielles” (Saint Jean de Monts, 1990), École Polytechnique, Palaiseau, 1990, Exp. no. XVI | Numdam | Zbl 0850.35024

[13] F. Golse - “Perturbations de systèmes dynamiques et moyennisation en vitesse des EDP”, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992) no. 2, p. 115-120 | Zbl 0749.58048

[14] M. W. Hirsch, S. Smale & R. L. Devaney - Differential equations, dynamical systems, and an introduction to chaos, Elsevier/Academic Press, Amsterdam, 2013 | Article | Zbl 1239.37001

[15] T. Y. Hou & X. Xin - “Homogenization of linear transport equations with oscillatory vector fields”, SIAM J. Appl. Math. 52 (1992) no. 1, p. 34-45 | Article | MR 1148317 | Zbl 0759.35008

[16] S. Müller - “A surprising higher integrability property of mappings with positive determinant”, Bull. Amer. Math. Soc. (N.S.) 21 (1989) no. 2, p. 245-248 | Article | MR 999618 | Zbl 0689.49006

[17] F. Murat - “Compacité par compensation”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978) no. 3, p. 489-507 | Numdam | Zbl 0399.46022

[18] Y. Nishimura - “Applications holomorphes injectives à jacobien constant de deux variables”, J. Math. Kyoto Univ. 26 (1986) no. 4, p. 697-709 | Article | MR 864471 | Zbl 0625.32022

[19] M. Reed & B. Simon - Methods of modern mathematical physics, I. Functional analysis, Academic Press, Inc., New York, 1980 | Zbl 0459.46001

[20] Y. G. Sinai - Introduction to ergodic theory, Princeton University Press, Princeton, NJ, 1976 | Zbl 0375.28011

[21] L. Tartar - “Nonlocal effects induced by homogenization”, in Partial differential equations and the calculus of variations, Vol. II, Progr. Nonlinear Differential Equations Appl., vol. 2, Birkhäuser Boston, Boston, MA, 1989, p. 925-938 | MR 1034036 | Zbl 0682.35028

[22] T. Tassa - “Homogenization of two-dimensional linear flows with integral invariance”, SIAM J. Appl. Math. 57 (1997) no. 5, p. 1390-1405 | Article | MR 1470929 | Zbl 0893.58047