Homogenization of linear transport equations. A new approach
Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 479-495.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.122
Classification: 35B27,  35F05,  37C10
Keywords: Homogenization, transport equation, dynamic flow, rectification
Marc Briane 1

1 Univ Rennes, INSA Rennes, CNRS, IRMAR - UMR 6625 F-35000 Rennes, France
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Marc Briane. Homogenization of linear transport equations. A new approach. Journal de l’École polytechnique — Mathématiques, Volume 7 (2020), pp. 479-495. doi : 10.5802/jep.122. https://jep.centre-mersenne.org/articles/10.5802/jep.122/

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