Homogenization of linear transport equations. A new approach

• Published in: Journal de l'École polytechnique. Mathématiques Vol. Tome 7
• Main Author: Briane, Marc
• Format: Journal Article
• Language: English
• Published: École polytechnique 01.01.2020
• Subjects: Analysis of PDEs; Mathematics; rectification Mots clés : homogénéisation; redressement Mathematics Subject Classification: 35B27; flot dynamique; homogenization; dynamic flow; rectification; 37C10; équation de transport; transport equation; 35F05
• ISSN: 2429-7100; 2270-518X
• DOI: 10.5802/jep.122

The paper is devoted to a new approach of the homogenization of linear transport equations induced by a uniformly bounded sequence of vector fields $b_\ep(x)$, the solutions of which $u_\ep(t,x)$ agree at $t=0$ with a bounded sequence of $L^p_{\rm loc}(\RR^N)$ for some $p\in(1,\infty)$. Assuming that the sequence $b_\ep\cdot\nabla w_\ep^1$ is compact in $L^q_{\rm loc}(\RR^N)$ ($q$ conjugate of $p$) for some gradient field $\nabla w_\ep^1$ bounded in $L^N_{\rm loc}(\RR^N)^N$, and that there exists a uniformly bounded sequence $\sigma_\ep>0$ such that $\sigma_\ep\,b_\ep$ is divergence free if $N\!=\!2$ or is a cross product of $(N\!-\!1)$ bounded gradients in $L^N_{\rm loc}(\RR^N)^N$ if $N\!\geq\!3$, we prove that the sequence $\sigma_\ep\,u_\ep$ converges weakly to a solution to a linear transport equation. It turns out that the compactness of $b_\ep\cdot\nabla w_\ep^1$ 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. The paper is devoted to a new approach of the homogenization of linear transport equations induced by a uniformly bounded sequence of vector fields bε(x), the solutions of which uε(t, x) agree at t = 0 with a bounded sequence of L p loc (R N) for some p ∈ (1, ∞). Assuming that the sequence bε · ∇w 1 ε is compact in L q loc (R N) (q conjugate of p) for some gradient field ∇w 1 ε bounded in L N loc (R N) N , and that there exists a uniformly bounded sequence σε > 0 such that σε bε is divergence free if N = 2 or is a cross product of (N − 1) bounded gradients in L N loc (R N) N if N ≥ 3, we prove that the sequence σε uε converges weakly to a solution to a linear transport equation. It turns out that the compactness of bε · ∇w 1 ε 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. Résumé Cet article propose une nouvelle approche de l'homogénéisation deséquations de transport linéaires induites par une suite uniformément bornée de champs de vecteurs bε(x) et dont les solutions uε(t, x) coïncident en t = 0 avec une suite bornée de L p loc (R N) pour un certain p ∈ (1, ∞). En supposant que la suite bε · ∇w 1 ε est compacte dans L q loc (R N) (q exposant conjugué de p) pour un champ de gradients ∇w 1 ε borné dans L N loc (R N) N et qu'il existe une suite uniformément bornée σε > 0 telle que σε bε està divergence nulle si N = 2 ou est un produit vectoriel de (N−1) gradients bornés dans L N loc (R N) N si N ≥ 3, on montre que la suite σε uε converge faiblement vers une solution d'uneéquation de transport. Il s'avère que la compacité de bε · ∇w 1 ε 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.