Homogenization of linear transport equations. A new approach

Briane, Marc

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 $b_{\epsilon }\left(x\right)$ et dont les solutions $u_{\epsilon }(t,x)$ coïncident en $t=0$ avec une suite bornée de $L_{\mathrm{loc}}^p\left({ℝ}^N\right)$ pour un certain $p\in (1,\infty )$. En supposant que la suite $b_{\epsilon }·\nabla w_{\epsilon }^1$ est compacte dans $L_{\mathrm{loc}}^q\left({ℝ}^N\right)$ ($q$ exposant conjugué de $p$) pour un champ de gradients $\nabla w_{\epsilon }^1$ borné dans $L_{\mathrm{loc}}^N{\left({ℝ}^N\right)}^N$ et qu’il existe une suite uniformément bornée ${\sigma }_{\epsilon }>0$ telle que ${\sigma }_{\epsilon }{b}_{\epsilon }$ est à divergence nulle si $N=2$ ou est un produit vectoriel de $(N-1)$ gradients bornés dans $L_{\mathrm{loc}}^N{\left({ℝ}^N\right)}^N$ si $N\ge 3$, on montre que la suite ${\sigma }_{\epsilon }{u}_{\epsilon }$ converge faiblement vers une solution d’une équation de transport. Il s’avère que la compacité de $b_{\epsilon }·\nabla w_{\epsilon }^1$ 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 $b_{\epsilon }\left(x\right)$, the solutions of which $u_{\epsilon }(t,x)$ agree at $t=0$ with a bounded sequence of $L_{\mathrm{loc}}^p\left({ℝ}^N\right)$ for some $p\in (1,\infty )$. Assuming that the sequence $b_{\epsilon }·\nabla w_{\epsilon }^1$ is compact in $L_{\mathrm{loc}}^q\left({ℝ}^N\right)$ ($q$ conjugate of $p$) for some gradient field $\nabla w_{\epsilon }^1$ bounded in $L_{\mathrm{loc}}^N{\left({ℝ}^N\right)}^N$, and that there exists a uniformly bounded sequence ${\sigma }_{\epsilon }>0$ such that ${\sigma }_{\epsilon }{b}_{\epsilon }$ is divergence free if $N=2$ or is a cross product of $(N-1)$ bounded gradients in $L_{\mathrm{loc}}^N{\left({ℝ}^N\right)}^N$ if $N\ge 3$, we prove that the sequence ${\sigma }_{\epsilon }{u}_{\epsilon }$ converges weakly to a solution to a linear transport equation. It turns out that the compactness of $b_{\epsilon }·\nabla w_{\epsilon }^1$ 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.

Classification : 35B27, 35F05, 37C10
Keywords: Homogenization, transport equation, dynamic flow, rectification
Mot clés : Homogénéisation, équation de transport, flot dynamique, redressement

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