Homogenization of linear transport equations. A new approach

Abstract

The paper is devoted to a new approach of the homogenization of linear transport equations induced by a uniformly bounded sequence of vector fields $b_{\epsilon }(x)$, the solutions of which $u_{\epsilon }(t,x)$ agree at $t=0$ with a bounded sequence of $L_{\mathrm{loc}}^p({\mathbb{R}}^N)$ for some $p\in (1,\infty )$. Assuming that the sequence $b_{\epsilon }\cdot \nabla w_{\epsilon }^1$ is compact in $L_{\mathrm{loc}}^q({\mathbb{R}}^N)$ ($q$ conjugate of $p$) for some gradient field $\nabla w_{\epsilon }^1$ bounded in $L_{\mathrm{loc}}^N({\mathbb{R}}^N{)}^N$, 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 $L_{\mathrm{loc}}^N({\mathbb{R}}^N{)}^N$ 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 $b_{\epsilon }\cdot \nabla w_{\epsilon }^1$ 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.