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Homogenization of linear transport equations. A new approach (1905.08985v1)

Published 22 May 2019 in math.AP

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 $Lp_{\rm loc}(\mathbb{R}N)$ for some $p\in(1,\infty)$. Assuming that the sequence $b_\epsilon\cdot\nabla w_\epsilon1$ is compact in $Lq_{\rm loc}(\mathbb{R}N)$ ($q$ conjugate of $p$) for some gradient field $\nabla w_\epsilon1$ bounded in $LN_{\rm loc}(\mathbb{R}N)N$, 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 $LN_{\rm loc}(\mathbb{R}N)N$ if $N!\geq!3$, 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 $b_\epsilon\cdot\nabla w_\epsilon1$ 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.

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