{"title":"Homogenization of linear transport equations. A new approach","authors":"M. Briane","doi":"10.5802/jep.122","DOIUrl":null,"url":null,"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 $L^p_{\\rm loc}(\\mathbb{R}^N)$ for some $p\\in(1,\\infty)$. Assuming that the sequence $b_\\epsilon\\cdot\\nabla w_\\epsilon^1$ is compact in $L^q_{\\rm loc}(\\mathbb{R}^N)$ ($q$ conjugate of $p$) for some gradient field $\\nabla w_\\epsilon^1$ bounded in $L^N_{\\rm loc}(\\mathbb{R}^N)^N$, 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 $L^N_{\\rm loc}(\\mathbb{R}^N)^N$ 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 $b_\\epsilon\\cdot\\nabla w_\\epsilon^1$ 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.","PeriodicalId":106406,"journal":{"name":"Journal de l’École polytechnique — Mathématiques","volume":"119 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2019-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de l’École polytechnique — Mathématiques","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5802/jep.122","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 4
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 $L^p_{\rm loc}(\mathbb{R}^N)$ for some $p\in(1,\infty)$. Assuming that the sequence $b_\epsilon\cdot\nabla w_\epsilon^1$ is compact in $L^q_{\rm loc}(\mathbb{R}^N)$ ($q$ conjugate of $p$) for some gradient field $\nabla w_\epsilon^1$ bounded in $L^N_{\rm loc}(\mathbb{R}^N)^N$, 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 $L^N_{\rm loc}(\mathbb{R}^N)^N$ 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 $b_\epsilon\cdot\nabla w_\epsilon^1$ 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.