线性输运方程的均匀化。新方法

M. Briane
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引用次数: 4

摘要

本文研究了由一致有界向量场序列引起的线性输运方程的均匀化的一种新方法 $b_\epsilon(x)$的解 $u_\epsilon(t,x)$ 同意 $t=0$ 的有界序列 $L^p_{\rm loc}(\mathbb{R}^N)$ 对一些人来说 $p\in(1,\infty)$. 假设序列 $b_\epsilon\cdot\nabla w_\epsilon^1$ 是紧凑的 $L^q_{\rm loc}(\mathbb{R}^N)$ ($q$ 的共轭 $p$)表示某梯度场 $\nabla w_\epsilon^1$ 有界的 $L^N_{\rm loc}(\mathbb{R}^N)^N$,并且存在一个一致有界序列 $\sigma_\epsilon>0$ 这样 $\sigma_\epsilon\,b_\epsilon$ 散度是自由的 $N\!=\!2$ 或者是外积 $(N\!-\!1)$ 中有界梯度 $L^N_{\rm loc}(\mathbb{R}^N)^N$ 如果 $N\!\geq\!3$,我们证明了这个序列 $\sigma_\epsilon\,u_\epsilon$ 弱收敛于线性输运方程的解。的紧度 $b_\epsilon\cdot\nabla w_\epsilon^1$ 是经典二维周期情况的遍历假设的替代,并允许我们处理任何维度的非周期向量场。均质化结果是由各种和一般的例子说明。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Homogenization of linear transport equations. A new approach
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.
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