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arXiv 1905.08985math.AP
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线性迁移方程的齐次化。一种新的方法

Homogenization of linear transport equations. A new approach

Marc Briane

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摘要：

研究了由一致有界向量场序列$b_ε(x)$诱导的线性迁移方程的均匀化问题，对于某些$u_ε(t,x)$的解在$t=0$上与$L^p_{\rm loc}(\mathbb{R}^N)$的有界序列一致.假设序列$b_ε\cdot\nabla w_ε^1$在$L^q_{\rm loc}(\mathbb{R}^N)$中是紧的（$p$的$q$共轭），且存在一致有界序列$\nabla w_ε^1$，使得如果$N\!=\!2$，则$σ_ε\,b_ε$是发散自由的，或者如果$N\!\geq\!3$，则$σ_ε\,b_ε$是$L^N_{\rm loc}(\mathbb{R}^N)^N$中$(N\!-\!1)$有界梯度的叉积，我们证明了序列$σ_ε\,u_ε$弱收敛到一个线性迁移方程的解。结果表明，$b_ε\cdot\nabla w_ε^1$的紧性取代了经典二维周期情形的遍历性假设，使我们能够处理任意维的非周期向量场。均匀化的结果说明了各种和一般的例子。

英文摘要：

The paper is devoted to a new approach of the homogenization of linear transport equations induced by a uniformly bounded sequence of vector fields $b_ε(x)$, the solutions of which $u_ε(t,x)$ 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 $b_ε\cdot\nabla w_ε^1$ is compact in $L^q_{\rm loc}(\mathbb{R}^N)$ ($q$ conjugate of $p$) for some gradient field $\nabla w_ε^1$ bounded in $L^N_{\rm loc}(\mathbb{R}^N)^N$, and that there exists a uniformly bounded sequence $σ_ε>0$ such that $σ_ε\,b_ε$ 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 $σ_ε\,u_ε$ converges weakly to a solution to a linear transport equation. It turns out that the compactness of $b_ε\cdot\nabla w_ε^1$ 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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