Steady Transport Equation in the Case Where the Normal Component of the Velocity Does Not Vanish on the Boundary
Steady Transport Equation in the Case Where the Normal Component of the Velocity Does Not Vanish on the Boundary
复制标题
速度法向分量在边界上不消失时的稳态输运方程
DOI：
10.1137/11082052x
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
J. Bernard
中科院分区：
文献类型：
--
作者：
J. Bernard
本文研究了一个定常输运方程的解在$-Mathbb{R}^d$的Lipschitz域中的解，该方程的无散度行驶速度为$H^1$。由于速度在边界上是完全非齐次的，解的存在唯一性需要一个边界条件。一个新的格林公式允许我们定义边界上$z\mathbf{u}$的法向分量，其中z表示应力，$\mathbf{u}$表示速度。本文的主要内容是研究空间中的一个截断算子的性质，其中z和$\mathbf{u}，.\，a z$都是$L^2$。利用这些性质，证明了稠密性结果，并利用一个从$L^2$到$L^2$的无界线性算子，建立了具有开部分边界条件的迁移方程解的存在唯一性，其中$Mathbf{u}$的法向分量为严格负.
This article studies the solutions in $L^2$ of a steady transport equation with a divergence-free driving velocity that is $H^1$ in a Lipschitz domain of $\mathbb{R}^d$. Since the velocity is assumed fully nonhomogeneous on the boundary, existence and uniqueness of solution require a boundary condition. A new Green's formula allows us to define the normal component of $z\mathbf{u}$ on the boundary, where z denotes the stress and $\mathbf{u}$ the velocity. A substantial part of the article is devoted to properties of a truncature operator in the space where z and $\mathbf{u}\,.\,\nabla z$ are $L^2$. By means of these properties, which allow us to prove density results, and by using in addition a nonbounded linear operator from $L^2$ to $L^2$, we establish existence and uniqueness of the solution for the transport equation with a boundary condition on the open part where the normal component of $\mathbf{u}$ is strictly negative.