Steady Transport Equation in the Case Where the Normal Component of the Velocity Does Not Vanish on the Boundary
| Autor: | J.-M. Bernard |
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| Rok vydání: | 2012 |
| Předmět: | |
| Zdroj: | SIAM Journal on Mathematical Analysis. 44:993-1018 |
| ISSN: | 1095-7154 0036-1410 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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