1 results
On the Cauchy problem associated with the Brinkman flow in
$\mathbb {R}_{+}^{3}$
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 152 / Issue 5 / October 2022
- Published online by Cambridge University Press:
- 07 September 2021, pp. 1089-1108
- Print publication:
- October 2022
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- Article
- Export citation
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In this study, we continue our study of the Cauchy problem associated with the Brinkman equations [see (1.1) and (1.2) below] which model fluid flow in certain types of porous media. Here, we will consider the flow in the upper half-space
under the assumption that the plane
\[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \]
$z=0$ is impenetrable to the fluid. This means that we will have to introduce boundary conditions that must be attached to the Brinkman equations. We study local and global well-posedness in appropriate Sobolev spaces introduced below, using Kato's theory for quasilinear equations, parabolic regularization and a comparison principle for the solutions of the problem.
