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 \[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R}^{3}\left\vert z\geqslant 0\right.\right\}, \] under the assumption that the plane $z=0$ 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.

在这项研究中，我们继续研究与 Brinkman 方程相关的 Cauchy 问题 [参见下面的 (1.1) 和 (1.2)]，该方程模拟了某些类型的多孔介质中的流体流动。在这里，我们将考虑上半空间中的流动 \[ \mathbb{R}_{+}^{3}=\left\{\left(x,y,z\right) \in\mathbb{R }^{3}\left\vert z\geqslant 0\right.\right\}, \] 假设平面$z=0$是流体不可穿透的。这意味着我们将不得不引入必须附加到 Brinkman 方程的边界条件。我们在下面介绍的适当 Sobolev 空间中研究局部和全局适定性，使用 Kato 的拟线性方程理论、抛物线正则化和问​​题解的比较原理。