We consider the Rayleigh--B\'{e}nard problem for the three--dimensional Boussinesq system for the micropolar fluid. We introduce the notion of the multivalued eventual semiflow and prove the existence of the two-space global attractor $\mathcal{A}^K$ corresponding to weak solutions, for every micropolar parameter $K\geq 0$ denoting the deviation of the considered system from the classical Rayleigh--B\'{e}nard problem for the Newtonian fluid. We prove that for every $K$ the attractor $\mathcal{A}^K$ is the smallest compact, attracting, and invariant set. Moreover, the semiflow restricted to this attractor is single-valued and governed by strong solutions. Further, we prove that the global attractors $\mathcal{A}^K$ converge to $\mathcal{A}^0$ upper semicontinuously in Kuratowski sense as $K\to 0$, and that the projection of $\mathcal{A}^0$ on the restricted phase space corresponding to the classical Rayleigh--B\'{e}nard problem is the global attractor for the latter problem, having the invariance property. These results are established under the assumption that the Prandtl number is relatively large with respect to the Rayleigh number.

中文翻译：

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微极与牛顿在 3D 中相遇。大 Prandtl 数的 Rayleigh-Bénard 问题

我们考虑微极流体的三维 Boussinesq 系统的 Rayleigh--B\'{e}nard 问题。我们引入了多值最终半流的概念，并证明了对应于弱解的双空间全局吸引子 $\mathcal{A}^K$ 的存在，对于每个微极参数 $K\geq 0$ 表示所考虑的偏差来自经典瑞利的系统-牛顿流体的 B\'{e}nard 问题。我们证明，对于每个 $K$，吸引子 $\mathcal{A}^K$ 是最小的紧凑、吸引和不变集。此外，限制到这个吸引子的半流是单值的，并由强解控制。此外，我们证明全局吸引子 $\mathcal{A}^K$ 在 Kuratowski 意义上收敛到 $\mathcal{A}^0$ 上半连续为 $K\to 0$，并且 $\mathcal{A}^0$ 在对应于经典 Rayleigh--B\'{e}nard 问题的受限相空间上的投影是后一个问题的全局吸引子，具有不变性。这些结果是在普朗特数相对于瑞利数相对较大的假设下建立的。