微极力学在三维空间中与牛顿力学相遇。大Prandtl数的Rayleigh-B{é}nard问题
Micropolar meets Newtonian in three dimensions. The Rayleigh--B{é}nard problem for large Prandtl numbers
摘要：
本文考虑微极流体三维Boussinesq方程组的Rayleigh-Bénard问题。我们引入了多值最终半流的概念，并证明了对于每一个微极参数$K\geq 0$，对应于弱解的两空间整体吸引子$\mathcal{A}^K$的存在性，该参数表示所考虑的系统偏离经典的牛顿流体Rayleigh-Bénard问题.我们证明了对于每个$K$的吸引子$\mathcal{A}^K$是最小的紧凑的，吸引的，不变的集。此外，限制到这个吸引子的半流是单值的，并由强解控制。进一步证明了全局吸引子$\mathcal{A}^K$作为$K\to 0$在Kuratowski意义下连续收敛到$\mathcal{A}^0$上，并且证明了$\mathcal{A}^0$在经典Rayleigh-Bénard问题相空间上的投影是后者的全局吸引子，具有不变性.这些结果是建立在假设的普朗特数是相对较大的瑞利数。
英文摘要：
We consider the Rayleigh--Bénard 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--Bénard 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--Bénard 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.
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