可压缩微极流体模型的数学问题研究

Micropolar fluids are fluids with microstructure characterized by an asymmetric stress tensor. They belong to a class of non-Newtonian fluids that we shall call polar fluids, and include, as a special case, the well-established Navier-Stokes model of classical fluids that we shall call ordinary fluids. They play an important role in biomedicine, ect. Moreover, those applications are greatly related to the well-poseness of solutions for the compressible micropolar fluids..In this project, we will employ the methods of Fourier analysis, spectrum analysis, harmonic analysis, weighted energy estimate and Lyapunov inequality, and so on. Then we may construct the new energy estimate. We will mainly be devoted to the two important and significant topics on the compressible micropolar fluids. The first is the decay estimates of the compressible micropolar fluids in Besov space. The second is the low Mach number limits for the compressible micropolar fluids in Besov space. Applied energy estimates in Besov space, we will obtain the decay estimates. We will discuss the solutions rate of convergence between the compressible micropolar fluids model and the incompressible model. Thus, the essential relationship between the two models can be founded. Therefore, it will develop new idea and technical innovation that are effective for the well-poseness of solutions for the compressible micropolar fluids. Furthermore, through the research on the compressible micropolar fluids model, it may offer the mathematical theory for potential applications.

微极流体是具有微观结构的极流体，是经典Navier-Stokes模型的推广，在生物医药等领域有重要应用，这些应用和三维可压缩微极流体模型解的适定性问题密切相关。.在本项目中，我们将借助于Fourier分析、谱分析、调和分析、加权能量法、Lyapunov不等式等工具，构造新的能量估计，从而对Besov空间下三维可压缩微极流体模型的两个重要问题展开研究：1.可压缩微极流体模型解的衰减估计问题，2.可压缩微极流体模型的低马赫数极限问题。我们将寻求Besov空间下解的有效的先验估计，建立可压缩微极流体模型的衰减估计，探索可压缩微极流体模型和其极限方程组解之间的收敛速率，给出这两种模型之间的内在联系，从而为解决三维可压缩微极流体模型解的适定性问题提供新思路和新方法，也为可压缩微极流体模型的广泛应用提供理论基础。

由于可压缩微极流体模型的数学问题在生物医药等领域中展现出日益重要的应用，这些应用和三维可压缩微极流体模型解的适定性问题密切相关。本项目对可压缩微极流体模型的相关数学问题进行研究，首先研究了可压缩微极流体模型的恰当弱解问题，其次研究可压缩微极流体模型的有限能量弱解问题，以及弱强唯一性问题。本项目取得主要结果如下：.1. 提出了可压缩微极流体模型的恰当弱解的概念。借助于经典的三层近似分析法，证明了可压缩微极流体模型的恰当弱解满足相对熵不等式，从而得到可压缩微极流体模型恰当弱解的全局存在性。进一步研究了可压缩微极流体模型的弱强解唯一性问题。.2.提出了可压缩微极流体模型的有限能量弱解的概念。借助于相对熵泛函和相对熵不等式，清晰刻画可压缩微极流体模型有限能量弱解和恰当弱解之间的关系。我们证明了可压缩微极流体模型有限能量弱解满足相对熵不等式，从而证明了有限能量弱解就是恰当弱解。结合相对熵、相对熵不等式，经典的嵌入定理和Gronwall引理证明了可压缩微极流体模型的弱强唯一性。.上述结果揭示了可压缩微极流体模型不同类型解之间的本质联系，为可压缩微极流体模型的衰减估计和低马赫数极限等数学问题成功解决奠定了一定的基础，也为解决三维可压缩微极流体模型的解的适定性提供新思路和新方法，还促进了经典Navier-Stokes方程组的数学理论发展，而且也可以探索不同领域数学模型之间本质联系，为研究更为复杂的流体提供可能。.