可压缩微极流体方程组解的性态研究

The compressible micropolar fluid model, a typical representative of compressible fluid models, describes the motion of the viscous compressible fluids with micro-rotation effects of particles. This model can be widely applied in physics and engineering, including rheologically complex liquids such as blood and suspensions. The investigation on this model from a mathematical aspect has gradually attracted the attention of mathematicians as well. This project aims to study the well-posedness of classical solutions to 3D Cauchy problem on compressible micropolar fluid model, as well as the vanishing viscosity coefficients limit and boundary layer to the 2D initial-boundary value problem.We expect a breakthrough in these aspects. When micro-rotation effects are neglected, the compressible micropolar fluid equations reduces to the compressible Navier-Stokes equations. Compared with the classical Navier-Stokes equations, the difficulty mainly comes from the micro-rotation effects. To overcome this difficulty, we need some new energy estimates.

作为可压缩流体力学方程组的一种典型代表，可压缩微极流体方程组描述了一类考虑微观粒子旋转效应的可压缩粘性流体的运动规律。它能很好的解释一些复杂流变液体(血液和悬浮液)的物理规律，在物理学和工程中有着广泛的应用背景，其数学理论的研究也引起了数学界的关注。本项目拟围绕以下两方面内容进行研究：1、三维等熵可压缩微极流体方程组Cauchy问题经典解的整体存在性；2、二维等熵可压缩微极流体方程组初边值问题粘性系数消失极限和边界层问题。当忽视微观粒子的旋转效应时，可压缩微极流体方程组就转变为经典的可压缩Naiver-Stokes方程组。因此与经典的Navier-Stokes方程组相比，要想在上述问题上有所突破需要发展一些新的能量估计来克服由微观粒子的旋转效应所带来的困难。

作为可压缩流体力学方程组的一种典型代表，可压缩微极流体方程组描述了一类考虑微观.粒子旋转效应的可压缩粘性流体的运动规律。它能很好的解释一些复杂流变液体(血液和悬浮.液)的物理规律，在物理学和工程中有着广泛的应用背景，其数学理论的研究也引起了数学界.的关注。本项目围绕以下两方面内容进行了研究：1、三维等熵可压缩微极流体方程组Cauchy.问题经典解的整体存在性；2、二维等熵可压缩微极流体方程组初边值问题粘性系数消失极限.和边界层问题。