不可压缩轴对称流体方程的定性研究

不可压缩轴对称流体方程是三维空间中一类具有特殊几何结构的流体方程，其解的行为涵盖了极为广泛的自然现象，例如圆管中的流动、机身的绕流问题等。有关这类方程的研究不仅具有重要的理论价值，而且在工程、空气动力学、天体物理等领域中有着极其广泛的应用，一直受到物理学家和数学家的关注。. 本项目将主要围绕以下三个方面展开研究：(1)不可压轴对称Navier-Stokes方程解的爆破率估计以及初值对奇性的影响；(2)不可压无旋轴对称Euler方程解的整体适定性问题以及在穿孔区域中的渐近行为；(3)不可压轴对称MHD方程特解的整体存在唯一性问题以及解的正则性准则。我们期望通过本项目的实施，对于这三类方程的轴对称结构有新的发现或者提出新的研究方法，改进现有结果并探讨新问题，得到一系列较成体系的理论成果，从而加深对于不可压缩轴对称流体方程在数学层面的认识和理解。

The incompressible axisymmetric fluid equation is a kind of fluid equation with special geometrical structure in three-dimensional space, the behavior of which covers a wide range of natural phenomena, such as the flow in the circular tube, the flow around the fuselage, and so on. The research on such equation not only has important theoretical value, but also has extremely wide application in engineering, aerodynamics, astrophysics and other fields, and has always been attracting the attention of physicists and mathematicians. . This project will mainly focus on the following three aspects: (1) Estimation of the blow-up rate of solutions for the incompressible axisymmetric Navier-Stokes equations and the effect of initial value on their singularity; (2) The problem of global well-posedness and asymptotic behavior in the perforated domain of solutions for the incompressible axisymmetric Euler equations without swirl; (3) The problem of global existence and uniqueness of special solutions and regularity criteria of solutions for the incompressible axisymmetric MHD equations. Through the implementation of this program, we look forward to get new discoveries to the axisymmetirc structures of the three kinds of equations or propose new research methods, improve some existing results, investigate some new problems, obtain a series of theoretical results, and therefore have better knowledge and understanding on the mathematical theory to the incompressible axisymmetric fluid equation.