一类流体与粒子相互作用模型的解的适定性和粘性消失问题

This project is concerned with well-posedness and vanishing viscosity limit of a fluid-particle interaction model: the bubbling regime. The following problems are considered. (1)The blow-up form of the strong solution to the spherically symmetric problem at the center of sphere; the global well-posedness of the strong solution to the spherically symmetric problem in a unbounded domain; the global existence of the weak solution with large initial value to the 3D Cauchy spherically symmetric problem. (2)The vanishing viscosity limit with Navier-slip boundary condition. This project intends to make a breakthrough in the above mathematical theoretical problems concerned by peers. The above study can enrich and perfect the mathematical theory of the compressible fluid dynamics equations, and the results, techniques, and methods from the study of these problems will not only promote some new methods to overcome the difficulties of nonlinear degenerate equations and mixed equations, but also provide important theoretic references for practical applications.

本项目主要研究鼓泡型的流体与粒子相互作用模型的适定性和粘性消失极限。主要考虑如下问题：（1）建立球对称问题的强解在球心处的爆破形式；在去掉球心的无界区域中研究球对称问题的强解的整体适定性；在三维空间上研究带有大初值的球对称弱解的整体存在性。（2）研究在三维Navier-slip边界下的粘性消失极限问题。本项目拟对同行关心的上述数学理论问题有所突破。通过对这些问题的研究所获得的结论、技巧及方法，有可能丰富和完善可压缩流体方程的数学理论，为解决非线性退化方程组和混合型方程组的相关问题提供新的思路以及对实际问题提供一定的数学理论依据和指导。

流体与流体中的粒子的相互作用广泛存在于自然界和工业过程，本项目主要研究流体与粒子相互作用模型的适定性和剪切粘性极限问题。主要考虑如下：（1）利用渐进性分析方法，推导了在磁场作用下的流体与粒子相互作用模型，进而研究了它的二、三维柯西问题的强解和经典解的存在唯一性；（2）在柱对称框架下，考虑了剪切粘性系数依赖于密度的流体与粒子相互作用模型，当剪切粘性趋于零时研究它的边界层现象。通过对这些问题的研究所获得的结论、技巧及方法，有可能丰富和完善可压缩流体方程的数学理论，对实际问题提供一定的数学理论依据和指导。