多组分反应流的若干数学问题

This project will study the well-posedness of two types of multi-component reaction flows from the perspective of mathematical theory. Firstly, based on the previous work of Fick diffusion, we further will investigate the more practical Maxwell-Stefan diffusion and prove the existence of the global weak solutions of isentropic compressible Navier-Stokes-Maxwell-Stefan equations; Secondly, we will improve the Zatorska’result about non-isentropic compressible Navier-Stokes coupling reaction diffusion equations and remove the unnatural cold-pressure assumption, meanwhile, we also will prove the existence of global weak solutions of non-isentropic compressible Navier-Stokes coupling reaction diffusion equations in the case of viscosity-dependent temperature; To solve these problems, the project has designed a series of feasible steps, the methods involved compactness tools, harmonic analysis, energy estimates, degeneration parabolic theory and so on. These research results can not only enrich the mathematical theory of multi-component reaction flows, but also play a guiding role for practical applications.

本项目将从数学理论角度来研究两类多组分反应流的适定性问题，一是在前面Fick扩散的工作基础之上，进一步考察更具实际意义的Maxwell-Stefan扩散，证明一类等熵可压缩Navier-Stokes-Maxwell-Stefan方程组的整体弱解存在性；二是改进Zatorska非等熵可压缩Navier-Stokes耦合反应扩散方程组的结果，去掉不自然的冷压假设，同时也分析一类粘性依赖于温度的非等熵可压缩Navier-Stokes耦合反应扩散方程组的整体弱解存在性; 为解决这些问题， 本项目设计了一系列可采取的步骤，所用的方法涉及紧性工具，调和分析，能量估计，退化抛物理论等等。这些研究成果不但可以丰富多组分反应流的数学理论，而且也为实际的应用起到指导性作用。

多组分气体混合物发生化学反应现象经常出现在很多工业中，比如航天,化学工程，燃烧理论等等。本项目研究了两类多组分反应流模型，即等熵可压缩Navier-Stokes方程组耦合Maxwell-Stefan扩散方程组和非等熵可压缩Navier-Stokes 方程耦合多组分反应扩散方程组，我们着重研究了这些模型的整体大初值弱解存在性，通过克服若干数学困难，给出详细的数学理论证明。