粘性液体-气体两相流模型解的适定性研究

The viscous liquid-gas two-phase flow model in 1 dimensional space can be viewed to be a kind of drifting model, which are widely used.to describe unstable compressible liquid-gas stream in pipeline; the viscous liquid-gas two-phase flow model in N dimensional space is widely used within the petroleum industry to describe production and transport of oil and gas through long pipelines or well. This research project will study specifically the following two directions: 1. Study 1-dimension viscous liquid-gas two-phase flow modal (initial vacuum) for fixed boundary value problem, discuss the global existence and uniqueness of classical solutions in general initial conditions. It is a generalizations of the existence of classical solutions for fixed fixed boundary value problem with sufficiently small energy. 2. Study the existence and uniqueness problem of spherical symmetry solutions for high dimension constant viscous liquid-gas two-phase flow modal in bounded domain. It analogy to the existing existence and uniqueness problem of spherical symmetry solutions for Navier-Stokes equation. This project is one of high-profile problem in modern partial differential equation. The research techniques of Navier-Stokes equation can be used for reference. At the same time, the new research methods are needed since the differents between two equations, which supply a new view for the existence research of solutions to Navier-Stokes equation.

一维空间中，粘性液体-气体两相流模型是一种“漂流”模型，用于模拟管道中不稳定的可压缩液体-气体流；在石油工业中，通常用N维的液体-气体两相流模型来描述管道和深井中油和气的生产和输运。本课题拟具体研究两个方面：1.研究一维粘性液体-气体两相流模型（初始可含真空）固定边界问题，讨论其在一般初始条件下经典解的整体存在性，这将是对能量充分小的情况下固定边界问题经典解存在性研究的一个推广；2. 研究高维常数粘性液体-气体两相流模型在有界区域球对称解的存在唯一性问题，类比于已有的Navier-Stokes方程组球对称解的存在唯一性的问题。研究课题是现代偏微分方程的热点问题之一，在研究上借鉴于Navier-Stokes方程组研究方法，由于方程差异同时也需要新的研究手段，这也为研究Navier-Stokes方程组解的存在性提供一个新的视角。

本项目我们主要针对一维粘性的液体-气体两相流模型大初始值整体解的存在性以及高维常数粘性液体-气体两相流模型在有界区域球对称解的整体存在性和唯一性进行了具体的研究。研究情况如下：一维粘性液体-气体两相流模型（初始可含真空）大初始值的固定边界问题，由于限制液体的质量上界 ，导致常规的证明方法都回归到必须进行先验假设，进而局限了能量的小性，我们还要进一步尝试其他办法，希望通过得到压力 L无穷模估计证明结果。对于高维常数粘性液体-气体两相流模型在有界区域的球对称解问题，我们通过构造新的函数，利用能量估计及重要不等式，证明得到液体质量的上界，进而证明了整个解的存在唯一性，文章整理中。按计划基本完成了本项目的研究。