The One-dimensional Viscous Liquid - Gas Two-phase Flow Model Free Boundary Problem

In this paper, we establish the viscous of two-phase flow model, we consider the existence and uniqueness of strong solutions when both the initial liquid and gas masses connect to vacuum discontinuously and the weak solutions when both the initial liquid and gas masses connect to vacuum continuously. This paper is organized as follows:In chapter one, we introduce the background and the basic knowledge of this paper.In chapter two, we establish existence and uniqueness of strong solutions for a viscous of two-phase flow model when both the initial liquid and gas masses connect to vacuum discontinuously. Applying techniques in studying Navier-Stokes equations and using a priori estimates, we get the positive upper and lower bound of m and n. Using the difference method to prove the existence and uniqueness of the global strong solutions. This improves the previous result of Evje, Karlsen and Yao, Zhu by enlarging the interval of β to β>0,γ> max{β＋1,2β}.In chapter three, we consider a free boundary value problem for two-phase liquid-gas model with mass-dependent viscosity coefficient, the gas is assumed to be poly tropic whereas the liquid is treated as an incompressible fluid, and the fluid velocities is unequal, i.e., ug≠ul. The local existence of weak solution is established when the initial gas masses connect to vacuum continuously.