The Existence Of Solution To Viscous Liquid-gas Two Phase Flow Model

We study the existence of the solutions to a viscous liquid-gas two-phase flow model in1dimensional space:where n=αgρg, m=αlρl denote gas mass and liquid mass respectively; ρl-,ρg, ul and ug denote density and velocity of liquid and gas respectively; P is common pressure for both phases; q represents external forces like gravity and friction;μ is viscosity coefficient。The unknown vari-ables αg,αl∈[0,1] denote gas and liquid volume fractions, satisfying the fundamental relation: αg+αl=1.The model (0.0.2) can be viewed to be a kind of drifting model, which are widely used to describe unstable compressible liquid-gas stream in pipeline, and where we assumed the fluids velocities are equal and have the common pressure, considered the viscous effect, neglected the effect of gas in the mixture momentum equation and the external force.For the model (0.0.2), we study the existence and uniqueness of the weak solution, as well as the existence and uniqueness of the classical solution, under approximate conditions, which have be divided into the following areas:· The free boundary problem about model (0.0.2) in one dimensional space has been in-vestigated, where the viscosity coefficient depends on the masses of fluid and gas, i.e. and both the two fluids connect to vacuum state continuously at the free boundaries. We get the existence and uniqueness of the weak solution, as0<β<1.· The fixed boundary problem (i.e. u=0) about model (0.0.2) in one dimensional space has been investigated, where the initial condition contains vacuum, and the viscosity co-efficient depends on the mass of liquid and satisfies μ∈C2[0,∞),μ(m)≥Mi>0. We obtain the existence and uniqueness of the classical solution.