The Flow For 1D Compressible Two-phase Fluids Of Immiscible Gas-liquid

In present paper,we consider the movement of the interfaces for the immiscible two-phase flow.But the classical description of boundary problem can’t be suited when the topological transitions such as droplet formation,coalescence or break-up of droplets occur.Navier-Stokes-Allen-Cahn(NS AC)is the equations describing the incompatible gas-liquid two-phase flow,and it regards the interface between the two-phase fluids of immiscible gas-liquid as a boundary layer with thickness,and derives the free energy of the interface.This paper is considered with the non-smooth free energy density for the viscosity compressible NS AC system with the periodic boundary value conditions in one dimensional space in the Eulerian coordinates.The main difficulty in this problem is non-smoothness of the free energy density.In the paper,we obtain the approximation function of the non-smoothness of the free energy density and get the existence and uniqueness of the local approximation solutions by the fixed point theorem.Then,we combine the extension method with the classical energy estimation to prove the existence and uniqueness of the global solutions.The main methods and conclusions are as follows:Using smoothing approximation method to get an approximation function of the non-smooth free energy density and structure approximate equations.By linearity the approximate equations,combining the fixed point theorem,we can prove the existence and uniqueness of the local approximate solutions.We use the extension method to obtain global approximate solution and get the existence of the global weak solution for the original equations system by calculating the uniform estimates about χ.Furthermore,using the higher order energy estimates,we prove the existence of the global solutions for the NS AC system and obtain the upper and lower bounds of the two-phase flow density and the component concentration difference respectively.In additionχ∈[-1,1].