The Analysis Of Gas-liquid Two-phase Fluid Flow

This paper is concerned with a class of immiscible compressible viscous gas-liquid two-phase fluids.Due to the complex of the fluid flow at the gas-liquid interface,the classical Navier-Stokes equations cannot describe it appropriately.Therefore,this paper introduces the Cahn-Hilliard equation to describe the interaction at the interface of two-phase flow,applying the compressible Navier-Stokes-Cahn-Hilliard(NSCH)equations with diffusion interface,and the selection of pressure introduces van der Waals equation of state replaces the traditional ideal multi-party gas model,and the article study the One-dimensional well-posedness of the periodic solution of the equation system.The main difficulties of this problem are the non-convexity of pressure and the determination of the upper and lower bounds of density.The article uses the knowledge of measurement theory and the monotone decomposition of pressure to obtain the prior estimate of the solution,and uses the existence and uniqueness of the local in time solution to extend the solution,so that the strong solution of the problem has the existence and uniqueness.Here are the concrete research methods and the main conclusions in this paper:(1)The existence and uniqueness of the local solution:By constructing an iterative scheme,we linearized the system of nonlinear equations,and then use fixed point theorem and compressive mapping theory,it is proved that the local existence of strong solutions is unique.(2)The existence and uniqueness of the global solution:A priori estimate of the solution is obtained through knowledge of inequalities,combined with the existence and uniqueness of the local solution,the global existence and uniqueness of the strong solution is obtained.From the embedding theorem,the solution of the equation is continuous,that is,the shock wave phenomenon does not occur.In addition,it is obtained that when the initial density does not contain vacuum,no vacuum phenomenon will occur.