The Investigation On The Optimal Decay Rates Of The Solutions To The Compressible Navier-Stokes Equations And The Related Model

This thesis is concerned with the compressible isentropic Navier-Stokes equations and the compressible gas-liquid two-phase drift-flux model.The Navier-Stokes equations are the basic equations describing the motion of the viscous fluid,and have a wide range of applications in science and engineering.The investigation on the mathematical theory of the compressible Navier-Stokes equations is one of the most popular directions in the field of pure mathematics.And now a lot of challenging problem remain open.While the compressible gas-liquid two-phase drift-flux model is firstly formulated by Zuber and Findlay et al.in 1965,and is often used to describe the motion of two-phase mixed fluid.In this thesis,we would like to prove the optimal decay rates as well as the decay estimates of the weak solutions to the free boundary problems of the compressible isentropic Navier-Stokes equations and the compressible gas-liquid two-phase drift-flux model.Precisely,In Chapter 2,we consider the free-boundary problem of the one-dimensional com-pressible isentropic Navier-Stokes equations with density-dependent viscosity coef-ficient.For the case when the density is connected to vacuum continuously,under appropriate smallness conditions on the initial energy,the optimal decay rate of the density function along with its behavior near the interfaces is studied.In the meanwhile,we obtain also sharper decay rates for the norms in terms of the velocity function.In Chapter 3,we investigate the free-boundary problem to the one-dimensional compressible gas-liquid two-phase drift-flux model,where the viscosity coefficient depends on the mass function,and the mixed fluid connects to vacuum continuously.When the initial data satisfies some smallness assumption,we otain the optimal point-wise upper and lower decay estimates on masses as well as the sharpest decay rates for the norms in terms of the velocity function.It was worth noting that according to the point-wise lower and upper bounds for the density(mass)function obtained in the present thesis,when the viscosity coefficient degenerates at the vacuum free boundary,the degenerate behavior of the density(mass)function near the free boundary is mutually related to its optimal long-time behavior.This is similar to the case of constant viscosity coefficient.Luo-Xin-Yang[45]considered the free-boundary problem of the one-dimensional Navier-Stokes equations with constant viscosity coefficient as well as density connected to vacuum continuously,and studied both the optimal decay rate of the density and the behavior of the density function near the interfaces.Our results can be viewed as generalization of the one by Luo-Xin-Yang.The proof of our main results is based on the standard finite difference method and the method of energy estimate.The key matter is to figure out the optimal decay rate of the density(mass)function as well as to obtain the lower bound of the density(mass)function involving the optimal decay rates.