Nonlinear Stability Of Viscous Shock Waves For One-dimensional Isentropic Compressible Navier-Stokes Equations With Density-Dependent Viscosity

In this dissertation,we study the large time asymptotic behavior of global solutions to the Cauchy problem of one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity.As we all know,the asymptotics of the global solutions of such a Cauchy problem can be uniquely determined by the structure of the unique global entropy solution of the corresponding Riemann problem of the Euler equations:If the unique global entropy solution of the above Riemann problem consists of rarefaction waves,shock waves,and/or their linear superpositions,then the large time behavior of the global solutions of the Cauchy problem can be described by the rarefaction waves,viscous shock waves and/or the composite wave patterns composed of them.In this dissertation,we study the nonlinear stability of viscous shock waves for one-dimensional isentropic compressible Navier-Stokes equations for a class of large initial perturbation.For the case when the viscosity coefficient is a positive constant,the results on the nonlinear stability of viscous shock profiles of the one-dimensional isentropic compressible Navier-Stokes equations are well-understood,those interested are referred to[28,29,33,38,41]and the references cited therein.On the other hand,when the compressible Navier-Stokes equations are derived from the Boltzmann equation through the Chapman-Enskog expansion,the viscosity coefficient is a function of the absolute temperature[3,23].For the isentropic flow,the viscosity coefficient can be regarded as a density-dependent function[24].When the viscosity coefficient depends on the density,[34]and[7]study the nonlinear stability of viscous shock waves for the Cauchy problem of the one-dimensional isentropic compressible Navier-Stokes equations under small initial perturbation and a class of large initial perturbation,respectively.Where,in[7]the pressure and viscosity coefficient of the fluid are assumed to satisfy p(?)=?^-?,?(?)=?^-? ?>1,??0 with ?>1 and 0??<1/2.The main purpose of this dissertation is to improve the results of[7]to ?>1,??0.Note that when ?=2,?=1,the one-dimensional isentropic compressible NavierStokes equations is reduced to the Saint-Venant shallow water wave equations,which have important applications in physics and oceanography[1,4,6].The key to study the above problems is to derive the uniform positive upper and lower bounds of specific volume(reciprocal of density)independent of time variable t.Note that in[7],the method of Kanel'[19]is used to derive the uniform positive upper and lower bound of the specific volume.When obtaining the upper bound of the specific volume,this method requires that 0??<1/2,?>1.For the problem under our consideration,we can first use the method of Kanel'[19]to derive the uniform positive lower bound of the specific volume.In order to expand the range of ?,we need to estimate the upper bound of the specific volume more precisely,which is the main innovation of this paper.After obtaining the uniform positive upper and lower bound estimates of the specific volume,we can extend the local solution of the Navier-Stokes equations step by step to the global solution by means of a well-designed continuation technique,and obtain the corresponding large time asymptotic behavior.This dissertation is mainly divided into four parts.In chapter 1,we introduce former results on the nonlinear stability of viscous shock waves of the one-dimensional isentropic compressible Navier-Stokes equations and then give the main results of this dissertation(Theorem 1).In chapter 2,we give some basic properties of viscous shock waves.In chapter 3,we prove our main result Theorem 1.Finally,in the fourth chapter,we give a summary of the thesis and list some problems to be further studied.