Global Stability Of Basic Wave Patterns For Compressible Navier-Stokes-Poisson System

The compressible Navier-Stokes-Poisson（denoted as CNSP in the sequel for sim-plicity）equations,which consist of the compressible Navier-Stokes equations coupled with the Poisson equation,can be used to describe the dynamics of the viscous and/or heat-conducting fluid under the influence of the self-induced electrostatic potential force,cf.[1,54]and the references cited therein.The study on the global well-posedness of strong solutions together with the precise description of their large time behaviors to its Cauchy problem and initial boundary-value problems,especially for the case when the asymptotics of the electrostatic potential is nontrivial,has been one of hottest topics in the field of nonlinear partial differential equations in recent years.For such a problem,the results available up to now with small initial data are quite complete,but for the case with large initial data,to the best of our knowledge,although the construction of global smooth non-vacuum solutions to the Cauchy problem of CNSP equations with large initial data and for a class of density and/or temperature dependent transport coefficients is carried out in[66],the problem on their large time behaviors is obtained only for the time-asymptotically nonlinear stability of non-vacuum equilibrium states under the assumption that the adiabatic exponentγ>1 is assumed to satisfy thatγ-1is sufficiently small.Thus a natural question is:For the Cauchy problem of CNSP equations,how to construct global smooth non-vacuum large solutions and to deduce its large time asymptotic behavior for any adiabatic exponentγ>1?Especially,how to obtain the nonlinear stability of non-vacuum constant equilibrium states and some basic wave patterns such as rarefaction waves,viscous shock profiles,viscous contact discontinuities and a composite wave pattern com-posed of rarefaction waves and viscous contact discontinuities,etc.under large initial perturbation?The main purpose of this doctoral dissertation is concentrated on such a problem.We begin with the study of the time-asymptotically nonlinear stability of rarefaction waves and viscous shock profiles of the Cauchy problem of isentropic CNSP equations under large initial perturbations.Then,we further study the Cauchy problem of the full CNSP equations and our main purpose focuses on the nonlinear stability of the non-vacuum constant equilibrium states and some basic wave patterns such as rarefaction waves,viscous shock profiles,viscous contact discontinuities and a composite wave pattern composed of rarefaction waves and viscous contact discontinuities,etc.with large initial perturbation.This dissertation is divided into the following six chapters:In the first chapter,we introduce the background of isentropic CNSP equations and non-isentropic CNSP equations,respectively,review some recent results closely related,list the main problems we plan to study and give the main results obtained.In Chapter 2,we study the nonlinear stability of weak viscous shock profiles of the Cauchy problem of isentropic CNSP equations for a class of initial perturbation which can allow the initial density to have large oscillation.For such a problem,the case with small initial perturbation is obtained in[4].The main idea in[4]is to use the a priori assumption that the H²-norm of the antiderivative of the perturbation function is sufficiently small,especially the smallness of the L^∞-norm of the perturbation function,together with the smallness of the strength of the viscous shock profiles to control the possible growth of the solutions of the Cauchy problem induced by the nonlinearity of the isentropic CNSP equations and the interactions between the solution and the viscous shock profiles.Our main observation is that we can indeed close the desired energy estimates by only using the smallness of the H¹-norm of the disturbance of the velocity function and the W^1,∞-norm of the potential function of the electrostatic field.Based on such an observation,we can indeed obtain the nonlinear stability of viscous shock profiles of the Cauchy problem of isentropic CNSP equations for a class of initial perturbations that can allow the density function to have a large amplitude by a well-designed continuation argument.In Chapter 3,we study the nonlinear stability of weak rarefaction waves of the Cauchy problem of isentropic CNSP equations with large initial perturbations.The key point in our analysis is to derive the uniform positive upper and lower bounds of the specific volume and the potential function of electrostatic field.Our main idea is to use the smallness of the strength of the rarefaction waves,the monotonicity of the rarefaction waves and temporal decay estimates of the smooth approximation of the rarefaction waves to control the possible growth of the solutions of the Cauchy problem of the isentropic CNSP equations induced by the nonlinearity of the equations and the interactions between the solutions and the rarefaction waves.Based on these estimates,we can then use Kanel’s technique to yield the desired uniform positive upper and lower bounds of the specific volume and the potential function of the electrostatic fields.In Chapter 4,we study the nonlinear stability of non-vacuum constant equilibrium states of the Cauchy problem of non-isentropic CNSP equations.Although the existence of global smooth large non-vacuum solutions to its Cauchy problem for a class of density and/or temperature dependent transport coefficients has been obtained in[66],the problem on time asymptotically nonlinear stability of non-vacuum constant equilibrium states is obtained only if the adiabatic exponentγ-1>0 is assumed to be sufficiently small.The main purpose of this chapter is to prove that the same nonlinear stability results holds for any adiabatic exponentγ>1 and large initial data.Our main idea is to make full use of the intrinsic structure of the non-isentropic CNSP equations to derive the relationship between the upper and lower bounds of the specific volume and the absolute temperature,especially to control the lower and upper bounds of the specific volume by the upper bounds of the absolute temperature.With these estimates in hand,three auxiliary functions are introduced and by proving that these auxiliary functions are uniformly bounded,we can then deduce the desired upper bound of the absolute temperature,and consequently deduce the desired uniform positive lower and upper bounds of the specific volume.Finally,by obtaining the H³norm estimates of the perturbation functions and by combining with the continuation argument designed in[71],we can then extend the well-established local solutions step by step to a global one.In Chapter 5,we study the nonlinear stability of a composite wave pattern consist-ing of rarefaction waves and viscous contact discontinuities of the Cauchy problem of non-isentropic CNSP equations with large initial perturbations.Compared with Chap-ter 4,the main difficulty we need to overcome is how to control the possible growth of the solutions induced by the nonlinearity of the equations,the interactions between the basic waves of different families,and the interactions between the solutions and the rarefaction waves and the viscous contact discontinuities.Our main idea is to use the monotonicity of the rarefaction waves,the temporal decay estimates of the viscous con-tact discontinuities and the smooth approximations of the rarefaction waves to control the above-mentioned possible growth of the solutions.In Chapter 6,we list some problems to be studied further.