| In recent years,the study on the time asymptotically nonlinear stability of certain elementary waves,such as viscous shock wave,rarefaction wave,viscous contact discontinuity and boundary layer solution,of some dissipative fluid dynamical systems,which include the compressible Navier-Stokes system as typical example,has been one of the hottest topics in the filed of nonlinear partial differential equations.For such a problem,for the case with small initial perturbations,the results available up to now are quite complete,but for the corresponding results with large initial perturbations,not so many results are available up to now.The purpose of this thesis focuses on the global solvability and the precise descriptions of the large time behaviors of solutions of certain compressible Navier-Stokes type equations with large initial perturbations and the results obtained can be summarized as in the following:ˇNonlinear stability of weak viscous shock waves of the inflow problem of the one-dimensional compressible isentropic Navier-Stokes equations with large initial perturbation which can allow the initial density to have large oscillations;ˇNonlinear stability of boundary layer solutions of the outflow problem of the one-dimensional two-fluid compressible Navier-Stokes-Poisson system with large initial perturbation;ˇGlobal solvability of the Cauchy problem of one-dimensional non-isentropic compressible Navier-Stokes-Korteweg system with large initial data.This thesis is divided into four chapters,in the first chapter,after introducing some progress on the related problems,we point out the problems we want to study and then state our main results.Chapter 2 is concerned with the inflow problem for the one-dimensional compressible isentropic Navier-Srokes system.For such a problem,a complete classification of its large time behaviors is carried out by Matsumura in[120]and for the mathematical justification of the above expected large time behaviors,for the case with small initial perturbations,the nonlinear stability of the boundary layer solution and the superposition of a boundary layer solution and a rarefaction wave is established by Matsumura and Nishihara in[127],while the nonlinear stability of supersonic rarefaction wave is obtained by Shi in[148].As for viscous shock wave,Huang,Matsumura and Shi study the nonlinear stability of the viscous shock wave and the superposition of a boundary layer solution and a viscous shock wave in[65].For large initial perturbation,the nonlinear stability of boundary layer solution for a class of large initial perturbation whose energy is small but the initial density can have large oscillations and the nonlinear stability of supersonic rarefaction wave with large initial perturbation is obtained in[29].Thus a natural question is whether is it possible to deduce the nonlinear stability of the viscous shock wave for a class of large initial perturbation?The main purpose of the second chapter is concerned with such,a problem and some nonlinear stability results on the weak viscous shock wave similar to those obtained in[29]are obtained for a class of large initial perturbation which can allow the initial density to have large oscillations by employing the energy method and the continuation argument.The key point in our analysis is to control the possible growth of the solutions of the inflow problem induced by inflow boundary condition.In chapter 3,we study the outflow problem of the two fluid one dimensional compressible Navier-Stokes-Poisson system.For such a problem,the nonlinear stability of the boundary layer solution,rarefaction wave and the superposition of a boundary layer solution and rarefaction wave is obtained in[26],while the convergence rate of the solution of the inflow problem to the boundary layer solution is further obtained in[186].It is worth to pointing out that the initial perturbation is assumed to be small in certain Sobolev norm in[26]and a more strong smallness assumption is imposed on the initial perturbation in[186],which asks that certain weighted Sobolev norms of the initial perturbation are small.In the third chapter,certain nonlinear stability results of the boundary layer solution of the outflow problem of the one dimensional two fluid compressible Navier-Stokes-Poisson with large initial perturbation are obtained.Based on these nonlinear stability results,the convergence rates of the global solution of the outflow problem to the boundary layer solution are further obtained.It is worth to pointing out that,for the non-degenerate case,to yield the desired convergence rates,in addition to the assrumption that the initial perturbation is assumed further to belong to certain weighted Sobolev space,the conditions we imposed on the initial perturbation is the same as the those conditions to guarantee the nonlinear stability results,while for the degenerate case,we do need to ask that certain weighted Sobolev norms of the initial perturbation to be sufficiently small.Compared with the case of the outflow problem of the one dimensional compressible Navier-Stokes,the key point in our analysis lies in how to control the possible growth,of the solution of outflow problem of the one dimensional compressible Navier-Stokes-Poisson system induced by the electric field.Chapter four is devoted to the construction of global smooth solutions to the Cauchy problem of the one-dimensional non-isentropic compressible Navier-Stokes-Korteweg system with large initial data.For such a problem,the results available up to now focus on the isentropic case(cf.[2,6,9,38,51,155]and the references cited therein)and to the best of our knowledge,no result is obtained for the non-isentropic case.In the forth chapter,a global solvability result to the Cauchy problem of the one-dimensional non-isentropic compressible Navier-Stokes-Korteweg system with large initial data is obtained.Similar to the case of the non-isentropic Navier-Stokes system,the key point in our analysis is to yield the positive lower and upper bounds on the density function and the temperature function and we had to deal with the analytical difficulties caused by the appearance of the Korteweg term. |
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