Nonlinear Stability Of Viscous Shock Waves For One-dimensional Nonisentropic Compressible Navier-stokes Equations With A Class Of Large Initial Perturbation

This thesis is concerned with the large time behaviors of solutions to the Cauchy problem of one-dimensional nonisentropic compressible Navier-Stokes systems.For such a problem,it is well-known that the precise description of the large time behaviors of its global solutions is completely determined by the structure of the unique global en-tropy solution of the resulting Riemann problem of the corresponding one-dimensional compressible Euler system:If the unique global entropy solution of such a Rieman-n problem consists of rarefaction waves,shock waves,contact discontinuities,and/or their superpostions,then the large time behaviors of global solutions of the Cauchy problem of the one-dimensional nonisentropic compressible Navier-Stokes systems can be exactly described by the rarefaction waves,viscous shock profiles,viscous contact discontinuities and/or their superpositions.Such a problem can reformulated as the problem of time-asymptotic nonlinear stability of some elementary wave patterns,such as rarefaction waves,viscous shock profiles,viscous contact discontinuities and/or their superpositions:If the initial data of the one-dimensional nonisentropic compressible Navier-Stokes systems is a perturbation of the elementary wave patterns taking val-ue at the initial time,does the Cauchy problem of the one-dimensional nonisentropic compressible Navier-Stokes systems admit a unique global solution which tends to the elementary wave patterns time-asymptotically?For the nonlinear stability of some elementary wave patterns to the one-dimensional nonisentropic compressible Navier-Stokes systems,for the case with small initial per-turbation,the results available up to now is quite complete except the cases when the elementary wave patterns are a linear superposition of one viscous shock profile and one rarefaction wave or viscous contact discontinuity(for the nonlinear stability of viscous shock waves,see[37,58]for results with zero mass assumption and[49]for the case with general initial perturbation,for nonlinear stability of viscous contact discontinu-ities,see[20,51]with zero mass assumption and[25]for general initial perturbation,for the nonlinear stability of rarefaction waves,see[39,50],while for the nonlinear stability of composite wave patterns consisting of either rarefaction waves and viscous contact discontinuities or viscous shock waves from different families,see[15,17]),while for the corresponding results with large initial perturbation,some results are obtained for rar-efaction waves(cf.[1,12,61,62,68]),viscous contact discontinuities(cf.[11,26])and their superpositions(cf.[13,23]),but for viscous shock profiles,to the best of our knowledge,no results are available up to now.The main purpose of this thesis is devoted to the nonlinear stability of viscous shock profiles to the Cauchy problem of one-dimensional compressible Navier-Stokes systems for a class of large initial perturbation.Our method is based on the elementary energy method and the continuation argument.By exploiting the smallness of the strength of the viscous shock profiles and by utilizing the underlying structures of the one-dimensional compressible Navier-Stokes systems fully,for a class of large initial perturbation,we can indeed control the possible growth of the solutions of the one-dimensional nonisentropic compressible Navier-Stokes systems caused the nonlinearities of system itself.In fact,we can first deduce the uniform positive lower and upper bounds on the density and the temperature,from which we can indeed deduce the desired nonlinear stability result.In our result,the oscillations of the initial density and initial temperature can be sufficiently large.It is worth to pointing out that the properties of the viscous shock profiles,the argument developed by S.Jiang in[28]to get the uniform positive lower and upper bounds on the density and the method developed by J.Li and Z.Liang in[47]to yield the desired uniform upper bound on the temperature play an important role in our analysis.