The Stability Of Waves To A Compressible Navier-Stokes-Poisson Equation

This thesis is concerned with the stability of waves for non-isentropic one-fluid and two-fluid Navier-Stokes-Poisson system. Firstly, we construct a rarefaction wave pro-file whose strength is not necessarily small, and then we prove that the non-trivial solution we construct globally exists and is time asymptoticly stable under small perturbations for the corresponding Cauchy problem on the two-fluid non-isentropic Navier-Stokes-Poisson system. Secondly, on account of the quasineutral assumption of the electric potential which may take distinct constant states at boundary, we construct a viscous contact discontinuity, and then we prove the stability of the con-tact discontinuity of the one-fluid Navier-Stokes-Poisson system with free boundary. Finially, we prove the nonlinear stability of the composite wave consisting of a viscous contact discontinuity and a 3-rarefaction wave of the one-fluid Navier-Stokes-Poisson system with free boundary.In Chapter 2, we are concerned with the large-time asymptotics of solutions toward the rarefaction wave of the quasineutral Euler system for a two-fluid plasma model in the presence of diffusion of velocity and temperature, i.e.Navier-Stokes-Poisson system. We mainly extend the results in [14] to the non-isentropic case under some extra assumptions about the temperature and mass of electrons and ions at both far fields.In Chapter 3, we are concerned with the study of the nonlinear stability of the contact discontinuity of the Navier-Stokes-Poisson system with free boundary in the case where the electron background density satisfies an analogue of the Boltzmann re-lation. We especially allow that the electric potential can take distinct constant states at boundary. On account of the quasineutral assumption, we first construct a viscous contact discontinuity through the quasineutral Euler equations, and then prove that such a non-trivial profile is time-asymptotically stable under small perturbations for the corresponding initial boundary value problem of the Navier-Stokes-Poisson sys-tem. The analysis is based on the techniques developed in [14] and an elementary L2 energy method.In Chapter 4, we are concerned with the study of the nonlinear stability of the composite wave consisting of a rarefaction wave and a viscous contact discontinu-ity of the non-isentropic Navier-Stokes-Poisson equations with free boundary. We first construct the composite wave through the quasineutral Euler equations, and then prove that the composite wave is time-asymptotically stable under small per-turbations for the corresponding initial-boundary value problem of the non-isentropic Navier-Stokes-Poisson equations. Only the strength of the viscous contact disconti-nuity is required to be small. However the strength of the rarefaction wave can be arbitrarily large. In our analysis, the domain decomposition is very important. By introducing this technique, we overcome the difficulty caused by the critical terms involved with the linear term which does not satisfy the quasineural assumption for the composite wave.