Some Analysis Results On Compressible Navier-Stokes-Poisson Equations In Half Space

It is well known that the compressible Navier-Stokes-Poisson equations,which can be used to simulate the transport of particle flow under the electric field force,have important applications in semiconductor physics and plasma physics.In terms of mathematical structure,the model is formed by the coupling of the classical Navier Stokes equations and the Poisson equation used to describe the effect of electric field,and it belongs to a class of typical hyperbolic-parabolic-elliptic mixed partial differential equations.This paper is divided into three parts:the unipolar isentropic model,the plasma model and the bipolar isentropic model.We study the initial boundary value problem of compressible Navier-Stokes-Poisson equations in half space.Considering the stationary electric field effect,we obtain the main analysis results in three aspects.First,we apply the manifold theory and the spectral analysis technique to establish the existence and uniqueness of stationary solution with the spatial decay property.Then,based on the standard energy estimate method,we prove the time asymptotic stability of the stationary solution.Furthermore,in the supersonic case,we successfully obtain the time decay rate of the time-dependent solution to the stationary solution by means of the time-space weighted energy estimate methodIn order to overcome the difficulty of solving higher-order stationary equations,we apply the manifold theory instead of the phase plane analysis method to find a new class of stationary solution.It is worth pointing out that the stationary solution is essentially different from those mentioned in other literatures because it contains a nontrivial stationary electric field.In the subsequent discussion of the large-time behavior of the solution,the electric field effect brings new difficulties in analytical techniques.We fully exploit the dissipative mechanism of the Navier-Stokes-Poisson system,including the spatial decay properties of stationary waves,the dissipative structure of the Poisson equation and the bipolar coupling effect,and finally succeeded in obtaining the last two main results after a series of careful estimates.