Stationary waves to the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line: Existence, stability and convergence rate

In this paper, we study the asymptotic behavior of solution to the initial boundary value problem for the two-fluid non-isentropic Navier-Stokes-Poisson system in a half line $\mathbb{R}_{+}:=(0,\infty).$ Our idea mainly comes from [10] which describes the large time behavior of solution for the non-isentropic Navier-Stokes equations in a half line. The electric field brings us some additional troubles compared with Navier-Stokes equations in the absence of the electric field. We obtain the convergence rate of global solution towards corresponding stationary solution. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proofs are given by a weighted energy method.