This is a survey paper on the study of compressible Navier-Stokes-Poisson equations. The emphasis is on the long time behavior of global solutions to multi-dimensional compressible Navier-Stokes-Poisson equations, and the optimal decay rates for both unipolar and bipolar compressible Navier-Stokes-Poisson equations are discussed.
In this paper, we consider the isentropic compressible Navier-Stokes-Poisson equations arising from transport of charged particles or motion of gaseous stars in astrophysics. We are interested in the case that the viscosity coefficients depend on the density and shall degenerate in the appearance of （density） vacuum, and show the Ll-stability of weak solutions for arbitrarily large data on spatial multi-dimensional bounded or periodic domain or whole space.
The multi-dimensional quantum hydrodynamic equations for charge transport in ultra-small electronic devices like semiconductors, where quantum effects （like particle tunnelling through potential barriers and built-up in quantum wells） take place, is considered in the present paper, and the recent progress on well-posedness, stability analysis, and small scaling limits are reviewed.
We consider the Cauchy problem for one-dimensional isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficient. For regular initial data, we show that the unique strong solution exits globally in time and converges to the equilibrium state time asymptotically. When initial density is piecewise regular with jump discontinuity, we show that there exists a unique global piecewise regular solution. In particular, the jump discontinuity of the density decays exponentially and the piecewise regular solution tends to the equilibrium state as t →＋∞
This paper is concerned with the free boundary value problem for multidimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local （in time） existence of a weak solution is established; in particular, the density is positive and the solution is regular away from the free boundary.
The compressible non-isentropic bipolar Navier-Stokes-Poisson （BNSP） sys- tem is investigated in R3 in the present paper, and the optimal time decay rates of global strong solution are shown. For initial data being a perturbation of equilibrium state in Hl（R3） （R3） for 1 〉 4 and s E （0, 1], it is shown that the density and temperature for each charged particle （like electron or ion） decay at the same optimal rate （1 ＋ t）-3/4, but the momentum for each particle decays at the optimal rate （1 ＋ t）-1/4-3/2 which is slower than the rate （1 ＋ t）-3/4-3/2 for the compressible Navier-Stokes （NS） equations [19] for same initial data. However, the total momentum tends to the constant state at the rate （1 ＋t）-3/4 as well, due to the interplay interaction of charge particles which counteracts the influence of electric field.
We consider the initial value problem for multi-dimensional bipolar compressible Navier- Stokes-Poisson equations, and show the global existence and uniqueness of the strong solution in hybrid Besov spaces with the initial data close to an equilibrium state.
