The Cauchy problem of the Navier-Stokes-Poisson equations in multi-dimensions (n 3) is considered. We obtain the pointwise estimates of the solution when it is a perturbation of the constant state. Then we have the optimal L p (R n ) (p 1) convergence rate of the solution.

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The Cauchy problem of the bipolar Navier-Stokes-Poisson system (1.1) in dimension three is considered. We obtain the pointwise estimates of the time-asymptotic shape of the solution, which exhibit generalized Huygens' principle as the Navier-Stokes system. This phenomenon is the the most important difference from the unipolar Navier-Stokes-Poisson system. Due to non-conservative structure of the system (1.1) and interplay of two carriers which counteracts the influence of electric field (a nonlocal term), some new observations are essential for the proof. We make full use of the conservative structure of the system for the total density and total momentum, and the mechanism of the linearized unipolar Navier-Stokes-Poisson system together with the special form of the nonlinear terms in the system for the difference of densities and the difference of momentums. Lastly, as a byproduct, we extend the usual L 2 (R 3 )-decay rate to L p (R 3 )-decay rate with p > 1 and also improve former decay rates in part.which was reconsidered in [18] by using the complex analysis method for the Green's function recently. In particular, they in [18,19] showed the solution behaves aswhere the first profile is called as the diffusion wave(D-wave) and the second profile is called as the generalized Huygens' wave(H-wave). Wang et al. [33,38] also discussed the pointwise estimates of the solution for the damped Euler system when initial data is a small perturbation of the constant state and showed the solution behaves as the diffusion wave since the long wave of the Green's function does not contain the wave operator. All of the results above showed the different asymptotic profiles in the pointwise sense for different systems. Indeed, the pointwise estimates of the solution play an important role in the description of the partial differential equations, since it can give explicit expressions of the time-asymptotic behavior of the solution. Moreover, one can get the global existence and optimal L p -estimates of the solution directly from the pointwise estimates of the solution. Here, we give some comments on the diffusion wave(D-wave) and the generalized Huygens' wave(H-wave) in (1.3) for the convenience of readers. As we know, for the L p (R n )-estimate of these two waves, when p < 2 the H-wave is the dominated part and when p > 2 the D-wave is the dominated part, and p = 2 is the critical case and the L 2 -decay rate of these two waves is the same as the heat kernel. In other words, the usual L 2 -estimates for some hyperbolic-parabolic coupled systems indeed conceal the hyperbolic characteristic of the solution. Besides, there are also other results in [21,22,23,24,25,27,39,40] on the related models for the L 1 -estimate, wave propagation pattern around shock wave by using the method of Green's function. Nowadays, for unipolar Navier-Stokes-Poisson system(NSP), there are also some results on the decay rate of the Cauchy problem when the initial data (ρ 0 , u 0 ) is a small perturbation of the constant state (ρ, 0). Li et al. [14] o...