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In this paper, we consider the initial value problem for the compressible Navier-Stokes-Poisson equations in three and higher dimensions. Firstly, global existence and new decay estimate of classical solutions are established. The proof is mainly based on the decay properties of solution operator to the compressible Navier-Stokes-Poisson equations, which may be derived from the pointwise estimate of the solution operator to a linear wave equation that is corresponding to the density function, while the pointwise estimate of solution operator to the linear wave equation may be established by an energy method in the Fourier space. The method is different from those in previous works on decay estimate of solutions to compressible Navier-Stokes-Poisson equations. Finally, asymptotic profile of the solution to the corresponding linear problem is also investigated.

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In this paper, we investigate three‐dimensional incompressible Boussinesq equations and establish some logarithmically improved blow‐up criteria of smooth solutions to the Cauchy problem for the incompressible Boussinesq equations. Copyright © 2011 John Wiley & Sons, Ltd.