On the Liouville type theorem for stationary compressible Navier-Stokes-Poisson equations in $\Bbb R^N$

Abstract: In this paper we prove Liouville type result for the stationary solutions to the compressible Navier-Stokes-Poisson equations(NSP) and the compressible Navier-Stokes equations(NS) in $\Bbb R^N$, $N\geq 2$. Assuming suitable integrability and the uniform boundedness conditions for the solutions we are led to the conclusion that $v=0$. In the case of (NS) we deduce that the similar integrability conditions imply $v=0$ and $\rho=$constant on $\Bbb R^N$. This shows that if we impose the the non-vacuum boundary condition at spatial infinity for (NS), $v\to 0$ and $\rho\to \rho_\infty >0$, then $v=0$, $\rho=\rho_\infty$ are the solutions.