On the Liouville type theorem for stationary compressible Navier-Stokes-Poisson equations in Rⁿ

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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 Rⁿ, N≥ 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 ρ=constant on Rⁿ. This shows that if we impose the the non-vacuum boundary condition at spatial infinity for (NS), v→ 0 and ρ→ ρ∞ >0, then v=0, ρ=ρ∞ are the solutions.