On the nonexistence of stationary weak solutions to the compressible fluid equations

In this paper we prove that under some integrability conditions for the density and the velocity fields the only stationary weak solutions to the compressible fluid equations on ${\mathbb{R}}^N$ correspond to the zero density. In the case of compressible magnetohydrodynamics equations similar integrability conditions for density, velocity and the magnetic fields lead to the zero density and the zero magnetic field.