  We consider the compressible Navier-Stokes system with variable entropy. The
pressure is a nonlinear function of the density and the entropy/potential
temperature which, unlike in the Navier-Stokes-Fourier system, satisfies only
the transport equation. We provide existence results within three alternative
weak formulations of the corresponding classical problem. Our constructions
hold for the optimal range of the adiabatic coefficients from the point of view
of the nowadays existence theory.
