Existence of weak solutions for compressible Navier-Stokes equations   with entropy transport
  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. 
David Maltese, Martin Michalek, Piotr B. Mucha, Antonin Novotny, Milan   Pokorny, Ewelina Zatorska
math.AP