This chapter contains a complete proof of the existence of global-in-time variational solutions for the full system of the Navier-Stokes equations of a viscous compressible and heat conducting fluid under suitable restrictions imposed on the constitutive laws. These restrictions are by no means optimal but, on the other hand, they seem to be in a good agreement with the underlying physical theory. Probably the most questionable hypothesis seems to be the necessity of the viscosity coefficients to be constant in all temperature regimes. On the other hand, as a by-product of our approach, we derive ‘optimal’ existence results for the barotropic flows with respect to the available (known) a priori estimates. Another novelty allowed by the present method is the possibility to consider general, not necessarily monotone, pressure-density constitutive equations arising in applications.
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