On Nash’s conjecture for models of viscous, compressible, and heat conducting fluids

We show a new blow up criterion for regular solutions of the Navier–Stokes–Fourier system in terms of uniform bounds on the density and integral bounds on the absolute temperature. In comparison with the existing results, we remove the technical conditions relating the values of the shear and bulk viscosity coefficients. The result can be seen as a rigorous justification of Nash’s conjecture concerning the character of possibly singularities in the equations of fluid dynamics.