Conditional regularity for the Navier-Stokes-Fourier system with Dirichlet boundary conditions

We consider the Navier-Stokes-Fourier system with the inhomogeneous boundary conditions for the velocity and the temperature. We show that solutions emanating from sufficiently regular data remain regular as long as the density $\varrho$, the absolute temperature $\vartheta$, and the modulus of the fluid velocity $\mathrm{\mid }\mathbf{\text{u}}\mathrm{\mid }$ remain bounded.