On Formation of Singularity for Full Compressible Navier–Stokes System with Zero Heat Conduction

Abstract

We prove that the maximum norm of the density and temperature controls the breakdown of smooth (strong) solutions to the two-dimensional (2D) Cauchy problem of the full compressible Navier–Stokes equations with zero heat conduction. The results indicate that the nature of the blowup for the full compressible Navier–Stokes equations with zero heat conduction of viscous media is similar to the full compressible Navier–Stokes equations. It also reveals that if the solution to 2D viscous compressible flows blows up in finite time, then either the mass of the compressible fluid will concentrate in some points or $$L^\infty (0,T;L^{2})$$ -norm of the temperature is bounded in the finite time. Furthermore, the initial vacuum states are allowed in our paper.