Blowup Criterion for Viscous Non-baratropic Flows with Zero Heat Conduction Involving Velocity Divergence

Abstract

In this paper, we prove that the maximum norm of velocity divergence 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 flow is similar to the barotropic compressible Navier–Stokes equations and does not depend on the temperature field. The main ingredient of the proof is a priori estimate to the pressure field instead of the temperature field and weighted energy estimates under the assumption that velocity divergence remains bounded. Furthermore, the initial vacuum states are allowed, and the viscosity coefficients are only restricted by the physical conditions.