Singularity formation to the 2D Cauchy problem of the full compressible Navier-Stokes equations with zero heat conduction

Abstract

The formation of singularity and breakdown of strong solutions to the two-dimensional (2D) Cauchy problem of the full compressible Navier-Stokes equations with zero heat conduction are considered. It is shown that for the initial density allowing vacuum, the strong solution exists globally if the density $\rho$ and the pressure $P$ satisfy $\|\rho\|_{L^{\infty}(0,T;L^\infty)}+\|P\|_{L^{\infty}(0,T;L^\infty)}<\infty$. In addition, the initial density can even have compact support. The logarithm-type estimate for the Lam{\'e} system and some weighted estimates play a crucial role in the proof.