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 $\Vert \rho {\Vert }_{L^{\infty }(0,T;L^{\infty })}+\Vert P{\Vert }_{L^{\infty }(0,T;L^{\infty })}<\infty$. In addition, the initial density can even have compact support. The logarithm-type estimate for the Lam{é} system and some weighted estimates play a crucial role in the proof.

15 pages