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

TL;DR

This paper investigates the conditions under which strong solutions to the 2D full compressible Navier-Stokes equations with zero heat conduction exist globally or develop singularities, considering initial vacuum states.

Contribution

It establishes global existence criteria for strong solutions with initial vacuum and allows initial density with compact support, using novel estimates for the Lamé system.

Findings

Global existence if density and pressure are bounded.

Initial vacuum states with compact support are permissible.

Logarithm-type and weighted estimates are key tools.

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.