A new blow-up criterion for the 2D full compressible Navier-Stokes equations without heat conduction in a bounded domain

TL;DR

This paper establishes a new blow-up criterion for the 2D full compressible Navier-Stokes equations without heat conduction, showing that the solution remains global if certain density and pressure norms stay finite, extending previous results.

Contribution

It introduces a less restrictive blow-up criterion involving $L^{p_0}$ norms of pressure, broadening the understanding of solution longevity in these equations.

Findings

Global existence under new criterion involving $L^{p_0}$ norm of pressure.

Extension of previous blow-up criteria to include vacuum initial conditions.

Applicable to bounded domains with Navier-slip boundary conditions.

Abstract

This paper is to derive a new blow-up criterion for the 2D full compressible Navier-Stokes equations without heat conduction in terms of the density $\rho$ and the pressure $P$. More precisely, it indicates that in a bounded domain the strong solution exists globally if the norm $\|\rho||_{{L^\infty(0,t;L^{\infty})}}+||P||_{L^{p_0}(0,t;L^\infty)}<\infty$ for some constant $p_0$ satisfying $1<p_0\leq 2$. The boundary condition is imposed as a Navier-slip boundary one and the initial vacuum is permitted. Our result extends previous one which is stated as $\|\rho||_{{L^\infty(0,t;L^{\infty})}}+||P||_{L^{\infty}(0,t;L^\infty)}<\infty$.