Refined blow up criteria for the full compressible Navier-Stokes equations involving temperature

TL;DR

This paper improves blow-up criteria for the 3D full compressible Navier-Stokes equations by allowing temperature in scaling invariant spaces, extending previous results and providing new regularity conditions.

Contribution

It generalizes existing blow-up criteria to include temperature in scaling invariant spaces and extends pressure-based regularity criteria from incompressible to compressible Navier-Stokes equations.

Findings

Enhanced blow-up criteria involving temperature in scaling invariant spaces.

Extended pressure regularity criteria from incompressible to compressible Navier-Stokes.

Provided conditions under which solutions remain regular.

Abstract

In this paper, inspired by the study of the energy flux in local energy inequality of the 3D incompressible Navier-Stokes equations, we improve almost all the blow up criteria involving temperature to allow the temperature in its scaling invariant space for the 3D full compressible Navier-Stokes equations. Enlightening regular criteria via pressure $\Pi=\frac{\text {divdiv}}{-\Delta}(u_{i}u_{j})$ of the 3D incompressible Navier-Stokes equations on bounded domain, we generalize Beirao da Veiga's result in [1] from the incompressible Navier-Stokes equations to the isentropic compressible Navier-Stokes system in the case away from vacuum.