A blow-up criterion for compressible viscous heat-conductive flows

TL;DR

This paper establishes a blow-up criterion for strong solutions of 2D compressible viscous heat-conductive flows, linking the potential singularity formation solely to the velocity gradient, similar to the Beale-Kato-Majda criterion.

Contribution

It extends the Beale-Kato-Majda criterion to compressible viscous heat-conductive flows, providing a new condition for blow-up based on velocity gradient.

Findings

Blow-up criterion depends only on the velocity gradient.

Criterion applies to 2D compressible viscous heat-conductive flows.

Results align with known incompressible flow criteria.

Abstract

We study an initial boundary value problem for the Navier-Stokes equations of compressible viscous heat-conductive fluids in a 2-D periodic domain or the unit square domain. We establish a blow-up criterion for the local strong solutions in terms of the gradient of the velocity only, which coincides with the famous Beale-Kato-Majda criterion for ideal incompressible flows.