A Blow-Up Criterion for Classical Solutions to the Compressible Navier-Stokes Equations

TL;DR

This paper establishes a new blow-up criterion for classical solutions to the 3-D compressible Navier-Stokes equations based on the velocity gradient, extending previous incompressible flow results and allowing for initial vacuum conditions.

Contribution

It introduces a novel blow-up criterion for compressible Navier-Stokes solutions that depends solely on the velocity gradient, accommodating initial vacuum states.

Findings

Blow-up occurs when the velocity gradient becomes unbounded.

The criterion is similar to the Beal-Kato-Majda condition for incompressible flow.

Initial vacuum conditions are incorporated into the analysis.

Abstract

In this paper, we obtain a blow up criterion for classical solutions to the 3-D compressible Naiver-Stokes equations just in terms of the gradient of the velocity, similar to the Beal-Kato-Majda criterion for the ideal incompressible flow. In addition, initial vacuum is allowed in our case.